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ip, derive ip (M odus Ponens) R 2) From
ip (a ), derive < p —> (\/x)ip(x) provided that a does not occur in
tp)->tp))->((tp->(tp->tp))->(tp->tp)) A xiom -schem a 2 2, ip—*((ip—*ip)—* +-+ tp ) is 1 iff the semantic value o f (p is identical to the semantic value o f tp. ip ) is a formula. (6) If (p and ip are form ulas, then (
A xiom -schem a 1
3, (
1, 2, M odus Ponens A xiom -schem a 1
5
3, 4, M odus Ponens ■
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As is evident from this proof, to write down actual derivations in this axiom atic system is rather cum bersom e W e will use theorem 2 in the p ro o f o f the law o f the excluded middle for predicate logic. (8 -4 9 )
T heorem 3, (V x)(ip (x) V ~(x )) P roof: 1, ~^?(a)—>~^(a)
by theorem 2
2,
1, definition o f disjunction
3 , (1/3(0) V
V ^ (a )))) A xiom -schem a 1 2, 3 , M odus Ponens
4, ((^/>—>(^/>—> ^ ))—>(1/3(0) V ~
5 ((ifi-*(ifi-*ifi))—> (y x ){ip (x ) V ~~
A
x
i
o
m
-
7, (Vx)(
8 .6 .4
s
c
h
e
m
a
t
a
1
5, 6, M odus Ponens
A b o u t com pleteness proofs
This axiom atization o f first-order predicate logic is com plete, i e ., all and only valid formulas are provable. A lthough this claim itself has been proven form ally b y Kurt G odel and later also by L eon Henkin, these proofs are quite technical and not directly useful for any linguistic purposes. But the n otion o f completeness is an im portant m eta-theoretical concept and to get an im pression o f its value we discuss the m ain ideas o f H enkin’s p r o o f semiform ally here. The m ain stages o f Henkin’s completeness p r o o f are the following three claims: (1) if a form ula
W e explain the three claims and the notions used in them non-technically:
■
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ad 1) Suppose that < p is not provable in predicate logic. W e reason in a reductio ad absurdum argument I f { ~ tp is provable. Since ~ ( ~ ip k ip ) is provable as well, it follows by M odus Ponens that ip is provable after all, which contradicts the initial assum ption that ip is not provable in predicate logic
So the additional assum ption that ~ tp is
inconsistent cannot be right, so ~ tp is consistent ad 2) A set o f formulas M “ is maximally consistent if M " is consistent and for every arbitrary form ula tp not in M " , M ” U { ^ } is inconsistent This means that there is no form ula which can be added to M * while keeping it consistent
A ny consistent set o f formulas M can be extended to a m axi
m ally consistent set M " which contains all formulas o f M
W e enumerate the
formulas
, tpn —■> ip is valid in predicate logic, then the p r o o f o f ip , , tpn must exist in our axiom atization o f predicate logic, ip is valid, then tpi —> (
(
and hence provable. W ith M odus Ponens applied n times we prove ip from
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since they aie sim ply semantic and syntactic counterparts Perhaps the most im portant and startling “ side effect” o f the research on completeness was the discovery o f negative results showing the incom pleteness o f some systems T o that top ic we turn briefly in Section 8 6 6
8 .6 .5
D ecid a b ility
W e have seen already that finding a p r o o f for a form ula with natural de duction rules often requires ingenuity and insight. There is no foo lp roo f procedure we can prescribe which yields a p r o o f for any provable formula In the semantics of the statement logic, however, there is the m echanical truth table m ethod which always answers the question whether a statement is a tau tology or not In predicate logic truth tables do only part o f the semantic jo b but we have to consider assignments to variables on possibly infinite domains as well There is no general procedure which yields a defi nite y e s/n o answer to the question whether a predicate-logical statement is valid or provable. N o success in finding a p r o o f may m ean either that we have to try harder or that the statement is not a theorem , but we never know which is the case! If you understand the workings o f the Beth Tableaux for predicate logic, you have seen that the construction o f some tableaux m ay just never com e to an end by closure or counterexample, Neither can we tell from the form o f a predicate-logical statement after how m any steps we will either find a coun terexample or obtain closure. All we know is that if' the statement is valid it will eventually end in closure This is why the tableau m eth od , even though it is more m echanical than natural deduction, does not constitute a general procedure answering ‘yes’ or ‘n o ’ to the question whether an arbitrary statment o f predicate logic is a theorem. The fact that mem bership in the set o f theorems o f predicate logic cannot be decided for every arbitrary form ula by such a m echanical procedure is called the undecidability o f predicate logic The statement logic, however, is decidable, because o f the existence o f the m echanical procedures o f truth tables or tableaux which characterize the valid statements. A nother subsystem o f first-order predicate logic, monadic predicate logic, where predicates can take only one argument, is also decid able, The p r o o f o f the decidability o f m onadic predicate logic is based on two facts: ( l ) that any m onadic form ula can be transform ed to a special prenex norm al form in which all existential quantifiers precede the universal quantifiers (see below on Skolem N orm al Forms) and (2) that a tableau for such formulas always ends after a finite num ber o f steps in closure or in a
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counterexam ple by enumerating the domain.
B oth claims can be proven,
but to do so here would lead us to o fax afield T he set o f theorems o f predicate logic is not decidable, but it is enu m erable in a system atic way.
T he B eth Tableaux m ethod provides such
an enum eration, since we know that every valid theorem eventually leads to closure The m ethod does not produce a y e s /n o answer for any arbi trary form ula, but it does produ ce “ yes” answers for all valid theorems We say that this constitutes an effective enum eration o f the set o f theorem s o f predicate logic. O f course, the com plem ent o f this set, i.e., the set o f all non-theorem s or invalid arguments, is not effectively enumerable. For if it were, we would have a decision procedure saying “ yes” to all theorem s and “ n o” to all non-theorems. T he completeness theorem is not in any conflict with the undecidability o f predicate logic, for if' an argument is valid, there is a finite p r o o f o f it, but we have no gener al m ethod o f finding such a p roof. W e return to the relations betw een decidability, effectively enum erable sets, different kinds o f functions, and com putability in Part E„
8 .6 .6
G o d e l’s incom pleten ess th eorem s
T he Peano axiom s form the fou n dation o f m athem atical number theory. W e m ay wonder whether it is com plete with respect to its intended interpreta tion, In 1931, just after proving the completeness o f predicate logic, Kurt G od el proved the startling theorem that Peano arithm etic is incom plete; i e ., there is a true statement about natural numbers which is not provable from the Peano axioms! The p ro o f o f what is know n as G od el’s first incom pleteness theorem con stitutes one o f the m ost sophisticated p r o o f techniques in logic, G od el’s first incompleteness result concerned the incom pleteness o f P eano arithmetic. His second incompleteness result stated that any consistent axiom atization which is strong enough to contain num ber-theoretic representations o f proofs does not yield a p ro o f o f its ow n consistency, The result is shocking when one realizes that it means that the p ro o f o f the consistency o f arithm etic is as questionable as its consistency itself. The first incompleteness theorem uses the natural numbers simultane ously as an object-level system and as its ow n m etalanguage in constructing a num ber-theoretic representation o f a statement which says about itself that it is false, The intutitive background stems from the ancient Cretan p aradox o f the liar, which is easily expressed in natural language b y the selfreferential statement, I am a liar A nyone making such a statement runs into
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semantic paradoxes, for if the statement is true, he is not a liar and hence what he says is false, but if the statement is false, then he cannot be a liar and what he says is true. T he vicious circularity resides in using a semantic truth predicate (being a liar = not speaking the truth) in the statement as well as in applying it to the statement itself T he form al p r o o f o f the incompleteness theorem is to o intricate to even outline here; the interested reader is referred to Hofstadter (1979) for a generally accessible exposition or to B ell and M achover (1977), E nderton (1972), or any other tex tb ook in m athem atical logic for all necessary details Another surprising consequence of the incompleteness theorem is that, since ZF C is effectively axiom atizable, com plete with respect to negation and thought to be consistent, its supposed consistency can never be proven. Even though the G odel results deeply shook the foundations o f m ath ematics and logic, m athem atical practice never took m uch notice and p ro ceeded as usual W e m ay take their m etaphysical im plications as saying that m athem atical reality is to o rich ever to get com pletely captured in a con sistent and com plete axiom atization; or we m ay conclude more soberly that form alizations have their limits too, and we had better be aware o f them
8 .6 .7
H ig h er-ord er logic
There are different ways in which we m ay enrich the language o f fiist-order predicate logic by adm itting quantification over predicates or functions. The system we introduced contained quantifiers restricted to individual variables, which constitute the first-order quantification. But there are m any nouns and verbs in natural language which are not properties o f individuals, but rather properties o f properties o f individuals. I f this vase is blue and blue is a color, we cannot infer that this vase is a color, but rather that this vase if o f a color T he predicate ‘is a color’ cannot properly be applied to ordinary individuals, but can be applied to properties o f them. Properties and relations o f these first-order objects are second-order objects, and so on, A logical system is an re-order logic when it contains at least one re-order variable (free or bou nd ). N ote that an re-order logic m ay contain quantifiers o f lower order as well as r elations m ixing arguments o f different orders. The interpretation o f second-order quantifiers must consider all or some assign ments o f subsets o f a domain to the second-order variables, since a first-order predicate is interpreted b y a subset o f the domain. T he interpretation o f ‘is a color’, for instance, is a set o f subsets o f colored objects. The semantics o f
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second-ordei quantifiers must hence em ploy the full power-set o f the domain o f the m odel W e will not develop a full-fledged syntax and semantics o f higher-order systems here but discuss some aspects o f second-order logic of interest in linguistic applications T w o examples o f fam ilial notions illustrate the expressive power o f secondorder logic (1) A well-ordering is an ordering relation in which every non-em pty set has a least element. This is expressed in second-order logic by ( V X ) ( ( 3 w )X w -+ ( 3 y ) ( X y k { \ / z ) ( ( X z k z ± y) -* y < z )) )
(2) The induction axiom in P ea n o’s axiom atization o f artithm etic can be translated into a second-order form ula It says that a set o f natural numbers which contains 0 and is closed under the successor fu n ction , constitutes the set o f all natural numbers. In a second-order formula, where X is a secondorder variable and S is the successor-relation, a relation betw een a set and a natural num ber, this is expressed as
(VX)(X(0)
k
(Vjf(X(jf) -
S ( X , y ) ) -> (Vy)X(y)))
A ny first-order form ula can be translated to a logically equivalent secondorder form ula which has a special form: it is prenex and all existential re order quantifiers precede universal first-order quantifiers. This is called its Skolem N orm al Form (SN F) after the logician Skolem who proved the gen eral theorem that such a norm al form exists for any first-order formula. We discuss some examples as illustration o f the m ethod. T he value o f Skolem N orm al Forms lies in their explicit representation o f quantificational depen dencies o f assignments to variables The simplest exam ple is the equivalence o f (V ® )(3y ) R ( x , y ) <=> ( 3 F ) ( y x ) R ( x , F ( x ) ) . The individual variable y is assigned a value by a function F depending on the assignment to the uni versally quantified x. To see that (3 F )( V x )R (x , F ( x ) ) implies ( ' i x ) ( 3 y ) R ( x , y ) , suppose we have an interpretation M with an assignment g which verifies (V ® )(3y ) R {x ,y ). W e know that for any a in the domain o f M there is at least one b such that [ # ( » , y)JM’3 is true. W e obtain a function / on the dom ain o f M b y choosing one such b for each a and taking f ( a ) = b. This uses the A x io m o f Choice, Then
F (x ))]sM'3 is true where g' is exactly like g with f substi
tuted for F . This function f is called a Skolem fu n ctio n , or a choice-function, for the form ula (V ® )(3y ) R ( x , y ) in the interpretation M , This illustrates the
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general m ethod: an existential first-ordei quantifier is translated to an ex istential second-order function quantifier, which binds a function taking as arguments all individual variables bou nd by universal quantifiers preceding it. One more exam ple o f this procedure m ay suffice as illustration, Suppose we have a form ula (3 y )(V x )(3 z )(V u )(V y )(3 w )R (y , z, w) (presum ably other variables occu r in the m atrix, but they are irrelevant for the present exam ple). The first existential quantifier precedes the universal quantifiers, so is already in SNF, Consider the remainder o f the formula:
(V®)(3z)(Vtt)(Vw)(3w)iJ(2/, z, w ) The second existential quantifier ( 3 z) depends on (V®), and we translate the form ula to
(3 F )(V x )(V u )(V v )(3 w )£ (y , F ( x ) , w) W hat remains is
(V x)(V u )(V v )(3 w )I£ (y , F ( x ) , w ) This is logically equivalent to (3 G )(V x )(V u )(V v )R (y , F ( x ) , G ( x , u, v )) Here G is a Skolem function with thiee arguments hence equivalent to the SNF
The original form ula is
( 3 y ) ( 3 F ) ( 3 G ) ( V x ) ( V u ) ( V v ) R ( y ,F ( x ) ,G ( x ,u ,v ) ) Linguistic applications o f Skolem-functions have been prim arily in the se mantics o f question-answering A multiple question like “ W hich student got which grade?” is interpreted as a set o f statements which constitute true an swers to it, which must give a specific grade for each student. The answer to the wh-quantifier “ which grade” depends on the answer to “ which student” . This dependency can be captured by a Skolem -function and im plem ented in a M ontague style semantics or in other m odel-theoretic frameworks
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The same considerations which showed that first-order predicate logic is undecidable apply to higher-order logics which contain first-order logic. There cannot be a procedure which answers ‘n o ’ in every case in which a form ula is not a theorem. But we have seen that the B eth Tableaux m ethod provid ed a procedure to enumerate the valid theorems
N o such effective
enum eration exists for the theorems o f higher-order logics, however. For if there was such an enumeration, it could b e turned in to a ‘n o ’-answering procedure for first-order logic by replacing names and predicate constants by appropriate variables, universally quantifying these and negating the entire second-order form ula W e know first-order logic is undecidable, so there cannot be such a ‘n o ’-answering procedure. So there is n o tableau m ethod for second-order logic, n o effective procedure identifying all theorems Second-order logic, and hence any higher-order logic, is incom plete. The p r o o f cannot be discussed here, but it is based on the fact that an incom plete fragment o f arithm etic can be represented in it, showing that G od el’s incompleteness result m ay be applied to higher-order logics. This means that there are valid theor ems which cannot be pr oven from any axiom atiza tion o f higher-order logic. It is an interesting open question just how much higher-order expressive power can be adm itted in to first-order logic while preserving its completeness. Type theory is the system of' logic with quantifiers and variable o f any order. M ontague Grammar is based on a type theory but usually employs only a lim ited fragment with third-order quantification in the interpretation o f natural language. New research is developing m ore flexible forms o f type theory which m ay have im portant applications in linguistics and com puter science
Exercises 1. Give a recursive definition of the well-form ed strings in the statement calculus in Polish parenthesis-free notation (Chapter 6, Exercise 13). Give a p r o o f CENApppp is a well-form ed string using your definition. 2. Let / b e a function that maps each n in Z = 0 ,1 ,2 ,3 ,. ,. into 2zn; e.g., / ( I ) = 22 = 22 = 4
Given a recursive definition o f / , and use it to
com pute the value o f /( 4 ) .
3 . Suppose we were to take the successor o f any positive num ber x as being x + 2. Show that the four Peano A xiom s (Section 8 4) would
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then specify the set o f natuial numbers as the set o f even numbers 0 ,2 , 4, 6 ,8 , ... W ou ld the Principle o f M athem atical Induction still be a reasonable rule o f inference when defined over this set? 4 . Prove by induction that the pow er set o f a set with ra m em bers has 2n m em bers, for any finite positive integer n. 5. Prove b y induction the generalized distributive law o f m ultiplication over addition; i.e., for all ra, a X (6j + b2 + b2) +
.
. . + bn) = (a X &i) + (a X
+ ( a X bn).
6. W hat is wrong with the follow ing inductive p r o o f that all hor ses are o f the same color? For a set containing only one horse, the base clearly holds, since that horse has only one color
Now assume that all sets o f
n horses contain only horses o f the same color. W e show that it follows that the same is true o f all sets o f n + 1 horses Choose a set o f ra + 1 horses and select any n o f them , disregarding the extra horse for the m om ent, B y assumption, these n horses are all o f the same color. Now replace one o f the n horses by the extra horse, form ing a new set o f ra horses These again, by assumption, are all o f the same color, and so the extra horse is the same color as all the others. Therefore, all hor ses are o f the same color. 7. Consider the follow ing axiom atic system. The “ alphabet” consists o f all well-form ed formulas in the statement calculus plus the symbols —», (, and ), There are three axioms: (A l) p -^ (q -> p ) (A 2 ) ( p - > { q - > r )) (A 3 ) ( ~ p
((p -> q) -> ( p -> r ))
q) -> (q ~ * p)
and tw o rule schemata: (R I ) From any tw o expressions o f the form A —* B and A , we can derive B (A and B are variables ranging over the w ff’s o f the statement calculus) (R 2 ) From any expression A we can derive B , where B is the result o f substituting a w ff x for every instance o f some atom ic statement, i.e ., p, 2 , r,etc., in A.
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T h e f o l l o w i n g is a p r o o f o f p —> p in th is s y s t e m :
1.
{ p - * ( q ~ * r ))
2
(p - » ((g - » p) - » r )) -> ((p - » (g - » p )) - » (p - » r)) 1, (R 2 ) (Substituting (q —> p) for g)
{{p
q) -> (p -> r ))
(A 2 )
3. 4
(p - » ((g ~ * p )~ * p )) - » ((p - » (g - » p )) - » (p p )) 2, (R 2 ) (Substituting p for r) p _» (2 _» p ) (A l)
5.
p _» ((2 _* p) _» p) 4, (R 2 )(S u bstitu tin g (g —> p) for g)
6. 7.
( p -* { q -* p ) ) - * { ? - * p) p -» p
3 ,5 ,(R 1 ) 4, 6, ( R l)
C onstruct a p r o o f o f ~ p —> (p —> ?) in this system, H int: Begin by substituting ( ~ g —
p) —> (p —> g) for p and ~ p for g in A l.
It
can be shown that the theorems o f this system are all and only the tautologous w ff’s o f statement logic [see, for exam ple, M assey (1970, pp, 125-159)],, The connectives & and V, which do not appear in this system , can b e defined in terms o f ~ and — 8 . R eform ulate the recursive difinition in Exercise 1 as an axiom atic sys tem having the w ff’s n Polish notation as its theorems. F ind an equiv alent sem i-Thue system 9 . C onstruct an extended axiom atic system whose theorems are all strings in { a } * o f length divisible b y 2 or by 3. For exam ple, aa, aaa, aaaa, aaaaaa, aaaaaaaa are theorem s, but a, aaaaa aaaaaaa are n ot. Can you see why there is no equivalent axiom atic system w ithout an aux iliary alphabet? 1 0 . Prove in the axiom atization o f predicate logic given in Section 6,3 the follow ing theorems. Y ou m ay assume that all predicate logical tautologies are provable in this axiom atization,
(a) P ( a ) - » (3 x ) P ( x ) ( b ) from premise (Vx)(ip —» P ( x ) ) ) that (ip —> (V x )P (x )) if x does not occur in tp ( c ) from premise (V ® )(P (® ) —> Q { x )) that (V®)P(®) —> (V x )Q (x ) 11. If the deletion o f a certain axiom from a negation com plete system makes it not negation com plete, then the axiom is independent. W hy?
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12. The follow ing systems all satisfy P e a n o ’s axioms under appropriate interpretations. For each case state what interpretation must be given to ‘ O’, ‘is a natural n um ber’ , and ‘is a successor o f ’. (a ) 0 , - 1 , - 2 , - 3 , . . .
(b) 5 ,1 0 ,1 5 ,2 0 ,2 5 ,.... (c ) + 1 ,- 2 ,+ 4 ,- 8 ,+ 1 6 ,- 3 2 ,.. 13. Consider the follow ing form al system W , P rim itives: set P o f objects called “p oin ts” , a set L o f objects called “lines” . A xiom s: (1) every line is a set o f points; i.e. L 6 p ( P ) (2) there exist at least tw o distinct points (3) if p and q are distinct poin ts, then there is one and only one line o f which p and q are b o th mem bers. (4) i f I is a line, there exists a p oin t not in I (5) i f Z is a line, and p is a point not in I, then there exists one and only one line containing p and disjoint from I.
( a ) F ind a m odel for W in which P has exactly four objects.
(b) P rove from the axioms that every point is in at least tw o distinct lines ( c ) Prove that the em pty set cannot be a m em ber o f L. ( d ) Show that there can be n o m odel for tw o members.
W
in which P has exactly
( e ) I f we added as sixth axiom “every line contains exactly one p o in t” would the resulting system be consistent? 14. Give the Skolem norm al forms for the follow ing form ulas: (a ) ( V x ) ( 3 y ) ( V z ) R ( x , y , z )
(b) ( V x ) ( 3 y ) ( 3 z ) R ( x , y , z ) ( c ) ( V x ) ( 3 y ) ( R ( x , y ) ^ ( 3 z ) ( S ( x , z) & S ( y , z ) ) ) ) (d )
(VxXVy)(3z)(R(x,y)&~P(z)^(Vu)S(z,y,x,u))
Appendix B-I
Alternative Notations and Connectives The logical language we have studied in this chapter is the best know n classical system o f logic o f statements. There are various alternative sym bolic notations for1the connectives we introduced, the m ost com m on o f which are listed below,
Alternative notation for connectives our sym bol
alternative(s)
&
~'> -> P A, • D (called ‘horseshoe’)
negation con ju n ction conditional
Table B .I .-l: Furthermore, there are tw o connectives which we have n ot in trodu ced, as they are not part o f the system m ost com m only used. One, called “ Q u ine’s dagger” , j is sufficiently pow erful to be the only connective in a system equivalent to the one given. Its truth table is shown in (B II.-2).
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240
B -I
A p p end ix
p
Q
1 1 0 0
l 0 1 0
p
IQ
0 0 0 1
Table B .I.-2 : Its nearest English correspondent is n eith er ,. , n o r, As an interesting ex ercise one can show that negation can be defined in terms o f this connec tive, and then that disjunction can be defined in terms o f negation and this connective. W e have already proven (C hapter 6, Exercise 7) that the fiveconnective system can be reduced to one containing just V and
Therefore
I suffices alone for the five connectives Similarly, there is another connective, written as |, which is called the “ Sheffer stroke” , whose by the truth table is as follows: p
Q
1
l l 0 0
0 1 0
p
\Q
0 1 1 1
Table B .I.-3 :
Appendix B-II
Kleene’s Three-valued Logic
TKe logic o f statements and the predicate logic are two-valued, since there are but tw o truth values and every form ula is either true or false. This is based on the assum ption that the semantic assignments used in an inter pretation are total functions. In some linguistic applications and especially in com putational contexts, that assumption seems m uch to o strong, since it requires that there is a clear semantic procedure which decides for any given x whether fvK®)! i s true or false for any arbitrary
In case a partial function is undefined for an argument it m ay be because we lack inform ation, or we m ay take it to mean that we disregard its value as it is does not m atter to our interpretation. The follow ing truth tables represent the strong Kleene semantics for the connectives, where 1 is ‘true’ , 0 is ‘false’ and * means the truth value is undefined. 241
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B -II
F rom this table we can see that if sufficient inform ation is available to verify or falsify a statem ent, the undefined part does n ot alter it But i f the value o f a part must b e know n to determine the value o f a com plex statem ent, the latter remains undefined until we know the value o f its parts This means that the value o f a com plex statement may be determ ined even when we do not know the value o f all o f its parts For instance, as soon as we know that the antecedent of a conditional is false, we know that the conditional is true, irrespective o f what the consequent m ay be K leen e’s operations are m on oton ic in the sense that any valuation function preserves its initial assignments when the dom ain is extended and new objects are added to the interpretation o f the partial predicates I f we understand the undefined cases as arguments o f which we have not yet determ ined the value, it is an unnatural consequence of these truth tables that a classical tautology such as p V ~ p remains undefined until we know the value o f p There are ways to escape such consequences and preserve the classical tautologies and contradictions in a partial truth definition, but we cannot go in to such systems here. Three-valued logics have prim arily been applied in linguistics in semantic theories o f presuppositions
There is a lively controversy surrounding the
analysis o f presupposition in philosophy and linguistics, which cannot be surveyed here. For illustration we define this n otion in sem antic terms: A ny statement p is a presupposition o f a statem ent q iff i f p is not true (but false or undefined), q is undefined The truth o f a presupposition p is in some sense an assertability condi tion for1 q. In our everyday use o f natural language we rely uncom m only
K
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’s
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lo g ic
243
often on such presuppositions. W e use names to refer to p eople under the assumption that they exist; we use definite NPs like the students who passed their exam s with the com m on understanding that we m ean to say something about a particular n on-em pty set o f students. We also presuppose that it is raining, when we say John knows it is raining or Jane does n ot regret that it is raining. Characteristic o f presuppositions is that if p is a presupposition o f q, p is also a presupposition o f ~ q. This means that presuppositions are preserved under negation, and this is captured in the fact that * is preserved under negation according to the truth tables above. It is a m ajor research question how the presuppositions o f com plex sentences are to be charac terized in terms o f the presuppositions o f its com ponent sentences and this is called the projection -problem o f presuppositions. K leene’s system goes a long way, since it recognizes that the presuppositions o f a com plex sentence may not b e ju st the sum o f the presuppositions o f its parts. Presuppositions o f sentences m ay for exam ple b e cancelled in sentences o f which they are a constituent Consider for instance the conditional I f there is a president o f the U.S., the president is elected. This sentence contains tw o sentences (1) There is a president o f the U.S. and (2) The president is elected. (1) is a presupposition o f (2), since if (1) is false, (2) must be undefined, because it does describe anyone in that case. T he Kleene interpretation o f the con ditional captures this nicely, since p —> q is true even when q is undefined and p is false. So the entire conditional sentence does not have (1) as pre supposition. But K leen e’s interpretation can be seen to lead to problem s in the follow ing sentence (3) I f revolutions are unconstitutional, the president is elected. Now ( l ) does seem to be a presupposition o f (3 ), since if (1) is false, the consequent o f (3) is undefined and hence (3) must be undefined The antecedent may, however, very well be false, when (1) is false, since they are independent. I f the antecedent is false, (3) is true. But if there is an assignment making (3) true and (1) false, (1) cannot be a presupposition o f (3). This seems wrong and not in accordance with our intuitions, Generally speaking, in a conditional with a contingent antecedent whose truth value is independent o f the presuppositions o f its consequent, the presuppositions of the consequent are incorrectly cancelled in the Kleene interpretation o f the connectives. There are ways to m end this problem , but none has yet found general acceptance. The suggested further reading for this chapter contains some m ain references to the literature on presuppositions..
teview Exercises 1. Suppose that P <— > Q is true, what is P V (a )
P
g?
(b )
p ^ q
2
_________________ (p & g ) V r
(c )
p - > ( g - > -----r )
~ g Vp (d)
pVg
P k
~ p\/ r
2 . .
(e )
p k (q —> (r V ~ ~ s ) )
(f )
2________________________ p fc (s v r ) 2. Prove:
p P - * g2 p P -->» ((g 2 --> » » r) 0 g —» (r —» s)
(S )
rV p
(h )
. p —> s
0)
P -2
V
r -> (p V s) g -»(«V t) ( ~ p & ~ t ) —> (~ r
(j)
pvg
qkr . . ~ p (k ) ~ r k ~ (s —> g) . . ~ p 3. Show that the follow ing set o f statements is inconsistent (a ) r fe ( p V g ) (b ) ~ (p k r ) ( c ) ~ (g & r)
246
R
e v ie w
e x e r c is e s
4 . D oes con ju n ction distribute over conditionals? I.e., is ( p k q) —» ( p k r) equivalent to p k (q —» r)? 5. Translate the follow ing expressions to predicate logic, ( a ) A ll horses are quadrupeds, but some quadrupeds are n ot horses.
(b)
D istinct utterances must have distinct ph onem ic transcriptions,
( c ) N ot all trees are deciduous ( d ) Some politicians are honest men ( e ) N o ducks are amphibious ( f ) E very cloud has a silver lining ( g ) Only Rosicrucians experience com plete happiness ( h ) Everything I like is im m oral, illegal or fattening ( i ) I like anything that is im m oral, illegal or fattening ( j ) Everyone wants everyone to be rich ( k ) E veryone wants to be rich 6. For each o f the follow ing formulas give an interpretation in a m odel which makes the form ula false, ( a ) ( ( 3 x ) F ( x ) k (3 e )G (2 :)) —> ( 3 e ) ( F ( e ) k G ( x ))
(b)
(\ /x)(3y)(\J z)B (x,y, z) -> ( 3 y ) ( V x ) ( V z ) B ( x ,y ,z )
7. Formalize and prove with natural deduction ( a ) A ll linemen for the Green Bay Packers weigh at least 200 pounds. M athilda weighs less than 200 pounds. Therefore, M athilda is not a linem an for the Green B ay Packers,
(b)
A ll cabdrivers and headwaiters are surly and churlish Therefore, all cabdrivers are surly.
8. C onstruct B eth Tableaux for ( a ) [ ( 3 e ) F ( e ) - » (V 2:)G (e)]
(b)
( V x ) ( F ( x ) -> G (a:))
* (V x )(G (x ) k ( ~ F ( x ) V B ( x ) ) ) ==> (V x )(G (x ) k
(V*)JET(*))
( 3 x ) ( F ( x ) —>
Part C
ALGEBRA
Chapter 9
Basic Concepts of Algebra 9.1
D e fin itio n o f a lg e b r a
A n algebra A is a set A together with one or m ore operations /*. W e may represent an algebra b y writing (9 -1 )
A = ( A , A , / 2, . . . , / „ )
or by using particular sym bols for the operations, such as (9 -2 )
A = { A , +, x )
The set A m ay finite or infinite, and there m ay be either a finite or an infinite number o f different operations. However, each operation must be finitary, i.e unary, binary, ternary ..... Each ra-ary operation must be a well-defined operation, i.e, defined for all ra-tuples o f elements o f A and yielding a unique element o f A as a value for each ra-tuple (cf. the m apping condition for functions in Section 2.3), These requirements on the operations can be stated in the form o f tw o axioms which each operation in an algebra m ust satisfy, For simplicity, the axioms are stated in terms o f a binary operation o; their generalization to arbitrary ra-ary operations is straightforward. A x i o m 1.
Closure: A is closed under the operation o, i.e. for any a ,b £ A
there is an elem en t c £ A such that a o b = c.
249
C
250
A x i o m 2.
hapter
9
Uniqueness: I f a = a1 and b — b' then a o b = a' o b'.
Closure and uniqueness in appropriate sets are ordinarily considered the m inim al requirements for well-behaved operations. A dm ittin g partial op erations in an algebra is com m on in universal algebra and category theory, which are beyon d the introductory scope o f this book. (See G oldblatt (1979), Gratzer (1971), M acLane and B irkhoff (1983) and for discussion in the con text o f M ontague grammar especially Janssen (1983).) W e shall n ot be concerned with operations that do not satisfy closure and uniqueness Vari ous kinds o f algebras can be obtained by adding farther axiom s to these two basic requirements. W e will study a number o f such algebras in this chapter. W e have already encountered m any structures which are algebras in this sense. T he syntax o f the logic o f statements, for instance, can be represented as an algebra based on the set o f well-form ed statements (S ) and the con nectives as operations: A = ( S, ~ , &, V, —*, *-* ). Similarly, the semantics o f the logic o f statements can be considered as an algebra, based on the set o f truth values and the truth tables, interpreting the connectives as opera tions: B = ( { 0 , 1 } , ~ , &, V, —», *-*), where the connectives are understood as operations on truth values, not as syntactic symbols. W e will see below that there is an im portant connection between the syntactic algebra and the semantic algebra o f such form al languages, which serve as m odels for the syntax and semantics o f natural languages. D
e f in it io n
9,1 A n algebra B is a subalgebra o f an algebra A =
{ A , f i ■ f% , .... , / f ) i f B satisfies the following conditions: B = { B , / f , / 2S , ... . , / n )> wiiere (i) B C A (ii) F or ev ery i, f f = / / TB ; i.e., f f yields the sam e values as ff" when restricted to elem ents o f B, (iii) B is closed under all operations f ?
■
9.2
P r o p e r tie s o f o p e ra tio n s
In Section 1.8 a num ber o f properties o f operations on sets were introduced. W e repeat certain o f these definitions here as properties o f operations in
Spe cia l
251
elements
algebras foi easy reference and add a num ber o f properties o f operations which are frequently encountered in algebraic operations A n operation
from A
o
X
A to B is associative if and only i f for all a, b, c in
A , (a o b) o c = a o [b o c). In an associative operation it is im m aterial in what order repeated applications o f it are m ade.
Set-theoretic union and inter
section and function com position are associative, as are logical con ju n ction and disjunction
Exam ples o f non-associative operations are set-theoretic
difference and division o f real numbers. A n operation
o
mA, aob=boa
from A
A to B is com m utative if and only if for all a, b
X
Familial' com m utative operations are logical conjunction
and disjunction; set intersection and union; and addition and m ultiplication o f real num bers Some non-com m utative operations are subtraction, division and function com position A n operation
o
from A
X
A to B is idem-potent if and only if for all a
in A , a o a = a. Set-theoretic union and intersection are idem potent, as are logical con ju n ction and disjunction. B ut most o f the operations we have encountered are n ot: addition, m ultiplication, subtraction, division, relative com plem entation and fu n ction com position are not idem potent operations. For tw o operations ox and
o 2
b o th from A X A to B ,
distributes over
o 2
if and only i f for all a, b, c in A , a oi (602 c) = (a ox b) 02 (a o i c)„ W e have seen that set-theoretic union distributes over intersection and vice versa. B ut, although arithm etic m ultiplication distributes over addition (a X (b + c) = (a X b) + (a X c)), addition does not distribute over m ultiplication, since in general a + (6 X c) ^ (a + b) X (a + c).
9.3
Special elements
The next three notions are special properties which certain mem bers o f a set may have with respect to some operation defined on the set. Given an operation 0 from A x A to B , an element e; is a left identity elem ent o f 0 if and only if for all a in A , ei 0 a = a. Similarly, er in A is a right identity elem ent o f 0 if and only i f for all a in A , a 0 er — a. As we saw in Section 2 4, for a function F : A —> B , if the operation 0 denotes function com position , then idg 0 F — F and F 0 id& = F .
Thus for the
operation o f com position o f functions the identity functions idg and id a are respectively a left and right identity element. Subtraction defined on the set o f integers and zero has a right identity element, nam ely zero itself, since
C
2 52
hapter
9
fo i all n, n — 0 = n. B ut there is no left identity element; i.e., there is no element m in the set such that for all n, m — n = n For com m utative operations, every left identity element is also a right identity element, and vice versa. To see this, consider a left identity e; . By definition (Va £ A )(e io a = a). Because the operation is com m utative, e ;o a = a o ei = a, for all a £ A, and so e; is also a right identity element. Similarly, every right identity is also a left identity for com m utative operations. A n element that is both a right and left identity element is called a two-sided identity or simply an identity elem ent W hile com m utativity o f an oper ation is a sufficient condition for every right or left identity to be tw o-sided, it is not a necessary condition; a tw o-sided identity m ay exist for some operations that are not com m utative. A n example o f this is fou n d in the operations of com position o f functions defined on some set o f functions F = {i*1, G , H , , , .}, each being a function in A
If id a is one o f these function, it is a tw o-sided
identity, since for each x £ F , id a o x — x o id a — z , but the operation o f com position o f functions is not in general com m utative. For addition the tw o-sided identity is 0, but for arithm etic m ultiplication it is 1, since for allra, r a + 0 = 0 + r a = r a and b x 1 = 1 x » = b. Given some collection o f sets, the identity element for intersection is U, the universal set, and for union it is the em pty set (verify!). Relative com plem entation has 0 as a right identity but in general it has no left identity,. It is provable that if for a given operation a tw o-sided identity exists, then this element is unique Given an operation o from A X A to B with a tw o-sided identity element e, a given element a in A is said to have a right inverse aT i f and only i f a o ar = e. A given element a in A is said to have a left in verse ai if and only if a; o a = e.
If a-1 is b o th a left and a right inverse o f a, i e.
a~l o a — a o a-1 = e, then a-1 is called a two-sided inverse o f a, W h en the term ‘inverse’ is used without further qualification, we m ean that it is twosided. N ote that inverses are always paired in the follow ing way. b is a right inverse o f a if and only i f a is a left inverse o f b, since b oth statements follow from a o b = e
One should observe also that the question o f the existence o f
an inverse can be raised with respect to each element in the set on which the operation is defined,. In contrast, an identity element, if it exists, is defined for the operation as a whole,. To illustrate, let addition be defined in the set Z o f all positive and negative integers and zero. As we have seen, 0 is the tw o-sided identity element for this operation. Consider now the num ber 3, and let us ask if it has an inverse in Z . Is there an element z in Z that when added to 3 yields 0? T he number —3 is such an element, and, furtherm ore, it is b oth a right and a left inverse, since 3 + ( —3) = ( —3) + 3 = 0, F fo m this it
M
aps
and
253
m o r ph ism s
also follows that 3 is a tw o-sided inverse o f —3 For addition, every mem ber o f Z has an inverse, since to each integer z, except 0, there corresponds a negative integer —z, such that z + ( —z ) = 0. The num ber 0 is its own inverse, since 0 + 0 = 0. Given an operation o from A x A to B , an element 0; is called a left zero o f o if and only if for all a in A , 0; o a = 0; Similarly, 0,. is called a right zero o f o if and only if for all a in A , a o 0 f = 0,.. A n element that is b oth a left and a light zero is called a two-sided z e r o , or simply a zero This term inology derives from the fact that the number zero functions as a zero element in arithm etic m ultiplication.
There is n o zero element for
subtraction or division. T he em pty set is a zero element for set intersection and the universal set U is the zero element for set union
9.4
M aps and morphisms
Relations betw een algebras m ay b e described by functions m apping one al gebra in another; F : A —» B
Such a map is injective if some function
F : A —» B is one-to-one, i.e, F (a ) = F (b ) im plies a = b. F : A —» B is surjective (or on to) sf { F ( a ) \a £ A } = B . A n d F : A —» B is bijective i f F is both injective and surjective (or on e-to-one and onto), A m orphism is a mapping F : A —» B con ceived o f dynam ically as a transform ation process o f A into B
I f A = {A , A , .. . , / n) and B = { B , g x, ., . ,g n ) then A and B
are isom orphic i f and only if there is a on e-to-one correspondence between their operations (we will assume for sim plicity that the correspondence is fi
9i)
a on e-to-one and on to function (p m apping A onto B such that
for all s , y, z, ., , in A and all i < n g i(ip (x),ip (y ),< p (z),.
„) = t p { f i { x , y ,z ,.,
)),
A h om om orphism is a correspondence betw een algebras with all the p rop erties o f an isom orphism except that the m apping from A to B m ay be m a n y-to-on e; the set B m ay be smaller than the set A. A n autom orphism o f an algebra A is an isom orphism o f A with itself. The identity m apping ( ^ ( s ) = x ) always provides an autom orphism for any algebra (the “ trivial” autom orphism ); the question generally asked o f a given algebra is whether it has any other ( “n on-trivial” ) autom orphism s. For instance, let A = {S, ~ , & ,V, —>, <-» ), and B = ( { 0 ,1 } , ~ , &, V, —>, <-»): as defined above in 9.1
2 54
C
hapter
9
A n y assignment o f trath-values to the statements in S is a h om om or phism F : A —> B . i.e distinct statements p ,q m ay be m apped to the same trath-value, but
F{ p k q )
=
F( p) k F(q)
F{ p V q) =
F( p) V F{q)
F( p -> q) =
F ( p ) -> F{q)
F( p h ? )
F(p)
=
F(~~ p) =
F(q)
~ F(p)
C onstruction o f truth tables for com plex statements can now be under stood as based on the fact that, given an assignment to the atom ic state m ents, the com position preserves the hom om orphism from the syntactic al gebra to the semantic one. This can be considered to be the algebraic coun terpart o f the Principle o f Com positionality, often also espoused in one form or another for the syntax and semantics o f natur al languages. T he principle requires the meaning o f a com plex expression to be a function o f the m ean ing o f its constituent parts and the way in which they are pu t together (See also Ch
13). H om om orphism s can, o f course, relate semantic algebras, e g.
by em bedding a given interpretation into an extension of that interpreta tion, Extensive applications are m ade o f these em beddings, for instance, in semantic theories based on dynam ic interpretations and in Kripke semantics (see Ch
12).
A simple exam ple o f an algebra A' which is isom orphic to A is a syntax o f the statment logic which uses instead o f p, q, r etc for statements, a different alphabet, say the Greek letters <£, tp, x etc , and possibly alternative symbols for the connectives I f alphabetic variance is the only difference between two logical systems they are isom orphic from an algebraic poin t o f view. Throughout the remainder o f this part o f the b ook we will encounter more interesting m athem atical examples o f hom om orphism s and isom orphism s. C ategory theory, a relatively recent and flourishing developm ent o f alge bra, studies properties o f algebras that can be expressed in terms o f morphisms. It provides a very abstract and universal perspective on the foun dations o f set theory, algebra and logic, in which cross-fertilization yields m any new insights and results, T he interested reader is referred to Goldblatt (1979) for an introduction.
E x e r c is e s
255
Exercises 1. Consider the operation o f intersection defined on some arbitrary col lection o f sets. ( a ) Is there a tw o-sided identity element?
(b)
W h ich sets have an inverse element?
2. Given an arbitrary collection o f sets, what elements, if any, have in verses with respect to the operation o f a) union and b ) symm etric difference? 3. I f for a given operation in an algebra a tw o-sided identity exists, it is unique
Prove this for the operation o f set-theoretic union.
Chapter 10
Operational Structures 10.1
Groups
A group G is an algebra which consists o f a set G and a single binary operation, which we will usually write as o, but which m ay sometimes be written + or X : G = { G, o )„ T o be a group, G must satisfy the follow ing conditions, the group axiom s: G l: G is an algebra (i e„ o com pletely defined and G closed under o). G2: o is associative. G3: G contains an identity element. G4: Each element in G has an inverse element.
N ote that a group operation does not have to be com m utative. A group whose operation is com m utative is a com m utative or Abelian group. W e are already acquainted with some m odels o f these group axiom s. a . The positive rational num bers with m ultiplication form a group: (G l) the prod u ct o f any tw o positive rationals is a unique positive rational, (G 2 ) m ultiplication is associative, (G 3 ) 1 is the identity element, and (G 4) every positive rational p/q has an inverse q/p. Furtherm ore this group is A belian since m ultiplication is comm utative, b . T he integers {0 ,1 , 2, 3 } form a group with the operation o f addition m odu lo 4. (T h e sum o f x and y m odu lo 4 is the rem ainder after dividing x + y by 4; e g. 3 + 7 = 2 (m odu lo 4) ) T he verification o f this will be left to the reader. 257
258
C
h a p t e r
10
c. The set of all even integers under addition form s a group, but the set o f all o d d integers does n ot, since it does not contain an identity element, and it is n ot closed under addition d . T he group o f ‘ symmetries o f the square’ is an exam ple o f a different sort, since the elements o f the set for this group are not num bers but the follow ing rigid m otions o f a square: RR'
a 90° clockw ise rotation abou t its center 0 - a 180° clockwise rotation about its center 0
R" I
-
a 270° clockwise rotation about its center 0 a 360° clockwise rotation about its center 0
H
-
a reflection in the horizontal axis through 0 (i e flipping the square about the horizontal axis)
V
-
a reflection in the vertical axis through 0
D D'
-
a reflection in the diagonal in quadr ant I and III a reflection in the diagonal in quadrants II and IV
The group operation is the successive perform ing or com position o f any o f these m otions: e g , R o R = R! This group is n ot com m utative, since, for instance, R o H = D while H o R = D '. The best way to com pute the products o f this group is to cut out a square o f paper and label its sides so that the m anipulations can actually be perform ed. First consider the front o f the square:
D
A
C
B
Figure 10-1,
Perform ing the operation defined as R will give the orientation shown in Figure 10-2:
G
259
r o u p s
c
D
B
A
Figure 10-2.
Starting from the original orientation and perform ing R! gives the orien tation shown in Figure 10 -3 ; i f we instead perform R " , the result is as shown in Figure 10-4.
B
C
A
B
A
D
D
C Figure 10-4.
Figure 10-3.
P erform ing the operation I from the original starting poin t o f Figure 10-1, or from any other configuration, does not change the orientation at all; in fact I is the identity operation for the group.
T he simplest way to
keep track o f these operations is to label the front o f the square as in Figure 10-5,
I
R
R"
R'
Figure 10-5.
C
2 60
hapter
10
A t this p oin t, the reader can verify such produ cts as R o R = R ', R o R ' — R " , R' o R ' = I. To label the back o f the square, perform each o f the reflections, starting each tim e from the I position, and label the side that comes out on top with the nam e o f the operation. The relevant axes are as shown in Figure 10-6: for V
Figure 10-6. T he back o f the square will then be labelled as in Figure 7, with V labelling the back o f the same side that I labels on the front: V
D
D'
H Figure 10-7. T hen the rest o f the products can be verified; e g ,, H o R! = V , V o D = R . N ote that when you have, for instance, perform ed V and then want to perform D , you must find what is then the appropriate diagonal axis and reflect the square (i.e., turn it over) through that axis; the p rod u ct is R, since the tw o operations in succession lead to the same orientation that R leads to directly. It is recom m ended that such a square actually be constructed since this exam ple recurs in several subsequent illustrations and problem s. From the group axioms we can prove the follow ing elem entary theorems.
G
T h e o re m
261
r o u p s
10.1 In any group, the equations x o a = b and a o y = b have the
unique solutions x
=
b o o ' 1 and y
=
a ~ 1 o b resp ectiv ely
■
Proof ■ ( b o a ~ 1) o a
=
b o (a T 1 o a)
associativity
b o ( a T 1 o a)
=
boe
d e f inveise
=
b
d e f identity
boe
Hence ( b o a - 1 ) o a = b, so x = bo aT 1 is a solution o f s o a = b. It is also the unique solution, since 8 =
D o c = j o ( a o
a ~ 1) = (x o a) o a ~ 1
So if ® o a = b, substitute b for ( x o a ) in the last m em ber o f the equality above and observe that x to a o y = b.
=
b o a -1
Similarly for the unique solution y = a ~ 1 o b ■
The theorem provides an answer to the follow ing general question: Con sider any two elements o f a group, a and b (they need not be distinct); will it be possible to find in the group more elements x such that x o a = bl The theorem says that there will always be exactly one such an element, namely whatever element is obtained by perform ing the operation on b and the inverse o f a The first part o f the p r o o f shows that b o a ~ 1 is indeed such an x, by putting b o aT1 in for x in the p rod u ct x o a and showing from the group axioms that the result must indeed be b. But this does not show that 6 o a -1 is the only such x. The second part o f the p r o o f does this, in an indirect way. First it is established that for a n y element x o f a group, x = ( x o a) o a ~ 1. Now consider an arbitrary element x for which x o a = b. (W e know already that there is at least one such elem ent, but so far there could be m ore than one )
T hen since x = (x o a) o a -1 for any x, i f we have an x for which
x o a = b, we see that for such an x we can deduce x = b o a - 1 ; i.e., any
solution o f x o a = b must be identical to the original solution, nam ely b o a - 1 , which is to say that the solution is unique, C o r o l l a r y 10.1
A group has on ly one id en tity elem ent.
■
P roof: By the group definition, there is at least one solution to e o x = e, i.e. x = e. B y T heorem 10.1, this is the only solution.
■
C
262
C o r o l l a r y 10.2
hapter
10
A group has on ly one inverse a 1 for each elem ent a
■
P r o o f: B y the group definition, there is at least one solution y = a -1 to a oy = e
T
B y T heorem 10 1, this solution is unique,
■
10.2 A group with 4 or few er elem ents m ust b e com m utative, m
heorem
P ro o f: case (i): 1 element - trivial case (ii): 2 distinct elements e and a, e o a = a o e = a (identity)
case (iii): 3 distinct elements e, a and b, e o a = a o e = a (identity) e o b = b o e = b (identity)
But a o b
^
a because then b would equal e; and a o b
^
b because then a
would equal e; hence a o b — e. Similarly for b o a hence a o b = b o a — e, case (iv ): 4 distinct elements e, a, b and c, e o a = a o e = a, e o b = b o e = b and e o c = c o e = c (identity) Consider any two non-identity elements, e.g., a and b.
The produ ct a o b
cannot be either a or b, as above, I f a o b = e, then b = a ~ l and hence b o a — e also. I f a o b = c, then b o a cannot be a or b (violation o f uniqueness o f identity elem ent), or e; in the last case b would be the inverse o f a, so a o b = e, but we already have a o b — c Hence b o a = c also. In either case the group is com m utative. ■ F rom the fact that the theorem only m entions up to 4-m em ber groups, it should not be inferred that groups with 5 or m ore members m ay be noncom m utative. In fact, it is provable, but tedious, that all 5-m em ber groups must also be com m utative. Groups with 6 or more mem bers need not be com m utative, however. T he operation on a finite group is often given b y a m atrix. Rows and columns are labelled with mem bers o f the set. The value o f a o b is placed in the cell at the ath row and the bth colum n. Because o f its similarity to
Su bg ro u ps,
sem ig ro u ps a n d
m o n o id s
263
the m ultiplication table of the natural numbers less than 10, such a m atrix is generally referred to as a ‘m ultiplication ta b le’. Finding the value o f a o b is called ‘m ultiplying a b y b’ even when the operation bears no resem blance to the operation o f m ultiplication
A word o f caution: when constructing
an exam ple o f a finite group by its m ultiplication table, it is quite easy to check for closure, the identity element and inverses by direct inspection o f the table. It is also straightforward to tell from the table whether a group is com m utative or n ot, i..e the table is sym m etric around the diagonal. B ut a group operation m ust also be associative, and there is no simple way to check associativity by inspection o f the table; rather, one has to check each instance o f the operation
10.2
Subgroups, semigroups and monoids
We define a subgroup G' as a subalgebra o f G which is itself a group give the follow ing examples o f subgroups as illustration,
We
a. T he group o f even integers with addition is a proper subgroup o f the group o f all integers with addition, b. The group o f all rotations o f the square ( { I , R , R ' , R " } , o) where o is com position o f operations as described above, is a subgroup o f the group o f all symm etries o f the square as in Exam ple (d ) above, c. T he system { { / , R , R '}, o) is n ot a subgroup o f the group o f all symm e tries o f the square; it is not a group and it is not even a subalgebra o f the original group, because the given set { I , R , R ' } is not closed under the operation o, d. The set o f all non-negative integers with addition is a subalgebra o f the group o f all integers with addition, because the non-negative integers are closed under addition. B ut it is not a subgroup because it is not itself a group: it is associative, and has an identity element (0 ), but all o f the m em bers o f the set except 0 lack inverses.
The order o f any group G is the num ber o f m em bers in the set G . A n im portant theorem o f group theory states that the order o f any subgroup exactly divides, i e, without rem ainder, the order o f the parent group. For instance, only subgroups o f order 1, 2 and 4 are possible for a 4-m em ber
264
C
hapter
10
group, these being the integral divisors o f 4. The theorem does not guarantee that every subset having the proper number o f mem bers will give rise to a subgroup - only that if a subgroup exists, its order is a divisor o f the order o f the group. A n im m ediate consequence o f this theorem is that a group G o f order 5 has only the trivial subgroups o f order 1 (the identity element itself) and o f order 5 (itself) as subgroup, since 5 has no other divisors. A second theorem , o f which we om it the p r o o f since it uses notions that are not introduced here, states that if a group is fin ite, then all its non-em pty subalgebras are also subgroups
The practical consequence o f this theorem is
that in checking whether a given system is a subgroup o f a finite group, one only needs to verify that the given subset is not em pty and is closed under the group operation, i.e. is a subalgebra. I f these tw o conditions are met, there will necessarily be an identity element and inverses for each element. E xam ple (b ) above is such a case. Exam ple (d ) above shows the failure o f a subalgebra to be a subgroup in an infinite case. A third theorem about subgroups will be proven here. Theorem
1 0 .3
The in tersection G ' D G " o f two subgroups
g r o u p G is its e lf a subgroup o f G ,
G'
and G " o f a ■
P ro o f: (i) I f a, 6 are in G ' D G " , they must b oth b e in b oth G ' and G ". G ' and G " are groups, so a o b is in b oth , hence a o b is in G ' fl G " . (ii) I f a is in G ' fl G " , it is in b o th G ' and G ". G ' and G " are groups, so a ~ 1 is in both , hence G ' D G " must contain a ~ 1, (iii) Since G ' and G '' are groups, they b oth contain e; hence G 'fl G ^ m u s t contain e. ■
There are some useful algebraic structures which are weaker than groups and satisfy only some o f the group axioms. A semigroup is defined as an algebra which consists o f a set and a binary associative operation ( G 1 + G 2). There need not be an identity element nor inverses for all the elements A m onoid is defined as a semigroup which has an identity element (G 1, G2, G 3). There need not be inverses for all the elements. A n Abelian monoid is a m onoid with a com m utative operation.
Su bg ro u ps,
sem ig r o u ps
and
m o n o id s
2 65
Given these definitions, any group is a subgroup o f itself, and a semigroup and a m on oid as well, Every m on oid is a semigroup, but not vice versa. Here are some telling examples, a. The set o f all non-negative integers with addition is an A belian m onoid, b. The set o f all positive integers (excluding zero) with addition is a semi group but not a m onoid. Since b oth ordinary addition and ordinary m ultiplication are associative, it can be deduced that addition and m ultiplication m od u lo any n are also associative. Therefore any system with addition or m ultiplication, either ordinary or m od u lo some n, is a semigroup if it is closed and is a m on oid if it also contains the appropriate identity element 0 or 1, c. The set o f all positive even integers with ordinary m ultiplication is a sem igroup but not a m on oid, since 1 is missing, d. The set o f all positive o d d integers with ordinary m ultiplication is a m onoid, (C losed since m ultiplication o f od d integers yields only odd integers) e. The set { 0 ,1 ,2 , 3 ,4 } with m ultiplication m od u lo 5 is a m onoid. f. The set o f all multiples o f 10 which are greater than 100, i.e. {1 1 0 ,1 2 0 ,1 3 0 ,.,,.} with ordinary addition is a semigroup, but not a m onoid. None o f the above examples are groups; in each exam ple, one or more elements lack inverses. Note that where m ultiplication (m odu lo n) is involved no system which contains 0 can be a group, since 0 has n o m ultiplicative inverse. Subm onoids are defined analogously to subgroups. M is a subm onoid o f the m on oid iff M is a m on oid and its identity element is the same as in M '. The stipulation that the identity elements must be the same is not necessary for subgroups, since there it is an autom atic consequence.
It is
possible to find subsets o f a m on oid that themselves form m onoids, however, but with different identity elements.
C
2 66
10.3
hapter
10
Integral domains
A n integral domain D is an algebra consisting o f a set D and tw o binary operations called ‘ addition’ and ‘m ultiplication’ , written a + b and a ■ b, respectively; D = { D , + , ), which satisfies the follow ing axioms: D l : D is an. algebra. D 2: The set D with the operation + forms an A belian group with identity 0,. D 3: The set D with the operation • forms an A belian m on oid with identity 1, and 1 / 0 D4: (C ancellation Law) I f c / 0 and c a = c
b, then a = b,
D5: (D istributive Law) For all a ,b ,c in D , a ■(b + c) — (a ■b) + (a
c).
The assum ption in D3 that 1 / 0 eliminates the ‘trivial’ case o f the set containing only 0, which would otherwise be an integral dom ain under ordi nary addition and m ultiplication. A x io m D 4, which says that m ultiplication obeys the cancellation law except in the case o f the additive identity 0, in effect recaptures a great deal o f the structure lost b y not requiring multi plicative inverses. In fact, whenever D is a finite set, A x io m D 4 insures that every element except 0 has a m ultiplicative inverse N ote that the distributive law is not sym m etric betw een • and + ; it will not in general be true that a + (b ■c) = (a + b) (a + c)„ Aside from the requirement that 1 / 0 , the distributive law is the only axiom which requires there to be some connection betw een the two operations. Because integral domains have two operations, we need to introduce some new notation for inverses. In a group, where there is only one operation, a -1 unambiguously designated the inverse o f a with respect to the given operation. For integral dom ains, we will use a-1 to designate the m ultiplicative inverse o f a (if it has one; since not all elements need have inverses, this n otation can be used only where it can be shown that an inverse exists), W e will introduce the n otation —a to stand for the additive inverse o f a, which by D2 always exists. Thus by definition —a is the element which when added to a gives 0. For all o f the infinite models m entioned below , this n ota tion corresponds to our ordinary use o f the minus sign, but it would be advisable to regard that correspondence as accidental (although clearly it is n ot) and throughout this section simply read ‘ —a ’ as ‘the additive inverse o f a'. In integral dom ains, as in ordinary arithm etic, the minus sign is also used as a binary operator, corresponding to the operation o f subtraction.
Integral
d o m a in s
We can define b — a as that element x such that x + a = b
267
B y Theorem
10.1 foi groups the equation x + a = b has the unique solution b + ( —a ), so the operation b — a is well defined. Thus the tw o uses o f the minus sign are closely related _ a as 0 — a
W e could also have defined subtraction first, and then define There is never any ambiguity, since subtraction is a binary
operation, whereas the sign for the additive inverse is always prefixed to a single element The standard m od el o f an integral dom ain is the set o f all integers, p o s itive, negative and 0, with ordinary addition and m ultiplication. Other ex amples o f infinite integral domains are the set o f all rational numbers and the set o f real num bers, again with ordinary addition and multiplication, A less obvious m od el is the set o f all rational numbers whose denom inator is 1 or a power o f 2„ Still other m odels can b e constructed. None o f these sets form a group with m ultiplication, because there is n o m ultiplicative inverse for 0 in any o f them , i.e n o num ber which when m ultiplied by 0 gives 1. In the dom ain o f the rationals or o f the reals, 0 is the only element without a multiplicative inverse; in the dom ain o f the integers, however, none o f the elements except 1 itself has a m ultiplicative inverse In Section 8 5.5, we introduced an axiom atic characterization o f several types o f orderings for sets in general. Here we will show a different approach to a linear or simple ordering which can be used only for integral domains, since it makes use o f the notions o f addition and m ultiplication. The relation < as defined b elow is a simple linear ordering, although we will not prove that assertion here. Not all integral domains can be ordered; thus the ordered integral d o mains are a m uch m ore restricted class o f systems that the integral domains. More restricted still are the ordered integral domains whose positive ele ments (to be defined below ) are well-ordered. These integral domains in fact turn out to be isom orphic to the set o f all integers with ordinary addition and m ultiplication and, o f course, to each other. For this reason, these n o tions o f ordering are o f central im portance in characterizing axiom atically our ordinary system o f arithm etic with integers D e fin it io n
10.1
A n integral domain D is s a id to b e ordered b y a relation
< i f the follow ing axiom s hold: (i) A d d ition law. F or all a,b,c and d i f a < b and c < d, then a + c < b + d (ii) M ultiplication law. I f a < b and 0 < c, then a c < b ■c
C
268
hapter
10
(iii) Law o f trichotom y. For any a and b, one and on ly one o f the following h old s: a < b, a = b, or a > b (also called con nected ness).
W e have already shown that subtraction can be defined in any integral dom ain: b — a is equal to b + ( —a). We now make use o f subtraction to define the properties o f being p ositive 01 negative for elements o f ordered integral domains D
e f in it io n
1 0 .2
A n elem ent a o f an ordered integral domain is positive
i f and only i f a > 0; a is negative i f and on ly i f a < 0,
T h e three basic
axiom s for the ordering relation < are reflected b y th ree similar properties o f p ositive elem ents: (i1) A ddition, T he sum o f two p ositive elem ents is positive. (ii') M ultiplication. T he p rod u ct o f two p ositiv e elem ents is positive. (iii1) Trichotom y. For any given elem ent a, one and on ly one o f th e following holds: a is p ositiv e, a = 0 or —a is positive,
The p r o o f o f these is left to the reader. As remarked above, not all integral domains can be ordered.
A m ong
the integral domains which can be ordered are the familiar infinite ones: the set o f integers, or the set o f all rational num bers, or the set o f all real num bers, all with ordinary addition and m ultiplication and < interpreted as ‘less than or equal t o ’ in the usual sense. I f we add to the axioms for ordering a further axiom for well-ordering for positive elements only, we will have a form al system all o f whose m odels turn out to be isom orphic to the integers with respect to addition, m ultiplication and < . D
e f in it io n
1 0 ,3
A su bset S o f a n ordered integral domain is well-ordered
i f each n on -em p ty subset S' o f S contains a sm allest elem ent, i.e. an elem ent a such that a < x for every x in S'.
■
The new axiom for well-orderings, which depends on the prior introduc tion o f the axioms for orderings and on the definition o f ‘p ositiv e’ already given, can be stated as follows:
In t e g r a l
2 69
d o m a in s
W e ll-o r d e r in g a x io m : The positive elements are well-ordered. T o illustrate the use o f the well-ordering principle, we prove that in any well-ordered integral dom ain, there is no element betw een 0 (the additive identity) and 1 (the m ultiplicative identity) For the standar d m odel, namely the integers, the theorem m ay seem obvious, but it is not so obvious how to prove it from the axioms for the well-ordered integral domains T
heorem
10.4
T here is n o elem en t betw een 0 and 1 in any well-ordered
integral domain.
■
P roof: Assum e, for a reductio ad absurdum p roof, that there is at least one element c with 0 < c < 1, i.e. the class o f such elements is not empty. B y the well-ordering axiom , there is a least element m in this class, and 0 < m < l , s o 0 < m and 0 < (1 —m ). B y the m ultiplication law for positive elements o f an ordered integral dom ain, 0 < m
(1 — m ), i.e. 0 < m — m 2,
so m 2 < m. B y the same axiom , 0 < m m , i.e, 0 < m 2 Then by the transitivity o f < , 0 < m 2 < m < 1. The m 2 is another element in the class o f elements betw een 0 and 1, which is, m oreover, smaller than m But m was by definition the m inim um element in the class, (contradiction!). So there is no element betw een 0 and 1.
T
heorem
10.5
■
A set S o f p o sitiv e integers which includes 1, and which
includes n -f 1 w henever it includes n, includes ev ery p ositiv e integer.
■
P roof: W e will prove by reductio ad absurdum that the set S', consisting o f those positive integers not in S, is empty. Assum e that S' is not em pty, then it contains a least elem ent m , B ut m ^ 1 by hypothesis. B y T heorem 4 there is no positive integer smaller than 1, so m > 1„ That means m — 1 is positive, Since m is the smallest positive integer not in the set S, m — 1 is in the set S, since m — 1 < m. Then, by hypothesis, (m — 1) + 1 is also in S, but (m — 1) + 1 = m , so m in in S. Contradiction! ■ W e can now prove directly that P ea n o’s fifth postulate (see Sections 8.4 and 8 5,7) holds for the positive integers. P r in c ip le o f F in it e I n d u c t io n , A ssociate with each positive integer n a statement P ( n ) which is either true or false. I f (i) P ( l ) is true, and (ii) for all k, P (k ) implies P (k + l ) , then P (n ) is true for all positive integers n.
C h ap ter
270
10
P r o o f: The set o f those integers k for which P ( k ) is true satisfies the hy pothesis, and hence the conclusion o f T heorem 10.5.
■
T o illustrate the application o f the principle o f finite in du ction in proofs, we will use it to prove one o f the laws o f exponents in integral dom ains, for which we first need to give a definition.
D
e f in it io n
10 4
A p o s it iv e p o w e r an o f a in any integral domain D is
defined recu rsively b y (i) a1 = a, (ii) an+1 = an ■a.
n
Using this definition, we can prove by induction that in any integral dom ain (a ■b)n — an bn for all n. First o f all, we view the statement o f the theorem as expressing a p roperty o f n; P (n ) = (a ■b)n = an bn , This is the first step in setting up any p r o o f b y induction, but it is often left im plicit The p r o o f itself has tw o parts: first we prove P ( 1), and then we prove that for arbitrary k, P (k ) implies P (k f 1). (f)
P (l) : (a -h )1 P ( k + 1) : (a ■b)k (a ■b)k+1
= =
a b a1 b1
b y definition
= =
ak •bk
cond. premise by definition
= -
by definition
(a - b)k ■(a ■b) ( ak ■bk) •(a ■b)
cond. premise
ak (bk ■(a
b))
=
ak (( bk ■a)
b)
= :: =
ak ■((a ■bk) b) ak (a - ( bk ■b)) (a k •a) ■( bk ■b)
com m utativity associativity associativity
=
ak+1 ■bk+l
by definition
[ ( a -6)* = ak ■bk} ->■ [(a . b)k+1 = ak+1 ■bk+1]
associativity associativity
cond
p ro o f
From (i) and (ii) it follows by the principle o f finite in d u ction that P (n ) is true for all positive integers n, i.e. that for all positive n, ( a ■b)n = an •bn. W e have introduced here the well-ordering principle, and hence induction, for the positive integers. A n alternative approach is to state b oth the wellordering principle and induction for the non-negative integers, in which case the first step in an induction p r o o f would be for the case o f n — 0. T he two approaches are interdefinable.
M
10.4
o r p h ism s
271
M orphism s
Since the notions o f hom om orphism and isom orphism are defined for algebras in general, it is possible to define a m orphism betw een algebras o f different sorts, as long as they have the same num ber o f operations. Here we give an exam ple o f a hom om orphism betw een a m on oid and a group, and then discuss some group isom orphism s and an isom orphism betw een tw o integral domains. Consider the m on oid M = (TV, + ) consisting o f all the non-negative inte gers with the operation o f ordinary addition M is not a group because o f the absence of inverses, Let the group G = (G = { 0 ,1 ,2 , 3 ,4 } , + m od 5). W e can define a h om om orphism from M to G b y the fu n ction F ( n ) which maps each non-negative integer in N onto the element o f G which is congruent with it m odu lo 5
(N um bers which are congruent m od 5 leave the same re
mainder after division by 5 ) For exam ple, ^ (1 6 ) = 1, F ( 23) = 3, F (4 5 ) = 0, etc. The fu n ction F establishes a hom om orphism , since F ( x ) + F {y ) ((m od 5)) = F ( x + y) The kernel o f a hom om orph ism F is defined as the set o f elements o f the dom ain o f F which are m apped on to the identity ele ment in the range o f F In this exam ple, the kernel o f the hom om orphism F : M —» G is the set o f all non-negative multiples o f 5 : {0 ,5 ,1 0 ,1 5 , „. ,}. The definition o f isomorphisms for groups is a direct application o f the definition o f isom orphism s for algebras in general. Since a group has only one operation, we can say simply that an isom orphism betw een tw o groups G = (G , o) and G' = (G ', o') is a one-to-one correspondence betw een their elements which preserves the group operation, which m ay be distinct oper ations in the tw o groups.
I.e., if a is m apped to a', and b to b' and vice
versa, then a o b is m ap p ed to a' o' b' and vice versa. P uttin g it m ore for mally, an isom orphism between tw o groups G = (G , o) and G ' = ( G ',o ') is a one-to-one correspondence F : G —» G' such that for all x , y in G, F (x ) o' F (y ) = F ( x o y ). Here are tw o examples o f such group isomorphisms. Exam ple 1: The group o f integers { 1 , 2 , 3 , 4 } under m ultiplication m o d ulo 5 is isom orphic with the group o f the integers { 0 ,1 , 2 ,3 } under addition m odulo 4, with the correspondence
1 <— ►0 ,2 <— ►1 ,3 <— ►3 ,4 <— » 2 This is best illustrated by the group tables
272
C
+
( m o d 4) 0
0
1
2
3
0
1
2
3
10
hapter
X
( m o d 5) 1
1
2
4
3
1
2
4
3
1
1
2
3
0
2
2
4
3
1
2
2
3
0
1
4
4
3
1
2
3
3
0
1
2
3
3
1
2
4
Exam ple 2: The group o f integers { 0 ,1 ,2 . 3 } under addition m od u lo 4 is also isom orphic with the group o f rotations o f the square: let 0 <— > 1 ,1 * — > R , 2<— > R ', 3 <— > R " W e can prove that isom orphisms are equivalence relations on the set o f all groups. T
heorem
10.6 The relation ‘is isom orphic t o ’ is a reflexive, sym m etric and
transitive relation betw een groups.
■
P ro o f: The reflexive property is trivial, since every group is isom orphic to itself by the identity mapping, As for the sym m etric property, let F be an isom orphic correspondence betw een G and G ’ Since F is on e-to-one, it has an inverse f 1-1 which is an isom orphism o f G' onto G
Finally, if F i maps
G isom orphically on to G', and F2 maps G 1 isom orphically on to G " , then the com position o f F 2 ° F i, the function whose value for a given argument a is the value o f F 2 applied to -Fi(a), is an isom orphism o f G with G " .
■
A n isom orphism betw een two integral domains D and D ' is a one-to-one correspondence o f the elements a o f D with the elements a' o f D ', which satisfies for all elements a ,b in D the conditions 1) (a + b)' = a' + b ' 2) (a ■bY = a1 b1 For an exam ple o f an isom orphism betw een tw o integral dom ains, let us start from the follow ing facts about reckoning with even and od d numbers. even + even = odd + odd = even even + odd = odd + even = odd even • even = even • odd = odd ■ even — even odd ■ odd = odd
E
x e r c is e s
2 73
W e can regard these identities as definitions o f operations o f ‘ addition’ and ‘m u ltip lica tion ’ in a new algebra o f the tw o elements ‘even’ and ‘o d d ’ . This algebra is isom orphic to the finite integral dom ain I 2 o f integers m o d ulo 2, with ordinary addition and m ultiplication m od u lo 2, under the corre spondence even <— > 0 and odd <— > 1.
Exercises 1. Show that the integers 0,1,2, and 3 form a group with the operation o f addition m od u lo 4, i e. show that each o f the four group axioms is satisfied. Y ou need not give a full dem onstration o f associativity— just 2 or 3 examples. 2. W h ich o f the follow ing are groups? ( a ) The integers 1,3,5,7,8 under m ultiplication m od u lo 11.
(b)
T he integers 1,3,4,5, 9 under m ultiplication m od u lo 11.
( c ) The system described b y the follow ing m ultiplication table: 0
a
b
a b c
a b c
b a
d
d
c
d
c
d c
d
d a
a
c
b
b
( d ) The system described by the follow ing table:
0
a
b
c
d
a
b
d
a
c
b c
d a
c b
b c
a d
d
c
a
d
b
( e ) The set o f all subsets o f S = { * 1 , 2: 2} with the operation o f set union.
(f)
The same set S as in (e) with the operation o f set intersection.
( g ) The set o f rigid m otions o f a square { I , H , V, R '} and the operation o f perform ing them successively. ( h ) T he set o f rigid m otions o f a square { / , D , R } and the operation o f perform ing them successively.
274
3.
C h ap ter
10
(a ) Draw the group operation table fo i the group o f symmetries o f the square { / , R , R\ R " , H ,V ,D , D '} ( b ) There are three different subgroups having exactly four elements.
Find them and draw their group op eration tables ( c ) There are five different subgroups having exactly tw o elements. F ind them and draw their tables, ( d ) Show which of the subgroups in (b ) are isom orphic ( e ) Show a non-trivial autom orphism for one o f the subgroups o f (b) ( f ) Show a hom om orphism o f one o f the subgroups o f (b ) with one o f the subgroups o f (c) 4 . Prove that the set consisting o f the identity element alone is a subgroup for any group 5.
( a ) Does the set {1 ,2 , 3, 4 ,5 } form a group with m ultiplication m od ulo 6? Justify youx answer
(b)
Show that the set {1 ,2 , 3 ,4 , 5 ,6 } forms a group with m ultiplica tion m odu lo 7,
( c ) Find three different proper subgroups o f the group in (b), ( d ) F ind a set o f integers which forms a group with addition m odulo some n which is isom orphic to the group in (b), ( e ) Can you find a general condition on n which will identify all those n ‘ s for which the set {1 , 2 ,, , „, n — 1 } form s a group with multi plication m odu lo n? Prove your assertion i f possible, otherwise explain why you think it is correct, 6 . Prove that if S is a subgroup o f S' and S' is a subgroup o f S ", then S is a subgroup o f S"„
7.
(a ) The set R o f all strictly positive rational numbers with multipli cation form s a m onoid which is also a group. F in d a sttfc-monoid R q o f R such that R q is not a group. ( b ) Is the set o f all rational numbers with m ultiplication a semigroup?
A m onoid? A group? 8. Determine whether the set-theoretic operation ‘ sym m etric difference’ is com m utative, associative and idem potent. Is there an identity element for this operation?
W hat sets, if any, have inverses?
G iven the set
E
275
xe r c ise s
A = {a , b} what sort o f operational structure is form ed b y the powerset o f A with the operation o f symm etric difference? 9. Let A = { a , b }
Show that {p (A ), U) and (p (A ), D) are b o th semi
groups but not groups 10.
Find an isom orphism betw een them
( a ) Prove that if a, b and c are any elements o f an integral dom ain D ,
then a + b = a + c implies b = c a has an additive inverse.)
(b)
Prove that for all a in D , aoO = Ooa = 0 (H int: Use a + 0 = a and the distributive law to prove that a o ( a + 0 ) = a o a and a o ( a + 0 ) = (a o a) + (a o 0)
(c)
(Hint: make use o f the fact that
Note: o is used here for ‘m ultiplication’ .)
Justify each step in the follow ing p r o o f o f ( —a) o ( —6) = a o b. (N ote: —a and —b are names given to the additive inverses o f a and b)
(1) [ aob + ao ( —6)] + ( —a) o ( —b) = ao b + [oo ( —b) + ( —a) o ( —6)]
(2)
(3)
(4)
[ a o b + a o ( —6)] + ( —a)
( —b) = a o b + [a + ( —a)]
o
[ ao b + a o ( —6)] + ( —a) o ( —6) = a
o
6 -)- 0
o
o
( —6)
( —6)
[ ao b + a o ( —6)] + ( —a) ° ( —b) = a o b
(5) [a o b + a o ( —b)} + ( —a) o ( —b) = a o [b + ( —b)} + (—a) o ( —b)
(6)
( 7)
(8)
[ a o b + a o ( —6)] + ( —a)
[aob + ao
( — 6)] +
o
( —6) = a o 0 + ( —a) o ( —6)
( — a ) o ( — 6) =
( — a) o
( —b)
( —a) o ( —b) = a o b
11. Prove the law o f transitivity for < in an ordered integral dom ain, i.e, for any a,b, and c, if a < b and b < c then a < c.
C
2 76
10
hapter
12. Using the definition o f positive elements, deduce the three basic laws o f positive elements, (i'), (ii') and (iii') in D e f 10 2, fio m the laws for < 13. The definition o f any positive power an o f a in any integral dom ain D is given by:
a}
=
a
( a ) Prove by induction that am o an = am+n in any integral domain. (Hint: Use in du ction on n; in the second part o f the induction, assume that am o an = am+n for all m and for n — k and prove that it must then hold for all m and for n = k + l ) .
(b) Prove by induction that in any integral dom ain (am)n = (an)m. (Hint: Y ou will probably want to make use o f the th eorem that an o bn = (a o b)n which was proven in this chaptei )
Chapter 11
Lattices 11.1
Posets, duality and diagrams
In the previous chapter we have seen that the arithm etical properties o f ele ments o f form al systems m ay be described in operational structures. Opera tions m ay serve to generate new elements from a given set o f basic elements, and thus we m ay view an operational or an algebraic structure naturally as a syntactic system which generates elements in a form ally precise way. The relation of this dynam ic con ception o f such systems and the linguistic notion o f a grammar which gener ates strings as elements o f a natural or form al lan guage will be explored in m uch more detail in Part E T he present chapter is concerned with certain ordering relations betw een elements o f systems or domains o f ob jects and the order-theoretic or ‘top o lo g ica l’ properties o f such ordered structures. We will see that the concepts introduced in this chapter provide a universal perspective on set theory and algebra in which im por tant correlations betw een the tw o mathem atical theories can be insightfully described. R ecently linguistic applications o f lattices have been made pri marily to sem antic topics such as plural N Ps, mass terms and events, using the ordering relations to structure the domains o f an interpretation o f a lan guage, The poten tial usefulness in linguistics o f syntactic applications o f lattice theory is explored in research on feature systems, for instance In this chapter we will introduce lattice theory w ithout paying attention to any particular linguistic applications or m otivations In Chapter 3 we p oin ted out the set-theoretic im portance of'partial or derings on sets, i.e.
sets o f ob jects ordered by a reflexive, anti-sym m etric
and transitive relation Here we will call any partially ordered set A together 27 7
2 78
with its ordering < , i.e,
C
hapter
11
( A , < ), a poset, often writing just A and tacitly
assuming the intended partial ordering, which is widely accepted practice, A poset which also satisfies the p roperty o f linearity (for all a, b £ A : a < b or b < a) is called a chain, or a fully or linearly ordered s e t, A d d i tional properties and operations m ay be defined on posets which constitute a stronger structure. Thus the real numbers form a poset, but also a chain, disregarding the arithmetical operations I f A is a poset and a, b £ A , then a and b are comparable elem ents or comparable objects if a < b or b < a I f a and b are not compar able, they are incomparable, written as a || b In a chain there are o f course n o incom parable elements. In an arbitrary poset A we define an upper bound o f B C A as an element a £ A , if it exists, such that for all b £ B , b < a. A n upper bound a o f B is the least upper bound o f B (abbreviated to lub o f B ) or the suprem um o f B (abbreviated to sup B ) if, for any upper bound c o f B , we have a < c. We often write a = V B , or a = su p B , since by antisym m etry o f the ordering relation we know that if B has a least upper bou n d , this is a unique least upper bound. I f ( A , < ) is a poset, then inversion o f the partial ordering preserves the poset characteristics, i,e. writing a > b for b < a in the given poset we have defined a new poset ( A , > ), Verification o f the three requirements on a partial order in this new poset ( A , > ) is straightforward: e.g., antisym metry holds since if a > b and b > a, the definition o f > gives us b < a and a < b, and we know that in ( A , < ) in that case a = b We call (A , > ) the dual of {A , < ), which is obviously a sym m etric relation betw een posets. This notion will com e in handy in proving statements about posets, since it allows us to replace all occurrences o f < in a true statement S b y > , thus obtaining the (equally true) dual S' o f S , without actually carrying out the entire p roof for the inverse o f the partial ordering. T o appreciate the im portance o f this duality in posets, we define the dual o f an upper bou nd o f B C A , called a low er bound o f B C A , as an element a £ A such that for all b £ B , b > a which is equivalent to a < b A lower bound a o f B is the greatest low er bound o f B (abbreviated to gib o f B ) or the infimum o f B (abbreviated to in f B ) if, for any lower bound c o f B we have a > c. W e write a = (\ B , or a = in f B Suprem um and iniimum are thus duals; hence whatever we m ay prove about one o f them holds also of the other. For instance, we proved above that if a subset B in a poset has a supremum, it has a unique supremum, so we know by duality that the infim um o f B , if its exists, is unique
P O S E T S , D UALITY AND DIAGRAMS
279
Partial orderings m ay be represented visually by so called Hasse dia grams
The diagram o f a poset { A , < ) represents the elements or objects
by o, and if the ordering relation holds betw een tw o elements, they are connected by a line, reflecting the order from b o tto m to top in the represen tation For instance, writing out the ordering set-theoretically, let the poset A = { ( 0 ,0 ) , (0, a ), (0 ,6 ), (0 ,1 ), (a, a ), (a, 1), (6 ,6 ), (6 ,1 ), ( l , 1 )}. Assum ing re flexivity and transitivity o f the connecting lines, we represent A by the dia gram in Figure 11-1 ( c f the im m ediate successor diagrams o f Section 3-5)
1
0 Figure 11-1: The diagram o f a poset A = ({0 , a, 6 ,1 }, < )
W e call a diagram planar or flat i f it does not contain any intersecting lines, as in Figure 11-1. In general greater clarity o f representation is ob tained if the number o f intersecting lines is kept as small as possible. F rom Figure 11-1 we can read o ff that 0 < 1 since we assumed transitivity o f the connecting lines, and also we generally know that x < x for any arbitrary element x. We say that a covers b (or that 6 is covered by a) if a > 6 and for no c : a > c > 6 (R ecall that a < b means a < b and a ^ 6.) B y induction on the length o f chains, we m ay prove that the covering relation determines the partial ordering in a fin ite poset. Diagrams usually represent finite posets, but infinite posets are sometime partially represented b y diagrams and need further explanation in the text Note that the real and the rational numbers, despite their essential order-
Chapter
280
11
theoretic differences, are represented b y the same linear diagram, due to the ‘p overty’ o f the covering relation which determines the diagram. D ualization o f a given poset turns the diagram upside down, but preserves the connecting lines Set-theoretic inclusion induces a natural partial order on the power set o f a given set A , i.e , p ( A ) is a poset. W e represent this inclusion relation on the power set for the set A = {a, b, c } in Figure 11-2
Figure 11-2: The diagr am o f the poset p ( A ) for A = { a, b, c}„
Intersecting Lines m ay not define an element
In Figure 11-3 a poset is
represented in which all pairs o f elements have an upper and lower bound, but these are not always unique. E .g., b o th c and d are upper bounds for { a, b}, but neither c nor d is a supremum for {a, &}.
11.2
Lattices, semilattices and sublattices
There is a special class o f posets, called lattices, which have proven to be very im portant in the general study o f a variety o f m athem atical theories includ in g analysis, topology, logic, algebra and geometry. T hey have led to many
L a ttic e s ,
sem ila t t ic e s
and
su b l a t t ic e s
281
Figure 11-3: The diagram o f a poset with non-unique upper and lower bounds
fruitful interactions and new results in various theories and to a p rodu ctive development o f universal algebra and more recently category theory. L in guistic applications o f lattice theory are currently being developed in syntax and semantics There are tw o ways o f defining lattices, one from a given poset and the other, m ore in line with the group-theoretic definitions, by requiring properties on operations W e present the tw o definitions in this order A poset { A , < ) is a lattice i f su p {a , b} and in f{a , b} exist for all a ,b £ A. We will introduce tw o new operations a K b — in f {a , b }, calling a A b the m eet o f a and 6, and a V b = s u p {a , 6 }, calling a V b the jo in o f a and b. In lattices the operations o f meet and join are always binary, i.e ., we m ay view them dynam ically as maps from A x A to A This allows us to characterize a lattice as an algebra, i e, as a non-em pty set with tw o operations with certain algebraic properties. T he four properties characteristic o f lattice operations are: (L I) a A a = a, aM a — a (L 2) a A b = b A a , a \ / b = b \ / a (L 3) ( a A b) A c = a A (b A c)
idem potent law com m utative law associative laws
(a V b) V c = a V (b V c) The im portant fourth p roperty o f lattice operations connects the tw o operations. N ote first that i f a < b, then in f {a , 6} = a, i e. a A b = a, and
282
C hapter
11
dually, i f a > b, then su p {a , 6} — a, i e , a V 6 = a.
Since a < a V b by
definition o f su p {a , b}, we let a V b substitute for b in the first equations to derive a A (a V b) — a, Similarly, since a > a A b by definition o f in f{a , b}, we derive from the second equations a V (a A b) — a. Thus we have the two absorption laws:
(L 4 ) a A (a V b) = a a V (a A 6) = a
ab so rptio n law
A ny algebra with two binary operations that have these fou r properties
( L l ) - ( L 4 ) constitutes a lattice. It will often be very useful to view lattices as algebras, since all that we know about algebraic structures can readily be transferred to lattices. In fact, we often make use o f the follow ing theorem, provable from ( L 1 ) - ( L 4 ) , about the connection betw een lattices represented as posets and lattices represented as algebras. T
heorem
11.1
(i) L et th e p o s e t A = (A, < ) b e a lattice. Set a h b = in f {a , b j and a\/b = s u p {a ,b } Then the algebra A a = (^4,A ,V ) is a lattice. (ii) L et the algebra A = {^4, A ,V ) b e a lattice. Set a < b if f a A b = a. Then A p = {^4, < ) is a p o s e t and the p o s e t A p is a lattice. (iii) L et the p o s e t A = {^4, < ) be a lattice. Then ( A a )p = A . (iv ) L et the algebra A = (A, A, V) b e a la ttice. Then ( A p ) a = A .
■ Proof, (i) W e leave it to the reader to verify that the meet and jo in operations as defined in (i) satisfy ( L l) - ( L 4 ) „ A bsorption becom es a = sup {a , in f {a , &}}, which is clearly true since in f{a , b} < a. (ii) Prom a A a = a follows a < a (reflexive). I f a < b and b < a then a A b — a and b A a = 6; hence a = b (anti-sym m etry). I f a < b and b < c then a A b = a and b A c = b, so a — a h b — a h (b he) = (a A 6 ) A c = a A c ; hence a < c (transitivity). I.e., < is a partial order on A. To show that this poset is a lattice we verify existence o f sup and in f for any a, b in A
From
a = a A (a V b) and b = b A (a V b) follows a < a V b and 6 < a V 6 . So a V i> is an upper bound o f both a and b. W e now want to show that it is a least u pper bou n d , i.e., if for some c, a < c and b < c, then a V b < c, Suppose
L
a t t i c e s
,
se m il a t t ic e s
and
s u b l a t t ic e s
283
a < c and b < c then a V c = ( a A c ) V c = c and similarly for b V c = c, so (a
V c)
V (b V c) = c V c = c
Hence (a V 6) V c = c. A bsorption gives us
(a V 6) A c = (a V 6) A [(a V b) V c] = a V 6, i.e. aV b < c. Thus a V b = s u p {a , &}. Dual reasoning gives us a t\b — in f{a , b} (iii) and (iv ) check to see that the orderings in (A a)p, A and (A p)a are the same.
■
These facts guarantee a sm ooth transition between lattices as posets and as algebras. W e m ay choose whatever perspective is m ost convenient for our purposes, while knowing that all results will be preserved when the same lattice is represented differently Duality in lattices as algebras is sim ply obtained b y exchanging the two operations in any statement about lattices. Next we consider parts o f the structure o f a lattice and we will see that the algebraic definition and the order-theoretic definition o f a lattice show some discrepancy concerning their substructures. I f L is a lattice and L ' is a n on-em pty subset o f L such that for every pair o f elements a, b in L ' b o th a A b and a V b are in L ' (where A and V are the lattice operations o f L ), then L ' with the same operations restricted to L ' is a sublattice o f L. I f L ' is a sublattice o f L , then for any a, b in L ' a < b is in L ' iff a < b is in L, But note that for a given lattice L there may be subsets which as posets are lattices, but which do not preserve the meets and joins o f L , and hence are not sublattices o f L. A n exam ple is given in Figure 11-4 where L = ({a , 6, c, d, e } , < ) and L ' = ({a , c, d, e } , < ' ), which is a lattice as poset but which is not a sublattice o f L, since in L c V d = b whereas in L' c V d = a In the next section we will com e to understand the reason for defining the sublattice n otion algebraically, rather than on the poset representation o f a lattice. For the present it suffices to note that the algebraic sublattice notion is stronger than the sub-poset which is also a lattice. It is im portant to realize that the above theorem about the equivalences betw een poset representation and algebraic representation o f a lattice may break down once we have to consider parts o f their structure. There are lattice-theoretic structures which are ‘weaker’ in the sense o f representing just parts o f a lattice with less o f its structure. The follow ing notions are special cases o f sublattices called semilattices A poset is a join semilattice i f s u p {a , b} exists for any elements a, b. Dually, a poset is a meet semilattice i f in f{a , b} exists for any a, b. In a diagram, conventionally, a jo in sem ilattice is represented top-dow n, and a meet sem ilattice b ottom -u p .
There are again equivalent
284
C
h a p t e r
11
a
Figure 11-4: The sub-poset L ' = ( { a , c , d , e } , < ) is a lattice, and a subalgebra, but not a sublattice.
algebraic definitions: I f ( A, o) is an algebra with one binary op eration o, it is a semilattice i f o is idem potent, com m utative and associative. T heorem 11-1 for poset and algebraic representations o f lattices holds with the appropriate m odifications for b o th kinds o f serrulattices. T
heorem
11.2
(i) L et th e p o s e t A = ( A , < ) be a jo in sem ilattice Then the algebra A a = (A , V) is a semilattice. (ii) L et th e algebra A
=
(A,oj
b e a semilattice.
S et a V b = s u p {a ,b }.
Set a < b iff a o b = b.
Then A p = { A , < ) is a p oset and th e p o s e t A p is a jo in sem ilattice. (iii) L et the p o set A = (A , < ) be a join sem ilattice
T hen ( A a) p = A .
(iv) L et th e algebra A = { A , V) be a sem ilattice. T hen ( A p )a = A .
T he p r o o f is deferred to the exercises
M
e
o r p h is m s
in
285
l a t t ic e s
b
Figure 11-5: A lattice with examples o f join and meet semilattices.
11.3
M o rp h ism s in la ttic e s
M appings from one lattice to another compare their structures, algebraically or order-theoretically. T w o lattices L i =
(L 1,A,\/) and L 2 =
(L 2, A, V) are (algebraically)
isom orphic if there is a bijection F from L i to L 2 such that for every a, b in L\ (i) F (a V b) = F ( a ) V F( b) , and (ii) F ( a A b) = F ( a ) A F( b) I f tw o lattices are isom orphic, F is called the lattice isom orphism , N ote that P -1 is also a lattice isom orphism , i f F is, and that if F : L i — > L 2 and F ' : L 2 — > L 3 are lattice isom orphism s, then their com position F ' o F : L i — > L 3 is also a lattice isom orphism . Isom orphism o f lattices as posets is defined b y requiring the bijection to be order-preserving. I f P^ - ( P1, < ) and P 2 = (P 2 , < ) are tw o posets and F : P i — > P 2 , F is called an order-preserving map if F ( a ) < F( b) holds in P 2 whenever a < b holds in P i , Sometimes an order-preserving map is called a m on oton e or an isoton e m apping. T
heorem
113
T w o p osets which axe lattices
= (L i , <
) and L 2 =
(L 2, < ) are (order-theoretically) isom orphic if f there is a bijection F : L i —> L 2 such that both F and P -1 are order-preserving.
C
286
hapter
11
Proof. ( = ^ ) I f F : L i —» L 2 is an. isom orphism and a < b holds in L i then. a = a A b, so F( a ) = F ( a A b) — F ( a ) A F ( 6), therefore .F(a) < F ( 6), and F is order-preserving. Dually, the inverse o f an order-preserving isom orphism is also order-preserving. (<£=) Let F : L i —» L 2 and its inverse i *1-1 be order-preserving. I f a, b in L i then a < a V b and b < a V 6, so F ( a ) < F ( a V b) and F( b) < F ( a V b), therefore F ( a ) V F( b) < F(a\/ b). Suppose F ( a ) V F( b) < c, then F( a ) < c and F( b) < c, and then a < F _ 1 (c) and b < F ~ 1 (c), so (a V b) < F ~ 1 (c ) and therefore F { a V b) < c. It follows that F ( a ) V F( b) = ^ ( a V 6) provable that F ( a ) A F ( 6) = F ( a A 6),
Dually, it is ■
The diagrams can represent such order-preserving m appings clearly. Fig ure 11-6 shows an order-preserving bisection F ( a ) = a , . . , , F( d) = d from a lattice to a chain which is not an algebraic isom orphism.
a
Figure 11-6: A n order-preserving bisection which is not an isom orphism
The follow ing notions are weaker than isom orphism s, and often suffice to characterize the structural similarity betw een tw o dom ains, especially when the m appings are intended to represent inform ation-preserving relations. A hom om orphism o f the semilattice
= {S 1 , 0) in to the semilattice
S 2 = {*52, °) is a m apping F : S i —» S 2 such that F (a 0 b) = F ( a ) 0 F(b). Since any lattice consists o f a join and a meet sem ilattice, this h om om or phism notion is split into a join hom om orphism and a m eet hom om orphism .
F
il t e r s
an d
287
id e a l s
A (full) lattice hom om orphism is a map that is b oth a join and a meet hom om orphism , i.e. a map F such that F (a V b) = F ( a A b ) = F(a) A F ( b ) .
F ( a ) V F( b) and
N ote that lattice hom om orphism s and join and meet hom om orphism s are order-preserving, but the converse is not generally true. In Figure 11-7 the three diagrams show the distinct notions; (11-7 1) is an order-preserving m apping that is neither a jo in nor a meet h om om orphism ( c f Figure 116 ), (11-7.2) a join h om om orphism that is not a meet h om om orphism and (11-7.3) a (full) lattice hom om orphism .
(7.1)
(7 2)
(7,3)
Figure 11-7: A n order-preserving m apping, a join hom om orphism and a lattice hom om orphism
Finally we define an embedding o f a lattice L i in to a lattice L 2 as an isom orphism F from in to a sublattice o f L 2 I f such an em bedding exists, L 2 contains a copy or an image o f as sublattice. This notion will be useful in determ ining whether a given lattice has a special structure, as we will see below in Section 11 5.
11.4
F ilte rs a n d ideals
In a lattice we may construct various special subsets with nice properties based on their closure under the ordering relation and the operations. A n ideal I o f a lattice L is a n on-em pty subset o f L such that b oth o f the follow ing hold: (i) i f a £ / , b £ L and b < a, then b £ I (ii) if a, b £ I , then (a V b) £ I
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11
A n ideal I is proper i f I ^ L and I is maximal i f it is not contained in another proper ideal o f L D ualizing these notions, we define a filter F o f a lattice L as a non-em pty set o f L such that both o f the follow ing hold: (i) if a € F , b 6 L and b > a, then b 6 F (ii) if a, b 6 F , then (a A b ) £ F A filter is proper if F 7 = L and F is m axim al if it is not contained in another proper filter of L M axim al filters are often called ultrafilters. The set o f ideals and the set o f filters o f a lattice are closed under finite intersection, and under arbitrary intersection in case the intersection is not em pty (p roof is an easy exercise). This finite in tersection p rop erty guaran tees existence o f the least ideal generated b y any non-em pty subset X C L, written as (X ], I f X is a singleton { » } C L , then we often write (» ] and call this a principal ideal (Dually, the filter [X ) generated b y X C L , etc ). I f L is a lattice and I (L ) the set o f all ideals in L , then I (L ) is a poset with set inclusion and constitutes a lattice, called the ideal lattice. Together with the (provable) claim that any n on-em pty subset of I (L ) has a supremum, which makes I (L ) a com plete lattice, we may prove that L can be em bedded in I ( L ) b y an em bedding function G ( x ) = (»]. Sometimes the im age o f this em bedding G is called the ideal representation o f a lattice (dually, filter rep resen ta tion ), The p r o o f appeals to a form o f the axiom o f choice but requires no ingenuity and can be found in any standard reference on lattices (e.g. Gratzer (1971)),, T o illustrate this notion o f an ideal representation, consider the following simple lattice L = ( { a , b , c , d } , < ) in Figure 11-8, T he set of all ideals in L , I (L ) , consists o f {a , 6, c, d}, { b, c, d}, { b, d} , {c, d} and the principal ideal {d }, (W h y is e.g. {a, b, d} not an ideal?) L can be em bedded into I (L ) by the follow ing em bedding function: G : L —> I(L ) such that G( a) = { a , b , c , d } G( b)
={b,d}
G( c ) = { c , d } G (d ) = { d } The ideal representation o f L is { { a , b, c, d } , { b , d} , {c, d}, { < i } } , The follow ing notions provide ‘b ou n d s’ to a lattice in a very general way. A n elem ent a o f a lattice L is join-irreducible if a = b V c implies that a — b
F
il t e r s
and
id ea l s
2 89
a
d Figure 11-8
01 a = c; dually, a is m eet-irreducible i f a = b A c implies that a = b 01 a — c. We call a lattice L = (L , A, V, 0 ,1 ) a bounded lattice if (i) (L , A, V) is a lattice (ii) x A 0 = 0 and x V 1 = 1, for any arbitrary element x, These notions will again be useful in Chapter 12 T he follow ing is an im portant theorem establishing a connection between join hom om orphism s and ideals. T
heorem
11,4 I is a p rop er ideal o f the la ttice L iff there is a. join h om o
morphism G from L on to the tw o elem ent chain C = { { 0 ,1 } , < ) such that I = G _1 (0 ), i.e I = { x \ G { x ) = 0}, ■ Proof. ( = » ) / is a proper ideal and let G be defined b y G ( x ) = 0 if x £ I and G ( x ) = 1 i f x
I, which obviously is a jo in hom om orphism ,
(•$=) I f G : L —» C is a jo in hom om orphism and I = G _ 1 (0 ), then for any a,b £ 1, G( a ) = G( b) — 0. So G (a V b) — G( a ) V G( b) = 0 V 0 = 0, hence (a V b) £ I If a £ I and x £ L with x < a, then G( x ) < G( a ) = 0, i e G( x) = 0, so x £ I. Furtherm ore, G is onto, so I ^ L, i.e I is p roper O f course T heorem 11.4 m ay be dualized for proper filters.
■
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11.5
hapter
11
Com plem ented, distributive and m odular lat tices
In this section we will discuss some particularly well-known lattices which have additional properties and operations providing m ore structure. In a bou nded lattice L we call the least element a, i.e a < b for any b £ L, the bottom or zero o f L , conventionally writing it as 0. Similarly, the greatest element in a bou nded lattice is called the top or unit o f L, conventionally written as 1. A bounded lattice L = (L, A, V ,0 ,1 ) is said to be complemented if for each a £ L there is a b £ L such that (C l) a V b = 1 (C 2 ) a A b = 0 and b is called the complement o f a In general an element in a lattice may have m ore than one com plem ent or none at all. A lattice L is relatively complemented if for any a < b < c in L there exists d in L with (R C l) b A d = a (R C 2 ) b V d = c and d is called the relative complement o f b in the interval [a, c], A lattice L is distributive if it satisfies either one of the distributive laws ( D l ) a A (b V c) = (a A b) V (a A c) (D 2) a V (b A c) = (a V b) A (a V c) Since ( D l ) entails (D 2 ) and vice versa (see exercises), satisfaction o f either one suffices for a lattice to be distributive It is im portant to realize that any lattice already satisfies the tw o lattice inequalities (L I l) [(a A b) V (a A c)] < [a A (6 V c)] (LI2) [a V (b A c)] < [(a V b) A (a V c)] Hence to check for distributivity o f a lattice it suffices to check the in verses o f these inequalities, which together entail ( D l ) and (D 2 ) A lattice L is modular if it satisfies the modular law (M ) a < b —» [a V (b A c) = b A (a V c)]
C
om plemented
,
d ist r ib u tiv e
and
modular
lattice s
291
A gain since any lattice satisfies a < b —> [& A (a V c) < aV (b A c)] checking the inverse inequality suffices to dem onstrate m odularity in a lattice. The follow ing theorem is str aightforward. T
heorem
11.5 E v ery distributive la ttice is m odular
Proof. I f a < b, a\/ b = b and use this in (D 2 ).
■ ■
N on-m odularity and non-distributivity o f a lattice can be verified by em bedding tw o special five element lattices in to it, represented b y the dia grams N M (N on -M od u larity) and ND (N on-D istribu tivity) in Figure 11-9. These results belong to the core o f lattice theory, and are due respectively to Dedekind and to the founder o f lattice theory, B irkhoff
NM
ND
Figure 1 1-9: Diagrams for non-m odularity and non-distributivity
h e o r e m 11.6 bedded in to L
T
(D ed ekin d ) L is non-m odular iff diagram N M can b e em ■
Proof. (<£=) In N M a < b, but it is not the case that a V (b A c) = b A (a V c), so L contains a cop y o f a non-m odular lattice and hence cannot itself be modular. ( = > ) Suppose L does not satisfy the m odular law, then we will construct a diagram isom orphic to N M as sublattice. For some a, b, c in L we have a < b, but a V (b A c) < b A (a V c)
Let a i = a V (b A c) and b\ = b A (a V c)
cAb T hen c A &i
and c V a i
= c A [b A (a V c)] = [c A (c V a)} A b
com m utative, associative laws
~ cA b
absorption
= c V [a V (b A c)] = [c V (b A c)j V a
com m utative, associative laws
= cV a absorption Since c A b < aj < bj we have c A b < c A
< c A J j = c A b, so
c A ai = c A bi = c Ab Similarly for c V bi = c V a! = c V a It is easy to see that the above diagram is isom orphic to and hence a cop y o f N M ■
T heorem
117
em b ed d ed in to
(B ir k h o ff)
L is a n o n - d i s t r i b u t iv e l a t t i c e i f f N D can b e
L
■
Proof. (^==}a V (b A c) = (a V b) A (a V c) does not h old in N D , so if ND can be em bedded in L, it cannot be a distributive lattice ( = ^ ) Suppose L is non-distributive, i e for some a,b,c € L , [(aAfe)V(aAc)] < [a A (b Vc)]. Assum e also that L does not contain a copy o f N M as sublattice, i e j £ is modular.
Define the follow ing elements, in order to construct a
sublattice in L which is isom orphic to ND, d =
(a A b) V (a A c) V (b A c)
e =
(a V b) A (a V c) A (b V c)
aj =
(a A e) V d
b\ =
[b A e) V d (c A e) V d
cx = Now d < a i ,b i ,c i < e
C
omplemented
,
d ist r ib u tiv e
and
modular
lattices
293
e
W ith absorption we derive from (a A
a A d = a A ((a
e) = a A (b V c),
that
A 6) V (a A c) V (6 A c))
Modularity allows exchanging a and (a A 6) V (a A c) = ((a A 6) V (a A c)) V (a A (6 A c)) = (a A 6) V (a A c) Now it follows that d < e To show that the above diagram is a copy of ND in L, we prove
b\ V Ci
a\ A b\ = a\ A
ci =
b1 A
Ci =
d
and
dj
V
b1
= % V Ci =
= d
W e prove this here for one case only, the others are similar. ®i A
b\ =
((a A
e))l_d) A
((6 A e) V rf)
e) V rf) V d ((a A e) A ((b V d) A e) V d ((a A e) A e A (6 V rf)) V d ((a A e) A (6 V d)) V ((a A e) A ((6 A
modularity modularity comm, assoc idempotent
(a A (6 V c) A (6 V (a A c))) V
absorption
(a A (b V 6 V c)
modularity
A a A c))) V
(a A (6 V (a A c))) V (a A c) V (6 A
a) \J d
a Ac < b\/ c modularity
d
■ The following theorems indicate clearly the force of complementation in distributive lattices and correlate it to the weaker notion of relative comple mentation.
11.8 In a
T
heorem
a
unique complement
distributive, complemented lattice each element a has a” ,
■
C
29 4
Proof.
hapter
11
Suppose there were tw o com plem ents a" and
bx
o f a, then a* =
a" A 1 = a" A (a V b“ ) = (a* A a) V (a* A &*) = 0 V (a* A &*) = a" A b* — a* A so a* = 6*.
similarly ■
distributive lattice relative complements are unique, if
T h e o r e m 11,9 In a
they exist Proof.
■
Let L be a distributive lattice with
b o th relative com plem ents o f
b in
d—
a < b < c in L [a, c], then
If
d
and
d1 were
the interval
dAc (b V d')
dA
d!)
(d A 6) V (d A (d A d')
Similarly, d' = (d A d1), so d = d'
distributive lattice, ifa has a complement, then it has a relative complement in any interval containing it. ■ T h e o r e m 11.10 In a
Proof.
Take
b<
a <
c and
a
Ax
aV
= a
x—
A
((d V
b) A c) =
a V ((d V
b) A c) —
[b,c],
T o prove a
((a A d) V (a A
(a V d V
b) A
A x — b and
b)) A c =
(0 V
b) A c x — c.
= (d V
a V
b) A c = b,
(a V c) = 1 A (a V c) = c
In a distributive lattice, if a and b have complements a“ respectively, then a A b and a V 6 have complements (a A b) ' and V 6)*, respectively, and the de Morgan identities hold:
T h e o r e m 11,11 and 6*,
(a
x
let d be the com plem ent of a and
the relative com plem ent o f a in
(i) ( a A bY (ii)
= a* V 6*
(a V 6)* = a ' A 6*
E xe r c ise s
Proof-
295
B y Theorem 11.8 we only need to prove (i) by verifying
(a A b) A ( a“V £>“) (a Ab) A (a* V 6“)
=
= 0 and
(a A b) V ( a “ V b“ )=
(a A 6 A a*) V (a
1
A 6 A 6") = 0 V 0 = 0 ,
(a A 6) V (a* V i>”) = (a V a” V T ) A (t V a “ V ii") = 1 A 1 = 1 The proof of (ii) is an exercise
Exercises 1. W hich of the posets in the diagrams of figures 11- 1 , 11-2 and 11-3 are lattices? 2.
(i) W hich o f the following sets of sentences can be formally repre sented as posets (each name corresponds to an element): (a ) Alan is a descendant of Bob and Carol Carol is a descendant of David and Eliza, Bob is a descendant of Fred and Gladys ( b ) as in (a) adding: Fred and Eliza are descendants of Henry and Isabella, ( c ) as in (a) adding: Everyone is a descendant of Adam, (d ) David and Eliza, who told Fred about it, were told by Bob and Carol, after Alan told them both, (e ) Jane told Jim and Joseph, who either told Jenny directly or she heard from Julius who had heard from them. (ii) Draw diagrams for the posets in (i) and indicate which are semi lattices and/or lattices, (iii) For the lattices in (i) compute all meets and joins
3 . Describe the poset formed by the power set of a four-element set and draw its diagram, W h a t corresponds to the set-theoretic operations in an algebraic representation of this lattice? Check whether they satisfy (L 4 )„
4 . Formulate and prove the dual of Theorem 11.2 for meet semilattices. 5 . Draw a diagram for a meet homomorphism which is not a join ho momorphism from a four-element lattice to a four-element chain and prove it does not represent a lattice homomorphism.
296
C
hapter.
11
6. Prove that the distributive laws ( D l ) and (D 2 ) are equivalent. 7. Prove that in a com plem ented distributive lattice a — (a * )’". 8. Prove the second o f the de M organ identities o f T heorem 11,11. 9. Supply the laws used for each o f the proofs o f Theorem s 11,8-11 11
Chapter 12
Boolean and Heyting Algebras 12.1
Boolean algebras
In this chapter we discuss two well-known algebras as specially structured lattices and prove some of their properties as well as present some semantic interpretations of these structures,
Boolean lattice B L = (L, A, V, *, 0 , 1 ) is a complemented distribu Boolean algebra is a Boolean lattice in which 0 , 1 and * (com plementation) are also considered to be operations; i.e B A = (B , A, V, *, A
tive lattice, A
0 , 1 ) where V and A are the usual binary operations, * is a unary operation and 0 and 1 are taken to be nullary operations, which simply pick out an element of
B.
For easy reference, we repeat and relabel the laws which a
B A = ( B, A, V, *, 0 , 1 ) obeys: (B O )
B A is an algebra
(B l)
A s s o c ia tiv e L aw s (i)
aA
(6 A c) = (a A
(ii) (B 2 )
b) V c
C o m m u t a t iv e L a w s (i) (ii)
( B 3)
b) A c
a V (6 V c) = (a V
(a A b) — (b A a) (a V b) = (6 V a)
D is tr ib u t iv e L aw s (i)
a A (6
V c) = (a A
b) V (a 297
A c)
298
C
a V (b A
(ii) (B 4 )
(a V b) A (a V
c)
T op and B o t t o m Law s
aA
(i )
1=
a
and
aV0
( ii) (B 5 )
c) =
12
hapter
=
aA0 = a and a
0 V 1= 1
C o m p le m e n t a t io n L a w s (i )
a A a* = 0
(ii)
a V a* = 1
Often in the literature a special two-elem ent B oolean algebra, called B O O L or 2 is used to represent the two truth values ‘false’ and ‘tru e’ where 0 < 1, 0 = 1* and 1 = 0* In a B A an element a is called an atom when a covers 0, The dual notion is less frequently encountered, but defined as an element a which covers 1, called the dual atom A B A is called an atomic B A when it contains an atom a for each o f its non-zero elements * such that a < x. A ny finite B A is atom ic and an atom ic B A m ay not be dually atom ic, W e prove some central theorems about B A which illustrate their power and structural elegance, T
heorem
12,1 In a B A an elem ent
is join-irreducible iffit
is an
atom.
■
Proof: (<£=) I f a is an atom then a = 6 V c means that 6 = a or 6 = 0; i f 6 = 0 then a = 0 V c = c; so a is join-irreducible, (=>) Suppose a is not an atom ox 0, then a > x > 0 for som e element x. W hen x < a, a = a A l = a A ( * V * * ) = (a A * ) V (a A * * ) = x V (a A **). Since a Ax* < a and a Ax* = a would im ply ® = a A * = a A * > F A * = 0 ,w e know a A x* < a, hence a would be join-reducible, ■ The definitions o f ideals and filters in a lattice given in C hapter 11 carry over directly to ideals and filters in B A , but note the additional fact that in a B A 0 is an element o f every ideal and 1 is an elem ent o f every filter Due to the strong notion o f com plem entation and the universal top and bottom element in any B A we have the follow ing strong correlation betw een ideals and filters T
heorem
12,2
(ii) for any F
C
In any B A (i) for any I C. B ,I is an ideal iffI* is a Rltev; B, F is a filter iffF* is an ideal. ■
B
o o l ea n
299
a l g e b r a s
Proof (i) Note that 0 £ I iff 1 = 0 * £ I * Take a,b £ I then a V 6 (a V £>)* = a * A 6* £ / * If a £ I, we know 6 < a iff a* < 6*; so 6 b* £ /*• The proof of (ii) is obtained dually. T h e o r e m 12 3 If F is a filter in b £ B either b £ F or b* £ F
Proof. {<=)
Suppose foi any 6 £
0 F,
F
c* £
F'
C
So
F'
I I
iff iff ■
then F is an ultrafiltei ifffor each
a BA,
■
B
be a filtei which properly contains c
£ £
either
F,
b £ F or b* £ F
ie
is not proper
Hence
F
F' to F' — F . Since
and take
there is some c £
is a maximal proper
filter, an ultrafilter. (=£•) Let
F
be an ultrafilter and take
not propei since
F
is already maximal
b 0 F.
intersection property and for some finite subset in f(X ) <
b*.
I n f(X ) is in F and hence
Set
F' = F
U {6 } which is
So F U {b } does not have the finite
X
of
F,
inf(Ar)
A b= 0
b* £ F
So ■
The following theorem is proven with a form of the axiom of choice, and shows the existence of a rich class of ultiafilters in any B A . T h e o r e m 12,4
(The Ultiafilter Theorem)
Each filter in
a BA
can be ex
tended to an ultrafilter.
■
Proof: Let P be the n on-em pty class o f all filters in some B A , partially ordered by set-theoretic inclusion. W e want to show that every chain in this ordering in
F has
and let C = U
an u pper bound. Let C I
f
- {C i
: i £ 1}
be a chain in
F
x , y £ C , then for some i , j £ I , x £ C{ and y £ Cj
Since C is a chain, either Ci < C j or C j < Ci', take Ci < C j Then x , y £ C j and since C j is a filter * A y £ C j £ C I f b £ B and x < b then b £ C j £ C Since 0 ^ C; for any i £ I , 0 0 C So C is a filter, which is the upper bound for C in F . W ith a form o f the axiom o f choice (called Z orn ’s L em m a) we derive that for any filter F, B A must contain a m axim al filter extending that filter.
■
There is an important connection between ultrafilters and homomorphisms, as indicated by the following theorem.
Let B A i and B A 2 be two Boolean algebras and consider homomorphism F : B A ] —> B A 2 If U is an ultrafilter of B A 2 , then i?_1(! 7 ) is an ultrafilter of B A ^ , ■ T h e o r e m 12,5
a
The proof of Theorem 12,5 is not given here, since it requires a number of algebraic concepts which have not been introduced.
300
C
12.2
h a p t e r
.
12
M odels o f B A
The B oolean laws m ay already have rem inded you o f the properties o f settheoretic operations, and, indeed, sets provide simple m odels o f Boolean algebras Starting from any non-em pty set X a m odel for B A can be con structed as follows: * Let B be p ( X ) , the pow er set o f X » Let V be set-theoretic union U
» Let A be set-theoretic intersection fl • Let * be set-theoretic com plem entation ' relative to X ® Let 1 be X » Let 0 be 0 W e m ay verify that all B oolean laws are true under this interpretation. (1 2 -1 )
Let X = { a , b , c } then B = { { a , 6 ,c }, { a , b } , { a, c }, { b , c } , { a } , {b} , { c } , 0}, Union and intersection are as usual and the complements are: {a , 6, c }* = 0, {a , b}* = { c } , {a , c }* = {6 } and {6, c }* = { a } .
N ote that by starting from a set with n elements, we construct a B A with 2n elements, Thus for every positive power o f 2 there is a B oolean algebra whose set has exactly that cardinality It can be proven, although we will not do so, that every finite B A has a cardinality o f 2 " for some positive n. In Section 3 we will prove that every finite m odel o f B A is isom orphic to a particular set-theoretic m od el based on the construction described above. Thus this fam ily o f models is particularly im portant. For infinite m odels the situation is not so simple. Every infinite set leads to a m odel foi B A by the given construction, but not every infinite B A is isom orphic to one o f these m odels There are, for instance, Boolean algebras o f cardinality Koi but Ho is not the cardinality o f the power o f any set, as we know from Cantor’s T heorem (see Section 4 4). W e can also show that the logic o f statements familiar from Part B constitutes a m odel o f B A . Let L be the logical language whose syntax was defined in Section 2 1, and S be the set o f statements generated by its syntactic rules For s and s' £ S we write s ~ s' when s and s' are provably
R
e p r e s e n t a t io n
b y
301
s e t s
logically equivalent in this logic o f statements relation on S
N ow ~ is an equivalence
For each s £ S we define the equivalence class Is] = ( s ' e S I s ~ s '}
Let B be the set o f all such equivalence classes o f logically equivalent state ments. Define a partial ordering on B by
Is] £ IV] iff (s —> s ') is valid Then { B , < ) is a B oolean algebra called the Lindenbaum algebra o f
L.
The
Boolean operations on B are defined by M A Is'J = Is
&
s'J,
Is]
V Is']] =
[s
V s '],
Is]* = I ~ s J .
Top and b o tto m are then respectively 1 = Is] for any tau tology s 0 = Is] for any contradiction s A n ultrafilter in the L indenbaum algebra o f
L can
be identified with a
maximally consistent set o f statements, which would constitute the first step in proving the com pleteness o f
L through
its ultrafilter representation. Such
topics belong to m ore advanced m odel theory and are beyon d the scope o f this b ook (R eference: Bell and M achovei (1977)).,
12.3
Representation by sets
The first exam ple we gave o f a m od el o f B A was the pow er set algebra o f a set, In this section we show that each B oolean algebra is isom orphic to a subalgebra o f a pow er set algebra, or, in other words, each B oolean algebra may be reprsented as a subalgebra o f a pow er set algebra. This im portant theorem is due to M .F, Stone and is called Stone Representation. W e first need to define tw o new notions: D e f i n i t i o n 12 1
(1) A ring o f sets is a fam ily o f subsets o f a set X which contains for any tw o subsets A and B o f X also A
UB
and A
ft B .
3 02
C
h a p t e r
.
12
(2) A field o f sets is a ring o f sets which contains X and the e m p ty set < and the com plem ent A ' o f any su bset A C X ,
F rom these definitions it is easy to see that a field o f subsets o f X is a B oolean subalgebra o f the B oolean power set algebra o f X , but that a ring o f subsets is a sublattice o f the power set algebra o f X , considered as a distributive lattice, W e will prove that any finite distributive lattice is isom orphic with a ring o f sets and that any finite B oolean algebra is isom orphic with the field o f all subsets o f some finite set. (T h e p r o o f follows essentially B irk h of and M acLane, 377-380), F rom Section 12,1 we need T heorem 12,1 and we define a set 5 (a ) = { * | x < a and x covers 0 } o f join-irreducible elements * for an element a in any finite lattice L 5 (a ), L e m m a 12 1
Consider the m apping F which assigns each elem ent a its
In any finite lattice L , F carries m eets in L into set-th eoretic
in tersection s: 5 ( a A b) = 5 (a ) fl 5 (6 ),
■
Proof. B y definition o f a A 6 we know that for any join-irreducible element x , x < a A b i f f x < a and x < b ,
■
L e m m a 12,2 In any finite distributive lattice L , F carries join s in L into set-th eo retic unions: 5 (a V 6) = 5 (a ) U 5 (6 ). ■
Proof.
Take any join-irreducible *, then * is contained in a V 6 iff * =
x A (a V 6) = (* A a) V (x A 6), Now x A a — x or x A 6 = x. So (a V b) contains * iff 5 (a ) contains * or 5 (6 ) contains x. The converse is obvious in any la ttice ,
■
These tw o lemmas show that F is a hom om orphism from L on to a ring o f subsets o f the set X o f join-irreducible elements o f L, T ogether with the result o f Exercise 3 at the end o f this chapter we know that F is also a on e-to-one onto hom om orphism , So we know T
heorem
sets.
12,6
A n y finite distributive lattice is isom orphic with a ring o f ■
H
In
e y t in g
3 03
a l g e b r a
the case o f a finite B oolean algebra we know from T heorem 12,1 that each
e le m e n t
a is the join o f the atom s x < a. W ith the above tw o lemmas we
know 5 (a )
fl 5 (a ') =
5 ( a A a *) = 5 (0 ) =
0
5 ( a ) U 5 ( a ') = 5 (a V a *) = 5 (1 ) = J where J is the set o f all join-irreducible element o f L, So [5 (a )]* = 5 (a ') and F as defined above is an isom orphism from any B oolean algebra to a field o f subsets o f join-irreducible elements o f L. W e still need to prove that this field contains all sets o f join-irreducible elements o f L. T H E O R E M 12 7 A n y finite B oolean algebra is isom orphic with the B oolean algebra, o f all sets o f its join-irreducible elem ents ■
Proof. We need to prove that for any two distinct sets o f join-irreducible elements o f L the joins o f each set are distinct. The claim that the jo in o f all elements in such a set contains all the join-irreducible elements o f that set and nothing else follows from Lemma
12,3
I f A is a set o f join-irreducible elem ents, and there is som e
join-irreducible elem en t a such that a < V { * I * 6 A } , then a £ A .
■
Proof, a = a A V { * I x 6 A } = V ( a A * ) and since a is join-irreducible for some such * £ A , a A * = a, so 0 < * < a. B ut then a = x. ■ The significance o f Stone Representations for representing inform ation and structuring m odels for the semantic interpretation o f natural language is discussed in Landm an (1986), The m athem atical im port o f B oolean algebras can be illustrated further by relating them to certain top ologica l structures and so called B oolean spaces, but the interested reader should consult the exposition o f such topics in Gratzer (1971) or Bell and M achover (1977)
12.4
H eyting algebra
Besides B oolean lattices and algebras which are used to represent such log ical systems as the classical logic o f Part B , there are other, weaker lattice theoretic structures and corresponding algebras which represent constructive reasoning based on a stricter n otion o f p r o o f by rejecting any use o f reductio arguments. In the semantics o f natural language attention has recently been
CHAPTER.
12
P (A )
x Y:
X'
X '- Y
Figure 12-1.
refocused on theories o f meaning based on verification/falsification condi tions, or m ore generally, assertability conditions, which subsume the classical truth conditions and analyze the inform ative content o f an expression in the context o f use.
A lth ou gh an account o f the philosophical and mathem at
ical foundations o f such constructivist logics and their p oten tia l linguistic applications is outside the scope o f this b ook , it is useful to present some o f the basic syntactic and semantic concepts to develop an initial understand ing o f m ore epistem ically flavored form al systems. W e need to define one additional lattice-theoretic notion. In a lattice L the ( relative) pseudo-complement o f an elem ent a relative to b is the greatest element c in L such that a Ac < b N ote that this pseudo com plem entation involves only the meet operation It is easy to see that c is the pseudo-com plem ent o f a relative to b precisely when (1 2 -2 )
for all * in £ , x < c i f f a A x < b
e.g., in the pow er set lattice with the inclusion ordering, { p ( A ) , < ), for any sets X , Y E p ( A ) , X ' — Y is the pseudo-com plem ent o f X relative to Y. (See Fig, 12-1.) In general, we write a =>■ 6 for the pseudo-com plem ent of a relative to b, and we say that a lattice L is relatively pseudo-com plem ented iff a =>■ 6 exists for every a and 6 in L, A relatively pseudo-complemented lattice { S , A , V ) is called a Heyting lattice (or a pseudo-B oolean lattice ) if it contains a bottom element 0 . Define
H
e y t in g
3 05
a l g e b r a
a* by setting a* = a => 0 and note that a* is the lub o f {x | a Ax = 0 } , i.e. a * is the pseudo-com plem ent o f a (relative to 0). The dual o f a H eyting lattice is called a B rouw erian lattice, which is studied in M cK insey and Tarski (1946). Correspondingly, an algebra (H , A, V, =>, 0 , 1 ) is a H eyting algebra ( H A ) and it has the follow ing properties: (H I)
{H, A,
V ) is a distributive lattice
( H 2 ) a A 0 = 0 and a V 1 = 1 ( H 3 ) a => a = 1 ( H 4 ) (a =>■ 6) A b = 6; a A (a => 6) = a A 6 T
heorem
12,8 A H eytin g lattice is distributive.
Proof. In every lattice, b and a A c
<
<
■
b V c and c < 6 V c, so a A b
<
a A (b V c). Hence (a A 6) V (a A c) < a A (6 V c),
a A (6 V c) W e know
a A 6 < (a A 6) V( a Ac ) , so in a Heyting lattice if 6 < a, then b < (aA b )V (aA c) and i f c < a, then c < (aAb)V(aAc) thus a A (b V c)
<
So if bVc < a then (fcVc) < (aAfc)v(aAc),
(a A b) V (a A c), i.e., a A (b V c)
=
(a A b) V (a A c).
■
The notion o f a filter in a H eyting lattice can be characterized as follows: T heorem
12,9 A su bset F C H is a filter o f a H eytin g lattice H if f (i) F
contains 1 and (ii) i f a £ F and a => b then b £ F .
■
Proof. (<=) Suppose 1 £ F and since in a relatively pseudo-com plem ented lattice a < 6 => (a A 6), we have a => (6 => (a A 6)) = 1, I f a,b £ F then b => (a A b) £ F and a A b £ F . I f a £ F and a < b then a => 6 = 1 £ F , so b £ F . This proves F is a filter. (=>) Suppose F is a filter, then 1 £ F .
I f a £ F and a => 6 £ F then
a A b = a A (a => 6) £ F , since a A (a => 6) < b (b y substituting a => 6 for c in the definition o f a relatively pseudo-com plem ented lattice) and b < a =$■ b, So b £ F .
m
The notion o f a filter in a H eyting lattice is used to capture closure o f a set o f statem ents under their im plications, as we will see in the follow ing section on the semantics o f Heyting algebras. W e can define a pre-ordering (reflexive and transitive) on the elements o f a filter in a H eyting lattice by a < p b iff (a => 6) £ F
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Logically equivalent statements a ~ 6 for which a < p b and b < p a form again an equivalence class determined by the filter F , A lattice ordering m ay be defined on elements o f such an equivalence class b y |a| < |6| iff (a =>■ 6) £ F and it is provable that the lattice obtained from this ordering is also relatively pseudo com plem ented
Details can be found in R asiow a and
Sikorski (1970), am ong others.
12.5
Kripke semantics
In 1965 Saul Kripke published a semantic interpretation o f intuitionistic logic based on H eyting algebras as a corollary o f his semantics for m od a l logics (see Chapter 15). Because some fruitful notions for a theory o f meaning for natural language based on inform ation conditions m ay be related to this Kripke semantics we sketch this semantics here inform ally w ithout entering in to the axiom atization or m etatheory o f intuitionistic logic T he core idea o f Kripke semantics is to relativize the truth o f a statement to tem poral stages or states o f knowledge. So a statement is not simply true but rather true at a stage or in a state o f knowledge, which we will generally call an in form ation-state W e take these inform ation-states to be ordered tem porally, i.e.
assuming a partial order which represents the
different alternative ways in which we m ay gradually acquire and extend our knowledge and inform ation. So the set o f inform ation-states does not just contain the past stages o f inform ation gathering but also all possible future states to which we m ay get from what we n ow actually know. We also assume that we share such inform ation-states as a com m unity o f (formal or natural) language users rather than thinking o f them as internal mental representations as the founder o f intuitionistic logic, L. B rouw er, suggested. A sentence which is true at an certain inform ation-state will always be true at later states, since once we have verified a statem ent we never lose that inform ation (this is the idealization o f the language user in seman tics!). Hence we require that truth o f a statement at an inform ation-state persists through all consecutive inform ation states. M ore form ally, the set o f inform ation-states in an interpretation form s a p oset under the tem poral order called a K ripke-fram e P = ( P, < ). The interpretation o f a statement corresponds to the subset o f inform ation-states at which the statem ent is true.
Persistence o f truth is tantam ount to requiring such subsets to be
filters on P or hereditarily closed under the tem poral ordering. The collection o f filters on P will be written P + . W e define a Kripke-
K
valuation to
r ip k e
se m a n t ic s
307
be a function from the set o f statements to the set o f filters
y ; S —» P + > assigning to each statement s £ S a filter F ( s ) C P , i,e. the ixxformation-states at which s is true. A model based on a Kripke-fram e is a pair M = ( P , F ) where V is a Kripke-valuation N ow we define recursively the notion o f tiu th at an inform ation-state for all com plex statements (we leave the syntax o f the statements im plicit as it is essentially the syntax o f the logic o f statements in Chapter 6) We define a notion M| —p s which is to be read as ‘the statement s is true at inform ation-state p in the m odel M ,,’ (1 )
M| = p s for s an atom ic statement i f f s £ F (s ),
(2 )
M| = P s & s' iff M| = p s and M| = p s'
(3 )
M|
(4 )
M| = p~ s iff for all p' such that p < p' not M| = pi s
(5 )
M| = P s i-> s' iff for all p1, p < p' i f M| = pi s then M| = p> s'
— p
s V s' iff M|
— p
s or M| = p s'
Thus at inform ation-state p the negation o f a statement s is true when the statement can never becom e true or verified at any later possible stage. A conditional statement is true when at all later stages which verify the antecedent the consquent is verfied as well, The advantage o f introducing inform ation-states is that they are used to quantify over possible extensions o f the actual inform ation-state in defining verification o f negative statements and con d itionals, Unrelativized truth can be defined as truth at all inform ation-states and a statement is valid on a Kripke-fram e i f it is true on all models based on that frame. I f we extend this definition o f truth at an inform ation-state to the full predicate loigc including the universal and existential quantifiers, we need to put some further conditions on the clauses for statements as well. I f we analyze the internal structure o f statem ents, we can only verify such quantificational statem ents when we know ‘what they are a b o u t,’ i.e., when we know the reference o f the terms that occur in such quantified statements, More inform ally speaking, we do not know how to verify (or falsify for that matter) an English sentence John is old when we do not know to w hom John refers or who John is. For similar reasons we require that (i) each inform ation-state is assigned a non-em pty set o f objects which act as ‘referents’ for individual constants
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and variables, and (ii) successive states only add referents.
Then clause
(1) above is extended to incorporate the requirement that F ( s ) picks only objects from this ‘referent’-set as interpretations o f the individual variables and constants occurring in s (an atom ic form ula o f a predicate log ic)
For
disjunction we require that all terms o f b oth disjuncts are interpreted by objects from the referent-set at the stage o f evaluation o f the disjunction and similarly for conditionals and negations. The quantifier clauses to be added are defined as follow s: (6 )
(7 )
M| —p (V z )^ iff for every p' such that p < p' M| = p<
From these recursive clauses we can see that the law o f double negation, al though valid in classical logic, is not valid in a Kripke semantics. N ote how ever that verification o f a form ula entails verification o f its doubly negated form but not vice versa (if M| = p s then M| = p~ ~ s), and a negative for m ula is equivalent to its triply negated form (M| = P~ s iff M| = p~ ~ ~ s). A lso the classical law o f excluded middle (every form ula or its negation is true) does not hold in Kripke semantics. On the basis o f a syntactic specification o f the intuitionistic logical sys tem and this Kripke semantics we can prove the im portant com pleteness and soundness theorem s, and Kripke semantics can also be form ulated by seman tic tableau rules (as it originally was) W e will not discuss such metatheoretic issues o f intuitionistic logic here, but rather poin t out the connection between Kripke semantics and Heyting algebras. Since the intersection and union o f two filters axe filters to o , the poset based on the set of filters with inclusion is a bou nded dis
P+ _ (p +, < )
tributive lattice with meets and joins given by intersection and union, P~ is indeed a H eyting algebra such that for any A, B £ P + A => B is the pseudo-com plem ent o f A relative to B. We have for all filters F, F < A => B iff A(~\F < B and A* = A => 0 Now a Kripke-valuation 7 : S h P + for a Kripke-fram e P is also a valuation for the H eyting algebra P + ,
This connection can be used to show that
ou y
Kripke-validity on the fram e P is the same as validity in a H eyting algebra P+ For a semantics o f natural language Kripke semantics m ay have to be extended beyon d its verification conditions to include recursive falsification conditions defining falsity o f a form ula in an inform ation-state. This would allow for a m ore interesting treatm ent o f positive and negative facts, as well as a m ore adequate analysis o f conditionals A n im portant current research question is how m odal verbs and conditionals m ay be analyzed as constraining the set o f possible extensions o f a given inform ation-state as rules or constraints which determine patterns o f inform ation growth or restrictions on possible ways o f gathering new inform ation. The interested reader will find further references in the suggested further reading for this chapter
Exercises 1. Prove that in any B oolean algebra i f a A b = 0 and a V b = 1 then a = 6* 2. Prove the idem potence o f the meet and jo in operations from the other B oolean laws. 3. Show that in a finite B oolean lattice every element is the join o f some join-irreducible elements. (H int: you need induction on n when n is the num ber o f elements * < a for given a, and the base o f the induction is the b o tto m elem ent). 4 . Prove for any join-irreducible element a o f a distributive lattice, if a < b V c then a < b or a < c 5. Prove for any relatively pseudo-com plem ented lattice with a top ele ment 1 that
(i) a => a = 1 (ii) a < 6 i f f a = > 6 = l (iii) b < a
b
6. Prove that any finite lattice is a Heyting lattice 7. I f { B , A, V, *, 0, 1 ) is a B oolean algebra then for any a ,b £ B define a i-> b to be a* Vfc. S h o w t h a t ( 5 , A, V,
0, 1 ) is a H eyting algebra,
Review Exercises 1. Following the procedure used in T heorem 10 1 prove that in a group, y = a~1 o b is the unique solution o f a o y = b. 2. Construct a com m utative group with five elements. 3. Let the operation * y be defined as * + (y — 3). Show that the set G o f all integers form s a group with respect to this operation and that 3 is the identity element o f this gr oup . W hat is the inverse element for an integer z? 4.
( a ) List four different subgroups o f the group o f all integers under ordinary addition. ( b ) For each o f these subgroups, state whether it is isom orphic to the original group. I f so, prove it and if not explain why not
5. Determine all the possible isom orphism s betw een the group o f the in tegers { 0 , 1 , 2 , 3 } with addition m od u lo 4 and the group o f rotations o f the square { I , R . R 1, R " }. 6. Let A = { a , b , c }
Find all the distinct one-to-one correspondences
from A on to A Construct com posites o f all pairs o f these one-to-one correspondences and express your results in the form o f a m ultiplication table. W hat sort o f m athem atical configuration is represented b y this m ultiplication table? 7. Let A ” be the set o f all strings form ed from some finite set A For all strings x and y m A* x is said to be a conjugate o f y iff there are strings u and v such that x = uv and y — vu ( a ) Show that conjugacy is an equivalence relation and describe the partition it induces on A *, For a string x n o f length n what are the m axim um and m inim um number o f strings in the equivalence class containing x n ? 311
31 2
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( b ) P iov e that if x is a conjugate o f y there is a string 2 such that x z = zy ( c ) Let T = { Ti , T2> I 3 , I 4 } be the set o f functions each o f which maps a string in A o f length 4 into one o f its conjugates, e.g,, Ti maps a ia 2a 3 a 4 in to 0 ^ 2 0 30 4 and T2 m aps 0 ^ 2 0 3 0 4 in to 020304 a!. Show that the operational structure consisting o f T and the op eration o f com position o f functions is an A belian group
8 . Each o f the follow ing is a system o f the form A = ( A, ® , 0 ) consist ing o f a set and two binary operations
For each system, answer the
follow ing questions: ( a ) W h ich o f the group axioms are satisfied b y ( A ,® )? ( b ) W h ich of the group axioms are satisfied by ( A , 0 ) 1 ( c ) Is A an integral dom ain? I f not, which axioms are not satisfied? (1) A = the set o f all integers ® = ordinary addition
0 = ordinary m ultiplication (2) A = the set o f all non-negative integers ® = ordinary subtraction
0 = ordinary m ultiplication (3) A = { 0 , 1 , 2 , ....,2 4 } © = addition m odu lo 25
0 = m ultiplication m odu lo 25 (4) A = { 1 , 2 , 3 , ..,9 ,1 0 } ® = addition m odu lo 11
0 = m ultiplication m od u lo 11 (5) A = { 1 ,5 ,7 ,1 1 } ® = addition m odu lo 12
0 = m ultiplication m od u lo 12 ( 6 ) A — the set o f all rational numbers p/q with 0 < p/q < 1 © = ordinary addition
0 = ordinary m ultiplication (7) A = the set o f all integers ® = ordinary addition
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0 defined by a © 6 = 0 for all a, b £ A 9. Prove the follow ing laws for integral domains. (Y ou m ay use the results o f previous problem s as well as the axioms and theorems from the te x t,) ( a ) (a + 6) •(c + d) = ( ac + be) + (ad + bd) for all a, b, c, d
(b) —0 = 0 (N ote: —x stands for “ additive inverse o f
* _1 stands
foi “m ultiplicative inverse o f * ” )
(c) I f a b = 0, then either a = 0 or b = 0 (d ) - ( - a ) = a 10. The follow ing correspondences are m any-one m appings o f the m ulti plicative group o f all n on-zero real numbers on part o f itself. W h ich are hom om orphism s? For those which are not, show why not. (*F or those which are, p rove that all the requirements are satisfied,,) ( a ) x i—^ [*[
(b) * i-> —* (c) x i—^ 2x
(d) x i-> 1/x ( e ) x >-+ x 2 11.* W hich o f the follow ing relations R are equivalence relations? For those which are, describe the equivalence classes. ( a ) G is a group, 5 is a subgroup o f G, and R is the set o f all ordered pairs (a, b) with a, b in G such that a -1 •b £ S.
(b) J is the integral dom ain o f all integers and R is the set o f all ordered pairs (a, b) with a ,b in J such that a + ( —b) is even,
(c) J is the integral dom ain o f all integers and R is the set o f all ordered pairs (a, b) with a ,b in J such that a + ( —b) is odd12. ( a ) C onstruct a group o f symm etries o f an equilateral triangle anal ogous to the group o f symmetries o f the square (There should be six elements altogether)
(b) Find all subgroups o f that group
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( c ) C onstruct a hom om orphism between the whole group and one of its proper subgroups. 1 3 . P rove using the axioms o f B oolean algebras that the set o f elements o f a B oolean algebra cannot form a group under the union operation, 1 4 . Prove in a relatively pseudo-com plem ented lattice (a ) a = 6 i f f a = > 6 = l = 6 = > a ( b ) 1 =» 6 = b ( c ) a < b => ( a A 6)
Part D
ENGLISH AS A FO R M AL LANGUAGE
Chapter 13
Basic Concepts Richard M ontague was the fust to seriously propose and defend the thesis that the relation betw een syntax and semantics in a natural language such as English could be viewed as not essentially different from the relation between syntax and semantics in a form al language such as the language of first-order logic W hile M on ta gu e’s claim was and is a controversial one, b o th the perspective he offered and the technical apparatus used in developing it have transform ed the study o f natural language semantics, In this section we focus first on the principle o f com positionality and its central role in articulating the relation o f semantics to syntax in a form al language. T he principle is also know n as Frege’s principle, and M ontague took him self to be formalizing a basically Fregean view point in adopting it. T he second topic of this chapter is the lambda calculus, invented by A lon zo Church in the 1930’s but in trodu ced to linguists m ainly through M on ta gu e’s work. T he lam bda calculus has no intrinsic connection to the principle o f com positionality, but it has proved to be one o f the m ost im portant and fruitful tools in the form al sem anticist’s to o lb o x , and w ithout it, it would be m uch harder to make a plausible case for the com positionality o f natural language semantics. For a linguist interested in semantics, we w ould suggest that a fam iliarity with the basics o f the lam bda calculus could be as im portant as a fam iliarity with first-order predicate logic
13.1
Com positionality
The term “ sem antics” is used in a variety o f ways in a variety o f fields; p ro b ably the only com m on denom inator am ong these is that semantics must 317
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be concerned with m eaning, but “m eaning” is i f anything an even vaguer term A m on g logicians, however, the term “ sem antics” has had a relatively precise usage, at least in the dominant W estern tradition reflected, say in D on ald K alish’s article in the E ncyclopedia o f Philosophy, and it is out o f that tradition that M on tagu e’s w oik came. If you rem em ber the discus sion of syntax and semantics in Part B : L ogic especially the presentation o f the syntax and semantics of statement logic in sections 6.1 and 6 2 , and o f predicate logic in 7,1 and 7,2, you might have n oted that b oth systems are syntactically disam biguated, i.e , no wellform ed form ula has m ore than one derivation, Quantifiers and connectives in form ulas always have a de term inate and fixed scope
Furtherm ore, the semantic interpretation o f any
statem ent oi predicate logical form ula is obtained via a system atic semantic procedure interpreting its parts and the logical sym bols connecting them. T he correspondence betw een the syntactic structure o f a form ula and its se m antic interpretation is in fact very tight, as we will see in this section The principle o f com positionality, oi Frege’s principle, is a way o f articulating the relevant notion o f correspondence. T h e P r in c ip l e o f C o m p o s it i o n a lit y The m eaning o f a com plex expres sion is a function of the meanings of its parts and o f the syntactic rules by which they are com bined. C onstrued broadly and vaguely enough, the principle is nearly uncontroversial, but M on tagu e’s precise version o f it places rather severe constraints on admissable systems of syntax and semantics. As the wording given above suggests, the exact im port of the com positionality principle depends on how one makes precise the notions o f m eaning, o f part, and o f syntactic rule, as well as on the class o f functions perm itted to instantiate the “is a function o f ” requirement In the specification o f form al languages, the com position ality principle is generally satisfied in the follow ing way: the syntax is given b y a recur sive specification, starting with a stipulation o f basic expressions o f given categories, and with recursive rules o f the follow ing sort: Syntactic Rule n: If a is a well-form ed expression o f category A and @ is a well-form ed expression o f category B, then 7 is a well-form ed expression o f category C, where 7 = Fi(a,j3). In such a rule, Fi is a syntactic operation such as concatenation; we will give illustrations o f such rules below
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319
The semantics is then given b y a par allel recursive specification, including a stipulation of the semantic values for the basic expressions and for each syntactic rule a single semantic rule o f the follow ing form : Semantic Rule n: If a is interpreted as a' and (3 is interpreted as /?', then 7 is interpreted as G ^ a 1, ^ ) In such a rule, G * is a semantic operation, examples o f which we will see below. Note the distinction betw een syntactic and semantic rules as schem atized above and syntactic and semantic operations, the Fi, G^ that appeal within the rules. W h en the com positionality requirement is taken as a constraint on the correspondence between rules, as it is heie, it does not by itself im pose any correspondence requirement on the operations that occur within the rules. One m ight, for exam ple, form ulate a rule o f Y es-N o Question For m ation for English utilizing a com plex syntactic operation built up out o f a com bination o f S u bject-A u x Inversion, D o-S upport, and the im posing o f an appropriate intonation contour, deriving a yes-no question from its declara tive counterpart. T he task o f giving a uniform semantic interpretation rule corresponding to such a syntactic rule is obviously very different from the (presum ably im possible) task o f giving a uniform semantic interpretation o f the operation o f S u b ject-A u x Inversion (w hich occurs in a semantically heterogeneous variety o f constructions in addition to Y es-N o Questions,)
1 3.1.1
A com position al account o f statem en t logic
To illustrate, let us recast a part o f the syntax and semantics o f the statement logic o f Chapter 6 and the predicate logic given in Chapter 7 in a form which makes the semantics explicitly com positional The syntax o f SL, the language o f statement logic originally presented in Section 6 , 1 , is restated below, (1 3 -1 )
T he prim itive vocabulary o f SL consists o f the follow ing: (i) a denum erably infinite set o f atom ic statem ents which we des ignate by the letters p, q, r, s, with primes or subscripts added when necessary, (ii) the follow ing logical constants: ~ , & , V, —>, <-+
T he same
sym bols are used in the metalanguage to designate these sym bols in the ob ject language.
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(iii) the punctuation sym bols (,), These aie also used as names o f themselves in the metalanguage (1 3 -2 )
T he set o f wffs o f SL is defined recursively as follows: (1) B a s ic C la u s e : Every atom ic statement is a w ff (2) R e c u r s i v e C la u s e s : (2.1) I f
(2 4) I f
+-+, ip,
and ) in that order is a wff.
T he somewhat pedantic way o f stating the recursive rules above is in tended to emphasize that the operations which apply to wffs to produce laiger wffs include the introduction o f logical constants and punctuation marks. Given this syntax, the only relevant parts o f a com plex w ff are its constituent wffs. In this syntax, the logical constants are n ot assigned to any syntactic category; the only category em ployed in the syntax is the category o f wffs.
T he semantics will be stated accordingly, assigning sem antic val
ues to com plex wffs as functions o f the semantic values o f theii constituent wffs and o f the rules by which they were constructed, (T h e same language could be given a different syntax in which the logical constants (bu t not the punctuation marks) aie indeed assigned to categories (e g, ‘ one-place con nective’ and ‘tw o-place connective’ ) and receive semantic values (which will be truth-functions); the semantic rules would then treat the connectives as well as the wffs as parts o f com plex wffs and the semantic rules would be stated in a correspondingly different way
See Dow ty, W all, and Peters
(1979) or Gam ut (1982, vol. 1) for explicit illustration and discussion o f this contrast. T he semantics o f SL, as stated in Section 6 1, assigns to every com plex w ff a truth value, either 1 (true) or 0 (false), based on the truth values o f
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its constituent wffs and the syntactic rules by which they were assigned
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A
specification o f the semantics that meets the com positionality requirement must therefore include a specification o f semantic values (in this case truth values) for the atom ic statements and a recursive semantic rule correspond ing to each o f the recursive syntactic rules. Such a semantics can be stated as follows (1 3 -3 )
Assignment o f semantic values to atom ic statements: Let / be a function which assigns to each atom ic statement o f SL one o f the tw o truth values 0 and 1
(1 3 -4 )
The semantic interpretation o f the set o f all wffs o f SL, given an initial valuation / for the atom ic statem ents, is defined recursively as follows (1) B a s ic C la u s e : I f <^>is an atom ic statem ent, then the semantic value o f 4> is f (). (2) R e c u r s i v e C la u s e s : (2.1) (unabbreviated) The semantic value o f the result o f pre fixing
~
to 4> is 1 iff the semantic value o f
(2.1) (abbreviated) The semantic value o f ~ & tp ) is 1 iff the semantic values o f (p and tp are b o th 1. (2 3) The semantic value o f ( V tp) is 1 iff the semantic value o f
Each o f the recursive semantic rules specifies an operation which applies to the truth values of the constituent wffs to determine the truth value o f the resulting w ff The operations are those familiar from the truth tables o f Section 6,2, although here they are defined im plicitly rather than explicitly
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It is clear from the statement o f the syntactic rules and o f the unabbre viated form o f the first recursive semantic rule that plain English makes a rather cum bersom e metalanguage and some succinct special-purpose nota tion cou ld prove useful. N otation in this area is not as standardized as in some other parts of mathem atics and logic, but we will introduce some fairly com m on notational practices below The syntax in (13-1) and (13-2) can be used in assigning each SL state ment a unique derivation tree similarly to the syntactic trees in a PhraseStructure com ponent o f a grammar of a natural language T he semantic rules in (13-3) and (13-4) will then give each tree a com position al interpre tation, follow ing the derivation n ode by n ode from the b o tto m up Here we only illustrate the kind o f tree for this syntax o f SL and its semantic inter pretation, and defer further discussion and alternative syntaxes to SL to the exercises o f this chapter. The derivation of the statement ~ (p V q) is as in (13-5), The tree is annotated with the number o f the (su b-) rule used in constructing each node. The com positional interpretation o f (13-5) is as given in (13-6), where it is assumed that the truth value of the atom ic statements is given: false for p and true for q N ote that the annotation remains exactly the same due to the rule-by-rule com positionality o f SL. Strictly speaking the semantics could either be defined on wffs the way we did in (13-1) and (13-2) with bracketing for disam biguation, or we could do w ithout the auxiliary bracketings and have the syntax generate the derivationtrees. These kinds of' derivation-trees would then constitute an equivalent but different way o f disam biguating a form al or natural language and the com position al semantics would have to take trees rather than form ulas as the objects to be interpreted
(1 3 -5 ) ~(pVg)
2.1
I (P V q) 2 3
/\ P,1
2,1
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(1 3 -6 ) 0
2.1
1
2.3
/\
0, 1
1 3 .1 .2
1.1
A com position al account o f predicate logic
The syntax and semantics o f predicate logic introduce several complexities related to the greater expressive pow er o f predicate logic com pared with statement logic. W e have to recognize the distinct categories o f predicates, terms, and form ulas, the distinction between constants and variables among the term s, and the semantics o f variables and o f the binding o f variables by quantifiers, The last o f these was presented in a somewhat inform al manner in Section 7.2, with the detailed form ulation deferred to this chapter.
It
is interesting to reflect on the fact that the category o f closed formulas, or statem ents, plays n o essential role in the com position al semantics. It is an insight that can be traced back to Frege that we can com positionally define semantic values for formulas from semantic values for formulas (and predicates and term s), and then define semantic values for statements b y an extra step, but that it is im possible to give a com positional semantics for statements using just statements as the recursive category.
W e return to
this issue after presenting the explicit syntax and semantics below , The syntax o f P L , the language o f predicate logic presented in Section 7.1, is restated in a different form below .
(1 3 -7 )
The prim itive vocabulary o f PL consists o f the follow ing: (i) A set o f individual constants, which we designate in the m eta language by the letters a, b, c, with and w ithout primes, (D if ferent ob ject languages have their own particular sets o f con stants, which m ay be finite or denumerably infinite,)
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(ii) A
h a p t e r
denum erably
.
13
infinite
set
of
individual variables, X0 , x i , X 2 , • The individual constants and the individual variables together constitute the term s. (iii) A set o f predicates, each with fixed arity, which we officially designate in the metalanguage b y PJ1 (for the «th n-ary pred icate) but m ore frequently represent as P , Q , R with primes or subscripts as needed and the arity clear from context As in the case o f the individual constants, particular ob ject lan guages will have their own particular choices o f predicates, (iv) The logical connectives o f SL: ~ , & , V, —>, <-+ , (v ) The quantifier sym bols V, 3. (vi) The parentheses ), (, [, ] Together these sym bols form the vocabulary o f P L ; m ore precisely, we should think o f P L as a fam ily o f languages, one for each choice o f individual constants and predicates (the non-logical constants ) Each such language is a language o f predicate logic; when we speak o f the language o f predicate logic we mean the form that all these languages share, assuming some arbitrary and representative set o f individual constants and predicates, (N ote that while different languages m ay use different sym bols for the individual vari ables, this is merely an alphabetic variation. A ll languages o f predicate logic have a denumerably infinite set o f individual variables, and their semantics is fixed by the recursive rules o f the semantics, not stipulated language by language as the interpretation o f the non-logical constants is ) Given the vocabulary, the set o f formulas o f PL is defined recursively as in (13-8) below. In the m etalanguage for talking about the syntax o f P L , we need variables that range over expressions o f various categories to use in the recursive rules, As in the statement o f the syntax o f SL, we use (f> and ip as variables over formulas W e use P as a variable over predicates, t, possibly with subscripts, as a variable over terms (including b oth individual constants and individual variables), and v as a variable over variables, (W e d on ’t need to introduce any variable over individual constants in our metalanguage because there are n o rules that apply only to individual constants ) Also in our metalanguage for writing syntactic rules, instead o f writing out the syntactic operations explicitly with m ention o f concatenation, prefixing, etc.,
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we follow the com m on convention already illustrated in Chapters 6 and 7 o f indicating the syntactic operation by writing its result (1 3 -8 )
The set P L o f formulas o f predicate logic is the smallest set satis fying conditions ( l ) - ( 8 ) below : (N ote: This “smallest set’! locu tion is equivalent to adding a condition (9 ) which says that nothing is in the set except b y virtue o f a finite number o f applications o f
(l)-(8)) (1) I f P is an n-ary predicate and Z1 : , , , t n are all terms, then P ( t ii • yin) is a form ula (These are the atom ic form ulas,) (2) If
is a formula.
(3) I f (p and ip are formulas, then (< p & ip ) is a formula. (4) I f (j) and ip are form ulas, then (
The definition just given is nearly identical to that given in Section 7,1 and yields exactly the same set o f formulas; the only difference is that we have expanded what was stated as one rule in to the five rules (2 )-(6 ) and another rule in to the tw o rules (7 ) and (8 ) We did that in order to be able to follow the letter o f the com positionality principle and have one semantic interpretation rule for each syntactic rule. W e cou ld n ’t literally do that if, for instance, rules (2 )-(6 ) were collapsed in to one rule, since each o f those operations yields a distinct interpretation In practice it is more com m on than not to collapse such rides anyway, and only if the com positionality o f a certain grammar is challenged or uncertain is it necessary to be very precise about the individuation o f the syntactic rules and their correspondence with the semantic rules, N ow we turn to the semantics o f PL , which was given in Section 7 2 but not fully formally. R em em ber that a m od el M for predicate logic consists o f a domain D
3 26
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13
o f entities or individuals and F an assignm ent-function which provides the interpretation o f the prim itive non-logical constants o f the language, i.e. a function which m aps each individual constant to an individual in D and each n -a iy predicate to P n to a set o f n-tuples o f individuals. (N ote that the dom ain m ay be indicated by other capitals; in 13.2.2 we use A, and E is also com m only fou n d.) Besides the interpretation o f the non-logical vocabulary, we need a separate function assigning individual variables, call it g : VAR—^D, The reasons why a separate variable-assignment is needed is that the recursive m anipulation o f these assignment functions plays a separate and distinctive role in the com positional semantic interpretation o f quantified formulas. In effect, we generally have to consider all possible assignments to the variable in a form ula in arriving at the truth conditions for the whole formula. Hence the semantic rules need to be able to refer to and change the variable-assignments in the course o f the interpretation. A ny elem ent o f the descriptive vocabulary, on the other hand, is interpreted once and for all independently o f anything else in the form ula in which it occurs, hence they are truly descriptive constants for individuals or for predicates T he interpretation o f the entire language o f predicate logic can now be form ulated com positionally as a rule-by-rule m apping defining the denotation o f an arbitrary com plex form ula relative to a given m odel and a given variable-assignment The denotation o f an expression a relative to a model M and an assign m ent g, sym bolized [a]]M’s , is defined recursively, in parallel to the syntactic rules given in (13-6) above, as follows:
(1 3 -9 )
(0) (a) I f a is a non-logical constant in CO N \, then [a]]M’s = F( a ) . (b ) If a is a variable in VARa, then [a]]M’s =
=
(3)
& tp^M’9 = 1 iff
(4) l<j) V tp1M’9 = 1 iff
= o = 1 and = 1 or
= 1 = 1
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(5) (<j> ->
= 1 iff M M’g = 0 or M M,g = 1
(6)
= 1 iff M M’g = M M,g
(7) If
j]Vv
= 1 iff for
all d £ D , M M'gd/u = 1 (8 ) If
= 1 iff there = 1
The rules just given define semantic values for formulas com positionally in terms o f the semantic value for their parts; in order to accom plish this, the semantic values for form ulas were taken to be truth with respect to a m odel and an assignment. Having done that, we can n ow define truth with respect to a m od el (independent o f assignment) for closed form ulas, or statements. The easiest way to do it will define truth with respect to a m odel for all form ulas, but it is for the closed formulas that we have the clearest idea o f what results the definition shall give, (See exercise 4 for an alternative definition to (13-10) which gives different results for som e o f the open formulas. (13-10)
(1) For any form ula <j>,
(i.e
(j) is a true sim pliciter with
respect to M ) iff for all assignments g, [4>]M 'S = 1. (2) For any form ula <j), 1,
— 0 iff it is not the case that
=
W hen working out the semantic interpretation o f a form ula, then, one first proceeds com position ally through the derivation, using rules from (139); then the last step is the application o f the rules o f (13-10) Since the semantic rules o f (13-9) m atch one-onto-one with syntactic rules we n ow have a rule-by-rule com positional semantics o f Predicate Logic, Let’s see how some simple examples o f derivations and their com positional interpretation work out, We will analyze the follow ing three examples: (i) M ary is reading a b ook .
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(ii) Every student is reading a book. (iii) N o student is reading a b ook K eeping the quantifier prefixes as close as possible to their corresponding predicates, their translations to predicate logic are: (i) (3 * ) (b o o k (* ) & read(m , * )) (ii) (V y)[student(y) —> (3 z )(b o o k (ic ) & rea d (y, sc))] (iii) ~ (3 j/)[s tu d e n t (j/)& (3 * )(b o o k (* )& r e a d (j/, *))] The deiivation tree for (i) according to the syntax in (13-8) is as in (1311), annotating each node with the number o f the rule used in constructing the expression on that n ode (the lowest nodes are all elements o f the primitive vocabulary given in (13-7)). (13-11) (3 * ) (b o o k (* ) & read(m , * )) , 8 (b o o k (z ) & re a d (m , * ) ) , 3 b o o k (* ),l / \ book
*
read (m , * ), 1 /N read m x
Given this derivation the interpretation o f the form ula proceeds from the b o tto m up the tree, determining the interpretation o f each higher node by the semantic rule corresponding to its syntactic counterpart. Let us assume we are given the follow ing m odel M and variable-assignment g for the interpretation M = (D , F }, where D = {m ary, jane, M M iL } F (m ) = mary F (stu d e n t) = {m ary, ja n e } .F (book ) = {M M iL } (i.e. this b o o k you are reading)
.F(read) = {(m a ry , M M iL }} g (x ) = jane g (y ) = mary Starting at the b o tto m we determine according to rule (0 ) in (13-9) that [bookJM,s = F (b o o k ) = {M M iL }, and
= g (x ) = ja n e
Furtherm ore,
we can determine Jbook (*)JM,s = 0 according to the semantic rule (1 ) in (13-9), since jane is not an element o f the denotation o f book , Next we proceed to the interpretation o f [rea d ]M,g = F (re a d ) = {(m a iy , M M iL }}, = F (m ) = m ary, and
= g (x ) =jane. A pplyin g rule (1 ) once
again we have [rea d (m ,*)]]M ’S = 0, since (m ary,jane) is n ot in [read]]M ’s , Now we have tw o false form ulas each with a free variable * as input to the next node where they are conjoined, and by rule (3) we determine that [(b o o k (z ) & rea d (m , * ) ) ] M,S = 0 too. The interesting part o f the com position al interpretation o f this form ula comes in the final n od e, where the variable is bou nd existentially b y an appli cation o f rule 8, A ccordin g to its semantic counterpart rule 8 [( 3 * )(b o o k (* ) & read(m , *))]]M ’S = 1 just in case we can find an assignment to the variable x o f an individual in D such that the m atrix (b o o k (* ) & read ( m ,x ) ) with that individual assigned to the variable * is true. It is clear that the given variable-assignment g does not fulfill these conditions, since we already found out that it makes the m atrix false. So n ow we are instructed to look be yond the given variable- assignment g that comes with the m odel, and search among all possible assignment-functions for an alternative g' to g which as signs to * another individual which is a b ook and is read by M ary. W e call a variable-assignment function g' an ^-alternative to g if it is identical to g with respect to all variables other than * , differing from g i f at all only in the value it assigns to Fortunately our simple m od el makes the search for such an ^-alternative o f g in this case easy. W e choose g (x ) = M M iL and determine subsequently that [( b o o k ( * )& read(m . ^:))]M ’S, = 1, since g' assigns M M il to * and M M il is an element in the denotation o f book , and is paired with m ary in the denotation o f the relation read. So by rule (8 ) we now have [(3 * )(b o o k (* ) & r e a d (m ,* ))] ]M ’S = 1, since we have shown that there is an ^-alternative o f g, nam ely g', which makes the subform ula (b o o k (* )& read (m , * ) ) true. Once we finish working up the tree, however, we still have to apply (13-10) to establish a truth value for the whole form ula independent o f the assignment. A ccordin g to (13-10), the form ula (3 * ) (b o o k (* ) & read(m , * )) is true with respect to M iff it is true for every assignment function. One can
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13
show (w ith a lot o f tedious work; it is m o ie com m on to just convince oneself and then asseit it) that the choice o f starting assignment had no effect on the ou tcom e in this exam ple. So, since it came out true for our original g we can be confident that it will indeed be true for all other possible starting choices o f assignments as well. So we conclude that our form ula (w hich is indeed a closed form ula) is true with respect to M , This concludes the com positional interpretation o f (i). N ote at this point that although we initially used the given variable-assignment g, in the end it did not play any role in the determination o f truth-value o f the entire formula. We will return this remarkable fact below. The second exam ple we want to consider is (ii) Every student is reading a book Despite the syntactic similarity o f the English sentences (i) and (ii), their translations to predicate logic are structurally different This difference is prim arily due to the fact that universal quantifiers require a conditional matrix. O f course, these differences will have their consequences for the interpretation o f the formula. (13-12) (V y)[student(y) —> (3 * )(b o o k (* )& r e a d (y , * ))], 7 V
stu den t(y) —> (3 * ) (b o o k (* ) & rea d y(y, * ) ) , 5 stu den t(y), 1
student
V
( 3 * ) (b o o k (* ) & read(j/, * ) ) , 8 x
(b o o k (z )& r e a d (j/, * )) , 3
b o o k (z ), 1 / \ book
*
read(y, z ) , l read
/N V *
W e will use the same m od el and variable-assignment as for (i) for the interpretation o f (ii). The style o f presentation is slightly m ore com pact here as an exam ple o f how lay-out m ay make it easier to follow the com positional
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interpretation procedure,
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331
Answering the exercises relating this material,
you will have an opportunity to develop your own style and im prove on this illustration Com positional interpretation o f (13-12): (1) like (13-11): fb o o k ]M,s = .F (b ook ) = {M M il}, and j[x]M 'g = g ( x ) =jan e, (2) Rule (1): fb o o k (* )J M,s = 0, as jane is not in denotation o f book, (3) Let U M 'S = g ( y ) = m ary, freadJM,s = F (re a d ) = {(m a ry, M M iL }} (4) Rule (1): fread(y, * ) ] M,S = 0, as (mary, jane) is not in |read]M,s (5) from steps (2) and (4), rule (3 ) gives 0 (6) Let g '{ x ) = M M iL , then [(3 x )(b o o k (x )& r e a d (y , x ) ) ] M 's' = 1 (7) fstudentJM,s = F (stu d en t) = {m a ry ja n e } (8) by rule (1), {[student(j/)]]M,s = 1 since g (y ) is in f student]M,s (9) from steps (8) and (6), rule (5 ) gives |student(y) —* (3 * ) (b o o k (* ) & rea d (j/, ^:))JM ’9, = 1 (10) Rule (7): [(V y)[stu d en t(y) —» (3 x )(b o o k (x ) & r e a d (y ,x ))]]M 'g = 0. Let g ’ (y ) = jane, then fstuden t( y ) ] M ’s' = 1, and given g '(x ) = M M iL and f b o o k ^ ) ] ^ 8, = 1, fread(y, x)JjM,g' = 0, and no z-alternative to g', keeping jane the referent o f y , gives a b o o k read by jane. Hence the form ula is false in the given model, (N otes: (1) There is nothing against assigning the same individual to two distinct free variables— functions m ay be m any-to-one! (2 ) T o evaluate a form ula containing a ‘there is ! within the scope o f a ‘for all’ you need to have access not only to all alternatives o f the given assignment, but also to all alternatives to any o f those. A lthough in small domains as in our example this remains still feasible, for larger domains the search can get rapidly m ore com plex, but for infinite domains the search m ay never end, In fact, on infinite domains even the evaluation of a simple universal quantifier may never end, as derivation by exhaustion is im possible, i.e., as long as you have not yet fou n d a counterexam ple you m ay be in one o f tw o situations; either' a counterexam ple is still to com e, or there is no counterexample. But you never know which situation you are in ,) The final exam ple we will discuss here in detail is (iii) No student is reading a book . The syntactic derivation is given in (13-13)
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13
(3j/)[stu dent(j/) & (3 * ) (b o o k (* ) & read(j/, * ))], 2 (3j/)[stu dent(j/) & (3 * ) (b o o k (* ) & read(j/, * ))], 8 V
student(y) & (3 * ) (b o o k (* ) & ready(j/, * )) , 3 s tu d e n t^ ), 1
student
V
(3 * ) (b o o k (* ) & ie a d (j/, * ) ) , 8 *
(b o o k (z ) & rea d (y, * ) ) , 3
b o o k (* ), 1 / \ book
read
*
V *
Using the same m od el and starting variable-assignment as before: (1) [ b o o k ( * ) & r e a d ( j/,* ) ) ] M,s = 0 (2 ) Let g '( x ) = M M il, then [b o o k (* ) & read(j/, a;))J'M'
= 1, so [(3 * )(b o o k (* )
& rea d (y , x ) ) ] M,g = 1 (3) {[student (j/)JM,s = 1 (4 ) Rule (3): [studen t(y) & (3 * )(b o o k (* ) & read(j/, *))| M,S = 1, (5 ) Rule (8): [(3 y)[stu d en t(y ) & (3 * ) (b o o k (* ) & read(j/, *))]]M ’S = 1. (6 ) Rule (2): J ~ (3 j/)[student(j/) & (3 * ) (b o o k (* ) & read(j/, * ) ) ] M,S = 0. (N ote: the logically equivalent form ula (V j/)[student(j/) — (3 * )(b o o k (* ) & r e a d (y ,* ))] would require m ore interpretation steps, since here we need to consider all ^-alternatives to g' to see whether they all falsify the formula in the scope o f this universal quantifier (one o f them will n ot, but the others will). P rom a semantic point o f view it is in general simpler to evaluate form ulas where the negation has widest scope, which can be obtained by applying the quantifier laws given in (7-7) through (7 -1 6 ),) We have seen n ow that in interpreting a quantified form ula you first work with the given variable-assignment, but then you are often required to search for an alternative o f it for a particular variable, This interpretive procedure requires not only that all alternatives for any variable to the given assignment are in some direct way available, but whenever the form ula con
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tains n successively nested quantifiers, an equal num ber o f distinct levels o f alternatives to alternatives needs to be accessed. D espite the com positional form ulation we gave this P redicate Logic, in the course o f an interpretation you m ay change what you had, if the given variable- assignment did not give you the right value in the first place (e.g, the application o f rule 8 in the interpretation o f (i))„ It is clear that rule-by-rule com positionality seems at first to be a very strong requirement on the relation betw een the syntax and the semantics o f a form al system.
Yet it seems to allow inform ation
which is gained fu st, to be lost later, In other words, it does not require by itself that the meaning o f the parts is still a recognizable part o f the final result This seems alm ost a flagrant contradiction o f the Fregean Principle o f Com positionality, i f it is interpreted in a static way as adding prim itive se mantic objects to com pose the meaning o f the whole without any interaction between the parts. C om positionality can be u n d erstood m ore dynamically as a requirement on the process o f interpretation, stating that the syntac tic structure will guide the semantics, while having access to all possible variable-assignments, Current research in semantic theory explores various forms o f interpretation processes which are com positional in different ways to different degrees. To conclude this subsection we should return to Frege’s insight m en tioned at the beginning. W e have seen in detail now how the com positional interpretation o f Predicate L ogic defines the denotation o f any quantified formula in terms o f the denotations o f a set o f form ulas in which the rele vant variable is free.
A t a tim e that Aristotelian syllogistic logic was still
the best available theory o f quantification on the market it constituted a tremendous innovation that universal and existential noun-phrases could be treated semantically on a par, if they were ‘ d ecom p osed ’ into a variable and a quantifier prefix and the crucial but small difference o f the conditional versus the con ju n ction in the m atrix. The conditions for this new understanding were created by the m athem atical development a more abstract concept of a function and the fundam ental separation o f the syntax o f a form al system from its semantic interpretation,
1 3 .1 .3
N atu ra l language and com p o sition ality
When it com es to natural languages such as English, there are obvious prima facie obstacles to com position ality that m ake it not surprising that b oth linguists and philosophers before M ontague doubted that English could be given a com position al semantics, One o f the m ost obvious obstacles is the
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phenom enon o f quantifier scope ambiguity in English: the semantic inter pretation is certainly not uniquely determ ined by the syntax if we take the relevant syntactic structure to be surface structure (m uch less deep structure or D -structure.) Indeed the treatm ent o f quantifier scope has been one of the m ost controversial issues within and across theories o f syntax and se m antics from the tim e o f the generative-interpretive semantics split in the late 1960’s down to the present.
Some theories posit a distinct syntactic
level (such as the LF o f M ay (1977) or the underlying representations o f generative sem antics) on which quantifier scope is disam biguated and on which a com positional semantics could potentially be defined (see Cooper and Parsons (1976)), M on tagu e’s rule-by-rule version of the com positional ity requirement makes possible a treatm ent in which the syntactic derivation rather than any level o f representation disam biguates quantifier scope (see P a itee (1975), D ow ty, W all and Peters (1981),,) Other theories (e,g
Cooper
(1 983)) opt for a more nearly context-free syntax and a weakening o f the com positionality requirement. We m ention these issues to give some indi cation o f the com plexity and controversy surrounding the application o f the com positionality principle to natural language semantics. The m athem atical form ulation o f the com positionality principle in M on ta gu e’s version o f it involves representing b o th the syntax and the semantics as algebras and the semantic interpretation as a h om om orphic m apping from the syntactic algebra in to the semantic algebra The technical working out o f this idea in the general case is somewhat com plex; the details are given in M ontague (1974) and explications can be found in Halvorsen and Ladusaw (1979), D W P (1981, Chapter 8), Link (1979), Janssen (1983), For the logic o f statem ents, whose syntax and semantics are quite simple, the relevant algebras and hom om orphism can be specified quite straightforwardly, (i) The syntax algebra: A = { A ,F 0,. . ,F 4} is defined as follow s: Let Xo be the set o f sentential constants {p ,q ,r ,p ',q ', and let F0,, , , be syntactic operations defined as follows: *bl; « )
=
~Q
F!\[a , 13) f 2\ [ a , 13) Fz(a>P) Fa1
=
Q & (3
= =
a V f3 a —> (3
=
a +-+ (3
(N ote: Here a and j3 are stants. These operations should be read as e.g. ’take a sentential constant and execute F q by prefixing it with the n egation -sym bol’ , etc.)
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Then the set A , the set o f all well-form ed expressions o f the logic o f state ments, is defined as closure o f the set X o under the operations F o ,. . , ,F±. Note that this gives exactly the same effect as the corresponding set o f five recursive rules, one for each o f the operations. This language is particularly simple by virtue o f containing only one syntactic category, that o f sentences, and in having a one-one correspondence between syntactic rules and syntac tic operations that allow us to conflate the tw o notions. (ii) The sem antic algebra
B = (B ,G q,
, G 4} is defined as follow s: B =
{ 0, 1 } , the set o f truth values. Go, ■ ,C?4, the semantic operations corre sponding to the syntactic operations F o ,,, . ,F^, axe the unary and binary functions from truth values to truth values defined by the usual truth ta bles for the respective operations Fo,. . . ,F i„ So Go, for exam ple, is the function from { 0 , 1 } to ( 0 , 1 } defined by (13-13), and G i the function from ( 0 , 1 } X ( 0 , 1 } to ( 0 , 1 } defined by (13-14). (1 3-14 )
G o( l ) = 0 G o(0) = l
(1 3-15)
G i(l,l)= l
G i ( l , 0) = 0 G i(0 ,l) = 0 G i ( 0, 0 ) = 0
(iii) Sem antic interpretation as a hom om orphism from A to B:
An
interpretation for the statement logic must assign a truth value to each sentential constant, and m ust recursively assign truth values to com plex statements on the basis o f their syntactic structure. Let / be an assignment o f truth values to the constants, i.e. a function from X 0 to ( 0 , 1 } , and let g be the interpretation function defined as follows: If
= =
1 1
iff iff
g{4>) g {4 )
= =
0 1 and g(tp)
=
1
g((
=
1
iff
g{4>)
=
1 o r g {ip )
=
1
9 ((<£->■ i ’ )) g((
= =
1 1
iff 9 W if f g(<j))
= =
0 o r g (tp ) g(ip)
=
1
W e can then see that g is indeed a hom om orphism from A to B every formula in A is m a p p ed by g onto a truth value in B , and the m apping is
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structure-preserving with respect to all the corresponding operators, G { in B corresponding with F{ in A ; this is established b y showing that (13-15) and (13-16) below are valid. (1 3 -1 6 ) g (F 0(
(1 3 -1 7 )
g{F{{
The hom om orphism requirements (13-15) and (13-16) m ay also b e schema tized by a pair o f diagrams as in (13-17) and (13-18), where g x g in (13-18) stands for the function m apping pairs o f formulas ($ , tp) on to the correspond ing pairs o f truth values (g (), g(ip)). (1 3 -1 8 ) *0
{ 0, 1}
{ 0 , 1}
(1 3 -1 9 ) A x A
----------------- ------------------ -
{0 ,1 } x {0 ,1 }
A
{ 0, 1}
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A further im portant property o f grammars that conform to the com posi tionality principle in this strong hom om orphism sense is that it is possible to think o f such grammars as generating expressions and their interpretations in tandem ; one need not conceive o f the syntax generating an entire expres sion “before” the semantics interprets it, The small piece o f a schematic interpreted derivation tree in (13-20) illustrates this point, (1 3 -2 0 )
(F2{F0(F3(a,P)),i),G2{G0( G M * ) A m , 9 b ) ) ) (F 0(Fa(a ,P)),G 0(G 3(g(a ) , g m ) )
(F3{a,0),Gs{g{a),gm) /\ (a,g(a))
(0,g(0))
(7,ff(r))
Returning to the issue o f com positionality in natural languages, we con clude this section with an inform al, intuitive discussion o f a core exam ple o f quantifier-scope am biguity and the way in which it is connected to anaphoric binding in discourse. The purpose is here merely to illustrate the im portance and extent o f the issue, rather than to legislate on the best possible solu tion. It should be clear from Chapter 7 and this chapter that the syntactic structure o f a logical form ula determines the set o f possible interpretations in which it is true, given an interpretation to the non-logical constants. B ut syntactically different formulas m ay have some m odels, though not all (un less they are logically equivalent), in which they are true in com m on. For instance, consider (i) and (ii) (i) ( V x ) ( 3 y ) ( P ( x ) - > R ( x ,y ) ) (ii) (3 2 /)(V * )(P (* )-^ iZ (* ,2 /)) In a m od el in which there is at least one variable-assignment to y such that every x which is a P bears the relation R to it, b o th (i) and (ii) are verified (cf, exercise 3). O f course, (i) is true in other models as well, since (ii) entails (i) but not vice versa, So (i) leaves in a manner o f speaking more open or undetermined as to what the m od el has to look like for it to be true, whereas
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(ii) is more demanding. If' we use an English sentence corresponding to (i), e.g
E very student loves som eon e (dom ain : P eop le), we may well mean that
every student loves the same person, although we cou ld have expressed that also with The students all love the sam e p erson or som ething like this. We should consider it a virtue rather than a vice that natural language allows us different ways o f describing the same situation, especially since further context often provides a clue as to which kind of situation is intended
For
instance, we could continue with M aria is her name N ow we understand that the students must all love the same person, since b o th the proper name and the possessive pron oun dependent, on it require a unique referent, (L et’s forego the far fetched interpretation in which M aria cou ld name any woman loved by a student as if it was used as a generic proper name for student loved w om en) So it is this second sentence that serves to ’disam biguate’ the first, and from a semantic p oin t o f view cuts down the set o f possible models in which just (i) is true to the subset in which (ii) is true as well Now we could require that the first English sentence should be repre sented as the logical form ula o f (ii), if it is follow ed by this second sentence. This would mean that the first simple English sentence would be translated to tw o distinct logical formulas If we also want to adhere to com positional semantics for English, this would have as a consequence that there should b e tw o distinct syntactic derivations o f this English sentence. This option is taken b y quite different contem porary theories o f quantifier scope e g. May (1985) and M ontague (1974), but not b y e g.
C ooper (1983),
W e should,
however, realize that this syntactic disam biguation o f English sentences is really a syntactic reflection o f a semantic process, nam ely determining the set o f m odels in which a sentence is true or the set of situations which can be correctly described by the sentence. Quantifier scope can be encoded syntactically in the order o f the quantifiers in the prefix or b y coindexing in various ways, but it is the choice o f variable assignments which d o the real w ork o f characterizing the dependencies between individuals.
13.2
Lam bda-abstraction
1 3 .2 .1
T y p e th eo ry
T he lam b da-operator, A, was introduced b y A lon zo C hurch (C h urch (1941)) to perm it the construction o f expressions which unam biguously and com posi tionally denote functions, W ithou t the A -operator, function names can only
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be in trodu ced by contextual definition, as in “ Let / be the function from R to R such that / ( * ) = x 2 + 3 ” There also tends to be equivocation in ordinary notational practice, using expressions like f ( x ) or x 2 + 3 to denote either a function or its value at argument * , an equivocation that is hard to avoid w ithout a system atic way o f building descriptive names o f functions, In the A-calculus, the function / such that f { x ) = x 2 + 3 would be denoted by the expression A * ,* 2 + 3 (or \ x [x 2 + 3] or ( \ x ( x 2 + 3 )); notation is not uniform ), and the application o f that function to an argument z would be expressed as (A*,,*2 + 3 )(z ), which as we shall see below is equivalent to the expression z 2 + 3, W hat has made A -abstraction so valuable a t o o lin contem porary seman tics is the recognition that m any o f the most basic syntactic constructions o f English can be interpreted com positionally as involving function-argum ent application, and that m any o f the apparently less basic constructions can be given a com position al semantics involving A-abstraction, We will illustrate some o f the linguistic applications at the end o f this section. As a preliminary, we need to introduce some basic notions o f type theory, (There exist b o th typed and untyped versions o f the lam bda calculus, but it is the typed version that has becom e familiar to linguists from M on tagu e’s work ) A type system is a system o f semantically m otivated categories de signed so that restrictions on well-formedness stated in terms of types can guarantee that any well- form ed expression will be sem antically well-defined, Russell in trodu ced the notion as a way o f regim enting the language of set theory so as to make the paradoxical “ sets” m entioned in Chapter 1 unexpressable. T he type theory we introduce here is just one among many, but is a com m on basic one. The set o f types is defined recursively as follow s: (13-21)
(1) e is a type. (2) t is a type, (3 ) I f a and b are types then (a, b) is a type.
Sometimes rule (3 ) in the form ation o f types is replaced by the more general rule (3 '): (3 ') I f a i , ...... On and b are types, then ( a i , . „ , , an , b) is a type.
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Since functions o f more arguments can be reduced to unary functions, the type (a j, ,a n ,b) m ay be replaced by ( a1, { a 2, {. , { an ,b ) ) , . . and there fore one does not need to postulate types for such n-ary functions explicitly. A typed language is a language each o f whose well-form ed expressions is assigned a type b y a com positional syntax whose semantics conform s to the follow ing principles, where D a stands for the set o f possible denotations of expressions o f type a. (1 3 -2 2 ) Let A be a given dom ain o f entities Then (1) D e = A (2) Dt = { 0 , 1 } , the set o f truth-values (3) D(^a ^ = the set o f functions from D a to Df,
In case an n-ary type is defined, its intended interpretation is: D/
VQ-1 ■- «
ft 10 1
the set o f functions from D ai x
, X D an to Df,
In other words, expressions o f type e denote individuals, expressions of type t denote truth values, and expressions o f type (a, 6) denote functions whose arguments are in D a and whose value is in Df,. Exam ples: (1) Suppose A , and hence D e, is the set o f real numbers. Then the expression m entioned in the previous section, (A * (* 2 + 3 )), is o f type (e, e), since it denotes a function from numbers to numbers (2) Sets in a typed language are identified by expressions denoting their characteristic functions, Consider, for instance, the set o f numbers greater than 7, Its characteristic function / is defined by (13-23), (1 3 -2 3 )
In A-notation, / can be expressed as (A * (* > 7)),. Since / is a function from numbers to truth-values, the type o f the expression is (e ,t ); this is the type for any expression denoting (the characteristic function o f) a set o f entities.
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T he predicate calculus can be given as a typed language; individual
constants and variables are o f type e, formulas are o f type t, one-place pred icates (w hich denote sets) are of type ( e ,f ) , tw o-place predicates are o f type ( e , ( e ,f ) ) , etc
1 3.2 .2
T h e syn ta x and sem antics o f A -abstraction
The A -operatoi and the rule o f A -abstraction give us a means for form ing com positionally interpretable names o f functions C om pare, for exam ple, the schematic letter name / with the structurally descriptive name (A * (* 2 + 3 ) ) . Now, how in general does one form an expression like the latter? Start with an expression that denotes the value o f the desired function for a variable argument: in this case, ( x 2 + 3) T o form the name o f the function which applies to an arbitrary num ber * and gives as value the num ber ( * 2 + 3), we “ abstract on ” the argument variable * and form the A-abstract (A * (* 2 + 3 )), consisting o f the prefixed X -operator Ax and the b o d y ( x 2 + 3). M ore generally, we have the syntactic form ation rule in (13-24). (1 3-24) X-abstraction, Syntax. If u is a variable o f type a and a is an expression o f type b, then (Awq) is an expression o f type (a, 6). T he A -operator, like the quantifiers, is a variable-binding operator. A n d like the quantifiers, its semantics makes crucial appeal to variable assign ments But note that the A-operator can change the type o f its argument, e g, from t to some functional type, whereas quantifiers take form ulas to make formulas and hence do not change their argument-type, (13-25) A-abstraction, Semantics: Given u of type a, a o f type b, [[(Awq)Jm,s is that function / from D a to Dj, such that: for any ob ject k in D a, f ( k ) = J a ]M,s , where g' is just like g except that g '(u ) = k.
To take our earlier exam ple, |(A* (* 2 + 3))JM,S is that function / such that for any num ber n in D e , f ( n ) — | ( * 2 + 3 )]M,S . T hat is, we find the
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value o f the function for a given argument n by evaluating the b o d y o f the A-abstract with respect to an assignment that assigns n to *, Suppose n = 5 , Then / ( 5 ) = [ ( z 2 + 3 )]M,S\ where g' { x) = 5; supposing the expressions in the form ula have their standard semantics, the value o f the expression (* 2 + 3 ) on the assignment o f 5 to * will be the num ber 28, where g' is the variableassignment which assigns exactly the same values to all variables other than x as g does, but assigns the number n to * T he process we have just illustrated is called lambda-conversion: it com putes the value o f the lamb da-expression for the argument the function is applied to It is im portant to realize that lam bda-conversion is fundamen tally different from the syntactic process o f substituting a constant for a variable in a formula, The form er is a semantic and com putational process o f executing an algorithm for a given argument, the latter should be viewed as a syntactically defined ’transform ation’ on formulas or a rule o f inference which supposedly guarantees that truth is preserved in a particular deductive system, Lam bda-abstraction gives us the im portant means to distinguish a function from the set o f its values, which has proven trem endously useful for linguistic applications, especially where intensional aspects o f meaning play a crucial semantic role (cf, section 13.2.5), There is an im portant restriction on lam bda-conversion that should be adhered to for (\ u a )(P ) to be logically equivalent to a ', where a ' is just like a but with the assignment o f j3 to the variable u. W e have to make sure that in case /3 contains any free variables o f some type, they do not accidentally get b ou n d b y any quantifiers occurring in a
For instance, we cannot convert
( A ^ z ( P ( z ) M ) )(£ (* ) to 3 x ( P ( x ) t Q( x) ) The reason why this conversion w ould not always be logically equivalent is that Q ( x ) is interpreted by the given variable assignment in the first formula, but could be interpreted by an ^-alternative to it in the second form ula re sulting in different truth-conditions. O f course, to avoid accidentally binding free variables one seeks recourse to logically equivalent alphabetic variants o f the argument the lam bda-term is applied to (cf, section 7 3 for the notion o f alphabetic variants), Similar restrictions apply when the form al language contains operators which have quantificational force, as the intensional lan guage defined in Chapter 15, But we will not discuss these any further here.
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Less accurately but quite m nem onically, one can say that (A u a ) denotes a function with % as argument and [ a ] as value,
1 3 .2 .3
A sam ple fragm ent
Drawing together the notions we have introduced thus far, we present the syntax and semantics o f a sample form al language-schem a T L , a schema for typed languages whose logical constants include all those o f the predicate calculus with equality, plus the lam bda operator Particular languages L falling under this schema differ in the choice of (non-logical) constants and their interpretation I. S y n t a x o f T L (1 3 -2 6 ) The set T o f types o f T L is the smallest set such t (i) e ,t £ T (ii) I f a ,b £ T , then ( a , b) £ T
(1 3 -2 7 )
The prim itive vocabulary o f TL consists o f the follow ing: (i) The connectives ~ , & , V , —♦,<-* (ii) The quantifiers V, 3 (iii) T he lam b da-operator A (iv) T he equality sym bol = (v ) The parentheses ), (, [, ] (vi) For every type a, a denum erably infinite set VARa containing variables b„ i0 for each natural num ber n (vii) For every type a, a (possibly em p ty) set CON% o f (nonlogical) constants o f type a
Note: the sym bols introduced in ( i )-( iv ) are called logical con stan ts; their meaning is fixed for all languages in the fam ily TL, T he non-logical constants of (vii) are language-particular, and their interpretation must be specified in a m odel for a particular language L„
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S y n t a c t ic ru le s o f T L T he set MEa o f meaningful expressions o f TL o f type a is defined recursively as follows: (i) For each type a, every variable in VARa and every constant in CONL a is in M Ea (ii) For any types a and 6, if a £ ME^ab^ and (3 £ M Ea, then a(0)eMEb. (iii) For any types a and 6, if u is a variable o f type a and a 6 then (Awq) 6 ME^aby (iv ) If (p and ip are in M E t (aie form ulas), then the follow ing are also in MEt'- ~
V0>
^0
(v ) For any type a, if
(v i) For any type a, if a and (3 are b o th in M Ea, then (a = ft) £ M Et.
As a particular instance o f this schema which we will use for examples in the text and in the exercises, let T L A ( “ typed language o f arithm etic” ) be syntactically defined b y the follow ing choice o f constants: (1 3 -2 9 )
C on sta n ts o f T L A . C 0 N TLA = { 0 , 1 , 2 , .... } C
O
con
N
= {e v e n , o d d , p r im e }
( , M ) =
COn TY \e,e/
= {s u c c}
C0 Nl l( i ) ) = {pIus} N ote that, e.g , 0 is a syntactic name, not a num ber. In (13-35) these
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n a m e s a re a s s ig n e d th e ir n a t u r a l r e fe re n ts
In practice it is com m on to drop outerm ost parentheses, and to drop other parentheses where there is no danger o f ambiguity, W e will also follow a com m on practice in using the follow ing, m ore readable, variables in place of the official ones: (1 3 -3 0 )
V a r ia b le c o n v e n t io n s Type
Variable
€
x , y , z (also with subscripts or prim es)
(e ,f)
P,Q
( e>(e>0)
R,S
(e >e)
f
Exam ples of well-form ed expressions o f various types o f TLA follow. E xpression (i) (ii) (iii) (iv) (v ) (vi)
o d d (z )V e v e n (z ) s u cc (s u c c (* )) g r (3) g r (3 ) (z ) (A z g r (3 )(x )) ( A * g r (3 )( * ))(s u c c ( 3 ))
Type t € (e ,f) t ( e, t) t
In order to parse a com plex A-expression, it is often helpful to construct a tree that displays its derivation according to the syntax, just like the derivation trees in 13.1.1 and 13 1.2 for the two logical languages Such a tree structure is sometimes called a derivation tree or an analysis tree. A derivation tree for (13-31) is given in (13-32). In the derivation trees for T L A , it is useful to annotate each node first with the type of the expression on that n ode and, second, with the number o f the rule that we applied in constructing that expression. The construction o f such a derivation tree can also b e valuable aid in working out the semantics of com plex expressions, since the semantic interpretation rules correspond rule-by-rule to the syntactic derivation rules. W e give the semantics for TL in general below , follow ed b y the language-particular interpretation o f the constant o f T L A
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(A* gr(3 )(* )), (e,t), 3
/\ _ x’e
gr(3)(x),t,2
/\ gr(3),{e,Z), 2 / \ gr ,(e,(e,t))
x,e
_ 3,e
I I . S e m a n tic s o f T L (1 3 -3 3 ) Given a non-em pty set A , the domain o f entities, the set D a o f pos sible denotations o f expressions o f type a is given by the following definition: (i) D e = A D t = {T R U E , FALSE (or 1 and 0 con ven tion a lly)} (ii) For any a,b £ T ,
= D f 0-, i.e. the set of all functions
fio m D a to Df,,
T he denotation o f an expression o f a language L o f the fam ily TL is defined relative to a model M and an assignm ent g o f values to variables. A model fo r L is an ordered pair M = (A , F ), such that A is a domain o f entities and F is a function which provides an interpretation to the primitive non-logical constants o f L, i.e. a function which maps each constant o f type a on to a denotation in D a, A n assignment g o f values to variables is a function which assigns to each variable un,a a value from the set D a(1 3 -3 4 )
S e m a n t ic r u le s o f T L The denotation o f an expression a relative to a model M and an assignment g, sym bolized is defined recursively, in parallel to the syntactic rules given in (13-28) above, as follow s: (i) (a) If a is a non-logical constant in C O N \, then
=
F (a ), (b ) If a is a variable in VARa, then |aJM,s = g( a) . (ii) If a
£
ME^ab^ and
M M’s ( M Mn ° ’
/3
£
M E a,
then
Ja(/3)]]M’S
=
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(iii) = the A-rule given in (13-25), (iv ) If d>, ip £ M Et, then:
[[~
= i iff
= o
tpJM’3 = 1 iff 14>]M,S = 1 and
= 1
V tp\M'9 = 1 iff \4>}M’g = 1 or {ip\M’g = 1 [[^ —> ^JM,S = 1 iff = 0 or = 1 ^(b <->
= 1 iff
= M M,S
(v ) I f 4> £ M E t, u £ F-AiZo, then (a) JVu<^]M,s = 1 iff for all d £ D a, = 1 (b ) l3u<j>}M,g = 1 iff there is at least one d £ D a such that = 1. (v i) If a,/3 £ M E a, then [a = /3}M’9 = 1 iff [ a ] M,s = 1/3}M'9.
N ote that the only language-particular part o f the interpretation of a lan guage L o f the fam ily TL resides in the choice o f a m odel M , i.e. in the choice o f a dom ain A o f entities and an interpretation F o f the language-particular, i.e. non-logical, constants. T he specification o f the set o f constants and their interpretation F can be thought o f as the lexicon of the language. Given the m odel, the semantic interpretation o f the infinite set o f expressions o f the language is fixed by the recursive semantic rules, which are the same for all typed languages of the fam ily TL. The semantics for the sample language T L A is fixed by the follow ing specification of a m odel: (1 3-35)
A m odel M fo r T LA M = (A , F ), where A = the set of natural numbers { 0 ,1 ,2 , 3 , . . . } and F is specified as follows: (i) F (0 ) = 0 , F ( I ) = 1, etc. (ii) .F (e v e n ) = (the characteristic function o f) the set o f even numbers .F (o d d ) = (the characteristic function o f) the set of od d n um bers. F ( p r i m e ) = (the characteristic function o f) the set o f prim e numbers.
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(iii) F ( g r ) ( “ greater than” ) is that function / such that /( a ) ( 6 ) = 1 iff 6 is greater than a Note that / ( a ) = the (characteristic function o f th e) set o f numbers greater than a, (iv) .F (s u cc) = the successor function (v ) F (p lu s ) is that function / such that / ( a ) ( 6 ) = a + 6; / ( a ) by itself is a function which takes a single argument and adds a to it.
We can now determine the denotation of the expressions given as exam ples in (13-31); this we do in (13-36). Denotations must be relative to model M and an assignment g; M was given in (13-35), and for g we will choose, arbitrarily, an assignment which assigns the number 2 to every variable. Details are left as an exercise to the reader. )
E xpression a (i)
(H) 'iii) 'iv)
1 3 .2 .4
o d d ( * ) V e v e n (* ) s u c c (s u c c (* ))
gr (3)
D en otation [ a ] ^ ’9 T R U E , independent o f assignment 4 the set o f num bers greater than 3 false, since 2 is n ot greater than 3
(v)
g r (3 ) (z ) (A* g r (3 ) (z ))
Vi)
(A* g r ( 3 ) ( z ) ) ( s u c c (3 ))
T R U E , since 4 is greater than 3
the set o f num bers greater than 3
T h e la m b d a -c a lc u lu s
T he pure lam bda-calculus is a theory o f functions as rules. It was devel oped around 1930 to serve as foundation for logic and m athem atics in its type-free version, in which any function m ay be its own argument.
(In
Zermelo-Fraenkel set-theory such self-application was ruled out b y the ax iom o f foundation.) Despite the paradoxes that arose in such unrestricted type-free systems, a consistent part o f the theory has turned out to be ex trem ely useful as a form al theory o f (partial) recursive functions. It has more recently been successfully applied in com puter science for instance in the semantics of program m ing languages. In linguistics it is not the entire system of lam bda-calculus, but rather its operation o f lambda-abstraction and conversion that have proven extrem ely useful as we will discuss in more
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detail in the next section. T he present section contains a very elem entary ex position o f the lam bda-calculus itself, illustrating how fundam ental a theory o f functions it really is. The syntax o f the lam bda-calculus is very simple. The set A o f lam bdateim s and formulas of the lam bda-calculus are defined in (13-37) recursively, (1 3 -3 7 )
S y n t a x o f th e la m b d a -c a lc u lu s Lexicon:
Rules:
A
-variable -lam bda-abstractor
(,)
-parentheses
(i) * 6 A (ii) if M 6 A then (Ax M ) 6 A (abstraction ) (iii) if M ,N 6 A then ( M N ) 6 A (application) (iv) if M , N 6 A then M = N is a formula
T he axiom atization o f the theory is given in (13-38), Here M , N and Z are arbitrary lam bda-term s (1 3 -3 8 ) A x i o m a t i z a t io n o f t h e la m b d a -c a lc u lu s (i) (Ax M ) N = M ' which is M with every occurrence o f * re placed b y N (conversion) (ii) = is reflexive, sym m etrical and transitive (iii) if M = N then M Z = N Z (iv) if M = N then Z M = Z N (v ) if M = N then (Ax M ) = (Ax N )
If M — N can b e proven from these axiom s, we say that M and N are convertible. It must be n oted that there is a certain asym m etry in for instance the equation expressed in (13-39). (1 3-39)
(A * s u c c (* )) (3 ) = 4
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The statement in (13-39) can be interpreted as “ 4 is the result of com puting the successor o f 3” , but not vice versa. This asym m etry is expressed overtly by calling the com putation a reduction o f (A * s u c c (e )) (3 ) to 4, One of the central theorems o f the lam bda-calculus, called the C hurch-R osser theorem , says that if two terms can be converted, then there is a term to which both can be reduced There are many more notions o f conversion and reduction definable in the lam bda-calculus than we in trodu ced here Our rule (13-25) is sometimes called /3-conversion, The lam bda-calculus it itself not a logical, inferential system , but we can define the usual predicate logical concepts in lam bda-term s as in (13-40). We give here the definitions from Henkin (1963), which are also presented in Gallin (1975) Here we use a typed language m erely for convenience and coherence with the previous sections: (p and ip are variables o f type t, i.e. form ula-variables, and / is a variable o f type (t,t). (1 3 -4 0 )
T R U E = (fc/ {(A
~ = * / (A^»(FALSE= k
= d e f ( \ < p \ i p ( \ f ( f ( < p ) = i p) =
A /(/(T R U E ))))
= d ef (A<M 1p((pk ip = <j>) v = def (A
Ip))
x aA ~ def [ Xx aA = A zaT R U E )
3xaA — j
~
xa ^A
In (13-40) the definitions of the disjunction and the existential quantifier are already quite familiar from Chapter 7
In developing an intuition for
what the other definitions do, we can understand for instance the definition o f conjunction as the operation o f assigning the truth-value T R U E to both formulas and of the conditional as stating that the inform ation that ip is true is already contained in the inform ation that
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paradigm atic program m ing language, laying out the applicative behavior o f the com pu tation al procedures in their barest form
But it is for most prac
tical purposes not a very handy program m ing language, exactly because o f its com pletely transparent structure. Yet several well-known program m ing languages have some features which resemble the lam bda-calculus. In LISP, for instance, procedures can be arguments o f procedures, as well as output o f procedures. Due to the fact that functions could take themselves as arguments, i e. the type-free character o f the pure lam bda-calculus the question arose what models o f the theory could look like A set X was needed in to which the entire function space X —* X could be em bedded, which was im possible in ordinary set-theory due to C an tor’s theorem (cf. Chapter 4), D ana Scott in 1969 developed models of the type-free lam bda-calculus b y taking only the continuous functions on X Only then did it becom e clear how a denotational semantics o f program m ing languages could be constructed. This has opened a vast area for innovative research in the com m on aspects o f the semantics o f program m ing and natur al languages, The interested reader is referred to Baxendregt (1984) for a com plete introduction to the m athem atical aspects o f’ lam bda-calculus and Scott domains.
1 3 .2 .5
L inguistic applications
In this section we will briefly discuss a few o f the many applications to natu ral language semantics that have made lam bda-abstraction a basic to o l in the developm ent o f form al semantics. In general, lam bda-expressions provide a means for giving com positionally analyzable names to fu n ction s of arbitrary types. This can be richly exploited in giving a com positional semantics for natural languages like English which have a large num ber o f syntactic cat egories and quite a variety o f recur sive mechanisms that end up em bedding expressions o f one category in expressions o f another or the same category. Lam bda-expressions can make explicit how the interpretation o f a relative clause is a function o f the interpretation o f a corresponding open sentence, how predicate negation or conjunction is related to sentential negation or conjunction, how the meaning o f a passive verb can be defined in terms o f the meaning o f a corresponding active verb, etc. Lambdas can also be used to make explicit the interpretation o f various gram m atical morphemes or function words, such as reflexive clitics and other “ argum ent-m anipulating” devices, logical determiners such as every and no, the com parative m or pheme -er, etc
W e will illustrate just a few of these applications below ;
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for other illustrations and references to more, see M ontague (1973), Cresswell (1973), Partee (1975), W illiams (1983), D ow ty, W all and Peters (1981), D ow ty (1979), Janssen (1983), and in fact much o f the form al semantics literature since the m id-1970’s
Lambdas becam e widely known to the lin
guistic com m unity through the work o f M ontague, but M ontague was not their inventor (A lon zo Church was: see Church (1941)), and one does not have to be a “M ontague gram m arian” or even a form al semanticist to find them useful. W h at does it mean to give com positionally analyzable names to func tions? The contrast is similar to that betw een proper names like John and definite descriptions like the oldest student in the sem antics class In much com m on m athem atical practice, and in m ost o f this b ook , functions are re ferred to with proper names, either coined on the spot or established names like M eet or + , or with typical variables in place o f names, like / and g In the language TL and other such typed languages that include lam bda abstrac tion, we have not only the possibility o f including as m any proper names o f functions as we wish but also the possibility o f building up structured function- denoting lam bda-expressions for which the semantics specifies just what function each such expression denotes.
I. Phrasal conjunction One very simple and elegant application o f lam bda-abstraction which M ontague exploited in P T Q is its use in defining the interpretation o f “ Boolean phrasal conjunction, disjunction, and negation in terms o f sentential conjunc tion, disjunction, and negation respectively, (T h e term “ B oolea n ” phrasal con ju n ction refers, for reasons the reader will appreciate, to the kind o f phrasal conjunction illustrated in the examples below , as opposed to e,g, the group-form ing NP con ju n ction illustrated b y “John and M ary are a happy cou ple” , or the irreducibly phrasal conjunction o f adjectives in “ a red and white dress” ,) T o illustrate this application, we give below a few syntactic rules in sim ple phrase-structure-rule form together with corresponding schematic trans lation rules W e will com m ent on the nature o f such rules after illustrating them.. Syntactic Rule S -* S and S S -* S or S
Translation In to TL & S'2 S [ v S'2
V P -> V P and V P
Xx ( V P ; ( * ) & V P ^ ( * ) )
NP
A P ( N P ;(P )V N P ^ (P ))
NP or NP
LAM BD A-ABSTR ACTIO N
First a note on the NP interpretations presupposed here:
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as we will
discuss later in this subsection, we interpret NPs as generalized quantifiers, o f type { { e , t ) , t } . W e use P and Q as variables o f type (e ,f). Second a note on the nature o f these rules: The syntactic rules are given in familiar phrase-structure form , but could easily be recast in the recursive form at specified earlier W hen dealing with simple context-free grammars, it is straightforward to convert a grammar o f one form to one o f the other. The translation rules are not semantic interpretation rules as defined above, since they specify expressions in T L as their output rather than m odel-theoretic objects The first rule, for instance, says that the translation into TL o f any expression form ed by application o f the first syntactic rule will be form ed by taking the translations o f the tw o parts and putting an & betw een them, (W e subscript the S’s in the translation simply to distinguish occurrences ) The resulting form ula o f TL will itself be semantically interpreted according to the semantic rules o f the language T L , which we do not repeat here (see Section 13.2.3), As M ontague showed, as long as the translation rules are com positional and the semantic interpretation o f the interm ediate language is com positional, the interm ediate language could in principle be eliminated and a com positional m odel-theoretic semantic interpretation given directly to the source language (in this case a fragment o f English ) But the use o f an interm ediate language is at least convenient; whether such an interm ediate language plays a role in capturing “linguistically significant generalizations” is a point o f some debate In any case we will continue to use the conventions illustrated above in giving short versions o f rules, including the convention o f using primes (') to indicate the translations o f the parts. In order to illustrate the workings o f the rules above, we need to com bine them with a few other rules for simple sentences; these follow M on tagu e’s treatment in P T Q except for supressing intensionality.
s NP NP
CN
NP'(VP')
AP ( 3 x ) ( C N ' ( x ] 1& P ( x ) ) \ P .(V x )(C N' ( x ) I ^ P ( x ) )
—
^
so m e/a CN every CN
—
^
walks, talks, , , „
AP . P ( j ) , etc walk', talk', , ,
man, wom an,
m an', w om an', ,
NP VP
NP VP
John, Mary,
. ,
Using these rules and lam bda-conversion, the reader can verify equivalences and non-equivalences like the following,
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is equivalent to:
(b ) John walks and John talks
(1 3 -4 2 )
(a) Some man walks and talks
is n o t equivalent to:
(b ) Some man walks and some man talks Before linguists had learned to exploit lam bdas, B oolean ph iasal con ju n c tion was treated syntactically b y positing some kind of “ con ju n ction reduc tion ” transform ation oi schema, which w ould, for exam ple, derive (13-41a) above from the full conjoined sentence (l3 -4 1 b ) B ut as (13-41) and (1342) together illustrate, syntactic conjunction reduction sometimes preserves meaning and sometimes does not It is now widely appreciated that the actual distributions o f forms and meanings in such cases o f B oolean phrasal conjunction can be more system atically explained by directly generating the phrasal conjunctions syntactically and using semantic rules like those for m ulated above with lam bdas to interpret them It thus appears that the intuition that phrasal con ju n ction is derivative from sentential conjunction is best regarded as based on semantic rather than syntactic generalizations. D irect syntactic generation com bined with com positional semantic interpre tation can, i f all goes well, capture the scopal interactions o f conjunction with other scopally relevant parts o f the interpretation, without any special stipulations. Further linguistic research has been and continues to be concerned with exploring the limits and generalizability o f such an approach to B oolea n con junction as well as analyzing the various sorts o f non-B oolean conjunctions (m entioned above) and their interrelations. One major generalization ex plored by Keenan and Faltz (1985), Gazdar (1980), and Partee and R ooth (1983) centers on the observation that the particular translation rules given above for V P -con ju n ction and N P-disjunction should not have to be stip ulated: it should be possible to predict them just by know ing (i) the in terpretation o f sentential conjunction and disjunction and (ii) the semantic types associated with the syntactic categories V P and NP, Partee and R ooth propose a schema for generalized conjunction, recursively defined, which has those properties, and which replaces the individual explicit rules given above, so that the lam bdas no longer play an overt role. B ut the functions that are denoted by those lam bda-expressions are playing the same central role in the resulting account, and the lam bda notation for denoting them compositionally helped to make the relevant generalizations findable
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II. Relative clauses A nother construction whose semantics has been much better understood since the in trodu ction o f lam bdas in to the linguist’s to o lb o x is the relative clause. W hile many open problem s remain concerning b oth the syntax and the semantics o f relative clauses o f various kinds in various languages, we can at least give one very str aightforward account o f one very basic kind o f relative clause construction as another g o o d sample application o f lam bdaabstraction
The account is probably approxim ately correct for some real-
life cases and at least suggestive o f how to proceed for others
W e will stick
to the syntactically simplest sorts of examples so as to be able to concentrate on the central aspects of the com positional semantics o f the construction There are tw o parts to the account o f relative clauses; their internal syntax and semantics, i e how they are form ed and interpreted, and the syntax and semantics o f the construction through which they com e to m odify nouns or noun phrases. L e t’s look at these in turn, W hat we call relative clauses are modifiers which play an adjectival sort o f role but which ar e form ed in some sense from sentences Lam bda-abstraction provides an explicit way to say exactly that, as long as we assume that the relevant “underlying sentences” are open sentences, i,e , sentences whose in terpretation contains a free variable in the “relativized p osition ” (a position which m ay end up occupied by a gap or a resum ptive pronoun depending on the construction)
So, for example, we might take all the relative clauses in
(13-43b) and (13-44b) below to be derived from som ething like the sentences (or alm ost sen ten ces)in (13-43a) and (13-44a), I f the translations o f the sen tences are as given in (13-43c) and (13-44c), the translations o f the relative clauses can be derived by lam bda-abstraction on the relativized variable, as shown in (13-43d) and (13-44d),
One can read the expression in (13-43d)
in quasi-English, a bit loosely, as “the property o f being an *3 such that M ary loves *3 ” (T his is loose because “p ro p e rty ” is norm ally an intensional notion and the literal interpretation is just the characteristic function o f a set; but when talking about the m odel-theoretic interpretations in English, explicit set-talk simply tends to be cum bersom e, and the misrepresentation is harmless as long as the context does not allow real, m isinterpretation.) (1 3 -4 3 )
(a) M ary loves e3 (b ) w h o(m ) M ary loves; that M ary loves; such that M ary loves him (c) lov e'(m , * 3 ) (d ) A* 3 .love^m , k3)
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(a) e 2 hates John (b ) who hates John; w ho 2 e 2 hates John; such that he /sh e hates John (c ) hate/ ( * 2, i ) (d ) Xx 2 h a te '(* 2, i )
The semantic type o f the open sentences is t, so when we do lam bdaabstraction with respect to a variable o f type e, the type o f the resulting expressions in (13-43d) and (13-44d) is ( e, t ) , the type o f functions from entities to truth values, i.e. the type o f characteristic functions o f sets o f entities So (l3 -4 3 d ) denotes the set o f entities that M ary loves, and (l3 -4 4 d ) the set o f entities that hate John
This is the same type that is assigned to
simple predicative adjectives like purple or carnivorous L am bda-abstraction on open sentences gives us an infinite stock o f such predicates. Note that we have not actually given explicit syntactic and semantic rules for form ing relative clauses but have simply specified what the effects of the rules should be. This is for tw o reasons. The obvious one is to avoid descriptive com plications and m aintain theoretical neutrality as regards the syntax o f relative clauses; the less obvious one is that in the im m ediately preceding example it was im portant to b e explicit about the interpretation o f NPs as generalized quantifiers, while in this example that would just present extraneous com plications (though no substantive p rob lem s), so we have presented the translations in the form they w ould have after some steps o f lam bda conversion had replaced the original translations o f the proper names and the gaps or pronouns b y expressions o f type e The one addition to the previous fragment that does need to be m entioned is the addition of the gaps or pronouns: the simplest way to add them for our purposes (though this is an area rife with com petin g approaches) is to assume that the syntax (via the lexicon) contains am ong its NPs an infinite supply o f indexed gaps (here represented as e i, e2, etc ) a n d /o r pronouns ( he\, he2, etc ), whose translations are ju st like the translations o f the proper nouns John, Mary, etc,, given in the previous fragment except containing variables like X i , x 2 , etc. instead o f constants like j,m . T he relative clause form ation rule has to b e sensitive to a choice o f relativized variable, since an underlying sentence m ight contain more than one, and the rule would not be com position al if with the same input it could give us tw o non-synonym ous outputs - it would then not be possible to determine the meaning of’ the whole as a function o f
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Aside
from that, the only com plications in the rule are the syntactic ones, the semantics being simple lam bda- abstraction Now how do relative clauses com bine with nouns or noun phrases? This is also an area o f controversy and possibly o f typological diversity as well B ut as discussed in Partee (1973), one can make the case that com positionality requires that relative clauses com bine with com m on noun phrases to form com m on noun phrases in old er for a determiner like no, ev ery , or m ost to have the relative clause as well as the head noun within its scope. W hile this claim can be challenged as well, we will assume that the basic external syntax o f relative clauses is as given in (13-45a); the corresponding semantic interpretation rule is then just a kind o f predicate conjunction, as given in (13-45b) The semantic type o f b oth constituent expressions and o f the result is (e, t), (1 3 -4 5 )
(a) CN -► CN REL (b ) A* (C N '(k ) k R E L '(* ))
W e can illustrate the com bined results o f the relative clause form ation rule and the rule in (13-45) by putting the relative clause in (13-44) together with the CN woman to form the CN phrase woman who hates John. The structure is shown in (13-46a) and the translation in (13-46b). The expres sion (13-46b) is o f a form to which lam bda-conversion can apply; applying it gives (13-46c) as an equivalent but simpler expression (1 3 -4 6 )
(a) [[woman] [who hates John]] (b ) A* (woman'(*) fc A*2 (hate/(*2Ij))(* )) (c) Xx (w o m a n '(* ) k hate'(® , j ) )
Since (13-46c) cannot be simplified further, it is an exam ple o f what is meant b y a “reduced translation” : an expression logically equivalent to the actual translation but simplified as much as possible b y applying lam bdaconversion (and any other relevant simplifying rules the given system may contain). W hether there is always a unique reduced form is a question that can be remarkably difficult and which depends in any case on what the full power o f a particular system is. Other constructions exhibiting unbounded dependencies also appear quite generally to have interpretations which include lam bda- abstraction on the
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position corresponding to the gap or pronoun introduced in the construc tion (if there is one)
One such construction is W H -questions (see especially
G roenendijk and S tokhof 1989)
As a fust step tow ard an interpretation,
we m ight say that the question in (13-47a) is translated as in (13-47b); o f course (13-47b) itself does not specify an interpretation until one provides an interpretation for the expression which man M u ch o f the debate about the semantics o f interrogatives concerns that last poin t; m ost theories agree on the aspects o f the interpretation represented b y (13-47b) (w ith im portant differences, however, concerning the possible scope interactions betw een op erators inside the b od y o f the question and expressions that are part o f the wfe-pkrase) (1 3 -47 )
(a) W h ich m an does M ary believe that Susan wants to marry e-i? (b ) (which m an)'(Aa:i (M ary believes that Susan wants to marry x i ) 1)
Other unbounded dependency phenom ena for which analyses using lam bda abstraction have been proposed include focus phenom ena, the comparative construction (Cresswell 1976), topicalization (Gazdar et al 1985)., and the pseudocleft construction (Partee 1984), Partee (1979) suggests the following generalization: (1 3 -48 )
A ll and only unbounded syntactic rules are interpreted semantically b y rules that bind variables which were free in one or more o f the input expressions
T he generalization as proposed allows that such variable binding m ay be either b y quantification or b y lam bda-abstraction; in terms o f primitiveness, it is w orth noting that it is possible to have lam bda-abstraction as the only prim itive variable-binding operator and define the quantifiers in terms o f it (see (1 3-40)), but not vice versa A lthough the generalization is to o the ory dependent to admit o f direct confirm ation or refutation, its plausibility suggests that lam bda-abstraction can provide a valuable to o l for capturing im portant semantic generalizations
III. Generalized quantifiers Generalized quantifiers will b e the topic o f the next chapter; here we just introduce them briefly as one o f the linguistically very im portant innovations
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that is crucially dependent on having higher types available than are found in predicate logic. L am bda abstraction provides a g o o d way o f giving the explicit semantics for generalized quantifiers and for the determiner meanings that m ay go in to building them up As indicated im plicitly in earlier examples, we can treat N P ’s as form ing a semantic as well as a syntactic constituent within a typed logic like M on tagu e’s, and this was one o f the m ost influential of M on ta gu e’s innovations in his classic paper “P T Q ” (M ontague 1973), The N P ’s every man, som e man, John, and M ary given in the earlier fragment are interpreted as generalized quantifiers; the reader should convince herself that their type is ((e, t ) t ) (e.g by drawing a derivation tree for the logical expression which translates each o f them and labelling each node with the type o f the expression ) Given those interpretations o f every man and som e man, it is possible to abstract on the CN position and get back a statement o f the meaning o f the D et W e d o this for som e in (13-49a-b) and for every in (13-50a-b)„
(1 3 -4 9 )
(a) (b )
some CN: some:
A P .(3a:)(C N /(a:) & P ( x ) ) \Q \ P . ( 3 x ) ( Q ( x ) & P ( x ) )
(1 3 -5 0 )
(a) (b )
every CN: every:
A P (Va:)(CN/(a:)-^ P (a:)) A<5,AP.(V e )(<5(2:)—> P (z ))
T he reader should im m ediately verify that applying the determiner m ean ing given for som e or ev ery to a CN translation like man! will indeed give an expression equivalent to the specified translation for som e man or' every man
One lam bda-conversion step should show the equivalence
W e can n ow replace the tw o earlier rules which introduced every and som e syncategorem atically (i e w ithout assigning them to any category) by the single rule “N P —> D et C N ” together with lexical entries for the specific Dets, M ore members o f the category D et will be discussed in Chapter 14 W e will go through the translation and sim plification via lam bda conver sion o f the sentence Som e man walks, whose syntactic structure is given in (13-51),
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(1 3 -5 1 )
NP Det
VP CN
walks
some
(1 3 -5 2 )
(i) (ii) m
some: man: som e man:
XQ X P . ( 3 x ) ( Q( x ) & P ( s ) ) m an' XQ. XP ( 3 x ) ( Q ( x ) & P (z )) (m a n ')
(reduce b y lam bda conversion before continuing:) iv (v j
walks: som e m an walks: (reduces to :)
X P (3a:)(m an'(a:) & P ( e ) ) walk' (A P ,(3 e )( C N '( s ) & P (a :)))(w a lk ') (3a:)(m an'(E ) & waLk'(z))
W e see from the last line that the translation o f S om e man walks in this system ends up, after lam bda conversion, the same as its translation in predicate logic.
T he difference is not in the interpretation but in whether
or not that interpretation is arrived at com positionally on the basis o f a reasonable natural language syntax In predicate logic there is no way to view the interpretation o f the NP as a c o n s t it u t e n t ; in a more rich ly typed system there is. The treatm ent o f NPs as generalized quantifiers has turned out to yield a very fruitful dom ain o f resear ch; some o f the work in this area is reported in Chapter 14. I V . V P -d e le t io n Y et another early recognized application o f lam bda-abstraction was in the analysis o f V P -deletion , and in particular to capture the distinction between the so-called “strict identity” and “ sloppy identity” readings o f sentences like (13-53), (1 3 -5 3 ) John believes he’s sick and Bill does too.
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The first step in the analysis is to recognize that the first conjunct, re peated in (13-54), is ambiguous even excluding the possibility o f he refer ring to som eone other than John, an ambiguity involving a b ou nd variable reading o f the pronoun, as represented in (13-54a), vs a “p ragm atic” or “ coreferential” reading o f the pronoun, represented in (13-54b) by simply translating the pronoun as the constant j
A lthough (13-54a) and (13-54b)
yield the same truth conditions, they involve differences in what corresponds to the V P believe h e ’s sick that do lead to truth-conditional differences in V P-deletion sentences like (13-53) (1 3-54 ) John believes h e’s sick (a) (Aa:.believe/(a:,sick/ (a :)))(j) (b ) believe'(,7, s ic k '(j)) The corresponding pair o f interpretations for the V P believe h e ’s sick are given in (13-55a-b), In effect, the V P in the interpretation in (13-55a) denotes the p roperty o f believing oneself to be sick, while that in (13-55b) denotes the p rop erty o f believing John to be sick. (1 3-55) believe h e ’s sick (a ) Xx believe7^ , sick^a:)): call this Pi (b ) A z.b e lie v e ^ z ,s ic k '^ ')): call this P 2
The am biguity in the original sentence (13-53) is then accounted for by the am biguity o f the antecedent V P, as represented in (13-55a-b), together with the principle that the missing V P is interpreted as semantically iden tical to the antecedent V P. T he reader is invited to work out that lam bdaconversion yields the tw o appropriate readings for Bill does too from the representations in (13-56). (13-56) B ill does to o (a) P , (6) (b ) P 2(6)
It has been n oted by m any semanticists, including D ahl, M cCawley, and Keenan in the early 1970’s, that the term “ sloppy identity” is a bit o f a misnomer resulting from the early syntactic approaches to the p roblem (the term was coined b y J.R R oss) and the possibility o f gender differences in
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examples like John finished his hom ework before M ary did and that in fact the syntactically “ sloppy” identity is a reflection of strict sem antic identity. The semantics o f V P -deletion is basically just identity o f V P interpreta tion: lam bdas sim ply help articulate the relevant V P interpretations.
V.
Passive
Lam bdas are also very useful for expressing the interpr etations o f various rules or operations which m anipulate the argument frames o f verbs or verb phrases and other predicates
T he passive construction makes an interesting
exam ple, since it was generally regarded as a transform ation o f a sentence structure within transform ational grammar' but within contem porary formal semantics the debate is m ore over whether passive is a phrasal or a lexical operation. W e will illustrate first the phrasal passive analysis o f B ach (1980) and then compare it with the lexical approach o f Dow ty, Bresnan and others. In order to focus on the relevant issues, we should make one simplification and rem ove one earlier sim plification. The sim plification we will make is to treat all N P ’s as o f type e, and restrict the N P ’s in examples to proper nouns so that no harm can be done by the simplification.
T he earlier simplifica
tion that we should now undo was never introduced officially but appeared in examples: in examples like (13-43), (13-44), and (13-45) we follow ed the practice o f predicate logic in representing transitive verbs as tw o-place rela tions that com bine with an ordered pair o f arguments, Here we will follow M ontague and other semanticists in interpreting a transitive verb as a func tion which applies to the interpretation o f the direct o b je ct to yield another function which applies to the interpretation o f the su bject to yield a truth value. So transitive verbs will be treated as o f type (e, (e, t)) (still assuming that we are treating all N P ’s as being o f type e.) T he rule for combining them with their ob ject to make a V P is given in (13-57)
(1 3 -5 7 ) V P -> T V P NP
Translation: T V P '(N P )'
A syntactic derivation tree for the simple sentence John loves M ary is given in (13-58a); a corresponding semantic derivation tree with its nodes labeled with the corresponding types and translations is given in (13-58b).
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John loves M ary ; S John ; N P
love M ary ; V P
love ; T V P
M ary ; NP
(b) lo v e '(m )(j); t j\ e
lo v e '(m ); (e, t) love'; (e, (e, t))
m ie
N ote that what is represented in predicate logic as lo v e '(j, m ) is now written lo v e '(m )(j) , reflecting the old er in which the function is com bined with its arguments rather than the order o f the N P ’s in the English sentence. It is custom ary in M ontague grammar to retain the form er notation as a conventional abbreviation o f the latter, however. N ow we can state the phrasal passive rule which derives a passive verb phrase from a transitive verb (phrase) as proposed by B ach (1980), The operation written E N in (13-59) is the operation which puts the first verb o f its argument in to the passive participle form , and adds a be before it. Since the passive rule is not a sim ple phrase structure rule, we have stated it in a form closer to the standard recursive form at o f the earlier sections o f this chapter, (13-59)
P h r a s a l p a s s iv e r u le fo r a g e n tle s s p a s s iv e s Syntax: I f a is a T V P , then E N (a ) is a P V P (Passive V P ), Semantics: I f a translates as a ', then E N (a ) translates as \ x . ( 3 y ) ( a ' ( x ) ( y ))
The passive verb phrase be loved is then translated as shown in (13-60) (13-60) be loved:
Xx (3 i/)(lo v e '(s )(j/))
In the exam ple ju st given, the T V P was a lexical transitive verb, love,
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T he rule is also intended to apply to what Bach analyzes as phrasal transitive verbs (m eaning phr ases which function as transitive verbs, not phrases which contain transitive verbs), such as the italicized (discontinuous) T V P ’s in (1361). (1 3 -6 1 )
Phrasal T V P ’s: give a b ook to John give John a book persuade John to leave buy this b o o k to read to the children
A com peting hypothesis is that passive is a lexical rule, as proposed by D ow ty (1978) and Bresnan. T he statement o f the rule could b e identical in its syntax and its semantics to the phrasal rule given above, but restricted to apply only to lexical items. T he difference betw een the lexical and phrasal versions o f the rule is subtle, m uch less than the difference betw een a syntac tic and a lexical treatm ent o f passive in earlier transform ational approaches. T he lexical rule in its m ost basic form does not apply to the phrasal verbs illustrated in (13-71). B ut it is not difficult to extend it to a schem a applying to lexical verbs o f all categories T V P /X , i.e all categories o f verbs which take some kind o f com plem ent to form a transitive verb (phrase). T he basic strategy for doing this is similar to the strategy used in defining phrasal con ju n ction in the beginning o f this subsection (In the term inology one encounters in the literature on categorial gram m ar and type-changing oper ations, we would want to “lift” the passive operation from an operation on T V P -ty p e interpretations to an operation on functions that have T V P -typ e interpretations as their values. T o write this out explicitly in detail is a bit to o com plicated to do here in this case, since T V P ’s are already o f a slightly com plicated type, the semantics o f the basic passive operation is slightly com plex, and the various complements and adjuncts that enter into phrasal T V P ’s are o f types we have not discussed.) W hether phrasal or lexical, lam bdas help provide an explicit and per spicuous statement o f the semantics o f the passive construction
V I. Meaning postulates and lexical rules D ow ty (1978, 1979) discusses a wide variety o f other governed rules which he proposes to treat as lexical rules, and lam bdas figure prom inently in the semantics. In discussing lexical rules, we should discuss meaning postulates at the same time, since the two are often closely related in form , and the
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choice of which is m ore appropriate to use in a particular case is generally an empirical linguistic issue. A m on g the phenom ena that have been analyzed in terms o f meaning postulates a n d /o r lexical rules are causative verbs, detransitivization, m o r phological reflexives, adjectival tin- prefixation, o b je c t raising, “ dative m ove m ent” , the relation betw een seek and try to find and that between want and want to have, and m any o f the m ore sem antically regular parts o f deriva tional morphology.
The best general reference, and the starting point for
much work in this area, is D ow ty (1979), W e will discuss just one o f these phenom ena, causative verbs, and use that discussion to illustrate b o th meaning postulates and lexical rules and the relation between them. This will also give us a g o o d opportunity to show how lam bda-expressions are arrived at by specifying what one wants them to do Consider the (m uch-discussed) relation betw een transitive and intransi tive break in English,
There are differences o f opinion as to just what the
semantic relation betw een the tw o is. M any linguists believe (I) that it is at least true that (13-62a) entails (13-62b) Some believe (II) that (13-62a) and (13-62b) entail one another, i e are logically equivalent, Others are doubtful about claim (II) but believe (III) that a similar claim holds true, but with an abstract and possibly universal causative operator “ C A U S E ” whose interpretation is not exactly identical to that o f the English word cause
Some believe neither (II) nor (III) but believe that (I) (or a variant
o f I with CA USE in place o f cause) is the strongest generalization that can be m ade; that is, that (13-62b) captures part o f the meaning o f (13-62a) but does not exhaust it. (1 3-62 )
(a) John broke the window, (b ) John caused the window to break,
In stating the relevant translations o f these sentences, we will use breal^ foi the transitive verb and breal^ for the intransitive; we translate the window simply as an e- type constant w ; and we will introduce w ithout m otivating it an operator T H A T which applies to a form ula to give a proposition- denoting expression. (D iscussion o f such an operator would need notions that will be introduced in Chapter 15; but nothing relevant in the present example depends on the occurrence o f this operator in it.) Then we can translate (l3 -6 2a) (uncontroversially) as in (l3 -6 3 a ) and (13-62b) (controversially) as in (l3 -6 3 b ). W e ignore tense throughout.
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(a) b r e a k ^ w X i) (b ) cause'(y, n L 4 r (b r e a k '1(w )))
Now we can state the three claims (I), (II), and (III) in terms o f three alternative “meaning postulates” one could write. (See Carnap (1947) for the introduction o f the term and Partee (1975) for its use to describe what M ontague introduced as restrictions on possible interpretations.) One way to think o f these meaning postulates is as capturing explicitly som e aspects of the lexical semantics o f the elements m entioned in the postulates; the cases linguists are generally m ost concerned to make explicit are those in which the given aspects o f the lexical semantics play a crucial role in some general inference patterns, especially where whole families of lexical items participate in com m on inference patterns (as is the case with causative verbs.) (1 3 -6 4 )
M P I:
(Va:)(Vi/)[bieak2( i/)( 2:) —> ca u se'(z, T iL 4 T (b ie a k i(y )))]
M P II:
(Va:)(V 2/)[break2( i/)( 2:) <-» ca u se'(z, T iL 4 T (b ie a k j(?/)))]
M P HI: (VE)(Vy)[break'2(y)(a:) «
C A U S E {x , T E A T(break^(?/)))]
W ith respect to the relationship between meaning postulates and lexical rules, the m ost im portant difference am ong the three meaning postulates above is that betw een M P I and M Ps II and III: only the latter tw o, involv ing full equivalence betw een the tw o sides, can be turned in to lexical rules defining transitive break as the causative o f intransitive break. W e will use M P II to illustrate how the conversion from meaning pos tulate to lexical rule can be done in such cases. First note that there is an asym m etry in the roles o f breaU^ and breaU± in the meaning postulate: the postulate im plicitly gives us a definition o f breatf2 in terms o f break^ but not straightforwardly vice versa To see how to turn the meaning postulate, already an im plicit definition, in to an explicit definition o f b r e a k look at breali, as a function which applies to tw o arguments in turn. T he right-hand side o f the equivalence tells us what the value o f that function is given in terpretations o f causd and o f break^. But that means the ligh t-h an d side is already perfectly suited to be the b o d y o f a lam bda-expression; all we have to do is abstract in the right order on the tw o argument variables o f break W hich order? W ell, since the transitive verb applies first to its direct object argument (the y argument in M P II), that should be the argument the first lam bda-operator binds, (I e. “ fust argument fu st” in the lam bda-notation
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we are using. Be careful, since this is not always uniform across different notational versions o f the lam bda-calculus.) The result is the definition in (13-65), (1 3 -6 5 ) break'2 = Xy Xx cause'(a:, T’iL4T’(b re a k i(i/)))
A ll we then need to turn this into a lexical rule is to abstract away from the specific lexical items breaki, breaks and cast this in to a rule applying to a certain class o f intransitive verbs to derive a causative transitive verb from them , with the similarly generalized form o f (13-65) providing the cor responding interpretation rule (1 3 -6 6 ) L exical Rule For Causative Verb Form ation (A ssum ing M P II)
F orm ation Rule: I f a is an intransitive verb with such-and-such properties, then F ( a ) is a transitive verb. (F in English is m ostly the identity fu n ction .) Sem antic Rule:
The translation o f the derived verb F ( a ) is A j/.A z.cause^z, T H A T (a '(y )))
T he reader is invited to go back to the earlier applications o f lam bda abstraction m entioned in this subsection and try to see where the lam bdaexpressions com e from. For the passive, for instance, it m ight be instructive to write down a bicon dition al statement stating the equivalence o f the agentless passive sentence to a corresponding active, and then see how the form o f the lam bda abstract given in the translation rule in (13-59) can b e fig ured out from there. (D o not be discour aged i f lam bdas seem difficult and a bit mysterious at first; like m any powerful tools, it takes time and practice before they feel natural to work w ith.)
Exercises 1.
( a ) Given the derivation tree for the follow ing statements o f SL ac cording to the syntax given in (13-1) and (13-2). (i) (( ~ p & g )V p )
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(ii) ( ~ ( p & g ) V p ) (iii) ~ ( ( p & g ) v p )
( b ) Give the com positional semantic interpretation o f the three trees o f (l a ), assuming that b oth atom ic statements are true. 2. Assum e the alternative syntax for SL, which has. besides the category for atom ic statements, tw o other categories for connectives: Neg for negation which combines with only one statement at a time, and Conn for all four other connectives which com bine with tw o statements at a time ( a ) W rite the tw o new syntactic rules for this syntax o f SL ( b ) Give the derivation tree for the three statements given under la . T o give SL with this syntax a com positional semantics we first have to interpret the connectives according to their truth tables as functions taking either one oi a pair o f truth values, which is/a re the value(s) o f their constitutive statem ent(s), and assigning a truth value. For instance, conjunction is interpreted b y the follow ing function: (1 ,1 )-> 1 (1 ,0 ) -> 0 (0 ,1 )-> o (0 ,0 ) -> 0
(a) Give the interpretations o f the other connectives o f SL as such functions. ( b ) W rite the two com positional semantic rules for SL with this syn tax. Hint: use the convention o f [PJ standing for the semantic interpretation o f P.
(c) Show how each o f the statements o f ( l a ) is interpreted compositionally according to the three trees you gave in (2 b )),
3.
(a) Give the trees and com positional interpretation o f the following formulas o f Predicate L ogic, according to (13-8) and (13-9) given in section 13,1.2. D efine your own m od el with two individuals and a non-em pty interpretation o f the predicates to use in the interpretation throughout this exercise which makes (i)-(iv ) true. (i ) (Va:)(3y)(P(a:) -> R ( x , y ) )
E
(ii)
( 3
j/
)(V
ie)
( P (
ie)
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x e r c is e s
->
R{x,y))
( i i i ) { 3 y ) ( P ( y ) -> ( V x ) ( R ( x , y ) ) ( i v ) ( V x ) ( 3 y ) ( P ( y ) -> R ( x , y ) )
( b ) Formulate four English sentences corresponding to (i)-(iv ) which would best express their scope differences 4 . In (13-10) we define a form ula or predicate logic as true in a m odel M i f it is true for all assignments in M , and false otherwise. A com m on alternative definition is given in (13-10') below . (13-10')
(1) ■^'or form ula ip, = 1 (i-e V5 is true sim pliciter with respect to M ) iff for all assignments g, (2 ) For any form ula
=
= 0 iff for all assignments g,
o,
(3) Otherwise,
is undefined.
T he tw o definitions give the same results for all closed form ulas, but they give different results for some open formulas. ( a ) F ind an open form ula which comes out true under b o th defini tions , ( b ) Find an op en form ula which comes out false under b o th defini tions. ( c ) Find an open form ula which comes out false according to (13-10) but undefined on this alternative (13-10'). ( d ) A re there any other cases, or does (c ) exhaust the ways in which the results o f the definitions can differ? State your reasons. 5. Give the derivation tree according to the SL syntax in (13-2) corre sponding to the ‘algebraic’ tree in (13-20) and show its com positional derivation which makes it true in a simple m odel o f you own choice. 6. A ssum e the follow ing constants and variables T yp e
Variables
Constants
e (e,t)
x,y,z, P ,Q
j, m walks, man
(e,(e,t)) ((e,t),t)
R P
loves som eone, everyon e
Check whether the follow ing expressions are well-form ed expressions
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o f TL I f so. specify their type. Draw derivation trees for at least three o f the w ellfoim ed ones. Sample:
(i) (ii) (iii)
Ax P ( x ) loves (m )
- yes, type (e, t) - yes, type ( e, t )
AP walks ( P )
- no
D erivation tree for (i): X x P ( x ) , (e, f), 3 x,e
P(x),t,2 P ,< M )
x,e
( a ) Xx loves ( m )(z ) ( b ) AR R ( m ) ( j ) ( c ) som eone(w alks) (d ) X P P ( j ) ( e ) Xx som eon e(z) (f) Xx(P(x)&Q(x)) (g ) A P (P (i)& P (m )) ( h ) (A P P (;))(w a lk s ) ( i ) Ax ^y loves ( x ) ( y ) ( j ) A P ( V P (P (P ) « P ( j ) ) ) ( ( j ) has extra parentheses, do not ju dge it illform ed on that ac count) 7. U sing the same variable-conventions as above, and the constants and semantic interpretation o f T L A given in 13,2.3, specify for each o f the follow ing the type and a description o f the semantic value for M and g as given in (13-36). O ptional abbreviations: P lu s (a , 6) for P lu s (6 )(a ) g r (a ,6 ) for g r (6 )(a ) Sample: (Aa:gr(a:, 5 ))(8 ) - type: t, value: T R U E ( a ) (Aicgr(iE,5))
E
x e r c is e s
( b ) (A a :(o d d (a :)& g r (s ,5 ))(8 ) ( c ) (A e P1 u s ( e , 3 ))(4 ) ( d ) ( Xx ( Xy g r ( y , z )))(5 )( 8 ) ( e ) ( Xx ( Xy g r ) y , z )) (5 ))( 8 ) ( f ) Ae 3 j/(P 1 u s ( j/, y) = x)
( g ) (A P P (8 ))(A zg r(iE , 5 )) Hint: one o f ( d ) and ( e ) is true, the other false
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Chapter 14
Generalized Quantifiers 14.1
D e te r m in e r s a n d q u a n tifie rs
The universal and existential quantifiers o f predicate logic introduced in Chapter 7 aie in tw o m ajor respects inadequate for the sem antic analysis o f the rich variety o f quantification in natur al languages, First o f all, as we have seen in translating from English to predicate logic and as was poin ted out again in Chapter 13, the syntactic structure o f quantified form ulas in pred icate logic is com pletely different from the syntactic structure o f quantified sentences in natural language. Q uantifying expressions o f natural language are typically full N Ps, where the noun (C N ) and possibly additional relative clauses provide essential restrictions on a quantifier. N ot ju st the determiner or specifier o f an NP binds dependent arguments or pronouns, but from a se m antic point o f view the appropriate scope-defining and binding category is the entile N P It will prove useful for linguistic purposes (to o ) to distinguish between quantifying over dom ains and binding arguments o f predicates — two jo b s conflated by the tw o standard first-order quantifiers o f predicate logic. Secondly, m any forms o f quantification in natural language are not expressible or definable in terms o f the first-order logical quantifiers. For instance, the NP m ore than half o f the CN is not expressible in terms o f just fixst-order quantifiers, since its interpretation requires a on e-to-one m apping between two finite or infinite sets dependent on a well-ordering b y cardinality (see Barwise and C ooper (1981) for a com plete p r o o f) These serious lim itations o f the analysis o f quantification in any semantic framework restricted to a first-order language can be overcom e b y general izing the notion o f a quantifier to a higher-order concept: 373
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quantifier is a fam ily o f subsets o f the dom ain o f entities.
R ichard M on
tague initiated this higher-order analysis o f quantification, as presented in the fragm ent o f Chapter 13, for the logical quantifiers in his fam ous paper ‘ T he P rop er Treatment of' Quantification in Ordinary E nglish’ (M ontague ( 1 9 7 4 ) , p. 2 4 7 - 2 7 0 ) , and recent research in m odel theoretic semantics has developed his insights considerably for linguistic purposes A lso genuinely new insights on the logical properties o f quantifiers have been obtained, e.g., a new notion o f first-order definability (see van B enthem ( 1 9 8 6 ) ) . In this chapter the main linguistic results o f the theory o f generalized quantifiers are presented as an introduction to a very fruitful ar ea o f linguistic seman tics where the m athem atical m ethods o f the previous chapters have found extensive em pirical applications. In the fragment o f Chapter 1 3, NPs are defined categorially as term s, and they are interpreted in the semantics as gener alized quantifiers,. T he seman tic type o f any term or NP is {{e, t ) , t ) , hence they all denote functions from sets o f individuals to truth values, T he fragment in Chapter 13 contained on ly the determiners every and som e, translated in the form al language with lam bda abstraction respectively as XQXP[ ' i x [ Q{ x)
P (x )]]
and X Q X P [ 3 x [ Q { x ) t P{ x) ) } In this functional perspective the subject NP denotes a function taking the interpretation o f the V P in the sentence as argument
For instance, every
man walks is true iff the set o f walkers is a member o f the fam ily o f sets interpreting every man. A lternatively we m ay view the interpretation o f this sentence in a ‘flat ten ed ’ version as a relation between sets, i.e., o f type ( ( ( e , t ) , ( e , t ) ) , t ) . A determiner is then analyzed as a relation between the set o f individuals in terpreting the CN o f the subject N P and the set o f individuals interpr eting the VP, This leads to the follow ing general definition: D
e f in it io n
14.1
A d e t e r m i n e r is a function D in a m od el M = ( ^ j [ ])
assigning to the domain o f entities E a binary relation b etw een subsets A and B . l D e t [ ^ l [ y P l ] E = [D e t(A , P ) ] g = D E A B
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In the follow ing we first present the main notions in their flattened rela tional versions for ease o f exposition, and discuss the relational analysis o f quantifiers further in Section 4. Switching back and forth betw een a hier archical functional perspective and a flattened relational analysis will prove heuristically useful since m any of the results o f the theory o f relations o f Chapter 3 can be brought to bear on generalized quantifier theory. T he m ethodological objectives and linguistic goals o f the theory o f gen eralized quantifiers are threefold: (1) to provide a sem antical characterization o f the class containing all and on ly the possible determiners o f natural language as a proper subset o f the logically possible det ermine.!, s, including a linguistically adequate classification o f types o f natural language determiners (2) to explain distributional data o f NPs in various contexts by notions definable in terms o f properties o f generalized quantifiers (3) to analyze conditions on binding, scope-dependencies, inference and other inform ational dependencies betw een NPs and other categories as semantic constraints on the process o f m od el theoretic interpretation as a form al m od el sim ulating hum an inform ation processing with its cognitive constraints The theory o f generalized quantifiers proves to b e a new fram ework for the semantic explanation o f linguistic data which have been studied exten sively in syntactic theory.
14.2
Conditions on quantifiers
In constructing gener alized quantifiers as families o f sets o f entities or in dividuals in a domain there are four universal conditions which set initial restrictions on a linguistically useful notion o f a quantifier.
A lthough not
all natural language quantifiers m eet these conditions, they serve us here to carve out an initial dom ain o f study, providing the foundation for a semantic analysis o f quantification. E xtensions o f the simple cases to m ore com plex forms o f quantification, including context-dependent or intensional determ in ers, can be obtained by dropping one or more o f these initial m ethodological assumptions.
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Foi the interpretation o f an N P the dom ain o f entities E is nevei entirely relevant to the determ ination o f its value, but rather just certain subsets o f E are. For instance, when we want to check whether every m an runs in a given m odel, we are not interested in the wom en 01 the books in that domain o f interpretation. T o express this rather obvious condition on the subsets which axe relevant in verification o f N Ps we define a first general condition on determiners in generalized quantifiers: C l . C onservativity (the ‘liv e-on ’ property) I f A ,B C E then D E A B <-* D E A ( A n B) A determiner that meets this condition is called conservative or is said to ‘live on ’ the set A. Subsets o f the dom ain that contain no m em bers o f A, no things with the p rop erty interpreting the CN in the subject NP, i.e. any ( X — A ) C E , hence fall outside the interpretation o f an NP. English examples that illustr ate this notion are the follow ing equivalences
M any men run <-> M any men axe m en who run Few wom en sneeze <-+ Few women axe wom en who sneeze John loves M ary <-*■ John is John and loves M ary
T he English expxession only, for instance, which is sometimes (we think erroneously) considered to be a determiner, fails to be conservative In eval uating for instance the sentence Only athletes run in a dom ain E we do have to consider the entire power-set o f E, since the sentence is verified or falsified after we have determ ined o f everything else in E whether it runs. This is an argument against treating only as a determiner, which finds additional support by the fact that other determiners m ay precede it, as in the only boy. T he relatively free distribution o f only in a sentence also sets it apart from the category o f determ iners, Consider (1 4 -1 )
Only athletes run / Every athlete runs Athletes only run / * A thlete every runs Athletes run only / * A thlete runs every
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But note that some determiners m ay “floa t” more easily, e g. all. The second general condition on the class o f generalized quantifiers that form our ob je ct o f study is the requirement that extensions o f the dom ain should not affect the interpretation o f the determiner
For instance, if we
know that som e men walk in a given m odel, then adding m ore book s to the domain o f that m od el will not have any semantic effect. This condition on determiners is defined as: C 2 . E xten sion (C on sta n cy ) I f A ,B C E C E ' then D EA B ^ D E, A B Determiners that satisfy this condition are called extensional or constant determ iners In principle the interpretation o f a determiner m ay vary with the extension o f the dom ain o f interpretation, but Extension says that in evaluating an NP the entities which fall outside the interpretation o f the CN and outside the interpretation o f the V P can be left out o f consideration. This means that the determiners under consideration have to be independent o f the size o f the dom ain; they are insensitive to dom ain-extensions In natural language there are, however, determiners which measure relative cardinalities and show an im portant dependence on the size o f the domain; they are the con text-dependent determiners like many or m ore than half. A lthough we return to context-dependent determiners below in section 5, it is easily seen that, for instance, there is at least one interpretation o f many men walk which means that the num ber o f m en who walk exceeds the number o f other things in the dom ain that walk. Under such a globally proportion al interpretation o f m any m en , addition o f m ore things to the dom ain m ay change the truth value o f the original sentence The tw o conditions Conservativity and Extension together perm it sup pression o f the param eter E in the interpretation o f an NP, which is defined as: C l '. Strong C onservativity If A ,B C E then D E A B ^ D a A { A
n B)
Strong conservativity tells us that the only sets which are relevant to the evaluation o f a determ iner are the C N -interpretation and the V P -in terpretatio For a p ro o f o f the equivalence o f conservativity plus extension to strong con servativity see W esterstahl (1985a), The third universal condition on quantifiers requires that on ly the num ber o f elements in the relevant sets determine the interpretation o f a quan tifier. To evaluate in a m od el M , for instance, whether several m en walk we
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only need to know how many men there are and how m any o f them walk. It is not relevant who they actually are, since the interpretation of that sen tence should not be affected if we were to substitute other m en for the given ones. N o quantifier hence m ay depend on a specific or particular choice o f individuals in the dom ain, i.e. the quantifiers under consideration are ‘topic neutral’ . This condition is form ulated as: C 3 . Quantity (Isom .) I f F is a bijection from
to M 2 , then D e 1A B <-+ D e 2F ( A ) F ( B )
Quantity requires sameness o f interpretation up to isom orphic models. The requirement m ay b e weakened to insensitivity to any perm utation tt on E or autom orphism F : M j —> M i o f the dom ain o f a given m odel, form ulated as: C 3 ; . W eak Quantity (P erm ) I f 7r is a perm utation o f E , then D ^ A B
D e { ^ { A ) ) ( tt( B ) )
I f we assume C onstancy, then Quantity and Weak Q uantity prove to be equivalent conditions (for a p r o o f see W esterstahl (1985a))
Hence in the
follow ing we disregard this difference and assume Q uantity as a constraint A quantifier D ( A ) which meets Quantity is said to be a quantitative quantifier. N ote that there is some potential term inological confusion here. Some authors (a,o.
van Benthem ) reserve the term ‘ quantifier’ for those
N P-interpretations that are conservative, extensional and quantitative, and others (a.o W esterstaM ) call this class the ‘logica l’ quantifiers. W e call here any N P -interpretation a (generalized) quantifier and use the universal conditions as properties o f subsets o f quantifiers A gain there aie expressions in English which are com m only considered to be determiners, but which fail to meet Quantity. This is not surprising, since in our ordinary use o f language reference to specific objects and particular things often m atters a great deal In logic it is custom ary to obliterate all that is specific, and in setting the stage for the study o f generalized quantifiers we chose to sim plify m atters first b y disregarding specificity for the m om ent as it brings in com plex contextual parameters. T yp ical specific determiners which are sensitive to the choice o f particular o b jects and let the interpretation o f NPs vary across isom orphic models are the possessive determiners such as John’s, ev ery m an’s.
One o f the
problem s to solve in the interpretation o f possessives is the fact that John’s
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book should be interpreted equivalently to the book o f John but every m an’s book should not necessarily be equivalent to the book o f every man.
See
for more discussion van B enthem (1986), Keenan and Faltz (1985), Partee (1984). The fourth and final universal condition on determiners requires that determiners must be active in the sense that they must have a detectable effect on different domains. The ‘universal’ determiner, which is true on any domain for any tw o sets, and the ‘em p ty ’ determiner, false in every domain and every tw o sets, axe excluded as trivial. This condition is form ulated as: C 4 . Variation For each dom ain E ther e is a dom ain E ' such that E C E ' , A , B and C C E ', such that D ^ 'A B and ~ D e ’ A C Variation says that, when you add entities to the dom ain, a set C can be found which is not related to A by the determiner. Hence natural language excludes determiners that ju st arbitrarily relate any or every tw o sets. A weaker condition m ay be called for when we do not want to exclude the interpretation of, for instance, at least two CN in a dom ain with only one single object, C 4 '. N on-triviality For n on-em pty domains E and E ' there exist A ,B C E and A 1, B ' C E ' such that D e A B and ~ D e ’ A 'B ' Again the difference betw een C4 and C4' m ay be disregarded for our purposes as b oth exclude the trivial quantifiers, and the precise strength o f the conditions concerns prim arily logicians who want to make the weakest possible gener al assumptions to obtain m axim al proof-theoretic elegance. Conservativity, Extension, Quantity and Variation together define a class o f determiners which exhibit linguistically interesting semantic properties We will study these properties in the next section o f this chapter
A s was
already remarked above, these four conditions together require that the de terminers under consideration depend only on the cardinality o f tw o sets A and B , respectively the C N -interpretation and the VP-interpretation, I f we assume finite domains, this allows tree-like representations o f these deter miners, called num ber-theoretic trees, which are outlined here just briefly, while relegating m ost results to the exercises Letting x = \A — B\ and y = \A n B\, the tree is built b y pairs ( x , y ) as in Figure 14-1.
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x + y
=
0
= =
1 2
=
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14
0,0 1, 0 2, 0 3,0
0, 1 1, 1
2 ,1
0,2 1 ,2
0 ,3
etc, Figure 14-1:
The num ber-theoretic tiee o f determiners
T o illustrate the use o f such num ber-theoretic trees briefly we can see directly that every man walks is true only on all right-m ost pairs on each line, since, for instance, if x + y = 3 all three elements must be in A n B , i.e. they all are men and walk. T he same argument applies indu ctively to any num ber x + y. There are m any other properties o f determiners expressible in this num ber-theoretic tree, but these results make g o o d exercises
14.3
Properties o f determiners and quantifiers
Assum ing the four universal conditions o f Conservativity, E xtension, Quan tity and V ariation, we now turn to some linguistically useful semantic char acterizations o f the types o f determiners which the NPs in this class may contain, R ecent research has uncovered a host o f interesting properties o f determiners and quantifiers, and we can present here only some o f the fun dam ental results T he role o f determiners and quantifiers in inferences is one o f the cen tral explananda o f the theory o f generalized quantifiers, T he fundamental question here is how inform ation expr essed in natur al language in inferential patterns is preserved or lost b y m anipulating models.
T he first property
o f determiners is based on increasing or decreasing the number o f entities in the relevant sets A (C N -interpretation) and B (V P -interpretation), If we have the inform ation that som e men walk, adding m ore walkers to the interpretation will not change the initial assignment. B ut if we have the inform ation that no m an walks, adding more walkers to the interpretation m ay very well change it, because some o f the walkers m ay turn out to be men. Similarly, in the first case, if we add m ore men we still m aintain the inform ation that some men walk. B ut, in the second case, if we add more m en, the inform ation that no man walks is not preserved. These facts are described b y attributing to the determiners that allow such addition in either A or B the p rop erty o f being respectively left m onotone increasing (in A ) or
P
r o p e r t ie s
o f
d e t e r m in e r s
and
q u a n t if ie r s
381
right m on oton e increasing (in B ) T he counterpart to increasing determiners are m on oton e decreasing determ iners, again in either A (left) or B (right). For instance, if few men walk, and we restrict the set o f men to a subset, it is still true that few men walk, since only fewer do walk then; so few is left m onotone decreasing A n d if we assume that n o men walk, and take a sub set o f the walkers, that set cannot contain any men, so the inform ation that no men walk is preserved and no is right m on oton e decreasing. The intu itive linguistic tests that discriminate the four different types o f determiners aie then the follow ing entailments in which the determiner is the same in antecedent and consequent but either the CN- or the V P-interpretation is increased or decreased
(1) left monotone increasing I f D C N l V P , then D C N 2 V P (where CN^ C C N 2) e.g. I f several wom en with red hair run, then several wom en run
(2) right monotone increasing If D C N V P i , then D C N V P 2 (where F P j C V P 2) e g. If every m an walks fast, then every man walks
(3) left monotone decreasing I f D C N 1 V P , then D C N 2 V P (where C N 2 C C N j ) e g I f all wom en run, then all wom en with red hair run
(4) right monotone decreasing If D C N V P i . then D C N V P 2 (where V P 2 C F P j ) e g. I f no man walks, then no m an walks fast These four related properties o f determiners are captured in the follow ing set-theoretic definition, P i . M on oton icity left m on oton e increasing: right m onotone increasing:
( D A B & A C A' ) —» D A ' B ( D A B S z B C B' ) —> D A B '
left m onotone decreasing: right m onotone decreasing:
( D A B & A! C A) —> D A 'B ( D A B & B ' C B ) —* D A B '
Note that left m on oton e increasing determiners are also called persistent, and the left m onotone decreasing ones anti-persistent. A syntactically simple determiner which is increasing in the left argu ment (A ) is increasing in the right hand argument (B ), i.e , left m onotone
3 82
C
hapter
14
increasing simple determiners are also right m onotone increasing. But left m on oton e decreasing simple determiners are not always also right m ono tone decreasing This m ay be verified easily b y considering the following ent ailments: Some bald men walk fast —> Some men walk A ll m en walk
A ll bald m en walk fast
The m on oton icity properties o f determiners provide a fundam ental clas sification o f inferential patterns. It in fact explains the pow er o f the Square o f O pposition o f Syllogistic L ogic (a pre-Fregean form o f predicate logic) with the four traditional logical quantifiers ev ery , som e, not ev ery and no (see van E ijck (1985)), A simple linguistic constraint on coordination o f NPs with conjunction and disjunction may now be form ulated as follows: Constraint on C oordination: NPs can b e coordinated b y conjunction and disjunction iff they have the same direction of m onotonicity, C oordination o f a decreasing determiner and an increasing one is hence excluded on semantic grounds, which conform s to the intuitive unacceptabil ity of, e.g ., *John or no student saw Jane *A11 the wom en and few men walk. O f course, coordination with the connective but requires N Ps o f differ ent direction o f m onotonicity, but note that this coordin ation, contrary to con ju n ction and disjunction, does not allow iteration freely. John but no student saw Jane A ll the w om en but few m en walk *A11 the wom en, few men but several students walk C onjun ction preserves m onotonicity, if it is interpreted as intersection o f the tw o CN-sets, But disjunction does not preserve m onotonicity, e.g., all or no is not m onotone. C om plex determiners are expressible equivalently as conjunctions of m on oton e determiners, even though they m ight themselves not be m on oton e, e.g., the n on-m onotone exactly two is equivalent to the m on oton e at least two and at m ost two
P
r o p e r t ie s
o f
d e t e r m in e r s
a n d
q u a n t if ie r s
3 83
T o obtain a sem antic universal stating that all syntactically simple natu ra l
language determiners (in the class m eeting the four universal conditions
on quantifiers) are characterized by a certain semantic property, we need a property which all right m onotone determiners share with num erical de terminers interpreted as exactly n form ulated as:
Such a weaker property is Continuity,
P 2 . Continuity D e A B 1 and D e A B 2 (where B i C B C B 2) —> D e A B Conjunction still preserves C ontinuity (the p r o o f is an easy set-theoretic exercise). Now we can state the first em pirical semantic universal o f natural language determiners: S I. Universal on simple determ iners (van B enthem (1984)) Every simple determiner in a conservative, extensional and quantitative NP o f any natural language is continuous The m eth od ological role o f such semantic universals is to dem arcate the class o f determiners that are realized in natural languages from the much wider class o f all logically possible determiners. Semantic universals are hence o f a genuinely em pirical nature and make falsifiable linguistic claims W e can state a general relation betw een increasing and decreasing deter miners if we define external and internal negation o f quantifiers as follows. Here Q e is an abbreviation for D e (A), D e f in it io n 14.2 I f Q e is defined in a m od el M with domain E, then: ~ Q e = { X C E \X 0 Q }(e x te rn a l negation) Q e ~ = { X C E | (E — X ) £ Q }(in tern a l n egation )
■ External negation o f an NP prefixes negation on the NP, e.g., N ot one man runs. This should not be confused with full sentential negation, e g , It is not the case that one man runs Internal negation corresponds to V P -negation, e g , One man did not run Now we can see that either kind o f negation reverses the m on oton icity direction o f a quantifier I e , if Q is m onotone increasing, then ~ Q and Q ~ are m on oton e decreasing, and if Q is decreasing, then ~ Q and Q ~ are
384
C
h apter
14
increasing. T o see that this is true, assume that Q is m onotone increasing and suppose that Y £ (Q ~ ) and X C Y (E — X ) , so (E — X ) £ Q , hence X
Then (E — Y ) £ Q and (E
£ (Q ~ )
—Y )
C
N ow suppose Y £ (~ Q)
and X C Y. Then Y $ Q so X £ Q so X £ ( ~ Q). The p r o o f o f the reversal o f m onotone decreasing quantifiers is similar. A lso, as we m ay expect from the set-theoretic definitions, it is provable (a simple exercise!) that Q — From this correlation we m ay conclude that any decreasing simple quantifier is a negated form o f an increasing simple one. This result m ay provide an explanation for the fact that not all N Ps can be negated, e g,, acceptable external negations are: (1) not every man (2) not m any men (3) n ot a (single) man whereas unacceptable are: (4) *not few men (5) *not n o man (6) *not John Note first that only right m onotone increasing quantifier s allow external negation Instead of (4) and (5) one could as well use the simpler m onotone increasing m any m en and som e m en respectively T o explain the unaccept ability of (6) we define the new notion o f a dual o f a quantifier. e f i n i t i o n 14,3 T he dual o f a quantifier Q e is the quantifier Q e = { X C E |(E — X ) Q Q } . I f Q = Q~~ then Q is called self-dual. ■
D
The NP every man turns out on this definition to be the dual o f some man. We call determiners D and duals if the quantifiers in which they occur are duals. Furtherm ore for any a £ E , { X C E j a £ X } is self-dual, so proper names, for instance, are self-dual
N ow we can form ulate the second
semantic universal: S 2 . Universal on negation (Barwise and C ooper (1981)) If a language has a syntactic construction whose semantic interpretation is to negate a quantifier, then this construction is unacceptable with NPs with m on oton e decreasing or self-dual determiners.
P
r o p e r t ie s
o f
d e t e r m in e r s
and
q u a n t if ie r s
This accounts for the un accept ability of (6)
385
Another linguistic illustra
tion o f this duality between quantifiers in natural language is the following. For instance, not m any m en did not leave is often taken to be paraphrased by quite a few m en left Hence quite a few CN should be the dual o f many CN, To show that this is untenable in general, we first prove a theorem of generalized quantifier theory THEOREM
14 1
(Barwise and C oop er (1981)) I f D is light m on oton e in
creasing, and D A B and D " A C , then B D C /
0.
■
Proof, (redu ctio) Suppose B n C = 0. Then B C (E — C ), so, b y m o n o tonicity, D A ( E — C) . B ut then ( D ~ )A £ 7 , so not ~ ( D ~ ) A C , resulting in a contradiction with the assum ption D ^ A C ■ M any is right m onotone increasing (on a non-context-sensitive, i e, extensional, interpretation); now in some m od el M, let B = \voted fo r Bush] and C = \voted fo r Dukakis]] I f many m en voted fo r Bush and quite a few men voted fo r Dukakis, then b y T heorem 14,1 we must accept that at least one m an voted for Bush and for Dukakis! This shows that m any and quite a few cannot sim ply be duals. P ossibly a m ore context-sensitive notion o f duality could solve this problem D uality is a fundam ental aspect o f b oth form al and natural languages The familiar negation laws o f predicate-logical quantifiers are a simple case o f reversal o f m on oton icity direction, which we also call polarity-reversal. But in natural languages the correlation betw een negation and duality is much m ore pervasive Negative polarity expressions, for instance, occur in various syntactic contexts, yet these prim a fa cie disparate and almost stylistic phenom ena can be given a universal semantic explanation (due to Ladusaw (1979)), The determiner any has been a notorious problem for semantic analysis, since it is sometimes equivalent to the universal every, which is called the free choice any as in A n y book is readable, But there are contexts where it is not equivalent to universal quantification and in other contexts it is simply unacceptable Closer scrutiny w ith a semantic eye o f the acceptable contexts where it is not equivalent reveals that they can be all interpreted by m on oton e decreasing sets Consider the data in (8)
(8) (a) John did not read any book s (a ') *John read any books
3 86
C
14
hapter
(b ) A t m ost ten students who read anything passed (b ') *A t least ten students who read anything passed (c) It is difficult to find any b o o k (c ') It is easy to find any b ook (d ) Before anyone enters, he must bow (d ;) * A fter anyone enters, he must bow (e) Never m ay anyone touch it! (e') * Always m ay anyone tou ch it! (f) I f anyone can do it, (then) you can! ( f ) A nyone can do it, so you can! (free choice ‘ an y’ ) (g) John left without any books (g ') *John left with any books (h) John left w ithout having read any books (h ') *John left (w ith) having read any books (i) N ot reading any books makes life boring (i') *Reading any books makes life borin g
The data in (8) dem onstrate polarity reversal across various categories, the first case o f negative polarity and the prim ed case o f positive polarity. W e see that contexts that admit negative polarity items (n .p,)
should be
interpreted as m onotone decreasing sets based on:
a) verbs for n.p. ob ject NP b) head NPs for relative clauses with n.p
NP
c) adjectival phrases for o b je ct n.p, NP in infinitival phrase d) locating tem poral adverbs for n.p, su bject NP e) frequency tem poral adverbial for n.p
subject NP
f) conditional for1n.p, subject NP preposition for n.p
NP in P P
preposition for n.p. NP in gerund negation for n.p, NP in nom inalization N ote that in (8 f) the negative polarity o f the con ditional antecedent conveys that if it can be done at all, you certainly can do it, i.e,, the task is difficult to do; whereas the positive counterpart ( 8 f ) admits the free choice
P
roperties
of
determ in ers and
387
q u a n tifie r s
any but changes the meaning drastically to express that since everyone can do it, you can, i.e. the task is a very easy one These facts show that m onotone decreasing sets serve in the interpretation o f expressions from all these different categories in a com positional m odel theoretic semantic theory, but it w ould lead to o far for our present purposes to develop a fragment in detail There is another im portant semantic property o f determiners which has been the subject o f m uch recent linguistic research, i.e., the contrast between definite and indefinite NPs.
The follow ing definition captures a relevant
semantic property: D e f i n i t i o n 14 4
(i) A d eterm iner D is positive strong i f for ev ery m od el M
and ev ery
A C E i f D (A ) is defined then D A A (ii) A determ iner D is negative strong i f for ev ery m od el M
and ev ery
A C E i f D ( A ) is defined then ~ D A A (iii) I f D is n ot p ositiv e or n egative strong, it is weak
■ Exam ples o f positive strong determiners are every, all, both, m ost, the ra; negative strong is neither, and weak are a, som e, at least ra, many, several, a few, few , no. A simple linguistic test for strength o f a determiner is to see whether a sentence o f the form
nri-KT j is a C N DCN < „ .T | are C N s comes out logically valid, contradictory or contingent In the first case, the determiner is positive strong, in the second case negative strong and in the third case weak. There is a simple theorem o f generalized quantifier theory which shows that positive strong determiners always include the entire dom ain E. Theorem
14 2
(Barwise and C oop er (1981)) I f D is a d eterm iner in a
conservative quantifier D ( A ) , then D A A i f f D A E .
■
3 88
C
hapter
14
Proof. Since D (A ) is conservative, D A E iff D A ( E n A ), but E n A = A, and we know D A A .
■
This th eoiem can be used to explain why only NPs with weak determin ers are acceptable in so-called existential contexts with pleonastic subjects there is/are NP For any positive strong determiner this context makes an uninform ative tautology, whereas for negative strong ones it gives contradic tions Furtherm ore, only weak determiners can be used when we deny the existence o f something with there is not/are not NP. B ut, as was p oin ted out in Keenan ( 1 9 8 7 ) , it is not straightforward to extend Definition 1 4 ,4 o f strong and weak determiners to ra-place determin ers, since it apparently makes the indefinite tw o-place determiners as many CN as CN, m ore CN than C N and few er CN than CN strong (e g
as many
students as students are students is tautologous, m ore students than students are students contradictory) Nor does this analysis account for the fact that There is every student is plainly ungram m atical, whereas E very student ex ists m ay be trivially true but not ungram m atical A m ore explanatory notion is the follow ing (K eenan’s work presents generalized quantifiers algebraically: for any tw o properties p, q, a determiner is interpreted as a fu n ction F from properties to sets o f properties, i e,, D A B is written q £ F (p ). Intersection is meet and union is join in a B oolean algebra o f properties. W e lim it ourselves here to extensional properties and give K eenan’s definition in our notation. It is an interesting research question whether there are em pirical arguments for or against using either set-theoretic or algebraic m ethods). D
e f in it io n
14 .5
A determ iner is e x is t e n t ia l i f f either it is a basic existential
d eterm iner (i.e., D A B i f f D ( A fl B ) E is true) or it is built up from basic existential determiners b y B oolean com binations, com position o f adjectives or the excep tion d eterm iner op era tor but N P.
I
This notion accounts for the acceptability o f the follow ing sentences: There are fewer cats than dogs in the yard There are just as m any female as male cats in the yard There is no cat but Felix (a cat) in the yard Note also that determiners which do not meet Variation m ay b e used ironi cally or with special meaning in such existential contexts, e g„ in: There are fewer than zero solutions to the problem
P
roperties
of
d eterm in ers an d
q u a n tif ie r s
3 89
L ook, there is either no or else at least one solution to the problem Even though tautologies 01 contradictions m ay not b y very inform ative from a logical p oin t o f view, a linguistic theory cannot ignore the fact that they can be used very m eaningfully in perfectly acceptable sentences (see Keenan (1987)), The relation discussed above betw een simple decreasing determiners and their positive simple counterparts can be specified here m ore precisely: there is a syntactically simple m onotone decreasing determiner D iff theie is a syntactically simple weak non-cardinal m on oton e increasing determiner D ' (Barwise and C oop er (1981)), From this general correlation betw een simple decreasing and increasing determiners we m ay predict that n o language will have syntactically simple determiners interpreted as not m ost, not every, not the or not two, since most, every and the are strong determiners and two is a weak cardinal determiner. Furtherm ore, only NPs w ith existential or weak m onotone increasing determiners seem acceptable antecedents o f unbound anaphora which escape c-com m and dom ains, and can be unselectively b ou n d by other quantifiers, as illustrated by the follow ing sentences (see H eim (1982) and Reuland and ter M eulen (eds ) (1987) for recent research on indefiniteness): (9) (a) P edro owns many/several/som e donkeys
He beats them .
(a ') *P edro owns every donkey He beats it. (b ) If Pedro owns a donkey, he beats it. (b ') *If Pedro owns every donkey, he beats it. (c ) Every farmer who owns a donkey, beats it (c ') *Every farm er who owns every donkey, beats it. Definite determiners constitute a subset o f the positive strong determ in ers, defined as follows: D
e f in it io n
14,6 A determiner is definite i f in every m od el M where D ( A )
is defined, th ere is a, set i? /
0, such that D ( A ) = { X C E ] B C X } .
■
N ote that for a definite determiner D the set D ( A ) is the principal filter generated b y B (see Chapter 11 for the general n otion o f a filter in a lattice) Determiners which are positive strong but not definite are every, m ore than half and at least half. Plural definite determiners are the only determiners that can be em bedded in partitive constructions, as we can see from (10).
C
390
hapter
14
(10) (a) most of the children (a ') *m ost o f some children (b ) several o f the five children (b ') ^several o f at least five children
Ladusaw (1982) added to this definiteness condition on determiners in partitives that the em bedded plural NP must be interpreted collectively, since one o f both is unacceptable although both is a filter generated b y a set with tw o elements
The two is a collective determiner, b u t both is distribu
tive, as can be seen from the fact that both boys lifted, the piano means that the piano was lifted tw ice, whereas the two boys lifted the piano does not specify how m any times the piano was lifted, as it is true also when the two b oys once lifted the piano together, i.e , collectively The analysis o f plurals as generalized quantifiers presents intriguing semantic problem s, m any still open, which have been explored in Link (1983), (1984) using lattice-theoretic domains, H oeksem a (1983), and elsewhere. Other restrictions on determin ers in partitives can also be described b y semantic properties in the general syntactic form o f partitives
D eti
of
D eti must b e pronom inal, i.e.
D et 2 D et 3
For instance,
CN
a determiner which can be interpreted in
isolation o f any CN such as som e, several, n on e, the three but not every, the, a, no (due to unpublished w ork o f J. Hoeksema; see also Westerstahl (1985)). Possessive or dem onstrative determiners are also pronom inal, but they should already b e excluded from D eti positions in partitives, since they do not occu r in extensional and quantitative NPs. For the D e t 2 position we find that only possessives, plural demonstratives and the are admissible, which are all definite once we allow context-sensitive filters (see Section 5 for context-sensitive determiners), D et 3 can be filled only with weak determin ers which im pose some measure or cardinality condition on the set A (the C N -interpretation). These results are typical descriptive sem antic facts of generalized quantifier theory, for which further explanation should b e sought in m ore procedural terms o f verification o f NPs in m odels, inform ation pro cessing, or com plexity-m easures o f quantifiers and determiners Other recent results in generalized quantifier theory include extensions to m any place determiners (few er m en than w om en) (K eenan and Faltz (1985), and Keenan and M oss in van B enthem and ter M eulen (1 9 8 5 )), exclusion
D
e ter m in e r s as
r elatio n s
391
determiners ( every man but John), determiners which rely on some order ing ( every other num ber), adjectival and num erical determiners, tem poral quantification, and com paratives for which we refer the reader to the further reading suggested for this chapter,
14.4
Determ iners as relations
In presenting the m ain concepts o f generalized quantifiers in the previous sec tion, we switched back and forth between the functional perspective, in which the V P is interpreted as element o f the second-order N P-interpretation, and the flattened relational perspective, where determiners are interpreted as re lations between the sets A and B, In this section we focus on the relational perspective and study properties o f relations as constraints on determiners, uncovering some im portant new semantic insights on the class o f possible natural language determiners, (T h e relational analysis is extensively stud ied in Zwarts (1983), whose exposition is follow ed closely in this section, and in van B enthem (1986) ) Let D be a binary relation in a m od el M = (E , [ ] ) on the sets A , B , C C E\ the follow ing list o f properties o f relations is useful for defining types o f determiners reflexivity:
DeAA
irreflexivity: sym m etry:
~ D e AA D e A B —> D e B A
asym m etry: anti-sym m etry:
DeAB —
connectedness: strong connectedness: alm ost connectedness: transitivity:
DeAB V DeBA V A = B
euclideanness:
(D e A B S z D e A C ) - ^ D e B C
anti-euclideanness:
( D e A B S z D e C B ) —> D e A C
circularity:
(D e A B & :D e B C ) —> D e C A
DeBA
D e A B & D e B A —> A = B DeAB V DeBA D e A B —> (D e A C V D e C B ) ( D e A B & D e B C ) —> D e A C
For exam ple, the determiner every is reflexive, transitive and anti-symm et-ric, and its external negation not every is irreflexive, alm ost connected and connected. The num erical determiner the rais transitive, anti-sym m etric but not reflexive. Sym m etric determiners are som e, no, at least n, at m ost n, exactly n B ut we will see that other properties o f relations like asymme-
3 92
C
hapter
14
tiy , strong connectedness, euclideanness or circularity are not properties of possible determiners in natural languages. W e assume the four universal conditions on quantifiers E xtension, Con servativity, Quantity and Variation set forth in Section 2 o f this chapter. There are some useful theorems on equivalences betw een relational proper ties and set-theoretic inclusion o f the sets A and B . T
heorem
1 4 .3
The following p a i r s o f statem en ts a re equivalent:
(1 ) (i) D is reflexive (ii) A C B —> D A B (2) (i) D is irreflexive (ii) D A B -> A % B (3) (i) D is antisym m etric (ii) D A B —> A C B (4 ) (i) D is connected (ii) A % B -> D A B
Proofs: (1) (i) —» (ii) Assum e D is reflexive, and take A C B .
Then D A A and
since A n B = A also D A (A fl B )., B y Conservativity then D A B . (1) (ii) —> (i) follows directly from the reflexivity o f inclusion, (2) D is irreflexive iff ~ D is reflexive. B y (1 ) D is irreflexive iff A C B —> ~ D A B . C ontraposition and double negation gives D A B —> A % B . (3) (i) —> (ii). Assum e D is antisym m etric and take for some dom ain E , sets A ,B C E such that D e A B . B y C onservativity D e A ( A fl B ). Extend E to E ', and take A' C E ' such that \A'\ = A' = Af\ B.
[A| and A fl
Then D e A ( A fl ^4') and by Extension D e 1 A (A fl ^4').
C onservativity gives D e 1 ^4^4'.
Now consider a perm utation -k o f E'
which yields identical sets for AC\ A* and E ' — (^4 U ^4') but permutes ^ 4 -^ 4 ' with A! — A. Quantity gives us D e i'k (A ),k ( A i), i.e., De< A! A. So we have now D e 1 A A! and D e 1A' A , and antisym m etry o f D gives A — A'. Since A n A' = A fl B it follows that A n B = A , so A C B .
D
(3) (ii) —> (i) so
eter m in e r s
Assum e (ii), let
as
DAB
393
r elatio n s
and
DBA.
Then
AC B
and
B C A,
A = B.
(4) B y definition
D
connected iff ~
is connected iff ~
DAB
— >
A
D is antisymmetric. W ith (3 ) D is D is connected iff A % B — i> DAB.
C B. So
Prom these relational characterizations we can now prove that some p rop erties o f relations cannot b e properties o f natural language determiners, given the four universal conditions we have assumed on quantifiers. Such results demarcate the class o f possible natural language determiners as a proper subset o f the class o f logically possible determiners. T
heorem
14 4 (van B en th em (1984)) T here a re n o asym m etric determiners.
Proof. B y definition o f asymm etry, to show that there are no asymm etric determiners amounts to showing that there are no irrefiexive antisym m etric determiners Suppose D is irrefiexive and antisym m etric, and assume D A B . From T heorem 14 3, (2) says that A % B but (3 ) says that A C B. C ontra diction! So D can only b e the trivial em pty determiner, but that violates Variety, ■ As a corollary to T heorem 14 4 it follows that there exist no irrefiexive and transitive determiners, since antisym m etry entails transitivity for de terminers (see exercises).
Since a binary relation which is irrefiexive and
transitive is called a strict partial ordering, we find that no determiners are strict partial orderings Similarly we can establish that no determiners induce a weak linear or dering on E (reflexive, transitive, anti-sym m etric and connected), since there are n o antisym m etric connected determiners T
heorem
terminers.
14.5
(Zwarts (1983)) There are no a ntisym m etric con nected de ■
Proof. Assum e that some D is antisym m etric and connected. From Theorem 14.3, (4 ) tells us that A <2 B —> D A B and (3) that D A B then A % B —i> A C B. C ontradiction!
A C B . But ■
In fact the only determiner which is reflexive and antisym m etric is the inclusion-determ iner every (and its plural counterpart all), since it is evi dent from T heorem 14,3 (1 ) and (3 ) that for reflexive and antisymmetric
394
Chapter
14
D , D A B *-* A C B . Hence even/ is the only determiner which is reflexive, transitive and anti-sym m etric, i.e. induces a weak partial ordering on E. W e prove tw o m ore non-existence results on natural language determiners here T
heorem
14,6
terminers
(Van B en th em ) There are n o sym m etric and transitive de ■
Proof. Since transitive determiners are positive strong (see exercises), and T heorem 14,2 showed that positive strong determiners always include the entire domain E , we have D A E Symmetry gives D E E , b u t that means that the only sym m etric and transitive determiner is the trivial universal one, which violates Variation, ■ Since equivalence relations are reflexive, sym m etric and transitive we can conclude from T heorem 14 6 that no natural language determiner will induce an equivalence relation on E Since euclidean determiners must be sym m etric and transitive (see Zwarts (1 9 8 3 )), we can also conclude that a fo rtiori there are no euclidean determiners in natural language T H E O R E M 14,7
(van B en th em ) There are no circular determiners,
■
Proof. W e show that circular determiners must be euclidean and hence can not exist in natural language because o f T heorem 14 6, T he reasoning is similar to the p ro o f o f T heorem 14,1 (3 ) (i) —> (ii) Assum e D is circular and take for som e dom ain E A, B ,C C E such that D e A B and D e A C . By C onservativity D e A ( A fl B ).
Extend E to E ', and take A ' C E ' such
that \A'\ = |.A| and A fl A' = A fl B. Then D e A ( A fl A ') and by Extension D e ' A ( A fl A'). Conservativity gives D e ' A A ' N o w consider a permutation it of E' which yields identical sets for A fl A' and E ' — (A U A ') but permutes A — A' with A ' - A. Quantity gives us D E n r(A )ir(A '), i.e
D e> A ' A- So we
have now D e ' A A ' and D e>A ' A , and circularity o f D gives D e ' A A and by Extension D e A A Prom this together with the assum ption that D e A C and circularity o f D we infer D e C A Given D e A B , the circularity o f D gives us D e B C , i.e., D must be euclidean ■ M ore negative but also positive existence results on possible natural lan guage determiners based on properties o f relations can be found in Zwarts (1983)
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q u a n t ific a t io n
Context and quantification
In studying the fundam ental properties o f NPs interpreted as generalized quantifiers we initially disregarded any form o f context-dependence o f the interpretation. Extension required NPs to be insensitive to dom ain exten sions, and Quantity made us disregard all that is specific in reference and quantification In this final section we indicate briefly how the theory m ay be extended to include the im portant forms o f context-dependence found in natural language, while seeking to extend the explanatory pow er o f this semantic analysis o f quantification in terms o f more dynam ic processes o f in terpretation and evaluation. (T h e exposition is partly based on W esterstahl (1985a) and (1985b) ) How m any is m any? W e already alluded to the fact that the determiner many violates Extension when interpreted by sets o f a cardinality greater than a least num ber fixed b y the size o f the dom ain E: manyE A B
=
manyE A (B
n
A ) where
\{B fl A)| >
f(E )
A lthough Extension is clearly violated in this interpretation, Conservativity and Quantity are n ot. Let us call this interpretation many1 for the m om en t. A nother interpretation o f many compares the num ber o f B ’s that are A ’s to some constant or norm al frequency. For instance, in a class o f twenty students we say that m any students got an A , when we com pare the num ber o f A-grades to the average score in other classes over some period o f time and find that five out o f twenty is more than that average ratio. Suppose that by coincidence there are as m any right-handed students as A -grade students, but they are not same individuals In that situation M any students got an A is true, whereas M any students are right-handed is false, i.e. D % A B but ~ D e A C although \A fi B\ = \A fl C |, violating Quantity. If we assume in addition that the students who got an A are the same individuals as the right-handed students, i.e. D e A B and ~ D e A C although A fl B = A fl C , we have a context where even C onservativity is violated W hat counts as m any m ay depend on various contextual parameters Although five A grades in a class o f twenty m ight be considered many, if five out o f twenty people are right-handed this is not considered to b e many. Different contextual parameters are at stake in this example, and we should ask which properties o f determiners are preserved under addition o f which
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parameters. If' we com pare the cardinality o f B fl A to some contextually fixed ratio o f A s, we get the follow ing interpretation:
m a n y^ A B — m any1A (A
fl
fl
B ) where |(A
5)1 < c
|^4|
and c is here a contextual parameter which m ay be 1 /4 for the A -students but 3 /4 for right-handed students. This interpretation saves Conservativity, Extension and Quantity, but introduces an external parameter which must be given as input to the interpretation. I f the dependence on the frequency o f 5 ’s in the dom ain E should be represented explicitly, the follow ing would be required:
m a n i^ A B
=
m a n i? A (A
fl B ) where
|(A
fl
IBI j
B)\ > —- ■|A|
\E\
Conservativity fails since E is now relevant to the interpretation, and hence E xtension fails as w ell Finally m any may be used to compare to the set o f B ’s, rather than A ’s, as in the follow ing examples from W esterstahl (1985a).
(1) M any winners o f the N obel prize in literature are Scandinavians (2) M any Scandinavians have won the N obel prize in literature (3) M any Scandinavians are N obel prize winners in literature
For (1) to be true a number o f external param eters, like the distribution o f nationality among N obel prize winners in general, will have to be con sidered, hence the interpretation m any2 is appropriate, T he sentence (2) is either equivalent to (1), or it counts among all Scandinavians the number o f them w ho won the N obel prize for literature, and that cannot turn out to be many. Sentence (3) expresses this latter interpretation o f (2 ), which is obviously false. I f (1) and one interpretation o f (2) are equivalent, many in these contexts is sym m etric, bu t, i f it still is based on an asym m etry in the cardinalities o f the respective sets, it must be interpreted as com paring to the set B , i.e. : m any%AB = manyAA (A
fl
B ) where |(A
n 5)1
> c
\B\
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397
q u a n t if ic a t io n
This interpretation satisfies Extension and Quantity, but not Conservativity. In Barwise and C ooper (1981) m any in any o f its interpretations is analyzed as requiring that |J5| / 0.
To gain m ore insight into the semantics o f m any, it should be contrasted to m ost, which appears to allow only two different interpretations.
M ost1
com pares the num ber o f A ’s that aie B ’s to the num ber o f A ’s that are not B ’s, equivalent (on finite sets) to the com plex determiner m ore than half.
m o stlE A B = m o s^ A ^ A n B ) where |(A fl 5)1 > |(A - B)\
This interpretation satisfies Conservativity, Extension and Quantity.
The other interpretation o f m ost is related to the universal quantifier and gives it a meaning o f almost all.
m ost2 E A B = m ost2A (A fl B ) where |(A fl J5)| > c ■ |A|
Here c is again some contextually given param eter like in many1, which determines in a given context how m any counts as alm ost all.
B y using
> instead o f > , most? A are B is always true when A = 0, just as for the universal determiner, but m ost1 A are B is always false when A = 0. In terms o f strength m ost2 is positive strong, whereas m ost1 is positive strong only when A / 0. Let us call this p roperty almost positive strong.
The follow ing table summarizes our findings on m any and m ost and lists their m on oton icity properties
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E x t.
C ons.
Q u a n t.
S ym m .
R ight incr.
L e ftin c r .
m any1
-
+
+
m any1
+ -
+ -
+
+ -
+ -
+ +
+ -
+ + -
+ +
-
many* m anyA m ost1 m ost2
+ +
+ +
+
+ +
+ -
Figure 14-2: Properties o f many and m ost The definite determiner the shares the property o f being alm ost positive strong with m ost1. In fact, all definite positive strong determiners require that |A| > 0 in order to be interpretable in a m odel R ecall that a determiner D is definite if it is interpreted as the principal filter generated b y a non em pty B in every m odel in which D ( A ) is defined. The requirement that A is not em pty is a presupposition o f definite determiners on the m odels in which their interpretation is defined (see also de Jong and Verkuyl (1985) on the presuppositions o f definite determiners). The set B determines the truthconditions or the interpretation o f the definite D , but the set A im poses a necessary pre-con dition on a m odel for its interpretation: prior context must have introduced elements o f the set A Indefinite determiners like three or few which are weak and intersective do not carry such a presupposition on A , but their interpretation, which is always defined, requires A to be o f a certain cardinality
Indefinite determiners serve to change the context so
that A is o f the required cardinality. B ut how do we represent such contextdependency in the notion o f a generalized quantifier? W esterstahl (1985b) suggests relativizing all determiners to a context set X C E b y the following universal condition.
Restriction. D g A B «-* D e ( X fl A )B . C onservativity and Extension are preserved under this R estriction, and a local form o f Quantity where the perm utation is restricted to X may even be m aintained as well.
M any o f the relational properties o f determiners
are preserved as well, but anti-sym m etry, for instance, is n ot.
N ote that
restriction to a context set X does not add another argument to a binary determiner, but serves to represent which elements o f the large domain of entities E have been contextually given. Definite determiners are analyzed as context set indicators requiring \X fl A\ > 0, and indefinite determiners extend a given, possibly em pty context set with new entities with property
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q u a n tif ic a t io n
399
The determiners which are neither definite nor indefinite, like the p osi
tively strong m ore than half or every, each and all together with their right m onotone decreasing counterparts are truly quantitative over E and do not depend on any context set. Note that this notion o f a context set solves a well-known p rob lem concerning the uniqueness o f the referent o f a singular definite description: A may contain other elements besides those in the rele vant context set X n A. E.g., interpreting the dog is running in the yard in a context where there are possibly m ore dogs around only requires X fl ffdog] to contain a single dog, but A C E m ay be o f any n on-em pty cardinality. In relation to this semantic analysis o f definite and indefinite determiners as context-sensitive generalized quantifiers it is w orth noting that quantifiers with definite determiners, whether plural or singular, have unique witness sets and quantifiers with indefinite determiners m ay have different witness sets.
D e f i n i t i o n 14,7 A witness set o f a conservative quantifier D ( A ) is any W C A such that D A W . ■
Since D A B with a definite m onotone increasing determiner is interpreted as principal filter generated by B , the witness set W for D ( A ) is always n on em pty and unique. Furthermore this W is a singleton in case the relevant C N is m orphologically marked as singular, and o f cardinality 2 or more in case the relevant C N is m orphologically marked as plural. Indefinite determiners introduce new elements o f the domain in to X fl A , hence there may be several disjoint witness sets for a quantifier D ( A ) with a weak determiner. The num ber marking o f the C N again determines whether the witness set is a singleton or not, but the determiner itself m ay in addition im pose an exact cardinality on X fl A and hence on any witness set for D (A ). In verifying a quantified sentence in a m odel we search for such witness sets, and if one is found, the statement is true in that m odel. Since quantifiers with definite determiners have only one witness set, the entire dom ain E may have to be searched in an attem pted verification, whereas a quantifier with an indefinite determiner is verified as soon as one witness is found, and the domain m ay contain m ore such witness sets. From this fact we m ay predict that experim ental verification o f definite m on oton e increasing NPs requires in general m ore search time than indefinite m onotone increasing NPs. A long these lines it m ay becom e feasible to give the theory o f generalized quantifiers in natural language em pirical foundations in psycholinguistic research.
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Exercises 1. Explain and illustrate in a Venn diagram with two sets A and B in a domain E what the com bined effects o f the four universal conditions on quantifiers are.
2 . Represent the following quantifiers and properties o f quantifiers in the num ber-theoretic trees b y describing the set o f pairs which verify the quantifier by shading the appropriate area in the tree, a n d /o r b y de scribing the general pattern which the condition im poses on the tree. ( a ) no men
(b)
not all men
( c ) neither man ( d ) at m ost tw o men ( e ) at least one man ( f ) exactly one man or exactly three m en do not walk ( B = walkers) ( g ) Variation ( h ) R ight m on oton e increasing (i) Symmetry
3 . Determ ine the m on oton icity properties o f the follow ing determiners: several, at m ost three, none, at least n, som e (sing ), these, neither, every, all, each, infinitely many, a finite num ber of, m ost, m any 4.
(a ) Show with some examples o f entailments for some simple left m onotone increasing determiners that they are also: ( i ) right m onotone increasing
( ii) continuous ( b ) Construct an exam ple o f a com plex English determiner which is left m on oton e increasing but not right m on oton e increasing. 5. Show in a m odel with four entities that John’s books is not a quanti tative quantifier, whereas every book is.
.
6 P rove set-theoretically that V ariation is preserved under external and internal negation, (i.e. if Q meets Variation, then ~ Q and
do).
E
x e r c is e s
401
7. Prove that ( a ) if Q is m onotone increasing, Q ~ is m onotone increasing
(b) for any Q , the dual o f Q ~ is Q itself ( c ) for any Q, Q ~ o
~ (Q ~ ) O ( ~ Q )
( d ) if Q is self-dual, then 8. Prove that every definite determiner is positive strong. 9. We define an intersective determiner as a determiner D for which in all m odels M and all B C E : D A B *-* D (A fl B )B . ( a ) P rove the follow ing equivalences for intersective determiners: D A B <-> D ( A fl B ) E *-* D B A
(b) Give natural language examples o f these semantic equivalences. 1 0. Prove in the relational analysis o f determiners that ( a ) every antisym m etric determiner is transitive ( b ) every connected determiner is almost connected ( c ) every transitive determiner is positive strong (i.e. if { { D A B t D B C ) —> D A C ) then ( D A B —> D A A ))
Chapter 15
Intensionality This chapter is o f a somewhat different nature than the rest o f this b o o k , since it does not present m athem atical tools for linguistic analysis, nor show successful applications o f such tools to linguistic problem s. It is concerned with some o f the m ost difficult issues in philosophical and linguistic semantics which for a long time have been and still are central to the theory o f meaning and interpretation o f natural language. Various analyses o f these issues have been p rop osed using different m athem atical tools, but at least in the present state o f the art there is n o single account o f these puzzles which is com m only received and recognized as the right solution. The core o f these issues is outlined here w ithout m uch form alization only to provide some initial understanding o f what is at stake. In Section 3 a simple m eth od is presented to analyze intensionality in natural language, and the discussion in subsequent sections m ay aid in appreciating the possibilities and limitations o f different m athem atical m ethods for linguistic analysis.
15.1
Frege’s two problems
Two semantic puzzles which p reoccu p ied Frege, the founder o f m odern logic, still constitute m ajor foundational problem s o f contem porary linguistic and philosophical theories o f meaning and interpretation, The first puzzle con cerns the inform ation expressed in identity statements with p referen tia l NPs, The classical discussion is based on the question why (15-1)
Hesperus is Phosphorus 403
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w ould have been an inform ative identity statement to the Babylonian as tronom ers who did not know it was true, whereas (1 5 -2 )
Hesperus is Hesperus
w ould to them be com pletely uninform ative, even though (15-1) and (15-2) are both true statements and the N Ps, all proper names, corefer to the same ob ject, the planet Venus,
The same puzzle is also often form ulated with
com plex referential NPs like definite descriptions, as in the inform ative (1 5 -3 )
The M orning Star is the Evening Star
and the uninformative (1 5 -4 )
The M orning Star is the M orning Star
I f coreferential expressions have the same semantic value, they must be substitutable for each other in any context without changing its semantic value. Substitution o f semantic equivalents is an im portant rule o f inference in any logical system. But how can (15-1) and (15-3) then be informative, whereas (15-2) and (15-4) in which coreferential expressions are substituted are com pletely uninform ative? I f a semantic theory is to account for such facts it must allow coreferential expressions to have different semantic values. For this purpose Frege introduced the fundam ental distinction between the referen ce (Bedeutung) o f an expression and its sen se (Sinn), P rop er names and other referential NPs m ay refer to the same ob ject or individual, but they differ in their sense, Since we use natural language to com m unicate our thoughts, the meaning o f any linguistic expression must at least in part be accessible to all its users. This objective part o f meaning is what Frege called the sense o f an expression Identity statements are inform ative when they contain expressions with different senses, and they are true when their NPs are coreferential. Conditions o f ‘inform ativeness5 hence cannot b e identified w ith truth conditions. Perhaps there is m ore to the semantic value o f an expression beyon d its sense and reference, like its psych ological associative pow er, connotation or ‘colo r,’ but that part o f its meaning will be subjective and should be disregarded in semantics, according to Frege, for it cannot be the source o f com m unicable inform ation. The sense o f an expression determines its reference in different situations, but even when the reference o f an expression in every situation is determined,
j- i t u i j D
o
X VV %»/
this does not fix its sense uniquely.
4U0
C I L U D L r j 1VL 3
I f we assume, as Frege did, that the
reference o f a sentence is its truth value, tw o sentences that necessarily have the same truth value in all situations, e.g. (1 5 -5 )
R ob in won the race
(1 5 -6 )
Everyone w ho did not com pete or lost in the race has done some thing R obin did not do
still differ in their Fregean sense (the exam ple is adapted from Bigelow (1978)). Similarly tw o distinct tautologies which are b oth always true m ay contain different inform ation In a p roof, for instance, we use the logical laws and inference rules to construct a sequence o f tautologies, yet new in form ation is inferred from its premises Reasoning in ordinary language is also based on the m anipulation o f old inform ation to gain new inform ation. If the semantics o f natural language is to account for coreference, inference and reasoning, it should contain a m athem atically satisfactory analysis o f the Fregean notions o f sense and reference. The second problem Frege presented as a central question to semantics is related to the first one o f inform ative identity statements. If such statements or any other tw o statements with the same truth values are em bedded as sentential com plem ents o f certain verbs, the resulting statements m ay differ in truth value. For instance, (1 5 -7 )
R ob in believes that Hesperus is Phosphorus
(1 5 -8 )
R ob in believes that Hesperus is Hesperus
(15-7) m ay be false, whereas (15-8) must be true even when R ob in knows nothing o f B abylonian astronomy, or when he is not even aware o f what the name ‘H esperus’ refers to. For Frege this meant that sentences em bedded in that-clauses do not refer, as they ordinarily do, to their truth value but refer indirectly, i.e.
they refer to their custom ary senses.
Substitution o f
p referen tia l or equivalent expressions in such that-claaises does not preserve the truth value o f the entire statement. Only if the believer knows that two NPs are p referen tia l can they be substituted in his b elief reports. Contexts where substitution o f p referen tia l or equivalent expressions does not preserve reference are called opaque, as op p osed to transparent contexts where the laws o f predicate logic hold w ithout restrictions.
To
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appreciate the extent o f this semantic problem in natural language consider the follow ing versions o f the same p roblem in different opaque contexts. (1 5 -9 )
The Babylonian astronomers did not observe or com pute that Hes perus was the same planet as Phosphorus
(1 5 -1 0 )
The Babylonian linguists did not realize that ‘H esperus’ and ‘Phos p horus’ are coreferential NPs, but they called Venus b oth ‘Hespe rus’ and ‘P hosphorus’ (based on Soames (1985))
(1 5 -1 1 )
Someone found out that the M orning Star and the Evening Star are the same planet
(1 5 -1 2 )
John told me that the Babylonians referred to Venus as Hesperus or Phosphorus
(1 5 -1 3 ) W e know which planet the M orning Star or the Evening Star is, but the Babylonians did not know that A nother source o f opacity was described originally in Quine (1956) and based on belief reports grounded in perception. R alph sees on one occasion a man in a brow n coat and says (o f him ) ‘He is a spy.’ On another occasion Ralph sees a m an in a grey coat and says (o f him ) ‘He is not a spy.’ Un beknownst to R alph, what he sees on these tw o occasions is the same man dressed in different coats. Ralph will report about the m an he has seen with the tw o sentences (1 5 -1 4 )
The man in the brown coat is a spy
and (1 5 -1 5 )
The m an in the grey coat is not a spy
Since the two NPs corefer, (15-14) and (15-15) are contradictions, But R alph is not aware o f their coreference, since from the two resource situations in which he used the NPs to refer to an individual he saw, he did not gain sufficient inform ation to identify their referent as the same man. Someone
F
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problems
407
who is aware o f this coreference w ould explicitly report R a lp h ’s beliefs as contradictory by using a proper name (or dem onstrative N P ) to refer to the ob ject o f R a lp h ’s beliefs. I f the man is called O rtcutt, R a lp h ’s beliefs would be described by (1 5 -1 6 ) R alph believes o f O rtcutt that he is and is not a spy.
The semantics o f b elief reports must on the one hand allow for people to have contradictory, incom patible or inconsistent beliefs, due to their mis inform ation or lack o f inform ation, but it must also be able to indicate on what grounds their beliefs are contradictory or incom patible. It must ex plain why Ralph believes (15-14) and (15-15), but does not assent to (15-16) nor to any other equivalent contradiction about the o b je ct o f his beliefs, even though (15-16) is entailed by the conjunction o f (15-14) and (15-15). This requires an account o f the sense o f expressions and the way their sense contributes com positionally to the reference o f a that-clause in an opaque sentence-em bedding construction. From a more linguistic point o f view we should note that NPs other than proper names or demonstratives inside opaque contexts cannot in general be extraposed w ithout changing the interpretation o f the entire statement. Consider the follow ing existential constructions, disregarding tenses for the m om ent. I f R alph has reported his beliefs with (15-14) and (15-15), someone who heard him m ay express what Ralph said with (15-17) or (15-18), to which R alph him self w ould still assent (1 5-17)
(1 5 -1 8 )
Ralph believes that there is a m an in a brown coat who is a spy and that there is a m an in a grey coat w ho is not a spy There is a m an in a brown coat o f w hom Ralph believes that he is a spy and there is a m an in a grey coat o f w hom Ralph believes that he is not a spy
Since the definite NPs in R a lp h ’s belief reports were grounded in his perception, we m ay infer with existential generalization to the existence o f the individuals he saw, now using indefinite NPs to report what Ralph saw. A lthough (15-18) and (15-19) may seem (near) equivalents, and Ralph should assent to both , someone who knows that R alph saw O rtcutt on both occasions will take (15-18) to be true but (15-19) to be false.
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15
There are tw o m en and Ralph believes o f the one that he is a spy and o f the other that he is not a spy
R alph cannot assent to tw o other sentences, (15-20) and (15-21), which each entail (15-17), a sentence to which he did assent. (1 5 -2 0 )
There is a man o f w hom Ralph believes that he is and is not a spy
(1 5-2 1)
O f O rtcutt R alph believes that he is and is not a spy
These sentences all differ in their sense, express different thoughts as Frege w ould say, but to provide a full-fledged com position al semantic anal ysis o f these differences in terms o f their inform ation value is an assignment that still constitutes a m ajor open research problem . It requires a m ath em atically satisfactory account o f equivalence o f ‘semantic value’ which is sufficiently fine-grained to explain when a statement expresses new inform a tion to someone in a particular context, and what that inform ation is,given the inform ation that is already available to him. T o see why ordinary predicate logic cannot account adequately for opaque contexts we should realize that they are characterized by (1 5 -2 2 )
(i) failure o f substitution o f predicate-logical equivalents (ii) failure o f existential generalization
A lthough the example o f R a lp h ’s beliefs did admit o f some forms of existential generalization, because his own reports were grounded in his (non hallucinating) perception, NPs in opaque contexts in general do not admit such existential generalization as we see in (15-22) and (15-23) (1 5 -2 3 )
(a) John wants to catch a fish and eat it (b ) There is a fish that John wants to catch and eat
(1 5 -2 4 )
(a) Every m an seeks a wom an who will always love him (b ) For every m an there exists a w om an w h o will always love him if he seeks her
In a situation where (15-23a) is true there m ay not be a particular fish that John wants to catch and eat, and from (15-24a) we cannot legitimately
F orms
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409
infer that for every man there is a wom an who will love him i f he seeks her Note that in each exam ple any situation in which the (b ) sentence is true must also verify the (a) sentence, so the (b ) sentences entail the (a ) sentences but not vice versa. The (a ) sentences are hence semantically ambiguous, i.e, open to various interpretations, in a way that the (b ) sentences are not. If opaque contexts like the (a) sentences are given an interpretation which is equivalent to their corresponding (b ) sentences, we call this interpretation its de re interpretation (from Latin, about the ‘res’ (thing or o b je ct) itself), otherwise the interpretation o f opaque contexts is called de dicto (literally, about what was said or abou t the w ord). In a com positional semantics these entailments will have to be reflected as a relation betw een the interpretation o f the sentences, and the difference in their structure will be im portant. In predicate logic quantificational ambiguities are dissolved by differentiating the scope o f quantifiers, but in m odels for opacity phenom ena quantifier scope m ay depend not only on other quantifiers but also on the interpretation o f the expressions that create the opacity.
15.2
F o rm s o f o p a c ity
O pacity phenom ena are inherent to natural language, and in fact contribute im portantly to the efficiency o f com m unication in ordinary language use. One and the same expression m ay be used in different linguistic or extralinguistic contexts and express different inform ation. A disam biguated fo r m al language is required for the purposes o f form alization, abstract represen tation and m athem atical analysis o f the semantics o f natural language, as we learned in practicing translations o f English to predicate logic. Interpretation in m athem atical m odels or translation to a form al language disam biguates natural language expressions, which is prerequisite for an account o f rea soning and inference in ordinary language use. Our choice o f m athem atical m ethods should be determ ined by the kind o f phenom ena we want to study: for some quantificational ambiguities predicate logic is suitable, but for op ac ity phenom ena different m athem atical m ethods m ay provide analytical tools suited to different kinds o f opacity. In the previous section various kinds o f opaque contexts were presented, and in this section we discuss m ore system atically but still inform ally what kinds o f expressions in natural language m ay give rise to opacity. The belief contexts in the previous section show that the epistem ic verbs to believe and to know give rise to opacity The examples were all based on
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sentential com plem ents in that-c\anses, but it is im portant to see that some epistem ic verbs also allow other constructions which m ay be partly opaque as well. Consider the follow ing sentences, (1 5 -2 5 ) R alph believes a man in a brow n coat to be spying
(1 5 -2 6 ) Ralph knows O rtcutt but does not believe (that) he is a spy
Sentence (15-25) does not contain a that-clause, but the o b je ct NP and infinitival clause m ay well be interpreted de d icto, i.e
dependent on, or in
the scope o f the belief-verb. Sentence (15-26) shows that an epistem ic verb m ay take an ob ject NP which is an antecedent for a coreferential pronoun in a coordinated sentence containing another epistem ic verb.
N ote that
even when O rtcutt is indeed a spy, (15-26) m ay well be true. A t least on some understanding o f what it is to know som eone, R alph m ay know Ortcutt although he does not know or even believe everything that is true o f him,. The verb to believe behaves differently in this respect, since to believe som eone is to believe that what he says is true. B elief in someone is again different, and borders on m etaphoric use, but it is also opaque in that for instance (15-27) (1 5 -2 7 ) R alph believes in a spy with a brow n coat
means that R alph believes that there is someone who is a spy and wears a brow n coat, but in reality there m aybe n o b o d y who actually has these properties. A lthough to believe and to know are typical epistem ic verbs, this class of verbs that give rise to opacity is much larger and includes the stative to be aware of, and to be conscious oi, besides action denoting verbs as to find out, to com pu te, to calculate, to discover, which are actions o f gaining inform ation and hence epistem ic. These verbs belong to the same class on criteria that are semantic in nature; they do not necessarily enter in the same syntactic constructions. The non-equivalence o f sentences (15-28) and (15-29) shows that coreferential NPs cannot be substituted in a context find out who NP is. (1 5 -2 8 ) R alph found out who the man in the grey coat was
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(1 5 -2 9 ) R alph fou n d out who Ortcutt was
Similarly a that-clause reporting the content o f a discovery or o f a com pu tation is opaque, as illustrated in (15-30), and the non-equivalent (15-31) and (15-32). (1 5 -3 0 )
A n astronom er discovered that Hesperus and Phosporus are the same planet
(1 5 -3 1 )
A n astronom er com puted that Hesperus is the planet Venus
(1 5 -3 2 )
A n astrom er com puted that Phosporus is the planet Venus
In (15-31) and (15-32) it becom es particularly clear that perform ing a certain operation on given objects is sensitive to the description o f the o b jects, i.e, the way the input is given to the operation. A class o f opacity creating verbs which is closely related to the epistemic verbs is the class o f percep tion verbs. However, certain constructions with percep tion verbs are transparent as we see from (15-34) and (15-35). (1 5 -3 3 )
R alph sees that the m an in the brow n coat is spying
(1 5 -3 4 ) R alph sees the man in the brown coat spy
(1 5 -3 5 )
R alph sees O rtcutt spy
In (15-33) we cannot substitute the coreferential O rtcutt for the definite NP em bedded in the that-clause reporting R a lp h ’s perception. But in the naked infinitive constructions in (15-34) and (15-35) coreferential NPs can be substituted w ithout distorting the content o f the report, (see Barwise (1981), H igginbotham (1983) and Asher and B onevac (1985) for more discussion) The class o f perception verbs includes visual and auditory perception , but also verbs like to n o tice, or perhaps even to catch.
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Epistem ic verbs and perception verbs directly concern the inform ation o f a person or a ‘system ’ , hence, for obvious reasons, com m unication verbs like to tell, to say, to announce, to inform, and to indicate give rise to opaque constructions, not only in that-clauses.
Consider the invalid inference in
(15-36) (1 5 -3 6 )
O r t c u t t t o l d R a lp h a b o u t h is p r o fe s s io n O r t c u t t is a p r o fe s s io n a l s p y O r t c u t t t o l d R a lp h th a t h e is a s p y
Telling som eone about or o f som ething or telling him what something is brings in prim e examples o f opacity, as (15-36) shows, since for the first premise to be true O rtcutt m ay have told Ralph anything ranging from true stories about his adventurous life in Casablanca to tales abou t his cover-up occu pation as real estate agent: the relation o f someone telling som ething can be true w ithout the relation truth-telling being true. T he verb to indicate has interesting semantic properties o f its own, but it does require a that- clause for a de dicto, opaque interpretation o f what is indicated as the invalid (15-37) and the transparent, valid arguments in (15-38) and (15-38') show. (1 5 -3 7 )
T h e t h e r m o m e t e r in d ic a t e s t h a t t h e t e m p e r a t u r e is n in e t y d e g r e e s F a h r e n h e it N i n e t y d e g r e e s F a h r e n h e it is t h e a v e r a g e s u m m e r te m p e ra tu re T h e t h e r m o m e t e r in d ic a t e s th a t th e t e m p e r a t u r e is th e a v e r a g e s u m m e r t e m p e r a t u r e
(1 5 -3 8 )
T h e t h e r m o m e t e r in d ic a t e s th e t e m p e r a t u r e T h e t e m p e r a t u r e is n in e t y d e g r e e s F a h r e n h e it T h e t h e r m o m e t e r in d ic a t e s n in e t y d e g r e e s F a h r e n h e it
(15-38')
T h e t h e r m o m e t e r in d ic a t e s n in e t y d e g r e e s F a h r e n h e it N in e t y d e g r e e s F a h r e n h e it is t h e a v e r a g e s u m m e r tem p era tu re T h e t h e r m o m e t e r in d ic a t e s th e a v e r a g e s u m m e r tem p era tu re
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413
Besides the epistem ic, perceptual and com m unication main verbs that give rise to opacity, the m odal auxiliary verbs may, must, and can create well known opaque contexts, as well as the m odal adverbials possibly and necessarily and the m odal constructions with pleonastic subjects that take sentential com plem ents it is necessary that and it is possible that. The logical aspects o f m odalities have been studied thoroughly in systems o f modal or intensional logic, that were developed after the extensional systems o f predi cate logic. (See Hughes and Cresswell (1968) and van B enthem (1985))
The
variety o f such intensional systems cannot be discussed here, but in the next section it is briefly indicated what general characteristics make them useful tools for the semantic analysis o f opacity phenom ena in natural language, In philosophical logic form al systems have also been designed for verbs o f perm ission or obligation, called ‘d eon tic’ verbs (from the Greek deon‘ d u ty ’) like to permit, or to allow, or the auxiliary verbs may and must. One im portant feature o f such deontic verbs is that the rule o f A dd ition (if p then p V q) which is valid in the logic o f statements cannot be a rule o f deontic logic, since the follow ing argument is clearly intuitively invalid, (1 5 -3 9 )
J o h n m a y / i s a llo w e d t o ta k e a p e a r J o h n m a y / i s a llo w e d t o t a k e a n a p p le o r a p e a r
In such deontic contexts disjunction apparently strengthens the inform a tion, rather than weakens it as it ordinarily does in transparent contexts and extensional inferences. Obviously, existential generalization is not valid either in deontic statements, since there is not necessarily a particular pear o f which it is said that John m ay take it. R elated to the class o f deontic verbs are the m ain psychological atti tude verbs to desire, to seek, to want and to search.. We have already seen some examples o f opaque contexts with such verbs. Finally there are the intentional verbs which describe a m ental state o f the subject like to try, to attem pt, to plan, including perhaps to promise. N ote that intentions are not to be confused with intensions; the form er are m ental states a subject can be in, whereas the latter are properties o f linguistic or logical expressions to which we return below . O ften the epistemic and perceptual verbs together with the psych ological attitude and intentional verbs are called the verbs o f (psychological) attitudes. This review o f opacity phenom ena in natural language is not intended as a com prehensive list, but it shows that natural language is far richer
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in expressive power than can be captured with the basic tools o f ordinary predicate logic. The next section introduces some new m ethods to account for scope ambiguities in opaque contexts and some other opacity phenom ena.
15.3
In d ic e s a n d a ccessib ility re la tio n s
In Section 12.5 the im portant idea o f relativizing a predicate-logical truth definition to possible states o f inform ation was introduced in Kripke models as semantics for H eyting algebras The same idea underlies the elementary system o f intensional logic which is introduced in this chapter
Instead of
the absolute notion o f truth (or falsity) in a m odel that was defined for predicate logical form ulas, we define here a n otion o f the truth value o f a form ula relative to an index. T he syntax o f the system o f intensional logic will be specified here briefly, but we will not discuss its m ethods o f proof. T he usefulness o f indexing truth values for natural language applications is illustrated with some examples o f English expressions translated into a system o f intensional logic, concentrating on the semantic aspects o f the intensional logic. E elativization o f truth values to an index is a generalization o f the inform ation-states o f Kripke models. A n index can be understood as an epistemic state representing the inform ation available to an agent, or as a ‘possible w orld 1 , i.e., an alternative to the actual world representing a way things could have been otherwise, or as a m ixture o f such interpretations, or we can give it any other interpretation we want to depending on the kind o f opacity to be explained by it. W e will understand the notion o f an in d ex in the most general way here, leaving aside any m etaphysical or philosophical reflection on its nature,
For sim plicity we start out taking indices as unanalyzed
prim itives o f the m odel theory and add a set o f indices to the predicate logical models I f we assume that a sentence expresses a statement at an index, the truth value o f that statement is either true or false at that index. W e assume for sim plicity that the reference o f an expression is always determ ined, although an interpretation could be defined partially, leaving the reference o f an ex pression undeterm ined at some indices is hence defined as its truth value
The Fregean reference o f a sentence
W e call the reference o f any expression
here its extension. To account for opacity the Fregean sense of a statement should also be defined As a first approxim ation o f Fregean senses let us define the intension o i an expression as a function or an operation from the
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set o f indices to the extensions o f the expression, For sim plicity we do not introduce in the syntax o f the intensional logic a particular sym bol for the intension o f an English expression, as is com m on in M ontague Gram m ar (see Dowty, W all and Peters (1981))
B elow it will becom e clear that this notion
o f an intension is not sufficiently fine-grained to account fully for Fregean senses, as it identifies the intensions o f expressions that should have differ ent sem antic value or senses, since they are not substitutable in all contexts preserving inform ative content Instead o f the partial order on inform ation states that characterizes Kripke semantics for Heyting algebras, in intensional logic any relation m ay be defined on the set o f indices.
Such relations are called accessibility re
lations, representing which indices can be reached from a given index in a m odel. The accessibility relations characterize the structure o f the m odels, and it is a sound m eth od ological strategy to im pose structure only when such is required for purposes o f natural language semantics, i e
depending
on the nature o f the opacity creating expressions, The syntax o f this system o f intensional logic is a simple extension o f predicate logic adding two operators that take form ulas to produ ce new formulas. To define the set o f form ulas o f intensional logic by extension o f the recursive set o f rules o f predicate logic the follow ing clause suffices: (i) if ip is a form ula then U
Oip,
Note that this syntactic clause generates for instance (V e)D ^ , (3 x )n {\ /y ) and the iterated n n {\ /x )ip and ('\/x)n<>ip , A lth ou gh the choice o f semantic prim itives is in principle open, we choose
for sim plicity to stay close to the predicate logical m odels and assume as prim itives o f the m od el theory a (non -em pty) set o f indices I, a dom ain o f individuals or entities D , an accessibility relation R C I x l , an interpretation fu n ction f , ,] and a variable-assignment function g A n intensional model is then a quintuple = (I, D , R, f ,,], g) W e define recursively the notion o f the extension o f a form ula in a m odel, relative to the assignment g and an index i, Let
be an intensional m odel, i £ I and g a variable assignment, then
an extension o f an expression at an index i in the intensional m od el is defined as: (i) for any constant c
£ D
4 16
C hapter
(ii) for any variable x
= g (x ) £ D
(iii) if P is an ra-ary predicate-letter, then (iv) if P ( t i , t 2,
.
15
C Dn
, tn ) is a form ula, then , t n) j u ^ = i i f f £ [P J ^ ® ’1; 0 otherwise
(v ) if tp and ip are form ulas, then [ ~ ^ J ^ ® ’1 = 1 iff = 0; 0 otherwise Ip k i p J ^ ® ’1 = 1 iff [i^jM ’®'1 = 1 and [^ J ^ ® ’1 = 1; 0 otherwise [y> V [ip —> [y> <->
= 1 iff [y>l^'®’1 = 1 o i — l ; 0 otherwise = 1 iff [^ J ^ ® ’1 = 0 or [t/jJM’®’1 = 1; 0 otherwise = 1 iff [y>|^'®’4 =
0 otherwise
[(V e )^ ]1^ ’®’1 = 1 iff for every d £ [ ( B e ) ^ ] ^ ’®’1 = 1 iff for some d £
= 1; 0 otherwise = 1; 0 otherwise
[ □ ^ J ^ 1®’1 = 1 iff for every i' such that iR i' [ ^ J ^ ’®’1 = 1; 0 otherwise JoysJ-^1’®’1- = 1 iff for some i' such that iR i' d ^ J ^ ’®’1’ = 1; 0 otherwise Formulas with the new intensional operators □ and o are interpreted as true by requiring truth o f the remaining form ula at all accessible indices or its truth at some accessible index respectively This resembles the universal quantifier, and the existential quantifier now ranging over indices instead o f ranging over individuals. These intensional operators consider the extension o f an expression in their scope on indices accessible from the index o f eval uation i. A ll the extensional predicate logical formulas are interpreted on i itself, but the extension o f intensional formulas depends on extensions of subformulas at other indices. As an illustration o f this interpretation in an intensional m odel let us evaluate the quantified form ula o (3 x ) ( P x fa n Q x ) in M * = ( { i o , *is *2}s {^Oi d i, d2, d s}, R — {{2 o ,^ o }j{^ lj^ l}){2*2j22}j{22)2* l}){^ lj^ 2 }}jt
‘1) }
given that
fpflM*,®*
_ = =
0) ie ] M*-®« = 0 {d2)ds}) = {4i4} {do)ds}) | q j M = {dud2>dz}
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To evaluate a form ula we pick an index o f evaluation, say ix, and see whether we can verify the formula, j[o (3 x )(P x & □
’9,n is true iff at some i' iR i' and (3 x ) ( P x & □ Q x )
is true; in the indices i\ and i2 are accessible from *1 ; le t’s pick i\ and see whether we can verify the remaining formula. j[(3 x )(P x & □ ’g’H is tiue iff for some g' which is like g except for assigning d to x, [ ( P x & □ Q x ) ] ^ ' ’3' ^ ^ ' 11 = 1 There are in fact two objects in the extension o f P at ix, so there are two distinct assignments g' and g " which verify the first conjunct P x But do they also verify the second conjunct? I f g '(x ) = d2 and g " ( x ) =
So for the _
Given this recursive definition o f the extension o f an expression at an index in an intensional m odel, tw o expressions are called coextensive at in dex i iff they have identical extensions at i. Coreference o f tw o NPs means that they are coextensive, so the notion depends on the index o f evaluation. For instance, the winner o f the race and Robin m ay be p referen tia l at some index, but not at others, W e call tw o expressions extensionally equivalent in an intensional m odel when they have identical extensions at every index o f that m odel. Tw o expressions are logically equivalent when they are exten sionally equivalent in every possible intensional model. The strongest notion o f equivalence says that two expressions are intensionally equivalent, when their intensions are identical, i e ., when their associated functions from or operations on the set o f indices to the extensions at an index are identical. This n otion o f an intension is best understood dynam ically as the procedure o f com puting the extension o f an expression at an index. For instance, the process o f com puting the reference o f the winner o f the race must involve evaluating the predicates w inner and race, but these steps do not play any role in the process o f com puting the reference o f Robin. Hence when the two NPs the winner o f the race and Robin are p referen tia l or even extensionally equivalent they are not intensionally equivalent Since the intensional logic does not contain an expression for the intension o f an expression this p ro cedural character o f intensions is not further analyzed in this m od el theory. Yet it is clear from the recursive definition o f the extension o f an expression in an intensional m odel by which procedure its extension is com puted.
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Now if we use these different notions o f equivalence o f expressions the follow ing criteria o f substitutability can be defined. 1. Expressions which are coextensive at i are substitutable in a context with preservation o f its extension only at i 2 Expressions which are extensionally equivalent are substitutable in a context with preservation o f its extension at any index in that m odel. 3, Expressions which are logically equivalent are substitutable in a con text with preservation o f its extension at any index in any m odel. 4. Expressions which are intensionally equivalent are substitutable in a context with preservation o f its intension T he translation o f English to the intensional logic will not be carried out here, but we assume that the opacity creating expressions discussed in the previous section will always involve other indices besides the index o f evaluation in the determ ination o f the extension o f an expression. For instance, to evaluate John believes that Robin is the winner o f the race belief is analyzed with an accessibility relation on indices, representing the different epistem ic alternatives John entertains and requiring that at least one o f these verifies the em bedded statement B elief contexts require verification o f the em bedded expression at som e index which is related to the index o f evaluation by the su b je ct’s epistemic accessibility relation (c f o ) , K now ledge contexts w ould require verification o f the em bedded expression at all epistem ically accessible indices (cf. □ ). Similarly exploiting the universal and existential quantification over indices in the analysis o f opacity, m od a l necessities require verification at all acces sible indices (c f O ), m odal possibilities only at some (cf. o). Some English examples which illustrate the four notions o f substitutabil ity are the follow ing inferences, (1) I f we evaluate sentential com plem ents o f belief contexts at an index related by an epistemic accessibility relation to the index o f evaluation, i.e., interpret all predicate letters and NPs at that b elief index, the following inference is valid, since the second premise states that at that index the two NPs are co-referential. (1 5 -4 0 )
John believes that Robin has red hair John believes that Robin is the winner of the race John believes that the winner o f the race has red hair
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419
Note that this inferential pattern is only valid with definite, referen tial NPs in the premise stating the co-reference at the belief-index. The difference betw een valid (15-41) and invalid (15-42) is based on ju st the defi niteness o f the anaphoric reference in (15-41) to the referent o f the indefinite NPs introduced in the first premise. (1 5 -4 1 )
J o h n b e lie v e s th a t a m a n h a s r e d h a ir J o h n b e lie v e s t h a t h e is th e w in n e r o f t h e r a c e J o h n b e lie v e s th a t th e w in n e r o f t h e r a c e h a s r e d h a ir
(1 5 -4 2 )
J o h n b e lie v e s th a t a m a n h a s r e d h a ir J o h n b e lie v e s th a t a m a n is th e w in n e r o f t h e r a c e J o h n b e lie v e s th a t th e w in n e r o f t h e r a c e h a s r e d h a ir
(2) P rop er names, i f translated to logical constants as in predicate logic, have a constant reference according to the first clause in the definition o f extensions in an intensional m od el Kripke in trodu ced the notion o f a rigid designator for this semantic characteristic o f proper names. Names do not contribute to the inform ation expressed in a statement, but serve m erely as ‘p e g s’ for properties a ttrib u te d to the referents. I.e., the extension o f a p ro p e r name is the same entity o f the dom ain at any index, so evaluating a proper name at any tw o indices gives the same extension. This means that a p ro p e r name in an opaque context can be extraposed outside the opacity creating expression to be evaluated at the index o f evaluation w ithout changing the truth value o f the entire sentence, (1 5 -4 3 )
J o h n b e lie v e s th a t R o b i n w o n t h e r a c e O f R o b i n J o h n b e lie v e s th a t h e w o n th e r a c e
Note that for (15-43) to be valid John him self does not necessarily report his beliefs with the proper name, but anyone describing J oh n ’s beliefs with the first premise invites the inference Because o f this rigid designation o f proper names, existential generalization is also valid when based on proper names. But for other N Ps, including definite descriptions, no such inferences are valid, since their interpretation may vary at the accessible indices from the index o f evaluation, as the invalidity o f (15-44) and (15-45) show (1 5 -4 4 )
J o h n b e lie v e s t h a t a m a n w o n t h e r a c e O f a m a n J o h n b e lie v e s th a t h e w o n t h e r a c e
420
C
(1 5 -4 5 )
h a p t e r
15
J o h n b e lie v e s th a t a m a n w o n t h e r a c e T h e r e is a m a n o f w h o m J o h n b e lie v e s t h a t h e w o n th e race
(3) Since predicate logic is contained in the system o f intensional log ic as defined above, the tautologies o f predicate logic are also valid in intensional logic, i.e., they are all logically equivalent, always true at any index in any m odel. This means that an inference is valid in intensional logic w hen a tau tology is conjoined in an opaque context to a true premise. (1 5 -4 6 )
J o h n b e lie v e s t h a t R o b i n w o n t h e r a c e J o h n b e lie v e s th a t R o b i n w o n t h e r a c e a n d t h a t h e d o e s o r d o e s n o t h a v e r e d h a ir
This is an unsatisfactory aspect o f the simple system o f intensional logic we in trodu ced here, since John m ay simply fail to have any beliefs about R o b in ’ s hair color if he believes that R ob in won the race. On the other hand no rational person can fail to believe a tautology when he realizes its necessary truth.
I f we m aintain the ordinary logical n otion o f proof,
inferences preserve truth, but do not necessarily preserve intension.
We
return to this p roblem below to suggest a solution. (4) Finally there are intensionally equivalent equivalent expressions, %vhich are equivalent because their extension at any index is calculated in the same way.
T o make this strong notion o f equivalence precise a form al account
o f the com putational procedures used in determining extensions should be given which specifies criteria o f sameness for such procedures. W e will not do so here, but a simple example illustrates the m ain idea. W e learned in Part B that conjunction can be view ed as a com m uta tive operation on statements. Hence when determining the extension o f a conjunction the order in which the extension o f each conjunct is com puted does not m atter In verifying p k q we m ay start with either p or q, know ing that the result will not be affected by our choice. L ook in g at predicate logical operations in this intensional way the logical equivalence o f p k q and q k p should be attributed to their intensional equivalence.
Some laws o f
extensional logic m ay b e viewed as determining sameness conditions on op erations, especially those that merely perm ute the order o f connectives or
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their arguments (A ssociative, Distributive and Com m utative Laws)
On the
other hand laws like the Com plem ent Laws, De M orga n ’s Laws or the Con ditional Laws are valid because o f the particular truth-functional meaning we assigned to the connectives in this system o f classical predicate logic, but there are extensions! systems in which they would not be valid. I f we would incorpor ate a for m al definition o f intensions along these pr ocedural lines and define which logical laws wer e valid due to the intensional equivalence o f their arguments, the follow ing statements could be considered logically equivalent on the basis o f intensional equivalence o f conjunction.
Premise and con
clusion are interchangable in any context because they are ‘inform ationally equivalent’ and never give rise to inform ative identity statements
(15-47)
John believes Robin John believes Robin
that Robin won the race and that has red hair that Robin has red hair and that won the race
To account for the full variety o f opacity phenom ena discussed in the previous section m ore sophisticated intensional models would be required, but the examples illustrate how some aspects o f these opaque contexts can be analyzed with these simple intensional models. It should be clear that the predicate logical laws o f existential generalization and substitution o f extensionally equivalent expressions do n ot hold universally in this interpretation in intensional m odels. T o conclude this section we discuss briefly some o f the lim itations o f this form alization o f intensional interpretations for natural language. As we rem arked earlier in this system o f intensional logic the predicatelogical tautologies are all logically equivalent, although not necessarily also intensionally equivalent, and similarly for contradictions This is partly due to the fact that we assumed that all functions are total, i,e defined for all arguments. I f that assum ption were dropped it would becom e possible to distinguish functions that are logically equivalent when defined, but which are not defined for the same arguments. For1 instance, the truth value o f the predicate-logical tau tology R obin did or did not win the race would only be defined if the name R obin had a reference at the same index, i.e. if the dom ain contained som eone called Robin, The interpretation o f names could still b e constant but partially defined, which is com patible with their being rigid designators.
The interpretation o f another tautology Jane did or did
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not win the race could similarly be partially defined, i e true at every index where Jane is interpreted. But the tw o tautologies would not necessarily be logically equivalent, for a m odel could be given which interpreted the tw o names at distinct sets o f indices. Partial functions in semantic interpreta tions yield already an im portantly finer-grained n otion o f logical equivalence. The second problem with these simple intensional models concerns the interpretation o f epistem ic verbs which give rise to opaque contexts. A ny logical consequence o f what is believed by someone must be believed by him as well.
For instance, if John believes that R obin w on the race, then he
must also believe that R ob in won the race and that two plus tw o is foux, assuming that names o f numbers always refer. This issue is often called the problem o f logical om niscience. W h at we know or believe is not ordinarily closed under its logical consequences, for we often discover new im plications o f inform ation we have had for a long time Only if totally defined functions with constant extensions were absolutely forbidden in an interpretation in intensional models could this problem be adequately, though not insightfully be solved
A more prom ising way would attem pt to analyze the subject
m atter o f beliefs and require that entailed beliefs must have the same subject m atter. A nother im provem ent would be to acknowledge the fact that on the one hand people have well understood beliefs which m ay be represented by a set o f statements closed under some sufficiently fine-grained n otion o f logical consequence, but on the other hand they have a chunk o f unanalyzed and only partially understood beliefs to be represented in a coarser way and not closed under any logical operation. The problem has not yet been solved in any definitive way for all epistem ically opaque constructions which would still allow for people who have inconsistent beliefs to be rational in not just believing anything The third aspect in which these simple intensional m odels need im prove ment is the interpretation o f proper names. E ven when we let the functions interpreting proper names be partial, this still would not account for two natural facts: first, different people m ay have the same name and only a nam e in a sufficiently specific context can b e said to refer uniquely, and second, people m ay use the same name intending to refer to the same indi vidual when in fact they refer to distinct individuals, for instance, because they were wrongly introduced or forgot their true names, A proper solu tion would require m ajor m odifications in the intensional m odels: a formal representation o f contexts and a general m ethod to incorporate parameters o f language use, in particular for cases in which speakers m ay hook up the same name to a different person. I f functions interpreting expressions could
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be speaker-sensitive, we could explain how three com m unicative situations differ essentially although what tw o speakers say, respectively ‘R ob in has red h a ii’ and ‘R ob in does not have red h air,’ is exactly the same. In the first situation they would contradict each other, in the second they would merely refer to two different individuals called R obin, and in a third situation one o f them or b oth could wrongly connect the name to the same individual or to different individuals The final p oin t in which these intensional models are to o simple is in their failure to account for genuine synonym ous predicates A lth ou gh there are not to o m any examples in natural language o f truly synonym ous descrip tive predicates, an interpretation would have to discriminate them, although they are logically equivalent and, if syntactically simple, even intensionally equivalent, For instance, John m ay believe that woodchucks are m arm ots, whereas he m ay not believe that groundhogs are m arm ots, although wood chuck and groundhog are synonyms.
The on ly option open to repair this
shortcom ing is to assume a set o f prim itive properties, as extensions o f pred icates, instead o f giving predicates an extensional set-theoretic interpretation as the set individuals in the domain with the property. These suggestions to im prove upon the intensional models outlined in this section are all currently subject o f new research. Various form al accounts o f interpretations are being developed m otivated b y these and other problem s that were encountered with the simple intensional models. The interested reader is referred to the readings suggested for this chapter.
15.4
Tense and time
A nother im portant linguistic application o f relativizing extensions o f expres sions to indices is the interpretation o f verbs, tenses and tem poral adverbs. In this section some elem entary aspects o f the semantics o f tem poral refer ence are outlined briefly and we discuss what m odifications the interpreta tion in intensional m odels requires in order to accom m odate some aspects o f tem poral reference The simple past tense inflection on a verb, for instance, creates an opaque context, requiring the sentence to be interpreted in its present tense form at an index in the past o f the index o f evaluation. E g , R obin had red hair is interpreted as true at an index i when there is an i' tem porally located before i at which Robin has red hair is true. This means that for the interpretation
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of tenses the set o f indices is ordered b y an asym m etric and transitive acces sibility relation representing the passage o f time.
The tenses expressed by
inflection and the tem poral auxiliary verbs require interpretation along this tem poral axis. Some classes o f tem poral adverbs are interpreted along this line as quantifying in the m eta-language over tem porally ordered indices. Always is a universal quantifier, som etim es an existential one and adverbs o f frequency like often, every week or twice a week require m ore com plicated forms o f quantification over in d ices, The set o f tem poral indices may be structured by additional assump tions m otivated b y linguistic considerations. Besides auxiliary verbs and the inflectional m orph ology on V Ps m ain verbs themselves describe changes in the world called events
A sentence like Robin won the race describes an
event which took place over a p eriod of' time in the past o f the in dex o f evaluation. W h en we consider the indices to represent periods o f tim e o f ar bitrary length, rather than smallest ‘p oin ts’ or atom ic m om ents, we m ay also want to assume a sym m etric tem poral overlap relation on the indices and construct smallest ‘p eriod s’ as m axim al sets o f pairwise overlapping periods (see van B enthem (1983), Kam p (1979) and (1980)). The tem poral adverb now has a special semantic function, since in any con text, transparent or heavily opaque, it refers to one and the same index, the tim e o f utterance, in an intensional m odel. Each intensional model has to contain one designated index representing the current tim e, i e, the ‘n ow ’-index, from which past and future coordinates are fixed
Consider for
instance (1 5 -4 8 ) Y ou will once be grateful for what I tell you now
The future auxiliary verb takes the interpretation from the index o f eval uation, the designated ‘n o w ’-index io, to a future index i\, at which you are grateful for what I told you at the original ‘n o w ’ index io, not for what I tell you at the new ‘n ow ’-index i\
So when tenses require the interpretation
to consider extensions at other indices, in any context the adverb now takes the interpretation back to the designated index,
A contextual parameter
‘time o f utteran ce’ can b e represented in an intensional m odel as a specially selected index, and n o matter at which index now is evaluated, its exten sion will always b e that designated index. The difference betw een the two sentences
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425
(1 5 -4 9 ) A child was born who would b e king
(1 5 -5 0 ) A child was born who will be king
can be analyzed by requiring for the interpretation o f (15-50) that the ‘n o w ’index be located betw een the past index at which a child is born is true and the future in dex at which the child is king is true, whereas the interpretation o f (15-49) does not put any requirements on its location The ‘n o w ’-index is identified in each m odel with the tim e at which the evaluated statement is uttered. B ut each intensional m od el represents then in fact a static picture o f the world at that time, and the past and future coordinates are fixed with respect to that index. I f we want to do ju stice to the flow o f time in an intensional m odel a series o f indices would have to be designated as consecutive ‘n o w ’-indices. A nd to interpret a discourse consist ing o f tem porally ordered utterances a chain o f such ‘n o w ’-indices would be required within each m odel, This cannot be incorporated straightforwardly into a simple intensional m od el if what was once true could later becom e false in one and the same m odel, and hence any analysis o f valid inference would be lost. The p roblem can be solved if we introduce in the form al lan guage variables for the indices which can then be referred to and quantified over at the level o f the ob je ct language. B ut a m ore satisfactory solution would take the need to represent contextual parameters seriously and define a context-sensitive n otion o f meaning and interpretation which does justice to the dynamics o f context change and processing inform ation. This would require syntactic and semantic adaptations which cannot be specified fur ther here, but in current research several frameworks are developed in which events and periods as their tem poral substratum constitute a dom ain o f o b jects in the interpretation (see D ow ty (1979), Kam p and Rohxer (1983), L o Cascio and Vet (1986), and Linguistics and Philosophy 9.1) The verbs which describe changing events create opaque contexts for some o f their arguments. A b y now classical form o f opacity, know n as M ontague’s ‘tem perature p u zzle’ is the following invalid inference: (1 5 -5 1 )
T h e t e m p e r a t u r e is n in e t y T h e t e m p e r a t u r e rise s N i n e t y rise s
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In this usage the verb to rise cannot have ordinary individuals in its extension, but must b e interpreted as a property o f intensional objects like functions or operations that determine extensions at an index. Verbs in the same semantic class include to change, to decrease, to increase, to grow, to diminish.
In contrast to names o f num bers, proper names o f individuals
apparently can take such change-denoting predicates notw ithstanding their rigid designation Rather than cop out o f the puzzles by treating these verbs as lexically am biguous, it would further our understanding o f intensionality and opaque contexts considerably if we analyzed how changes in an indi vidual must be distinguished from changes in measurable properties such as volum e, size, weight, warmth or relative loca tion A nother class o f verbs which create interesting opaque contexts is formed b y verbs describing actions o f creation or destruction, e g, to build, to write, to construct, to cook, and to destroy, to demolish, to devastate, to burn. The object NPs o f these verbs are opaque since the extension o f these NPs gradually comes into existence or ceases to exist during the period at which the action takes place, and existential generalization cannot be valid in an unrestricted form
The progressive tense is typically used to describe such
actions as they are going on. The interpretation o f these verbs must hence add new objects to the domain o f the interpretation, and withdraw objects or parts o f them from it. The simple intensional models were based on one static dom ain o f individuals which were taken to ‘exist’ at any index in the m odel. A n analysis o f this form o f opacity must allow the dom ain of objects over which quantifiers range to depend on the index. Verbs o f creation shift the index and add new objects to the dom ain, and verbs o f destruction shift the index and drop objects from the dom ain. B ut if m ore structure is im posed on the domains to represent part-w hole relations betw een objects and a related relation between events, the invalidity o f the follow ing inference can be analyzed
(15-52)
John was writing a poem John wrote a poem
and contrasted to the validity o f the inference
(15-53)
John was writing poetry John wrote poetry.
In d e x ic a l it y
427
These inferential patterns know n as the ‘Im perfective P a rad ox’ receive much attention in recent research on tense and aspect. In early work on tem poral reference by the logician H, Reichenbach (Reichenbach (1947)) used besides the time o f utterance, and the index at which the event took place, a third tem poral index, the reference time to repre sent the tem poral ‘point of view ’ o f the speaker. The difference betw een the simple past tense sentence Robin won the race and the present perfect tense sentence Robin has won the race is accordingly analyzed by requiring the interpretation o f the simple past tense to shift the reference tim e as well as the event tim e to a past index relative to the time o f utterance, whereas the present perfect tense only shifts the event tim e, leaving the reference time at the tim e o f utterance Note, however, that not all simple past tense sentences shift the reference tim e but only those which describe ‘b ou n d ed ’ or com pleted events
This leads in to the semantic analysis o f aspect and verb
classes which is an im portant focus o f ongoing research. In frameworks o f dynam ic interpretation the notion o f reference tim e has com e to be crucial in accounting for tem poral dependencies and tem poral anaphora (see Kam p and R ohrer (1983) and Partee (1984))
15.5
Indexicality
In the previous section it was poin ted out that in any linguistic context the tem poral adverb now refers to a fixed contextual param eter, the tim e o f utterance. The class o f expressions in natural language which serve such a semantic function, the indexical expressions, is m uch larger and includes the personal pronouns I, you, and we, besides the locative expressions here and there The dem onstrative or deictic expressions this, that, these, those, and the deictic pron oun he are also often included in this semantic class. The semantic interpretation o f indexicals is essentially dependent on the extra-linguistic situation o f use. A n indexical refers in any linguistic context, i.e., on any index, directly to a value o f a contextual param eter, e g, /re fe rs to the speaker who utters the sentence, here to the location o f utterance and that to what is being poin ted at b y the speaker But I cannot be interpreted synonym ously with the speaker since the reference o f a definite description is always dependent on the index. So on the one hand the reference o f an indexical does not depend on its linguistic sentential context, on the other hand it shifts when the context o f use changes. The value o f a contextual parameter, i.e., w ho is the speaker at an index, where som ething is said and
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what is being poin ted at changes constantly. T he context o f use determines the con ten t o f an indexical expression. For instance, when I say (1 5 -5 4 ) I am speaking
what I said differs from when you utter (15-54)
The com pon en t o f the
Fregean sense o f an expression which determines how the content is deter m ined by the context o f use is called the character o f the expression This is what com petent speakers o f a language have to know if they axe to un derstand (15-54) in any context o f use The content o f an expression may be identified with what we called intension in the previous section, i.e., the context-independent procedure which determines its extension at an index. Now we may explain that any utterance o f (15-54) is true, but that b y utter ing (15-54) I am not expressing a logically necessary truth, but a contingent statem ent, whose truth value varies from index to index Non-indexicals have a constant character and express the same content in every context. T o incorporate this account o f the semantics o f indexicals and their direct reference to elements o f the context o f use, the interpretation by indexing the extension o f an expression in an intensional m odel is extended b y contexts, i.e., a set o f contextual parameters representing elements o f a context o f use and containing at least parameters for the speaker, the hearer, the utterance time and the utterance location : context c = ( sc ,h c, t c, l c). O f course, the set o f contextual parameters can be extended upon need. This is called the fleshing out strategy, supply contextual parameters when indexical linguistic expressions refer to them, In such a double-indexing interpretation the indexical I, for instance, is interpreted by a function from contexts to the param eter sc, the speaker at that con text, and the extension o f sc is determined at an index b y identifying it with the extension o f the predicate speaking at that index why (1 5 -5 5 )
The speaker is the one who is speaking
is an uninform ative identity statem ent, whereas (1 5 -5 6 ) I am the one who is speaking
This explains
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xe r c is e s
4 29
m ay express new inform ation to others beside oneself Proper names interpreted as Kripkean rigid designators are expressions with constant contents T h ey are also independent o f contextual parameters so their character is also constant
T he identity statement
(1 5 -5 7 ) I am R obin
is inform ative in tw o different ways: first, when it gives a hearer a new way o f rigidly referring to the speaker when he is not present and second, when the hearer already has inform ation about someone called R ob in , but is not acquainted with him from his ow n experience. In the first case, the inform ation stabilizes the character, in the second case it hooks up a stable content to the external context o f use T o provide a satisfactory m athem atical m odelling o f these issues lies at the heart o f contem porary research in the semantics o f natural language. We have attem pted to show that the tools offered in the first three parts o f this b o ok provide a choice o f m athem atical m ethods to tackle such problem s in a linguistic theory o f meaning and interpretation.
Exercises 1. I f the intensional m od el M * defined in Section 15.3 had an irreflexive accessibility relation, is the form ula J(3x ) ( P x & O Q x )}M .‘r’g,n still verifiable? I f so, define the assignment.
2 . C om pute in M * defined in Section 15.3 the truth value o f ff(3 x ) { P x t Q x ) -* o(Vx) ~ -Pa:)jM*’s’io 3. I f the follow ing formulas are assumed to be valid in a system o f inten sional logic they characterize a p rop erty o f the accessibility relation. Find out which p roperty b y drawing relational diagrams o f small in tensional m odels, m aking the form ula true at each index and verifying it. ( i ) Dtp - + ip (ii) Dtp —> O D tp (iii) o O ip —> ip
Part E
L A N G U A G E S, G RAM M ARS, AND AUTOM ATA
Chapter 16
Basic Concepts 16.1
Languages, gramm ars and autom ata
A t one level o f description, a natural language is sim ply a set o f strings— finite sequences o f words, m orphem e, phonem es, or whatever. N ot every possible sequence is in the language: we distinguish the grammatical strings from those that are ungrammatical,
A grammar, then, is some explicit
device for making this distinction; it is, in other words, a means for selecting a subset o f strings, those that are gram m atical, from the set o f all possible strings form ed from an initially given alphabet or vocabulary. In this chapter we will consider two classes o f form al devices which can function as grammars in this very general sense: ( 1 ) autom ata, which axe abstract com puting machines, and ( 2 ) string rewriting systems, which gen erally bear the name “gram m ar” or “form al gram m ar” .. The latter will be familiar to linguists inasmuch as grammars in this sense have form ed the basis o f m uch o f the work in generative transform ational theory. W e begin by considering certain properties o f strings and sets o f strings. Given a finite set A , a string on (or o v er) A is a finite sequence o f occurrences o f elements from A . For exam ple, if A — {a , b, c }, then acbaab is a string on A. Strings are by definition finite in length. (Infinite sequences o f symbols axe also perfectly reasonable ob jects o f study, but they axe not suitable as m odels for natural language strings.) The set from which strings axe form ed is often called the vocabulary or alphabet, and this too is always assumed to be finite. T he length o f a string is, or course, the num ber o f occurrences o f symbols in it (i.e., the num ber o f tokens, not the num ber o f types). The string acbaab thus is o f length 6 . 433
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Because we are dealing with tokens o f an alphabet, there is an im por tant difference betw een the linearly ordered sequences we call strings and a linearly ordered set I f the set A = { a , b ,c } were linearly ordered, say, as b —* a —» c, each element o f A would occu p y a unique place in the ordering. In a string, e g , , acbaab, tokens of a, o ccu r in the first, fourth, and fifth positions T o be form al, one could define a string o f length n over the alphabet A to be a function m apping the first n positive integers into A, For exam ple, acbaab would be the function { ( l , a ) , { 2 , c ) , { 3 , 6 ) , { 4 , a ) , { 5, a ),{ 6 , 6 )} There is little to be gained in this case b y the reduction to the prim itives of set theory, however, so we will continue to think o f strings sim ply as finite sequences o f symbols, A string m ay b e o f length 1, and so we distinguish the string 6 o f length 1 from the sym bol 6 itself. W e also recognize the (unique) string o f length 0 , the em pty string, which we will denote e (som e authors use A ). T w o strings are identical if they have the same sym bol occurrences in the same order; thus, acb is distinct from abc, and strings o f different length are always distinct A n im portant binary operation on strings is concatenation, which amounts simply to juxtaposition. For exam ple, the strings abca and bac can be concatenated, in the order m entioned, to give the string abcabac. Some times concatenation is denoted with the sym bol
thus, abca'~'bac. Con
catenation is associative since for any strings a , / ? , 7, ( a /-' j0 ) /-' 7 = 7 ), but it is not com m utative, since in general a^/3 ^ The em pty string is the identity element for concatenation; i e ., for any string a , a ^ e = e ^ a = a. Given a finite set A , the set o f all strings over A , denoted A*, together with the operation o f concatenation constitutes a m onoid. C oncatenation is well-defined for any pair o f strings in A* and the result is a string in A*; the operation is associative; and there is an identity element ( A * , ) fails to be a group since no element other than e has an inver se: n o string concatenated with a n on-em pty string x will yield the em pty string. Since concatenation is n ot com m utative, ( A*, "- ' ) is not an A belian monoid, A frequently encountered unary operation on strings is reversal.
The
reversal o f a string x, denoted x R , is simply the string form ed b y writing the symbols o f x in the reverse order. Thus (acbab)R = babca T he reversal o f e is just e itself, T o be form al, we could define reversal b y in d u ction on the length of a string:
Languages,
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16.1
grammars
and
4 35
automata
Given an alphabet A:
(1) I f x is a string o f length 0, then x R = x (i.e., e R = e) (2) I f x is a string o f length k + 1, then it is o f th e form wa, where a £ A and w £ A *; then x R — (w a )R — awR.
m C oncatenation and reversal are connected in the follow ing way:
For all
strings x and y , (2'^ ''y )R = y R '~'x R. For example, (1 6 -1 )
(bca'~' ca )R = (c a )R ^ ( b c a ) R = a c ^ a c b = acacb
Given a string x, a substring o f x is any string form ed from continguous occurrences o f sym bols in x taken in the same order in which they occu r in x For example, bac is a substring o f abacca, but neither bcc nor cb is a substring. Formally, y is a substring o f x iff there exist strings z and w such that x = z'~'y'~'w. In general, z or w (or b oth ) m ay be empty, so every string is trivially a substring o f itself. (N on-identical substrings can be called p roper substrings.) The em pty string is a substring o f every string; i.e.., given x we can choose z in the definition as e and w as x so that x = e
e
x.
A n initial substring is called a prefix, and a final substring, a suffix. Thus, ab is a (proper) prefix o f abacca, and cca is a (p rop er) suffix o f this string. W e m ay now define a language (over a vocabulary A ) as any subset o f A ” . Since A * is a denum erably infinite set, it has cardinality
its power
set, i.e., the set o f all languages over A , has cardinality 2^° and is thus nondenumerably infinite. Since the devices for characterizing languages which we will consider, viz., form al grammars and autom ata, form denumerably infinite classes, it follows that there are infinitely m any languages-in fact, non-denum erably infinitely m an y-w hich have no gram m ar. W hat this means in intuitive terms is that there axe languages which axe such m otley collec tions o f strings that they cannot be com pletely chaxacterized b y any finite device, The languages which are so characterizable exhibit a certain amount of order or pattern in their strings which allows these strings to be distin guished from others in A * by a grammar or autom aton with finite resources. The study o f form al languages is essentially the investigation o f a scale of
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com plexity in this patterning in strings. For exam ple, we m ight define a language over the alphabet {a , 6} in the follow ing way: (1 6 -2 )
L = { x \x contains equal numbers o f a ’s and 6’s (in any order)}
W e m ight then com pare this language with the follow ing: (1 6 -3 )
L i = { x £ {a , 6}* | x = anbn(n > 0 )} , i.e , strings consisting of some number o f a ’s follow ed by the same num ber o f 6’ s L 2 = { x £ {a , 6}* |x contains a num ber o f a ’s which is the square o f the num ber o f 6’s}
Is L i or Lz in some intuitive sense more com plex than L I
M ost would
p robably agree that L% is a m ore com plex language than L in that greater effort would be required to determine that the mem bers o f a ’s and 6’s stood the “ square” relation than to determine merely that they were equal In other words, a device which could discriminate strings from non-strings o f L 2 would have to be more powerful or more “intelligent” than a device for making the com parable discrim ination for L, W h at o f L i and L I Here our intuitions are much less clear. Some might think that it would require a less pow erful device to recognize strings in L reliably than to recognize strings in L i, others m ight think it is the other way around or see no difference. As it happens, the particular scale o f com plexity we will investigate (the so-called Chom sky Hierarchy) does regard L 2 as more com p lex than L but puts L\ and L in the same com plexity class. A t least this is so for the overall com plexity measure. Finer divisions could be established which m ight distinguish L\ from L One linguistic application o f these investigations is to try to locate nat ural languages on this com plexity scale. This is part o f the overall task o f linguistics to characterize as precisely as possible the class o f (potential and actual) natural languages and to distinguish this class from the class o f all language-like systems which could not be natural languages. One must keep clearly in m ind the lim itations o f this enterprise, however, the principal one being that languages axe regarded here simply as string sets, It is clear that sentences o f any natural language have a great deal m ore structure than sim ply the concatenation o f one element with another. Thus, to establish a com plexity scale for string sets and to place natural languages on this scale may, because o f the neglect o f other im portant structural properties, be to classify natural language along an ultim ately irrelevant dimension. Extend
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rammars
ing results from the study o f form al languages in to linguistic theory must therefore be done with great caution.
16.2
Gram m ars
A form al grammar (or simply, gram m ar) is essentially a deductive system o f axioms and rules o f inference (see Chapter 8), which generates the sentences o f a language as its theorems. B y the usual definitions, a grammar contains just one axiom , the string consisting o f the initial sym bol (usually S ), and a finite number o f rules o f the form ip —* us, where ip and us ar e strings, and the interpretation o f a rule is the follow ing: whenever ip occurs as a substring o f any given string, that occurrence m ay be replaced by us to yield a new string. Thus if a grammar contained the rule A B —* C D A , we could derive from the string E B A B C C the string E B C D A C C Grammars use tw o alphabets: a term inal alphabet and a non-term inal alphabet, which are assumed to be disjoint.
The strings we are interested
in deriving, i.e., the sentences o f the language, are strings over the term inal alphabet, but interm ediate strings in derivations (p roofs) by the grammar m ay contain sym bols from b oth alphabets. W e also require in the rules o f the gram m ar that the string on the left side n ot consist entirely o f term inal sym bols. Here is an example o f a grammar m eeting these requirements: (1 6 -4 )
Vt Vn
(the term inal alphabet) (the non-term inal alphabet)
S
(the initial sym bol— a m em ber o f Vn ) S
R
(the set o f rules)
={a,6} — {S ,A ,B }
—> A B S
5 -> e A B -> B A B A -> A B A
—►a
B
6
A com m on n otational convention is to use lower case letters for the term inal alphabet and upper case letters for the non-term inal alphabet A derivation o f the string abba by this gram m ar could proceed as follow s: (1 6 -5 )
5 = > A B S ==» A B A B S ==> A B A B = > A B B A = > A B b A = > a B bA = > abbA = > abba
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Here we have used the sym bol “ =>” to m ean “ yields in one rule application,” N ote that abba is not subject to further rewriting inasmuch as it consists entirely o f term inal sym bols and n o rule licenses rewriting strings o f term i nals. T he sequence (16-5) is said to be a derivation ( o f abba from S ), and the string abba is said to be generated by the gram m ar. T he language gener ated by the grammar is the set o f all strings generated. Here are the formal definitions:
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e f in it io n
16.2
Let £ = V t U V n • A (form al grammar G is a quadrupJe
{ V t , V at, S, R ), where V t and Vjv are finite disjoint sets, 5 is a distinguished
m em ber o fV jv , and R is a finite set o f ordered pairs in £*VatIT x S ‘ .
■
W e have written ip —> w above for clarity instead o f (ip, w). T he last condition simply says that a rule rewrites a string containing at least one non-term inal as some (possibly em pty) string.
D
e f in it io n
16.3
Given a gram m ar G = {V t,V iv , S, R ), a derivation is a
sequence o f strings x i , x 2 , . ■ ., x n (n > 1) such that x\ — S and for each (2 < i < n ), Xi is obtained from X i-i by one application o f som e rule in R.
m
T o be com pletely form al, we would spell out in detail what it means to apply a rule o f J? to a string. T he reader m ay want to d o this as an exercise.
D
e f in it io n
16.4
A gram m ar G generates a string x £ V f i f th ere is a
derivation x \ ,. , . , x n b y G such that x n = x.
■
N ote that b y this definition only strings o f term inal sym bols are said to be generated.
e f i n i t i o n 16.5 T he language generated b y a gram m ar G , d enoted L (G ), is th e set o f all strings generated by G. ■
D
T he language generated b y the grammar in the exam ple o f (16-4) is { 2: £ { a , b } ” |x contains equal numbers o f a’s and 6’ s }
T
rees
439
Figure 1 6-1: A typical constitutent structure tree
16.3
Trees
W h en the rules o f a grammar are restricted to rewriting only a single non term inal sym bol, it is possible to contrue grammars as generating constituent structure trees rather than simply strings, A n exam ple o f such a tree is shown in Fig. 16-1. Such diagrams represent three sorts o f inform ation about the syntactic struc ture o f a sentence: 1, T he hierarchical grouping o f the parts o f the sentence into constituents 2, The gram m atical type o f each constituent 3, T he left-to-right order o f the constituents For exam ple, Fig. 16-1 indicates that the largest constitutent, which is la beled by S (for Sentence), is made up o f a constituent which is a N (ou n ) P (h rase) and one which is a V (e rb ) P (h rase) and that the noun phrase is com posed o f tw o co n stitu e n ts : a D et(erm iner) and a N (ou n ), etc. Further,
C h ap ter
440
16
in the sentence constituent the noun phrase precedes the verb phrase, the determiner precedes the noun in the noun phrase constituents, and so on. T he tree diagram itself is said to b e com posed o f n od es, or poin ts, some o f which are connected by lines called branches Each n ode has associated with it a label chosen from a specified finite set o f gram m atical categories (S, NP, V P , etc.) and form atives (m y, sister, etc ), As they are custom arily drawn, a tree diagram has a vertical orientation on the page with the nodes labeled b y the form atives at the b o tto m
Because a branch always connects a higher
n ode to a lower one, it is an inherently directional con nection This direc tionality is ordinarily not indicated b y an arrow, as in the usual diagrams o f relations, but only by the vertical orientation o f the tree taken together with the convention that a branch extends from a higher n od e to a lower node.
1 6 .3 .1
D om in an ce
W e say that a node x dom inates a node y if there is a connected sequence o f branches in the tree extending from x to y This is the case when all the branches in the sequence have the same orientation away from x and toward y. For example, in Fig 16-1 the node labeled V P dom inates the n ode labeled A rt, since the sequence o f branches connecting them is uniform ly descending from the higher n ode V P to the lower n ode Art, T he n od e labeled V P does not dom inate the node labeled Poss, since the path b y which they are joined first ascends from V P to S and then descends through NP and Det Given a tree diagram, we represent the fact that x dom inates y b y the ordered pair ( x , y) . T he set o f all such ordered pairs for a given tree is said to constitute the dom inance relation for that tree. D om inance is clearly a transitive relation.
I f x is connected to y b y a sequence o f descending
branches and y is similarly connected to z, then x dom inates z because they are also connected by a sequence o f descending branches, specifically, b y the sequence passing through y , As a technical convenience, it is usually assumed that every node dominates itself, i.e., that the dom inance relation is reflexive. Further, if x dominates y , then y can dom inate x only if x — y\ or in other w ords, dom inance is antisym metric. Thus, the relation o f dom inance is a weak partial ordering o f the nodes o f a tree. I f x and y are distinct, x dominates y, and there is n o distinct node betw een x and y, then x im mediately dom inates y. In Fig. 16-1, the node labeled V P im m ediately dominates the node labeled V but not the node labeled found. A node is said to be the daughter o f the n od e im mediately
T
rees
441
dom inating it, and distinct nodes im m ediately dom inated b y the same node are called sisters. In Fig. 16-1, the node labeled V P has tw o daughters, viz., the n ode labeled V and the rightm ost node labeled NP. T he latter tw o nodes are sisters. A n ode which is m inim al in the dom inance relation, i.e., which is not dom inated b y any other node, is called a root
In Fig. 16-1 there
is one root, the node labeled S, M axim al elements are called leaves, and in Fig 16-1 these are the nodes labeled b y the form atives, my, sister, etc. Note that a tree diagram is ordinarily drawn upside dow n since the root is at the top and the leaves are at the bottom .
M athem aticians sometimes use the term tree for a configuration with m ore than one root, e.g., that shown in Fig. 16-2. For linguists, however, a tree is invariably singly rooted , the configuration in Fig.. 16-2 being considered a “forest” o f trees. W e shall adhere to linguistic usage and accordingly we have the follow ing condition:
T h e S in g le R o o t C o n d i t io n : In every well-form ed constituent structure tree there is exactly one node that dominates every node. The ro o t n ode is, therefore, a least element (and necessarily also a minim al elem ent) in the dom inance relation. W e note, incidentally, that the Single R o o t C ondition is m et in the trivial case o f a tree that has only one node, which is sim ultaneously ro o t and leaf. T he con dition w ould not be m et b y an “ em p ty” tree with no nodes at all, since it asserts that a node with the specified p rop erty exists in the tree.
1 6 .3 .2
P receden ce
T w o nodes are ordered in the left-to-right direction just in case they are not ordered b y donim ance. In Fig. 16-1 the node labeled V precedes (i.e., is to
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16
the left o f) its sister node labeled NP and all the nodes dom inated b y this NP node; it neither precedes noi follows the nodes labeled S, V P , V , and found, i.e., the nodes that either dom inate or are dom inated b y the V node. It is not logically necessary that the relations o f dom inance and left-to-right precedence be m utually exclusive, but this accords with the way in which tree diagrams are usually interpreted. Given a tree, the set o f all ordered pairs ( x , y ) such that x precedes y is said to define the precedence relation foi that tree. T o ensure that the precedence and dom inance relations have no ordered pairs in com m on, we add the E xclusivity C ondition: T h e E x c lu s iv it y C o n d i t io n :
In any well-form ed constituent structure
tree, for any nodes x and y, x and y stand in the precedence relation P , i.e., either ( x , y ) £ P or ( y , x ) £ P, if and only if x and y do not stand in the dom inance relation D , i e., neither {x , y ) £ D nor ( y , x ) £ D Like dom inance, precedence is a transitive relation, but precedence is irrefiex ive rather than reflexive. T he latter follows from the E xclusivity C ondition, since for every node x , ( x , x ) £ D and therefore ( x , x ) 0 P. I f x precedes then y cannot precede x , and thus the relation is asym m etric. Precedence, therefore, defines a strict partial order on the nodes o f the tree. One other con dition on the dom inance and precedence relations is needed to exclude certain configurations from the class o f w ell-form ed trees.
An
essential characteristic o f a tree that distinguishes it from a partially ordered set in general is that no node can have more than one branch entering it; i.e., every n ode has at m ost one n ode im m ediately dom inating it. T he structure shown in Fig, 16-3(a) has a node d with tw o im m ediate predecessors, b and c, and therefore it is not a tree. A nother defining p rop erty o f trees is that branches are not allowed to cross.
Figure 16-3(b) illustrates the sort o f
structure that is forbidden. B oth types o f ill-form edness can b e ruled out by adding the N ontangling Condition: T h e N o n t a n g lin g C o n d it io n : In any w ell-form ed constituent structure tree, for any nodes x and y , if x precedes y , then all nodes dom inated b y a precede all nodes dom inated b y y. T he configuration in Fig, 16-3(a) fails to meet this con d ition because b precedes c, 6 dom inates d, and c dominates d, and therefore d ought to precede d.
This is im possible, however, since precedence is irrefiexive. In
Fig 16-3(b), b precedes c, 6 dominates d, and c dom inates e. Thus, b y the Nontangling C ondition, d should precede e, but in fact the reverse is true.
T
a) '
443
r e e s
a
b
c
d
b
c
e
d
Figure 16-3: Structures excluded as trees b y the N ontangling C ondition
1 6 .3 .3
L abelin g
T o com plete the characterization o f trees we must consider the labeling o f the nodes. It is apparent from Fig. 16-1 that distinct nodes can have identical labels attached to them , e.g., the tw o nodes labeled NP. Since each node has exactly one label, the pairing o f nodes and labels can be represented b y a labeling function L, whose dom ain is the set o f nodes in the tree and whose range is a set (in syntactic trees, a set o f gram m atical categories and form atives). The m apping is, in general, an into function. In summary, we have the follow ing definition: D
e f in it io n
16,6 A ( c o n s t it u e n t s t r u c t u r e j tr e e is a m athem atical configu
ration {N , Q , D , P , L ), where N is a finite set, the set o f nodes Q is a finite set, the set o f labels D is a weak partial order in N X N , th e dom inance relation P is a strict partial order in N X N , the precedence relation L is a fu n ction from N in to Q , the labeling function and such that the follow ing conditions hold; (1) (3 x £ N ) ( V y £ N ) ( x , y ) £ D (Single R o o t C ondition) (2) ( Vx , y £ N ) ( ( ( x , y ) £ P V ( y , x ) £ P ) (E xclusivity C ondition)
({x , y ) $ D k { y , x ) $ D) )
444
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16
hapter
(3) { y w , x > y , z £ N ) ( ( { w , x ) £ P & ( w ,y ) £ D & (x, z) £ D ) -> (y, z) £ P ) (N ontangling C ondition)
G iven this definition, one can prove theorems o f the follow ing sort: T
heorem
16 1 Given a tree T = (N , Q , D , P , L ), ev e ry pair o f sister nodes
is ordered b y P. P ro o f:
■
Take x and y as sisters im m ediately dom inated b y some n od e z.
B y the definitions o f ‘sister1 and ‘im m ediate d om ination ,’ x . y , and z must all be distinct. As an assum ption to be proved false, let x dom inate y. Therefore, x must dom inate z, since z im m ediately dom inates y B ut z also dominates x, and x and z are distinct, so this violates the condition that dom inance is antisym metric. Therefore, x cannot dom inate y. B y a sym m etrical argument, we can show that y does not dom inate x. Thus, { x , y ) $ D and ( y , x ) $ D , and b y the E xclusivity C ondition it follows that ( x , y ) £ P V ( y , x ) £ P\ i.e., x and y are ordered b y P. ■
T
heorem
16.2
Given a tree T = { N , Q , D , P, L ), the leaves are totally
ordered b y P .
■
P r o o f : Let M be the set o f leaves, and let R be the restriction o f the rela tion P to the set M ; i.e., R = { ( x , y ) £ M
X
M | { x , y ) £ P } . R is a strict
partial order, since if there were any ordered pairs violating the conditions o f irreflexivity, asym m etry, and transitivity in R , then because R C P , these pairs would also appear in P , and P would not be a strict partial order. B y definition, a leaf dominates no n ode except itself, and therefore for every pair o f distinct leaves x and y , ( x , y ) g D and ( y , x ) $ D. Thus, b y the E xclusivity C ondition ( x , y ) £ P V ( y , x ) £ P. Since x and y are leaves, ( x , y ) £ R V ( y , x ) £ R, b y the definition o f R , and thus R is connex. There fore, R is a strict total order.
■
E very statement about the form al properties of' a constituent structure tree can be form ulated in terms o f the dom inance and precedence relations and the labeling function. For exam ple, one useful predicate on trees is that o f belonging to A node will be said to belong to the next highest S node that dominates it. Formally, the definition is as follows:
T
D
e f in it io n
r e e s
445
16 7 Given a tree T = {N , <3, D, P, L), node x belongs to node
y iff
(1) x ^ y (2) { y , x ) £ D
(3) {y, S ) £ L (4) ~ (3 w £ N ) ( ( w , S } £ L k w ^ y k w ^ x k ( y , w ) £ D k {w , x ) £ D ),
■ Parts 2 and 3 o f this definition specify that the node to which x belongs is labeled S and dominates x. Part 4 prohibits any S node from standing betw een x and y in the dom inance relation, and part 1 excludes the case o f an S node belonging to itself. To illustrate, let us consider the tree in Fig 16-4. T he n ode Prn belongs to the circled S node since this is the next highest S n ode dom inating it. Prn does not belong to the highest S (i.e., the ro o t) o f the tree because the circled S node is betw een the ro o t and P m in the dom inance relation W ith this definition we can easily define some other predicates.
Two
nodes are called clause mates iff neither dominates the other and b o th belong to the same node. In Fig 16-4 the nodes labeled John and him aie clause mates since neither dominates the other and b oth belong to the circled S node, Fred and him are not clause mates since they do not belon g to the same node, and P rn and him are not clause mates since P rn dominates him. If we let B ( x , y ) denote ‘x belongs to y ,’ we can state the definition o f clause mates as follows: D
e f in it io n
16.8 Given a tree T = { N , Q , D , P, L), nodes x and y are clause
mates i f f (x, y) 0 D k ( y , x ) 0 D k (3 z £ N ) ( ( x , z ) £ B k ( y , z ) £ B ,
■
A node x is said to command a node y iff neither dominates the other and x belongs to a node z that dominates y (Langacker, 1969), In Fig, 16-4 the n ode labeled Fred com m ands the n ode labeled him since neither dominates the other and Fred belongs to the root node S, which also dom inates him The n ode him does not com m and Fred, however, since the n ode to which him belongs— the circled S n ode— does not dom inate Fred. N ote, further, that John com m ands him and vice versa. Formally, the definition is as follows
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16
NP
N
Fred
Figure 16-4: Tree illustrating the definitions o f ‘belonging t o ’ and ‘com m a n d ’
D
e f in it io n
16.9 Given a tree T = { N , Q , D , P , L ), node x com m ands node
y iff ( x , y ) $ D & (y, x) $ D k. (3 z £ N ) ( ( x , z ) £ B k ( z , y ) £ D).
■
Problem: P rove that tw o nodes are clause mates iff each com m ands the other,
16.4
Gram m ars and trees
As we have said, if a grammar has only rules o f the form A —> ip, where A is a nonterm inal sym bol, there is a natural way to associate applications o f such iules with the generation o f a tree.
For exam ple, if the grammar
contains the rule A —» a B c, we can associate this with the (sub)tree in Fig. 16-5. in which A im m ediately dominates a ,B , and c, and the latter three elements stand in the precedence relation in the order given. Further, if the grammar
G
rammars
and
447
trees
A
a
B
c
Figure 16-5.
Figure 16-6
also contains the rule B —> ba, we can apply this rule at the node labelled B in the preceding tree to produ ce the tree shown in Fig. 16-6 Let us define the yield o f a tree as the string form ed b y its leaves ordered according to the precedence relation,. The yield o f the tree in Fig. 16-6, for exam ple, is abac; that o f Fig, 16-5 is aBc. W e can now say: e f i n i t i o n 16 10 A gram m ar (having all rules o f the form A ^ ip) generates a tree iff all the following hold:
D
(i) the root is labelled with the initial sym bol o f the grammar (ii) the yield is a string o f terminal symbols A (iii) fo r each subtree o f the form
a i .... a n
in the tree, where A immedi-
a tely dominates a i ....a n , there is a rule in the grammar A —» or
, , a n.
■
C hapter
448
16
Figure 16-7
Thus the grammar G — { { a , b } , { S , A , B } , S , R ) where (S -^ A B B - » Bb ) R = < A -*a A b B -+ b I
J
generates trees such as those in Fig. 16-7. W e can further say that a string is generated b y such a grammar iff it is the yield o f some tree which is gen erated, T he language generated is, as usual, the set o f all strings generated. For giam m ars in which there is only a single sym bol on the left side o f each rule, this definition and the earlier definition o f generation o f a string turn out to be equivalent: a string is generated (b y the earlier definition) iff it is the yield o f some generated tree. Problem: W hat language is generated b y the above grammar? Such grammars have interested linguists precisely because o f the possi
G
rammars
and
trees
449
bility o f specifying a constituent structure tree for each string generated In attem pting to write such grammars for natural languages, however, linguists have n oted that often such rules are not universally applicable but m ay be allowed only in certain contexts. For example, a rule rewriting D et(erm inei) as many m ight be applied only if the follow ing noun were a plural form. Such considerations led to the investigation o f form al gram m ar rules o f the form A —► , where the “ / ” is read “in the con text” , and where marks the p osition o f the A, The interpretation o f such a rule is that the sym bol A can be replaced b y the string ip in a derivation only when the string a im m ediately precedes A and the string /? im m ediately follows A T he context specifications are not necessarily exhaustive: additional sym bols m ay occu r to the left o f the a and to the right o f /?, For example, if the rule were A —> a B c / C _ D c , then the string B E C A D c b A could be rewritten as B E C a B c D c b A
Such rules are called context sensitive in contrast to rules o f the form A —> ip, which are called context free A context free rule, thus, is a context sensitive rule in which the context is null
A context sensitive rule A —» ip/a_fi can also be written as aA(3 —» aipfi in con form ity with the schem a for grammar rules generally. So long as we regard these grammars as string rewriting systems the notations are interchangeable: in either case we m ay replace A by ip when we find the substring aAfl, However, if we want to think o f context sensitive rules as generating trees, the tw o representations m ay not be equivalent.
For
exam ple, the rule C A B D —►C A a B D could be construed either as A —> A a j C _ B D or as B —> a B / C A _ D , and the associated trees w ould obviously differ depending on whether an A node or a B node was expanded.
A nother p roblem which arises is how the context restriction is to be satisfied by the tree.
I f we think o f the rules as specifying how one tree
is to be converted in to the next in a derivation, then does a rule such as A —» a B c / C _ D mean that the C and D must be leaves im m ediately to the left and right, respectively, o f A when the rule is applied, or is it sufficient that the C immediately precede the A and the D immediately follow, w ithout necessarily being leaves along with A^ Under the latter interpretation, the follow ing derivational step would be allowed, but b y the form er it would not
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16
Note also that in the definition o f tree derivation b y means o f context free rules in D e f 16-10 above, we essentially thought o f the trees being somehow given in advance and then checked for well-formedness b y the grammar rules. That is, the rules served as so-called “ node adm issibility conditions” rather than as directions for converting one tree into another
In the context free
case, the tw o points o f view are equivalent, but this is not the case for context sensitive rules. For exam ple, the grammar
(1 6 -6 )
S -► A B A —» a/_b B -► b/a_
will generate the tree
Figure 16-8
if the rules are interpreted as node adm issibility conditions but not if they are interpreted as tree generating rules (the p rob lem being that the A cannot be rewritten until the B has, and vice versa.
T
16.5
he
C h o m sk y H ie r a r c h y
451
The Chom sky Hierarchy
B y p u tting increasingly stringent restrictions on the allowed form s o f rules we can establish a series o f grammars o f decreasing generative power. M any such series are im aginable, but the one which has received the m ost attention is due to C hom sky and has com e to be know n as the Chom sky Hierarchy A t the top are the m ost general grammars o f the sort we defined above in Section 16.2, There are n o restrictions on the form o f the rules except that the left side must contain at least one non-term inal sym bol (A ctually, even this restriction could be eliminated in favor o f one which says sim ply that the left side cannot b e the em pty string. The form ulation we have chosen is essentially a technical convenience)
Chom sky dubbed such grammars ‘ T ype
0 ,’ and they are also sometimes called unrestricted rewriting systems (urs) The succeeding three types are as follows: T y p e 1: each rule is o f the form a A fl —> a i p f } , where ip
e.
T y p e 2: each rule is o f the form A —> ip. T y p e 3: each rule is o f the form A —» x B or A —> x In the above a , (3, and ip are arbitrary strings (possibly em pty unless oth erwise specified) over the union o f the term inal and non-term inal alphabets; A and B are non-term inals, and x is a string o f term inal sym bols T y p e 1 grammars are also called context sensitive; an equivalent form u lation is to say that each rule is o f the form ip —* u , where u is at least as long as ip (i e ., the rules are “non-shrinking” ).
T ype 2 grammars are
called context free, and T ype 3 grammars are called regular or right linear for reasons which will becom e apparent in the next section, N ote that these classes o f grammars do not form a strict hierarchy in the sense that each typ e is a subclass o f the one with the next lower number. E very T ype 1 grammar is also a T ype 0 grammar, but because rules o f the form A —►e are allowed in T ype 2 grammars, these are not properly con tained in T yp e 1, T yp e 3 grammars, however, are properly contained in the T yp e 2 grammars, It is nonetheless apparent, technical details concerning the em pty string aside, that the hierarchy represents a series o f generally increasing restrictions on the allowed form o f rules. T he question then arises whether the languages generated b y such gram mars stand in an analogous relationship. W e say that a language is o f T ype
Chapter
4 52
16
n (n = 0, 1, 2, oi 3) iff it is generated b y some grammar o f T yp e n. For exam ple, we saw in Section 16.2 that I x contains equal numbers o f a ’s and b’s} is o f Typ e 0 inasmuch as it is generated b y the gram mar given in 16-4 But one m ight wonder whether it could also be generated b y a grammar o f some other type— say o f T yp e 2 This is indeed the case; this language is generated b y the follow ing T yp e 2 grammar. (1 6 -7 )
G = { { a , b } , { S , A , B } , S , R ) where S -+ e S -► aB S -+ b A
A —» a A - » aS A bAA
B -► 6
B —> a B B
B -+ b S This fact im m ediately establishes this language as T yp e 0 also, since every T yp e 2 grammar is perforce a T yp e 0 grammar. (It does not at the same tim e establish it as a T yp e 1 language since the given grammar is not T y p e 1, because o f the rule S —■> e In fact, this language could not be T yp e 1 since T y p e 1 languages can never contain e„) Is this language also T yp e 3? It turns out that it is n ot, but to prove this is not a simple m atter. One must show somehow that no T y p e 3 grammar, however elaborate, can generate this language. W e will consider techniques for provin g such results in later sections, N ote that if one has tw o classes o f grammars Gi and G j such that G ; is prop erly contained in G j, it does not necessarily follow that the correspond ing classes o f languages stand in the proper subset relation. Because every T y p e i grammar is also a T y p e i + 1 grammar it does follow that every T ype i language is also a T yp e i + 1 language, i e., Li C L{+\, But it m ight also be the case that every T y p e i + 1 language happens to have some T yp e i gram mar which generates it. In such a case Li is a subset o f Li+i but not a proper subset. A m on g the earliest results achieved in the study o f form al gr ammars and languages were proofs that the inclusions among the languages o f the C hom sky hierarchy are in fact proper inclusions, Specifically, (i) the T yp e 3 languages are properly included in the T y p e 2 languages; (ii) the T yp e 2 languages not containing the em pty string are properly included in the T yp e 1 languages; (iii) the T yp e 1 languages are p roperly included in the T y p e 0 languages.
L anguages
and
automata
453
Some o f the proofs will be sketched in the follow ing chapters
16.6
Languages and a u to m a ta
As we m entioned at the beginning o f this section, languages can also be characterized by abstract com puting devices called automata. Ultim ately we will define a hierarchy o f autom ata and establish correspondences between them and the grammars o f the Chom sky Hierarchy. This gives us yet another point o f view from which to examine the notion o f ‘com plexity o f a language’ which we hope eventually to put to use in characterizing natural language. Before turning to the detailed study o f the various classes o f autom ata, it w ould be well to make a few general remarks about these devices. A n autom aton is an idealized abstract com puting m achine— that is, it is a m athem atical ob ject rather than a physical one A n autom aton is charac terized b y the manner in which it performs com putations: for any autom aton there is a class o f inputs to which it reacts, and a class o f outputs which it produces, the relation betw een these being determ ined b y the structure, or internal organization o f the autom aton
W e will consider only autom ata
whose inputs and outputs are discrete (e.g., strings over an alphabet) rather than continuous (e ,g , readings on a dial), and we will not deal with autom ata whose behavior is probabilistic Central to the notion o f the structure o f an autom aton is the concept o f a state.
A state o f an autom aton is analogous to the arrangement o f
bits in the m em ory banks and registers o f an actual com puter, but since we are abstracting away from physical realizations here, we can think o f a state as a characteristic o f an autom aton which in general changes during the course o f a com putation and which serves to determine the relationship betw een inputs and outputs. We will consider only autom ata which have a finite num ber o f states (c f a com puter whose internal hardware at any given m om ent can be in only one o f a finite number o f different arrangements o f l ’s and 0 ’s„) A n autom aton m ay also have a memory. For the simplest autom ata, the m em ory consists sim ply o f the states themselves. M ore pow erful autom ata m ay be outfitted with additional devices, generally “ tapes” on which the m achine can read and write sym bols and do “ scratch w ork ,” Since the amount o f m em ory available on such tapes is p oten tially unlim ited, these machines can in effect overcom e the limitations inherent in having only a
Chapter
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finite num ber o f states. W e will see that the m ost pow erful autom ata, Turing machines, are capable in principle o f perform ing any com pu tation for which an explicit set o f instructions can be given A utom ata m ay be regarded as devices for com puting functions, i.e., for pairing inputs with outputs, but we will norm ally view them as acceptors, i.e., devices which, when given an input, either accept or reject it after some finite amount o f com putation. In particular, if the input is a string over some alphabet A , then an autom aton can be thought o f as the acceptor o f some language over A and the rejector o f its com plem ent. As we will see, it is also possible to regard autom ata as generators o f strings and languages in a manner similar to grammars
Chapter 17
Finite Automata, Regular Languages and Type 3 Grammars 17.1
F in ite a u t o m a t a
A finite autom aton (fa), or finite state autom aton (fsa), is an abstract co m puting device that receives a string o f sym bols as input, reads this string one sym bol at a tim e from left to right, and after reading the last sym bol halts and signifies either acceptance oi rejection o f the input. A t any point in its com pu tation a fa is in one o f a finite num ber o f states. The com putations o f a fa are directed by a “program ,” which is a finite set o f instructions for changing from state to state as the autom aton reads input symbols. A com pu tation always begins in a designated state, the initial state. There is also a specified set o f final states; if the fa ends up in one o f these after reading the input, it is accepted; otherwise, it is rejected. It m ay help to visualize a finite autom aton as com posed o f (1) a control b o x , which at any p oin t in the com putation can be in one o f the allowed internal states, and ( 2 ) a reading head, which scans a single sym bol o f the input.
In Fig, 17-1 we have represented a fa in its initial state, qo, at the
beginning o f its com pu tation o f the input string abaab. Let us suppose further that the set o f states for this fa is { ? o , ? i } and that the set o f final states is { 51 } (W e could have m ade the initial state a final state also, but we have not chosen to do so here,) W e specify the program 455
C
456
a I)
hapter
17
& & b
Jo]
Figure 17-1
for this fa as a set o f triples o f the form ( q i , x , q j ) , where qi and qj are states and x is a sym bol o f the alphabet— heie, {a , 6}. Instructions are interpreted in the follow ing way: when the fa is in state
( ? oj
go)
(?o > M i)
(?i> a .? i ) (? i,M o )
Thus, from the initial situation shown in Fig, 17-1 the fa would fir st execute the instruction ( q0, a, qo) and find itself in the follow ing situation: (1 7 -2 )
Now the instruction ( 50 : b, Si) is applied to produ ce: (1 7 -3 )
and so on. Y ou should now b e able to verify that after reading the final sym bol 6, the fa is in state qo, and since this is not a final state, the input
F in it e
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457
is rejected. It should also be easy to determine that the input ab would be accepted, while aa is rejected. Problem: Describe the set o f all tapes accepted b y this fa W h at w ould happen if the fa o f our example were given the em pty string as input? In such a case, the input tape has no sym bols on it, and so no instructions can be applied— there being nothing to read. Thus, the initial situation is identical to the final situation, and since the initial state, qo, is not a final state, this input is rejected. N ote that to say that a machine accepts the em pty string as part o f its language is far different from saying that the language accepted is em p ty The latter means that it accepts neither the em pty string nor any other string A n autom aton w ith no final states w ould, for exam ple, accept n o strings and thus would be said to accept the em pty language. One m ight wonder how a finite autom aton w ould behave if there were no instruction applicable at a particular point or if there were more than one instruction which could be applied. Such questions will arise with the socalled non-deterministic autom ata, which we consider below For now, we will be concerned only with deterministic fa ’s, in which there is one and only one instruction for each com bination o f state and scanned symbol. A s the name suggests, the behavior o f such an autom aton is com pletely determined, given the input tape and the initial state.
1 7 .1 .1
S ta te diagram s o f finite au to m a ta
A convenient representation for a fa, called a state diagram, can be con structed in the follow ing way. Each state is represented b y a circle labelled with the name o f the state For each instruction ( & , x, qj) an arrow is drawn from the 5; circle to the qj circle and labelled with sym bol x. Final states are enclosed by an additional circle, and the initial state is marked b y a caret T he state diagram for our exam ple fa is shown in Fig, 17-2. W ith such a diagram it is easy to trace the steps o f a com putation like that for abaab in the exam ple above The fa starts in state qo and returns to qo reading an a. The 6 takes it to state 51 ; the next tw o a ’s leave it in ? i; and the final 6 returns it to 50 Since qo is non-final, the string is not accepted. It is also somewhat easier to see from the state diagram than from the list o f instructions in (17-1) that this fa accepts exactly the strings over the alphabet {a , 6} containing an o d d number o f 6’s
458
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a
Figure 17-2,
1 7 .1 .2 D
Form al definition o f determ inistic finite a u to m a ta
e f in it io n
17 1 A determ inistic finite autom aton (dfa) M is a quintuple
( K , 2 , 6 , q 0 , F ) , where K is a finite set, the set o f states S is a finite set, the alphabet qo £ K , the initial state F C K , the set o f final states S is a function from K x Y l in to K , the transition fu n ction fo r next-state function ).
■ N ote that the p rop erty o f determ inism is expressed in this definition b y the fact that 8 is a function; that is, foi all q £ K and all cr £ S , 6 (q,
F in it e
head, respectively
459
automata
In this notation, the sym bol being scanned is the left
m ost sym bol, if any, o f y. Thus, the diagram in (17-3) would be represented as (ab, 5 i, aab), and the sequence o f situations in the com putation o f abaab would be as follows: (1 7 -4 )
( e , q 0 ,abaab) h ( a ,q 0,baab) (a ba a ,qi,b) h (abaab, qo, e)
b
( a b ,qi,aab)
h
( aba,qx,ab)
h
Here we have used the sym bol h (the ‘turnstile’) to indicate that one situation leads to another by a single m ove o f the autom aton, W e will define this form ally below, Note that at the beginning o f the com putation the string to the left o f the reading head is empty, and likewise for the string to the right at the end o f the com putation, In form al terms, a situation o f a dfa is defined as follows: e f i n i t i o n 17.2 Given a, dfa M = { K , Y , S , q o , F ) , a situation o f M is a triple ( x , q , y ) , where q £ K and x , y £ ■
D
This definition allows situations for a given M which are not actually at tainable in the course o f any com putation by M . For example, (aa, qlt abb) would be a situation o f our example dfa, but it is not a situation which can be reached from the initial situation (e, qo, aaabb) b y the given transition fu n ction o f the automaton. It is convenient, nonetheless, to define the notion o f situation in this overly broad way and to focus on attainable situations in our definition o f acceptance. Let us define, for a given dfa M , a binary relation on situations which we will call produces-in-one-move. Situation A produ ces situation B in one m ove just in case b y applying one instruction in 6 to A we produ ce situation B. A n y tw o adjacent situations in (17-4) would stand in this relation, for example. Formally, we have: D e f i n i t i o n 17 3
Given a dfa M = { K , £ , 8 , qo, F ) , a situation (x , q, y)
produces situation ( x 1, q', y') in one move iff ( 1 ) there is a cr £ Yl such that
y = cry' and x' = xcr (i.e., the reading head moves right b y one symbol), and ( 2 ) 5(q,cr) = q' (i.e,, the appropriate state change occurs on reading cr). Problem: In general, is the produces-in-one-m ove relation reflexive? m etric? transitive?
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A s n oted above, we indicate b y the turnstile that two situations stand in this relation; thus ( x , q , y ) H (x ' ,q ' ,y ' ) Once again, this definition is permissive in that it allows pairs o f situa tions to stand in the produces-in-one-m ove relation whether or not either is attainable from some initial situation in the course o f a com pu tation
For
exam ple, in the dfa above, ( a a ,qi,abb ) 1- (a a a ,q i,b b ), despite the fact that neither situation could arise from (e, qo, aaabb). As a final step before giving a form al definition o f acceptance, we extend the previously defined relation to a new binary relation: “produces in zero or m ore steps,” W e say that a situation A produces situation B in zero or more steps (or sim ply A produces B ) iff there is a sequence o f situations So I- Si H , . . I- Sit such that A = So and B = Sk (k > 0 ) (If k ~ 0 , there is only one situation in the sequence, and A = B\ thus, every situation p roduces itself in zero or more moves ) This relation is reflexive (as we have just seen) and transitive; in fact, it is in form al terms the reflexive, transitive closure o f the produces-in-one-m ove relation. This is just the produces-inone-m ove relation (for a given dfa) with enough pairs added to it to make it a reflexive and transitive relation. W e will denote this relation b y h ( ‘turnstile star ’ ) W e m ay also add a subscript M to this or to the turnstile if necessary to emphasize the fact that the relation is defined w ith respect to a particular autom aton M ; thus, 1^ or (1 7 -5 )
Referring again to 17-4, we see
(a, ?o> baab) ^ (aba,qi,ab) and (ab, q\, aab) ^ (ab,qi, aab)
but neither o f these would be true if 1- were replaced b y h A ccepta n ce o f a string by dfa is now easy to define form ally: D
e f in it io n
17 4
Given a dfa M = ( K , h , 8,qo, F ) and a string x £ £ * , M
accepts x iff there is a q £ F such that (e, qo , x ) Ijf ( x , q , e ) ,
■
A nd finally: D e f i n i t i o n 17,5 Given a dfa M = ( K , S , 8 , qo, F ) , t i e language accepted b y M , denoted L ( M ) , is the set o f all strings accepted b y M ■
1 7 .1 .3
N on -d e te rm in istic finite a u to m a ta
W e now consider what happens if we relax the requirement that the next m ove o f a fa always be uniquely determ ined. Departures from determinism can occu r in tw o ways:
F
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461
(i) for a given state sym bol pair, there m ay be more than one next state (ii) for a given state sym bol pair, there m ay no next state at all There is an additional generalization from the determ inistic case which, while it is strictly speaking not a departure from determinism, is often included in the definition o f non-determ inistic fa ’s: (iii) transitions o f the form (& , w, qj) are allowed, where w £ £ * ; i.e., the fa can read a string o f sym bols in one m ove; and in particular, (iv ) transitions o f the form (qi, e, qj) are allowed; i.e., the fa can change state w ithout m oving the reading head. A n exam ple o f a non-determ inistic fa which illustrates all four o f these con ditions is shown in Fig
17-3.
The behavior o f a non-determ inistic fa is defined as follow s: an input tape is accepted iff there is som e path through the state diagram which begins in the initial state, reads the entire input, and ends in a final state. In the fa o f Fig. 17-3, for exam ple, abb is accepted b y virtue o f the path from ?o (reading ab) to qi and then to qi (reading 6). The fact that there is also a path reading abb which ends in ?i is irrelevant; only the existence o f at least one accepting p ath is required. On the other hand, ba is not accepted since there is no path through the state diagram which succeeds in reading the entire string. Likewise, aba is not accepted. N ote, however, that the string a is accepted b y the path leading from qo to q2 (reading no in pu t) and then again to qi (reading an a).
C h ap ter
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1 7 .1 .4
17
F orm al definition o f n on -d eterm in istic finite a u to m a ta
Formally, a non-determ inistic fa is identical to a dfa except that the transi tion function becom es a relation D e f i n i t i o n 17 6
A non-deterministic finite automaton (nfa) M is a quin tuple { K , E , A , q o , F ) , where K , S , qo, and F , are as for a dfa, and A , the
transition relation, is a finite subset o f K X £ * X K
(i.e., a relation from
K X £ * into K ) .
a
The fact that A is a relation, but not necessarily a function, allows for conditions (i) and (ii) above. The fact that it is a relation from K X S ’” rather than from K X S allows for condition (iii) and its special case (iv). Because o f the infmiteness o f £ * , a relation from K X £ * to K is itself potentially infinite; we stipulate that it must be a finite subset o f K x X K in order to retain the notion o f a finite machine; i.e., an autom aton with a finite number o f states and a finite number o f instructions in its program . N ote that by this definition d fa’s are a p ro p e r subclass o f n fa’s The definitions o f situation, produces-in-one-m ove, etc.
are similar to
those for d fa ’s. D e f i n i t i o n 17.7 A situation is any tripie in
£ * x K xE*. ( x , q , y ) \ - (x ',q',y')
is true iff there exists a string z £ £* such that x' = x z , y = z y ' , and (q ,z ,q ')£ A .
■
D e f i n i t i o n 17,8 Produces, i.e., I- , is the reflexive, transitive closure o f the
\-
reiation, and, as before, an nfa M accepts a string x £ £ * iff(e,q o ,x )\ iI ( x , q , e )
for som e q £ F .
■
T he language accepted is, o f course, the set o f all strings accepted.
1 7 .1 .5
E quivalence o f determ inistic and n on -d eterm in istic fi nite a u to m a ta
One m ight expect that when fa’ s are allowed the extra degrees o f freedom inherent in non-determ inism , significantly m ore pow erful devices would be the result. Surprisingly, this is not the case, nfa’ s accept exactly the same class o f languages as dfa’s; oi in other w ords, for every nfa there is an
F
in it e
463
a u t o m a t a
a
equivalent dfa— equivalent in the sense that b o th accept exactly the same set o f strings. (T h e equivalence in the other direction is trivial, every dfa being a fortiori a nfa.) A n exam ple o f a dfa which is equivalent to the nfa in Fig, 17-3 is shown in Fig, 17-4. A dfa will typically have m ore states that an nfa to which it is equivalent. The dfa works b y essentially keeping track, in its states, o f the set o f states that the nfa could be in if it followed all possible paths sim ultaneously on a given input. There is in fact an algorithm for converting any nfa in to an equivalent dfa, but it is to o long to be included here. It can be found in C hom sky and Miller (1958), R abin and Scott (1959), H opcroft and Ullman (1979), and Lewis and P apadim itriou (1981). In view o f the equivalence o f d fa ’s and nfa’s, one m ight wonder why we bother to consider n fa’s at all. For one thing, n fa’s are generally easier to construct than d fa’s Thus, it m ight be simpler to show that a given language is o f the sort accepted b y a fa b y devising an nfa which accepts it. We will in fact make use o f this convenience in proving certain theorem s about fa ’s below, For another thing, determinism and non- determinism are notions which arise in connection with other classes o f autom ata to be considered later, and, as we will see, the tw o varieties are not always equivalent in these cases,
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17.2
17
Regular languages
W e will say that a language is a finite automaton language (fal) just in case there is some fa which accepts it. W e know, for exam ple, that { x £ {a , 6}* | x contains an od d number o f b's} is such a language b y virtue o f the fact that we exhibited an fa accepting this language in F ig
17-2 above. Consider,
however, the general problem of’ deciding whether a given language L is a fal or not. Suppose we try to construct a fa accepting L, but all our attempts result in failure. W e would not b e justified in concluding that L is not a fal, o f couse, since we m ight succeed in oui attempts with renewed persistence or perhaps a bit o f luck. It would b e useful if we had another way to characterize this class o f languages which does not depend on our ingenuity, or lack o f it, in constructing fa ’s. To that end, we define a class o f languages, called regular languages, which turn out to b e provably identical to the class o f fa l’s, Since these languages are defined (recursively) b y reference to operations on sets of strings rather than to acceptance b y autom ata, we have another interesting and poten tially useful approach to these languages. In the next section, we prove a theorem which is used prim arily to show that a given language is not an fal W e will need a prelim inary definition:
D e f i n i t i o n 17,9 Given two sets o f strings, A and B , the concatenation for set prod u ct j o f A and B , denoted A B fo r ju s t A B ) , is the set o f strings
{ x ^ y | x £ A and y £ B } .
■
For exam ple, (1 7 -6 )
if A ~ {a , 6} and B = {cc, d}, then A B — {a c c, ad, bcc, bd}
N ote that the concatenation o f tw o sets o f strings is itself a set o f strings, in contrast to the Cartesian prod u ct o f tw o sets (o f anything), which is a set o f ordered pairs. W e should also note that, according to the definition, if one o f the sets is empty, the concatenation is also empty. R ecall also that the n otation A* is used to denote the set o f all strings form ed over the alphabet A. This is a special case o f an operation called closure or K leen e star on a set o f strings: given a set o f strings A , the Kleene star or closure o f A , denoted A * , is the set form ed b y concatenating mem bers o f A together any num ber o f times (including zero) in any order and allowing repetitions For exam ple,
R
(1 7 -7 )
e g u l a r
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465
{a , 66}* = { e , a, 66, aa, abb, bba, 6666, aaa, aabb, abba,, .,} .
Note that our original notation treated the members o f the alphabet as strings o f length 1. We are now ready to give the definition o f the regular languages. D
e f in it io n
17 10
Given an alphabet 5L
1 . 0 is a regular language. 2. For any string x £ £ * , { x } is a regular language. 3 I f A and B are regular languages, so is A U B . 4. I f A and B are regular languages, so is A B . 5. I f A is a regular language, so is A " .
6 . N othing else is a regular language unless its being so follows from 1-5.
■ For exam ple, (1 7 -8 )
Let £ = {a, b,c}. Then since aab and cc axe mem bers o f £ * , {aa& } and { c c } are regular languages. So is the union o f these tw o sets, viz., {aab, c c }, and so is the concatenation o f the tw o, viz., { aabcc} . Likewise, {aa b}*, {c c }* , and {aab, cc}* , axe all regular languages, etc.
A nother way to state the definition is to say that the regular languages (over a given alphabet) are just those which can be obtained from the em pty language and the ‘u n it’ languages (those containing just one string) b y re peated application o f the operations o f union, concatenation, and Kleene stax. Thus, to show that a given language is in fact regular, we indicate how it can b e built up out o f em pty or unit languages b y these operations. For exam ple, the language { x £ {a , 6}* | x contains an odd number o f 6 ’s} is a regular language since an equivalent representation o f this language is { a } * ' {&} { a } * - ( { 6} {a } * { 6}"{a }* )* „ In writing such expressions, it is usual to render them less cum bersom e b y suppressing the braces around sets and the dots in concatenation; extra parentheses can also b e dispensed with in view o f the associativity o f union and concatenation. The previous expression in
466
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h a p t e r
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this pared-dow n notation would be: a'cbax(ba’‘ ba’‘ Y
Such expressions are
called regular expressions W e note also that the set o f all strings in {a , 6}* which contain exactly tw o or three 6’ s is a regular language since it can be represented (as a regular expression) as a^ba^ba* U a*ba"ba>cbax , or equiv alently as a* 6a’' 6a“ (e U ba~). Note, finally, that { e } is a regular language since it is equal to 0 “ H aving thus characterized the regular languages, we want to show that they are in fact identical to the finite autom aton languages h e o r e m 17 1 (K leen e) A set o f strings is a finite automaton language i f and only i f it is a regular language, m
T
W e will sketch the p r o o f o f one half' o f this theorem ; i.e , that every fal is a regular language, The p r o o f o f the converse is to o com plex to give here, but can b e found in works such as H opcroft and Ullman (1979) and Lewis and P apadim itriou (1981). First, we show that the em pty language and the unit languages (for a given S ) are fal’s, T he em pty language is accepted b y the one-state fa in Fig. 17-5(a), and foi each x in £ * , a fa o f the form show n in Fig, 17-5(b) accepts the language { z } , (N ote that these fa ’s are in general n on-deteim inistic )
?o
Figure 17-5
N ext we show that the fal’s are closed under the operations o f union, concate nation, and Kleene star. From this it will follow that the fal’s are included in the regular' languages. W e will indicate how , given any tw o fa’s accepting languages L\ and L 2, we can construct fa ’s accepting, respectively, L\ U L 2, L\L2, and L\, Suppose, for exam ple, we are given the follow ing fal’ s: Li = a 6*a; that is, all strings beginning and ending with an a with any number o f b’s betw een them,
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L 2 = all strings in {a , 6}* containing exactly tw o 6’s. These are fal’s since they are accepted b y the fa ’s in Figs. 17-6 and 17-7, respectively
To form a fa accepting the union o f L\ and L 2 we first relabel the states o f one o f the fa’s so that all have distinct names— let us suppose that we add prim es to the states o f M 2 , Now we introduce a new start state, q'J, and establish e-transitions, i.e., changes o f state reading the em pty string, from 5 q to the old start states, go and q'0 , Everything else, including the final states, remains the same. The resulting autom aton M 3 is shown in Fig, 17-8, Mz o f course is non-determ inistic. From its initial state it can go without reading any input to qo, from which point it acts like M i, or it can go to q 0 ' and then behave like M 2 . Given a string x which is in Li U L 2, there will be an accepting path in M 3 corresponding to one (or b o th ) o f these possibilities. If x is n ot in Li U L 2 , it is not accepted on any path in M3. It should not b e difficult to see that the m ethod o f construction is general and can b e applied to any tw o fa ’s. This is the basis o f the p r o o f that the
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fal’ s are closed under union, The dem onstration that the fal’s are closed under concatenation is sim ilar, except that the autom ata are hooked together “ in series” rather than “ in parallel,” To construct a fa accepting L\
L 2, we relabel the states o f
M 2, if necessary, to make them distinct from those o f M 1 and then run etransitions from all final states o f M\ to the initial state o f M 2. Final states o f M i now becom e non-final, but final states o f M 2 remain. T he result for our examples M i and M 2 above, would b e as shown in Figure 17-9,
Figure 17-9
R
M 3 accepts Li
e g u l a r
L 2 as follows
l a n g u a g e s
469
G iven x, it will be accepted just in case
x = w z , where w would be accepted b y M i (in state 52 ) and 2 would be accepted b y M 2 (going from state q '0 to state q'2 ) I f x is not in L\ L 2, there will be no factorization o f x in to w z such that w would be accepted b y M i and z accepted b y M 2 and thus n o accepting p ath through M 3 , Again, the m ethod is general and does not depend on the particular characteristics o f autom ata M i and M 2 Note, however, that if M i had m ore than one final state, we would have e-transitions from each o f them to the initial state o f M 2 , and all these states would becom e non-final in the resulting fa Finally, we want to show that the fal’ s are closed under the Kleene star operation; that is, we want to take an autom aton accepting L and convert it in to an autom aton accepting L “ . The strategy here is to establish etransitions from all the final states back to the initial state so that the new fa can “recy cle,” accepting an input string X1 X2 . . . x n just in case the original fa would have accepted x i and x 2 .
and x n . There is a slight difficulty,
however, in connection with the acceptance o f the em pty string, which is o f course a m em ber o f L * for any language L. One would naturally want to insure that e is accepted b y making the initial state a final state if it isn ’t already However, in certain cases this can lead to trouble Consider, for exam ple, the follow ing fa accepting the language a(b U baa)*b:
I f we were sim ply to add an e-transition from qi to qo so that the fa could “recycle” and make qo a final state, we would have:
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M ' accepts e, as required, but it also accepts aba, which is not in (a(b U baa)*b)* Rather, what we should do is add a new initial state to M , which will also b e a final state, and establish an e-transition from ? 2 back to the old initial state, qo-
R eturning now to the examples M i and M 2 above, we see that this m ethod o f construction would produce the fa ’ s in F ig. 17-13, accepting L\ and LJ, respectively. This com pletes our inform al dem onstration that every finite autom aton language is a regular language. As we have said, we will not attem pt to show the converse here. It relies essentially on a procedure for extracting from any given fa a representation o f the language it accepts. This representation can
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be shown to involve only the em pty and unit languages together with the operations o f union, concatenation, and Kleene star, and thus, that every fal is a regular language.
1 7 .2 .1
P u m p in g T h e o re m for fa l’ s
C onsider an infinite fal L B y definition, it is accepted b y some fa M , which, again b y definition, has a finite number o f states. B ut since L is infinite, there are strings in L which are as long as we please, and certainly L contains strings with more sym bols than the num ber o f states in M . Thus, since M accepts every string in L, there must be a loop in M — in particular, a loop which lies along a p ath from the initial state to some final state. In other w ords, in accepting a string longer than the num ber o f states o f M , M must enter some state m ore than once, and a path leading from such a state back to that state constitutes a loop along an accepting path. Let i b e the string o f sym bols which M reads on going from its initial state to the state at the beginning o f some loop (call it qi), Let y b e the string read b y M in going around the loop once, i e , from qi to the first
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re-entry to 5,-. Finally, let z be the string read on going from ?,■ to some final state. Thus, x y z is accepted b y M But now notice that any loop lying along an accepting p a th can b e tra versed any number o f times— zero or more— and the result will still b e an accepting path. Therefore, if M accepts x y z in the way indicated, it will also accept x z , x y y z , x y y y z , . . . , in fact all strings o f the form x y nz for n > 0 . Finally, we observe that if I, is an infinite language (so that there is no upper b ou nd on sentence length), there has to be some loop along an accepting p ath in M such that y ^ e Otherwise, M could not accept strings o f length greater than the number o f states o f M . These observations are summarized in the following theorem , know n as the Pum ping Theorem (for fal’ s) because the string y is said to be “p u m p ed ” , i.e ., repeated with each traversal o f the loop recognizing it. T
heorem
1 7 .2 I f L is a n in fin ite fal o v e r alphabet S , then there a re strings
x , y , z £ £* such that y ^ e and x y nz £ L for all n > 0.
■
As an example, consider the language (1 7 -9 )
{ s £ {a , 6}* |x contains an od d number o f 6 ’ s }
Since this is an infinite fal, the P um ping T heorem applies. H ence, there exist strings x, y, and z (y e) such that x y nz £ L for all n > 0 M any examples o f such strings could be found; to take just one, let x = e, y — 66, and z — ab. Then it is true that ab, bbab, bbbbab, bbbbbbab,. . .., are all in L\ that is ( 66)n a 6 £ L for all n > 0. For some choices o f x , y , and z, this will not be true, but that doesn’t m atter: the theorem guarantees on ly that at least one choice for z , y, and z exists such that the specified condition holds. The usefulness o f this theorem lies in its application to languages which are not fal’s. Suppose we have a language L which is infinite and for which we could somehow show that for no choice o f x , y, and z (y ^ e) whatsover is it the case that x y n z £ L for all n > 0. In such a situation we would be justified in concluding that L is not a fal. Here we are using the theorem in its contrapositive form. A s stated, it is a conditional: I f L is an infinite fal, then so-and-so. The contrapositive is: if n ot-so-and-so, then L is not an infinite fal. For example, consider the language (1 7 -1 0 )
L = { a nbn |n > 0 }
T
ype
3
grammars
and
fin ite
automaton
languages
473
and let us show that it is not a fal using the P um ping Theorem , I f L were a fal, there would b e some x , y, and z (y /
e) such that x y nz £ L for all
n > 0 W e show that no such x, y, and z exist. The string x y z would have to be in L , so what could y consist o f?
It
ca n ’t b e em pty, so it would have to consist o f (1) some number o f a ’s, or (2) some number o f 6’s, or (3) some number o f a ’s follow ed b y some number o f b’s. It is easy to see that (3) is im possible because any string that contains more than one repetition o f y, e.g., x y y z , will contain 6!s preceding a’s— the b’s at the end o f the first y and the a ’s at the beginning o f the next— which could not be a string in L , So such a choice o f y is not pum p able. W h at about case (1)? Here all the b’s are contained in the z part, and as y is p u m p ed , the num ber o f a ’s in the string will increase while the number of b’s remains constant
Thus, we will continually b e producing strings which
have more a ’s than b’s in them , which cannot be in the language anbn. Case (2) is par allel, but here the number o f b’s outstrips the number of a’s. These are the on ly logical possibilities for the choice o f y, and since none meet the con dition laid down in the Pum ping T heorem , we conclude that no such x, y, z exist for this language
Conclusion: anbn is not a fal.
W h a t we have ju st done, then, is to show that there is no fa accepting anbn w ithout actually attem pting to construct such an automaton. One should also note that the Pum ping Theorem does not yield p a rtic ularly useful inform ation when one shows that the consequent o f the con ditional is true, I f we were to consider the language L = { x £ {a , 6}* | x contains equal numbers o f a’s and b’s in any order } and observe that for x = e, y = ab, and z = e it is the case that x y nz £ L for all n > 0, this w ould tell us nothing about whether L is a fal or not. G iven A —» B and B , we can conclude nothing about the truth or falsity o f A. A s it happens, the language just m entioned is not a fal, T he m oral is that the Pum ping T heorem m ay b e useful in showing that certain languages are not fa l’s but m ay not prove useful in other cases, even though the languages in question are in fact not fa l’s,
17.3
T ype 3 grammars and finite autom aton lan guages
W e now want to examine the fal’s from the poin t o f view o f grammars which generate them. R ecall our previous discussion o f form al grammars as con sisting o f Vt , the term inal vocabulary; Vn , the non-term inal vocabulary; S,
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the initial sym bol; and R , a set o f rules or productions. The various types o f grammars differ in the form o f productions they m ay contain, and here we want to focus our attention on T yp e 3 grammars, also called right linear grammars, in which each prod u ction is either o f the form A —> x B or A —» x, where A and B are in Vn and x is any string in V f. That is, each rule o f a T yp e 3 grammar has a single non-term inal on the left side, and on the right a string o f terminals (possibly em pty) follow ed b y at m ost one non-term inal sym bol A n exam ple is shown in (1 7 -1 1 ) (1 7 -1 1 )
G = { V t , V n , S , R ) , where V t = {a , b}', Vjv = { S , A , B } \ and ' s - » aA A -► aA A - » bbB B -► bB B -* b
A n exam ple o f a derivation b y this grammar is shown in ( 1 7 - 1 2 ): (1 7 -1 2 )
S => aA => aaA => aabbB => aabbbB => aabbbb
The phrase-structure tree associated with this derivation is shown in Pig, (1 7 - 1 4 ),
It is evident from this tree w hy the T ype 3 grammars are also called right linear: the non-term inal sym bols form a single linear sequence down the right o f the tree. (There is also a class o f grammars called left linear in which every rule is o f the form A —» B x or A —> x. The m ore general class o f linear grammars has every rule o f the form A —» x B y or A —» x, i.e., the right side o f each rule has at most one non-term ini sym bol but it need not to be left-m ost or right-m ost in the string. W h at we will say here about right-linear grammars could equally well b e form ulated in terms o f left-linear grammars— they are equivalent in generative capacity. Linear grammars, however, generate a larger class o f languages.) N ote that in a derivation b y a right-linear grammar there is exactly one non-term inal sym bol at the right end o f each string until the last one, at which point a rule o f the form A —» x is applied and the derivation terminates.
This observation suggests an analogy to finite autom ata: the
one non-term inal sym bol at the right side o f a string is like the state o f a fa
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in that the future course o f the derivation or com putation can depend on ly on the identity o f that state or sym bol and the given produ ction s o f the device in question. In particular, the past history, i.e., the string already read b y the fa or the string already generated b y the gram m ar, has no influence on the future course o f events. Let us then associate with each rule o f a T yp e 3 grammar o f the form A —» x B a transition in a (non-determ inistic) fa from state A to state B reading x. Further, let us associate each rule o f the form A —> x with a tranisition from state A reading x to a designated final state F , The initial state o f the autom aton will, o f course, be S. Carrying out this construction for the grammar in (17-11) gives: It should be reasonably easy to convince oneself that this fa accepts the same language as that generated b y the grammar in (17-11),
M oreover,
the m ethod is general and can be applied to any given T yp e 3 grammar to produ ce an equivalent fa. ‘E quivalent,’ o f course, means that the language accepted b y the fa is the same as the language generated b y the T yp e 3 grammar. On further consideration, we see that there is no reason w hy we should
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consider fa ’s on ly as acceptors o f languages, W e m ight as well think o f an fa as a language generator which starts in an initial state, moves from state to state em itting, or writing, symbols on an output tape, and halting at will in either a final or non-final state. I f the fa halts in a final state, the output string is said to be generated; otherwise, it is not genei ated. The state diagram o f a fa is the same whichever way we want to look at it, and the same language would be accepted b y a given fa acceptor as that generated b y the fa regarded as generator. From this point o f view, then, non-determ inistic fa ’s and T ype 3 grammars are virtually isom orphic representations. Problem: How could one construe a T yp e 3 gram m ar as accepting rather than generating strings, i.e., how would the rules and derivations be inter preted? W h at we have just argued (w ithout giving a form al p r o o f) is that every T yp e 3 language is a fal.
T o show the converse is equally easy, but the
construction proceeds in the opposite direction. G iven a fa, we use its in structions to create the rules o f a T ype 3 grammar in the follow ing way. For each transition (
Finally,
for each transition ( q { , x , q j ) where qj is a final state, we also add to the grammar the rule 5; —►a.
I f we carry out this construction on the fa in
(17-1), we get the following grammar: (1 7 -1 3 )
G = { VT , V N , q 0, R ), where Vt = {a,b}\ VN = { g o , ? i } ; and
f
qo -» a q0 qo~*bqi
?i -> b q0 1 qn -* b >
Qi~* a Qi
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A derivation o f the string aaba b y this grammar would b e as follows: (1 7 -1 4 )
qo => aqo => aaqo => aabqi => aaba
The reader m ay find it instructive to com pare this with an accepting com pu tation for aaba b y the fa in (17-1). T o be rigorous we would have to give a p r o o f that the m ethod o f construc tion ju st outlined does indeed p rodu ce a grammar equivalent to the original fa, but we will not do so here since the equivalence is intuitively evident The main poin t is that we now have three quite different char acterizations o f the same class o f languages: the languages accepted (or generated) by (determ inistic or non-determ inistic) finite autom ata, the regular languages, and the languages generated b y T y p e 3 grammars It is always useful to view m athem atical objects from different perspectives; our understanding is enhanced, and new m ethods o f p r o o f are opened up.
W e also com e to
realize that we are dealing with a coherent, and in some sense “ natural” m athem atical class.
1 7 .3 .1
P ro p erties o f regular languages
W e also gain in understanding o f m athem atical objects when we ascertain their behavior under various sorts o f operations. Since languages are sets, it is natural to ask how they behave when su bjected to certain set-theoretic manipulations.
W e have already seen, for exam ple, that the class o f fa l’s
( = regular languages = T y p e 3 languages) is closed under the operation o f union: i.e., the union o f any tw o fa l’s is also a fal. Similarly, we know that the fa l’s are closed under concatenation and Kleene star. W h at about the operations o f com plem entation and intersection? G iven a regular language L £ E “ , its com plem ent, i e., £ * — L, is also regular H ow can we show it? G iven our equivalent characterizations o f this class o f languages, we can make use o f whichever one is most convenient for what we want to prove. In this case, the desired result is easiest to show with finite automata. Let M be a determ inistic fa accepting L. Construct a new fa M ' from M b y interchanging final and non-final states. That is, M ' is identical to M except that all final states are now non-final and vice versa, M ' is also deterministic. Now M and M ' read any input string in the same way, in the
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sense that for a given string they go through the same state tran sition s. The on ly difference is that when M accepts (ends in a final state), M ' rejects (ends in a non-final state), and when M rejects, M ' accepts.
Thus, M ’
accepts the com plem ent o f L , which is therefore also a fal. For example, applying this construction to the fa o f (17-1), we obtain the follow ing deterministic fa which accepts { x £ {a , 6}* I ® contains an even num ber o f 6 ’s}
This is clearly the com plem ent o f the original language.
a
a
Problem; W h y w ouldn’t this procedure work in general if the original fa were not deterministic? It now follows that the regular languages are also closed under inter section, since for any sets X and Y , X fl Y — ( X ' U Y ’ ) ’ b y D eM organ ’s Laws In m ore detail, if X and Y are regular languages, then so are their com plem ents, X ' and Y ' , as we have ju st seen T he union o f the latter, X ' U Y ', is also regular, and the com plem ent o f the last set is also regular, i.e., ( X ' U Y ')', which is equal to X fl Y , Given then that the regular' languages are closed under union, intersec tion and com plem entation, and that the em pty language and E x (for any given alphabet S ) are regular, we have the result that the regular languages over any fixed alphabet form a B oolean algebra (see Ch, 12), This gives us some inform ation about the class o f regular languages but does not provide a com plete characterization since there are other sets o f languages which also form B oolean algebras (e.g., the set o f all languages over a given alphabet) which are not regular. Another frequently encountered problem concerning m athem atical o b jects is this: W hat questions about them can b e answered b y algorithm? Or to put it another way, do there exist procedures which can b e applied m echanically to any instance o f one o f these objects to yield an answer to a particular sort o f question in a finite time?
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A n exam ple o f this sort o f consideration as applied to finite autom ata would be:
Given a fa M and a string x, can it be determ ined whether
M accepts x or not?
The answer in this case is yes
One procedure for
answering this question would b e the follow ing. Given M , convert it to an equivalent deterministic fa M ' (if M is not already determ inistic,)
There
is an algorithm for perform ing this conversion, as we m entioned earlier I f x $ £ x, we know it cannot be accepted, since it contains sym bols not in the alphabet o f M '
I f x £ £ x, trace the com putation o f x b y M '
There
is a unique path through the state diagram o f M ' reading x which ends in some state 5* I f & is a final state, x is accepted; otherwise, x is rejected. Since the first part o f the algorithm will yield an equivalent deterministic fa in a finite amount o f time, the num ber o f states o f M being finite, and the second part will be accom plished in a finite number o f steps, viz , the num ber o f sym bols o f x , the procedure outlined is guaranteed to terminate after a finite time with the correct answer This is an algorithm ic solution to the so-called membership question for fa l’s, Another exam ple o f a question about fa ’s which has an algorithm ic so lution is the emptiness question, given an fa M , does it accept any strings at all?
One could proceed as follows
I f M is not determ inistic, make it
so. The result is M ' , which necessarily has a finite number o f states and a finite num ber o f transitions, M ' accepts at least one string ju st in case there is a path in its state diagram from the initial state to some final state Furtherm ore, if there is any accepting path in M ', there is an accepting path without loops in it (A n y path with loops can also b e traversed b y going through each loop zero times.) Since the number o f states and state con nec tions is finite, there are only a finite number o f loop -fiee paths to examine to determine whether any ends in a final state One could imagine system atic ways o f looking at the paths, but the essential part here is not the relative efficiency o f the process but only that it is a finite task, Thus, there is an algorithm ic solution to the emptiness question foi fa ’s. Similarly, one might ask o f a given fa, does it accept all strings in E~? This can be reduced to the previous question b y noting that £ x is the com plem ent o f 0, Given M , make M determ inistic if it isn ’t already, to produce M '„ Interchange final and non-final states o f M ' to p rod u ce M " . A pply the algorithm for answering the emptiness question to M " , M " accepts 0 iff M ' accepts
,
Problem: Given tw o fa’s, M i and M 2, show that there is an algorithm for answering the question, is L(M \) C L ( M 2) (H int: X C Y iff ( X fl Y ') = 0.)
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Is there an algorithm ic solution to the question o f whether tw o fa ’s accept the same language?
1 7 .3 .2
In adequ acy o f right-linear gram m ars for natural lan guages
Is English a regular language? W e can prove that it is not, using the Pum ping T heorem and the fact that regular languages are closed under intersection. W e assume that all the follow ing are gram m atical (although in some cases suiely incom prehensible) English sentences: (1 7 -1 5 )
( 1 ) The cat died, (2) The cat the dog chased died (3) The cat the dog the rat bit chased died. (4) The cat the dog the rat the elephant adm ired b it chased died, etc
These are all o f the form: (1 7 —16)
( the + com m on n ou n )n (transitive verb )n_1 intransitive verb
Let us take some finite set A o f com m on noun phrases o f the form the + com m on noun: (1 7 -1 7 )
A = {th e cat, the dog, the rat, the elephant, the k a n g a ro o ,, , }
Let us also choose a finite set B o f transitive verbs: (1 7 -1 8 )
B = {chased, bit, admired, ate, b e frie n d e d ,, , ,}
T hus, the strings illustrated in (17-15) are all o f the form : (1 7 -1 9 )
x ny n ~ 1 died, where x £ A and y £ B.
T he language L consisting o f all such strings, o f which the sentences in (1715) are m em bers, is easily shown not to be regular.
The p ro o f uses the
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P um ping Theorem and is very similar to the p r o o f that { a nbn |n > 0 } is not regular L is the result o f intersecting English (considered as a set o f strings) with the regular language A " B x{ d i e d } , Since the regular languages are closed under intersection, if English were regular, L w ould be also. Thus, English is not regular The dem onstration that English is not a finite autom aton language was one o f the first results to b e achieved in the nascent field o f m athem atical linguistics (C hom sky. 1956; 1957, Chapter 3), although Chom sky did not use this particular m ethod o f p roof, nor did he focus on the particular subset o f English exemplified in (17-15), R ather, he pointed out that English has a certain number o f constructions such as e i t h e r . , , or, i f . , then, and the agreement betw een the subject o f a sentence and the m ain verb, which can be thought o f as obligatorily paried correspondences or dependencies, (In sentences o f the form Either Si o r S2, we cannot substitute then or and for or, for exam ple, and similarly, we cannot replace then in I f Si then Si by and or or, etc , and produ ce a gram m atical sentence ) Further, these depen dencies can be found in gram m atical sentences nested one inside the other and to an arbitrary depth, Chom sky and M iller (1963) cite the following exam ple in which the dependencies aie indicated b y subscripts: (1 7-20)
A nyonei who feels that if2 so-m any 3 m ore4 studentss w hom we6 haven’tg actually adm itted ares sitting in on the course thaii4 ones we have that3 the ro o m had to be changed, then 2 probably auditors will have to b e excluded, isi likely to agree that the curriculum needs revision.
This structure o f nested dependencies finds an analog in the strings o f a language like { x x R |x £ {a , 6 }* } (recall that x R denotes the reversal o f the string x )
In strings of this language, the ith sym bol from the left must match
the i th sym bol from the right as indicated in the diagram o f Fig, 17-17: The language x x R can be shown not to be regular b y first intersecting it with the regular language aa'bbaa* to give { a nb2an |n > 1 } and showing that the latter is not regular b y means of' the P um ping Theorem, This result illustrates one sort o f p ractical result that can sometimes be obtained from the study o f form al grammars and languages.
A linguistic
theory proposes that the grammar' o f every natural language is drawn from some infinite class Q o f generative devices,
(This is ju st to say that the
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a b b a b b a b b a
u
Figure 17-17
linguistic theory specifies, as it should, the form that grammars m ay take.) Such a theory is supported, but not o f course proven true, b y each succesful prediction, i e„, whenever we are able to show that a grammar from Q is adequate for some natural language. On the other hand, repeated failure to find an adequate grammar in Q for, say, Swahili m ight raise doubts but would not suffice to prove the theory wrong Since Q is an infinite set, as it will be in all interesting cases, failure after a finite num ber o f attempts m ay reflect only ineptness or b ad luck. In certain cases, however, we m ay be able to dem onstrate conclusively that a linguistic theory is inadequate for one or more natural languages, as we ju st did for the theory o f right linear grammars vis-a-vis English, W e are then justified in concluding that the theory is inadequate in principle and can be rem oved from consideration as a viable proposal
Exercises 1. Consider the follow ing state diagram:
Figure 17-18,
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( a ) W h ich o f the follow ing strings are accepted b y the machine? (i) 01011 (ii) 0011 (iii) 11001101 (iv) 01010111111 ( b ) Describe as sim ply as possible the language accepted b y the au
tom aton. Consider the follow ing set o f transition rules: (5 0 .0 ) —» So (5 3 ,0 ) —» So
(50.1) —» S i
(53,1) -* S 4
(5 1 .0 ) - * S2
(5 4 ,0 ) —» So
( 51. 1 ) - 5 3
( 5 4, 1 ) - 5 b
(5 2 .0 ) —» So
(5 s ,0 )^ 5 o
(5 2 . 1) -* Si (5 5 , 1) —» S 5 Final states: Si, S5 . ( a ) Draw a state diagram for this autom aton ( b ) Describe the set o f input strings accepted b y the automaton.
( c ) Draw a state diagram for an autom aton which is equivalent to this one but which has four states. C onstruct state diagrams for finite autom ata which accept the follow ing languages using as few states as possible: ( a ) The set o f all strings containing a total o f n l ’s, where n is co n gruent to 1 (m odu lo 3) (i.e., the remainder when n is divided by 3 is 1). (b)
The set o f all strings containing a tota l of exactly two l ’s.
( c ) The set o f all strings which contain a b lock o f at least three con secutive l ’s, e.g., 010111001 but not 001111100 1101110111
0101011011 0011011000 1101100101
(N ote that once such a block occu rs, it is irrelevant what comes later.)
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( d ) T he set o f all strings which contain n o block o f m ore than one consecutive 0 nor any block o f m ore than one consecutive 1, e.g., 0 e
but not
0101101
11 11010
101
101010 0101 ( e ) T he set o f all strings which contain the substring 101 anywhere within them ( f ) * The set o f all strings in which the total number o f 0’s is congruent to the total num ber o f l ’s m odu lo 3 (see part ( a ) above) Consider non-determ inistic finite autom ata whose input alphabet is {the, old, man, men, is, are, here, and}. (a ) C onstruct a state diagram for an autom aton w hich accepts the follow ing language: {the man is here, the m en are here}. ( b ) D o the same for the follow ing language: {t h e man is here, the m en are here, the old man is here, the old m en are here, the old old man is here, the old old men are here,. . ( c ) Construct a state diagram for an autom aton which accepts all the sentences in (b ) plus all those form ed b y conjoining sentences with and, e.g., the old man is here and the old old m en are here and the men are here. ( a ) C onstruct a state diagram for an autom aton w hich accepts the term inal language o f the follow ing grammar . (T h e input alphabet is {a ,b , c }; it does not include S or C ) 5 -> SS S -^ b b S -> a Ca
C -> C c
S -*b C b
C -► c
S —> aa ( b ) Draw a diagr am for an autom aton whose language is that o f part ( a ) plus the em pty string Show h ow , given any finite autom aton, you can construct an equiva lent one which has n o transition arrows leading to the initial state
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A ll autom ata asked for in exercises 7 and 8 m ay be non-determ inistic, and o f course must be finite state. The input alphabet is to be { 0 , 1 }
7.
(a ) C onstruct an autom aton A which accepts any string which con tains n o l ’s. ( b ) C onstruct an autom aton B which accepts any string which con tains an o d d num ber o f l ’s, with any num ber (including zero) o f 0 ’s ( c ) C onstruct an autom aton C which accepts the union L ( A ) U L (B ). ( d ) C onstruct an autom aton D which accepts the com plem ent L (C )'. ( Caution: first find a deterministic equivalent o f C .) Describe L ( D ) in words
8.
( a ) C onstruct an autom aton A which accepts any string which con tains no block o f four or more consecutive l ’s ( b ) C onstruct an autom aton B which accepts any string which con tains no block o f three or m ore consecutive 0 ’s ( c ) C onstruct an autom aton C which accepts the intersection L ( A ) fl L (B ),
9. F ind a regular expression for the language of: ( a ) Exercise 4a (a b ove) ( b ) Exercise 4b. ( c ) Exercise 4c. ( d ) Exercise 3b. ( e ) Exercise 3e. 10. Draw a state diagram for a finite autom aton corresponding to the follow ing regular expressions. (a ) 010*1 ( b ) (010*1)* ( c ) (010*1)*1 ( d ) (010*1)*1(0 U 1)* (e ) 1(010*1)*
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1(010‘“1)*(0 U 1)“ (Hint: this one can be done by a simpler m a chine than any o f the others.
11. Construct T yp e 3 grammars that generate each o f the follow ing lan guages Assum e a fixed terminal vocabulary Vr = {«,& }. ( a ) Li = {a a ,a b ,b a ,b b }
(b)
L 2 = { x \x contains any num ber o f occurrences o f a and b in any or d e i}
( c ) Lz = { x contains exactly two occurrences o f a , n ot necessarily contiguous} ( d ) L 4 = { x | x contains exactly one occurrence o f a, or exactly one occurrence o f b, 01 b o th } ( e ) L 5 = { x \x contains an even num ber o f a ’s and an even number o f 6’s } (Zero counts as even ) ( f ) L q — Lz fl Lz
1 2 . C onstruct finite autom ata (non-determ inistic) accepting each o f the languages in Exercise 11.
Chapter 18
Pushdown Automata, Context Free Grammars and Languages 18.1
Pushdown autom ata
W e turn next to a class o f autom ata which are m ore powerful than the finite autom ata in the sense that they accept a larger class o f languages. These are the pushdow n autom ata (p d a ’s). A p d a is essentially a finite autom aton with an auxiliary tape on which it m ay read, write, and erase sym bols. This tape is organized as a stack or pushdown store similar in principle to the spring-loaded devices for holding plates seen in cafeterias. B oth w ork on the basis o f “last in, first o u t;” that is, the m ost recently added item is the first one to be removed. Items below the topm ost ones cannot be reached w ithout first rem oving items above them on the stack P d a ’s, like finite autom ata, lea d their input tapes from left to light and have a finite num ber o f internal states. There is a designated initial state, and a set o f final, or accepting, states. T he transitions o f a pda, however, allow the top sym bol o f the stack to be iea d and rem oved, added to, or left unchanged
We can represent these transitions schem atically as
( q i , a , A ) —* (qj,"f), where qi and qj are states, a is a sym bol of the input alphabet, A is a sym bol o f the stack alphabet (w hich need not be the same as the input alphabet), and 7 is a string o f stack sym bols 487
Such an instruction
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is interpreted as follows: when in state q;, reading a on the input tape, and reading A at the top o f the stack, go to state qj and replace A by the string
7
I f 7 were, for exam ple, the string B C , the A would be rem oved and
B C added to the stack (in the order B first, C n ext) so that C would now becom e the top sym bol on the stack
In case 7 is e, the em p ty string, the
net effect is to remove ( “p o p ” ) A from the stack. The sym bol next below A , if any, would then becom e the top sym bol I f 7 were, for exam ple, A B , the effect would be to add ( “push” ) a B on top o f the A. I f 7 were A , the transition would leave the stack unchanged W e also allow e to appear in the position o f A in the above schema. In this case the transition does not depend on the contents o f the stack since e can always be r ead at the top o f the stack whatever it m ay actually contain. N ote that the e here does not indicate that the stack must b y em pty If A = e and 7 = B , for example, the transition would push B on to whatever was already on the stack. T he stack is assumed to be em pty at the beginning o f a com putation w ith the p d a in its initial state and the reading head p osition ed over the left-m ost sym bol o f the input A n input tape is accepted if the com putation leads to a situation in which all three o f the follow ing are simultaneously true: (i) the entire input has been read (ii) the p d a is in a final state (iii) the stack is empty One could define acceptance by em pty stack or final state or, as we have done, b y both , and the resulting classes o f autom ata turn out to be equivalent This choice is convenient for our purposes. T he follow ing is an exam ple o f a pda which accepts the language {a nbn |n > 0 }: (1 8 -1 )
States:
K - {q a,q i }
Input alphabet: S = {a , b} Stack alphabet: T = { A } Initial state:
go
Final states:
F =
Transitions:
{<7o j <7i }
P
aabb _L
u sh d o w n
aabb
g o — * j_ _ J
aabb
go
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a u t o m a t a
aabb
%
A A
stack
aabb
?i - A A
I
- U
Figure 18-1
Fig
18-1 shows how this p d a would accept input aabb,
Since the entire input is read and the p d a halts in a final state with an em pty stack, this input is accepted. The leader should also be able to verify the follow ing statements about the behavior o f this pda: (i) ba is rejected: the p d a blocks in state go and fails to read the entire input (ii) aaabb is rejected: the p d a halts in state gi with A on the stack (iii) aabbb is rejected: the p d a fails to read the last b since there is n o A on the stack (iv) e is accepted: the com putation begins and ends in g0 with an em pty stack This machine works by using its stack as a counter for keeping track o f the num ber o f a ’s in the initial part o f the input string Once a b is found, it switches to state qi and begins p op p in g an A from the stack for each b encountered. Only if the num ber o f b’s equals the num ber o f a’s will the stack be em pty at the end (and the entire input string read). Problem: W h y is it necessary to go to a new state when the first b is en countered, i.e., w hy not stay in state go? H eie is an exam ple o f a p d a which accepts the language { x x R |x £ {a , 6 } ” } (1 8 -2 )
K = { g 0 , ?1} , E = {a, b}, T = { A , B } , Initial state = g0 , F = {g 0 , g i } f (go, a, e) —> (g0, A ) A = S (go, b, e) —> (g0, B ) [ ( g o , A)
( g i ,e )
(g0, b, B ) —> (g i, e) )
(g i, a, A ) —> (g i, e) (q !,b ,B )
( g i ,e ) } J
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This machine works by putting an image o f the left h a lf o f its input string on the stack (in the form o f capital letters), and then, after non-determ inistically “ guessing” that the m iddle o f the string has been reached, com paring each input sym bol in the right half against the top sym bol o f the stack
I f the
sym bols correspond, the stack sym bol is rem oved; if n ot, the m achine blocks. Since symbols com e o ff the stack in the reverse o f the order in which they went on, the stack will be em ptied ju st in case the right h a lf o f the input is the reversal or “m irror im age” o f the left h alf This p d a is non-determ inistic since there is m ore than one m ove available to the autom aton in certain situations. If, for exam ple, it has ju st read an a in state qo (and therefore put an A on the stack) and is now reading another a, it could execute either the first or the third instruction above T he former corresponds to the “pushing” m ode in which the im age o f the left h alf is being placed on to the stack; the latter corresponds to a decision that the m iddle o f the string has just been encountered and that it is tim e to switch from pushing to poppin g m ode. As with the finite autom ata, we say that a non-determ inistic p d a accepts if there exists at least one com putational p ath on which the input is accepted; an input is rejected if there is no such accepting path. B y this definition, the above m achine accepts the language { x x R |x £ {a , 6 }“ }, The pda in (18-1), by contrast, is said to be determ inistic, in that it has at m ost one m ove available to it for any situation. Unlike the finite automata, we do not insist that deterministic p d a ’s have one m ove available for every situation; rather, that in n o situation is more than one m ove allowed One can determine from inspection o f the instructions o f a p d a whether it is deterministic or properly non-determ inistic. A p d a is determ inistic iff there are no two distinct instructions ( q i , a , A ) —* ( q j , j ) and ( q i , a , A ') —* (?fc,fi) such that A = A' or A = e or A' = e , Thus, as w ith the finite automata, the deterministic machines are a proper subset o f the non-determ inistic. The question then naturally arises as to whether determ inistic and nondeterministic p d a ’s are equivalent, The answer is that they are n ot, although it is not a simple matter to give a p r o o f o f this fact. W e will sim ply note that this result agrees with oux intuition that no determ inistic p d a could be devised to accept { x x R |x £ {a , 6 }“ } , there being n o way in general for a p d a reading strictly left to right to tell with certainty when the center o f the input string has been reached In contrast, the language { x c x R |x £ {a , 6 }“ } in which the center o f the string is marked b y the c is easily acceptable by a deterministic pda.
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It is often a m atter o f some practical interest to be able to tell whether a non-determ inistic p d a language is also accepted b y some deterministic pda. This is so because many program m ing languages— languages used for writing instructions for computers to execute— belon g to the class o f nondeterm inistic p d a languages. W hen program s are com piled, i e ., translated in to sequences o f l ’s and 0 ’s for execution by the com puter, the com pilation process is carried out by what is in effect a p d a , and if this p d a can be made determ inistic, then the process can b e m ade more efficient b y avoiding back tracking or the pursuit o f alternative paths. M ost program m ing languages in current use are in fact deterministic p d a languages (or nearly so), but it is an unfortunate fact that there is no way to tell in general whether any arbitrarily given non-determ inistic pda language is also a deterministic p d a language. W e end this section by giving form al definitions o f p d a ’s and the related notions o f situation, acceptance, and so on. Note that in the follow ing we have generalized the n otion o f a transition o f a p d a to allow the possibility that a string o f input sym bols and a string o f stack sym bols can be read on a single move. This does not affect the pow er o f the autom ata and has the advantage o f bringing the definitions in to a form parallel to those for nondeterm inistic finite automata, A non-determ inistic finite autom aton will thus appear form ally as a non-determ inistic p d a which never makes use o f its stack. D
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A n o n - d e t e r m in is t ic p u s h d o w n a u t o m a t o n is a sextuple
{K , £ , r , A , s, F ) , where K is a Unite set o f states, £ is a Unite set (the input alphabet), T is a Unite set (the stack alphabet), s £ K is the initial state, F C K is the set o f final states, and A , the set o f transitions, is a Unite subset o f K x £ * x T " x K x T * . ■ D
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A situation o f a pda is a quadruple ( x , q , y , z ) where q £
K , x , y £ £ * , and z £ T*.
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The intended interpretation is that the p d a is in state q with x to the left o f the reading head, y to the right o f the reading head with the left-m ost sym bol o f y currently being scanned, and z is the contents o f the stack. e f i n i t i o n 18 3 Given a non-deterministic p d a M , we say that situation ( x , q , y , z ) produces ( x ' , q', y', z') in-one-m ove if f x' — x a , y = a y', z = j w ,
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18.4 Produces is the reflexive, transitive closure o f the produces-
in-one-m ove relation,
% These are denoted, as usual, b y 1^ and 1^, respectively,
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Given a p d a M , a string x E S* is accepted i f f ( e , s , x , e )
\ ~ M ( x , q , e , e ) for som e q £ F. The language accepted by M is the set o f all strings accepted. D
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18.6 A p d a is deterministic iff for no pair o f distinct transitions,
-*■ and (qk-,xk, y k) -> (
■ 18.2
Context free grammars and languages
Non-determ inistic p d a ’s accept exactly the languages generated b y context free (T y p e 2) grammars. Recall that in a context free gram m ar every rule is o f the form A —> tp, where A is a non-term inal sym bol and ip is any string, possibly empty, from the union o f the term inal and non-term inal alphabets. It follows from this definition that every right-linear grammar is also a context free grammar, and therefore that the regular languages are contained in the context free languages. This containm ent is proper since, as we have seen, { anbn |n > 0 } is n ot a regular language, but it can be generated by the simple context free grammar containing only the tw o rules S —> aSb and S —> e. T he p ro o f o f the equivalence o f context free languages and non-determ inistic p d a languages is too long and com plex to give here. To give something o f the flavor o f this proof, we will show an algorithm for1 constructing from any given context free grammar an equivalent non-determ inistic p d a (but we will not prove form ally that the constructed p d a is actually equivalent). For the construction in the reverse direction— from non-determ inistic p d a to equivalent context free grammar— we refer the reader to H opcroft and Ullman (1979) or Lewis and Papadim itriou (1981), which also contain references to the original sources Given a context free grammar G — (Vn ,V t ,S , R ), we construct an equiv alent non-determ inistic p d a M as follows. T he states o f M are go and qi, with go being the start state and gi being the on ly final state. T he input alphabet is Vt and the stack alphabet Vpr U V r
T he transitions o f M aie
constructed out o f the rules o f G in the follow ing way:
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(i) M contains the instruction ( g0, e , e ) —> (g i, S) (ii) For each rule o f the grammar A —> ip, M contains an instruction (q 1 } e , A )
(qi,rp).
(iii) For each sym bol a £ Vt , M contains an instruction (g j, a, a) —> (g i, e). As an exam ple, let us take G to be as follows: (1 8 -3 )
VN = { 5 } ; VT = {a , &}; R = { S ^ aSb, S -> e }
(This is the gram m ar we referred to above which generates { a nbn | n > 0 },) A ccordin g to the construction procedure ju st given, M will contain the follow ing: (1 8 -4 )
K = {g 0 !g i} ; £ = { a , b } ; T = {S , a, b}\ s = g0; F = { g i } ; (go, e, e ) —> (g i, S) ( ? i , e , 5 ) ->• (q i,a S b ) A =
( g i , e , 5 ) -»• (g i ,e ) (g i,a , a) -»• ( g i ,e )
. ( ? i , 6, 6)
(?i,e)
,
M accepts the input string aabb by the following com putation: (e, go, aabb, e) I- (e, gi, aabb, S) I- (e ,q i,a a b b ,a S b ) I- (a ,q i,a b b ,S b ) I- (a,qi,abb,aSbb) I(aa, qi, bb, Sbb) h (aa ,q i,b b,b b ) h (aab, q\,b, b) h (aabb, gi, e, e) M works by loading S on to its stack and then sim ulating a derivation there by m anipulations which correspond to the rewriting rules o f G, W hen a term inal sym bol appears at the top o f the stack, it is p op p ed o ff i f it m atches the sym bol being read on the input; otherwise the com putation blocks. W h en a non-term inal appears at the top o f the stack it is rewritten in a way licensed b y the rules o f G. Thus, M carries out what is in effect a left-m ost derivation (one in which the left-m ost non-term inal sym bol is rewritten at each step) according to the grammar. I f the derived terminal string m atches the input, the stack will be em ptied, the entire input read, and the string accepted. Note that p d a ’s constructed in this way will in general be non-determ in istic since there m ay be in the gram m ar more than one rule rewriting some non-term inal A.
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Pumping Theorem for cfl’s
There is a P um ping Theorem for the cfl’s which is similar in form to the P um ping T heorem for fal’s , It is useful prim arily in showing that a particular language is not context free. T he theorem makes use o f the fact that a derivation b y a cfg can be naturally associated with a paise tree (see Section 16 4 ), and the fact that the m axim um width o f any such parse tree is constrained by its height. Let us see what this means in more detail. Given a cfg G , there is some m axim um number o f sym bols on the right hand side o f any rule. Suppose, for exam ple, that no rule has a right hand side longer than 4 symbols.
This means that in one rule application, the
w idth o f the tree (the number o f sym bols in its yield) can have increased by at most 3. I f each o f the 4 sym bols just introduced should happen to be expanded into 4 sym bols, then b y these steps the single node could have grow n into 16 sym bols, but no m ore than 16 symbols. I f we define the height o f a tree to be the length o f the longest path in it extending continuously downward from the root, then what we have said is that (for a given gram m ar) the m axim um w idth o f a parse tree is bou nd ed by its height. More specifically, i f n is the m axim um length o f a right side o f the rules o f G , then the m axim um w idth o f a parse tree generated b y G o f height h is just nh. (It m ay in fact be considerably less than this, depending on the exact nature o f G , but we are interested here on ly in setting an upper b ou n d ) To phrase this another way, i f a parse tree for some gr ammar G has w idth greater than nm , where n is the m axim um length o f the right side o f rules o f G , we can be sure that the height o f the tree is greater than to. Let us now suppose that G has to non-term inal sym bols in its alphabet. If we find a parse tree generated by G o f w idth greater than n m, then there must be some continuously descending path in the tree o f length greater than to, and thus some non-term inal sym bol must appear at least tw ice along this path (there being on ly to symbols to choose from ).
Let us represent this
situation b y the diagram in Fig. 18-2, The repeated non-term inal is called A.
S dominates a term inal string
w, and so each non-term inal in the tree must also dom inate some terminal string. Let x be the terminal string dom inated b y the lower A , and v x y the term inal string dom inated b y the upper A. (x must be a substring o f this string since, b y hypothesis, the lower A is dom inated b y the u pper A .) Let u and 2 be term inal strings dom inated b y S to the left and right, respectively,
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Figure 18-2
o f v xy . This general situation must obtain whenever there is a repeated non-term inal along some path in a parse tree, But note that the lower subtree rooted b y A could have s too d in the place in the tree where the upper A -rooted tree stands. The rules o f G are, after all, context free, so if it is possible to rewrite A in one position ultim ately to yield x, the same is possible in any position in which A appears. Thus, the tree in Fig, 18-3 must also be generated b y G.
S
Figure 18-3.
Further, the lower A in Fig. 18-2 could have been rewritten as the upper
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one was, to produce the tree shown in Fig. 18-4.
Figure 18-4.
Since we know that the derivation can be term inated b y rewriting the lowest A , ultim ately to give x , it follows that the string u v v x y y z must also b e generated b y G. T he process just illustrated cou ld, o f course, be carried out any finite number o f times; thus, u v l x y 1z is generated for all i > 0. W e are now ready to state the P um ping T heorem for context free languages. 18,1 I f L is a n i n f i n i t e c o n t e x t f r e e l a n g u a g e , t h e n t h e r e is s o m e K s u c h t h a t a n y s t r i n g w in L l o n g e r t h a n K c a n b e f a c t o r e d i n t o
THEOREM c o n s ta n t
s u b s tr in g s w
= u v x y z su ch th a t v and y a re n o t b o th e m p ty and u v 'x y 'z € L
f o r a ll i > 0 .
■
N ote that if L is an infinite language then it is guaranteed to contain strings longer than any given constant K W hat is K 1 It is a num ber which depends on the grammar for L and which is big enough to ensure that any strings longer than K have derivation trees with a repeated non-term inal along some path. There is always such a K for any cfg since G contains a finite num ber o f non-term inal sym bols, and there is some m axim um to the degree o f branching allowed on the right sides o f rules.
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W h at about the stipulation that v and y are not b oth em pty? I f they were, we could get from A to A on a branch o f a derivation tree without generating any term inal sym bols either to the left or right. This situation cou ld arise, since rules o f the form A —* B , B —> C , C —* A are allowed in a cfg. However, not all the non-terminals in the gram m ar could appear only in rules o f this form or else the grammar could n ot generate an infinite language. T hat is, there must be som e non-term inal sym bol A which can be repeated along some path, i.e., A p- v A y , such that not b o th v and y are em pty, and .further, such a non-term inal must appear in the derivation o f any string longer than K . Note that like the Pum ping Theorem for fa l’s, this theorem is a con di tional but not a biconditional. G iven a language L, it is not particularly inform ative to find strings u , v , x , y , z such that u v lx y 1z E L for all i > 0. R ather, we use the theorem in its contrapositive form : i f there do not ex ist strings u , v , x , y , z such that, , „, then we can conclude that L is not an infinite cfl. Let us see how this can be done in the case o f the language L = { a n bn cn |n > 0 }. Suppose L were context free Since it is infinite, the P um ping T heorem m ust apply, and there would be some constant K such that any string in L longer than K — let us choose aK bK cK , for exam ple— would be factorable in to u v x y z such that v and y are not b oth em pty and the v and y are pum pable. We show that no such factorization can exist. First, v cannot consist o f b o th a ’s and V s, because when it is pum ped, it w ould produce strings containing 6’s before a ’s, which cannot be in L, Similarly, v cannot consist o f b o th 6’s and c’s, and the same argument applies to the other pum pable term, y. Therefore, the only possibilities remaining are for v to consist o f just a’s or ju st b’s or ju st c ’s. T hen no m atter how we choose y, the result o f pum ping v and y sim ultaneously gives strings n ot in L. Suppose, for exam ple, that v consists o f a’s and y o f b’s. Then on pum ping v and y the a’s and b’s increase but not the c ’s, and we get strings not in L. The other cases are similar. We conclude that there is n o choice o f u , v , x , y , z m eeting the conditions o f the P um pin g T heorem for this language; therefore, L is not context free.
18.4
Closure properties o f context free languages
G iven the class o f context free languages (identical to the non-determ inistic p d a languages), we want to investigate whether this class is closed under
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operations such as union, intersection, com plem entation, etc. W e will see that unlike the fal’s the cfl’s are not so conveniently and tidily closed under all these operations
Union: Given tw o cfg ’s G i = (V /v,, , S i , i i i ) and G 2 - {Vn 2, Vt 7, S2, R2), we form grammar G in the follow ing way If the non-term inals o f G i and G 2 are not disjoint sets, we make them so (b y appending primes to every sym bol o f G 2 , say). T he start sym bol of G we take to be S, and G contains, in addition to Ri and R 2, the rules S —> Si and S —> S 2 G is context free, and it generates L ( G i ) l l L ( G 2) since the start sym bol m ay be either rewritten as S i, whereupon G behaves like G i, or as S 2 , whereupon G behaves like G 2- A string x E (Vt\ U V t2T is generated b y G just in case it is generated b y G i 01 b y G 2 (or b o th ) Since this m ethod o f construction is general (and, as the reader will have noted, quite similar to that used in showing that the fal’s are closed under union; see Section 17.2), we conclude that the cfl’s are closed under union
Concatenation (or Set Product): The m ethod o f construction is similar to that for union except that instead o f the tw o rules m entioned there we add to G the single rule S —> S 1 S 2
Thus, G will generate all strings o f
the form x y such that x £ L (G 1) and y £ L ( G 2)
Further, G will generate
only such strings, and, again, since the m ethod o f construction is general, we conclude that the cfl’s are closed under concatenation,
Kleene Star: Given G = (V n , V t , S , R ), we construct G* as follows: The start sym bol o f G * is S ', and G contains, in addition to all the rules in R, the rules S' —> e and S' —> S 'S , G* generates all strings in (L ( G ))* since by application of the rules rewriting S ', G “ produces strings Sn for all n > 0. Each such S can be rewritten to produ ce a string in L (G ) , and e is produced b y the rule S' —> e. Further, all strings in (L (G ))* can be generated in this way. Thus, the cfl’s are closed under Kleene star. Problem: W h y was it necessary to introduce the new start sym bol S '? W hy not ju st add the rules S —» S S and S —» e?
Intersection: T he cfl’s are not closed under intersection
T o see this, we
note that the languages { a%bl c7 |i , j > 0 } and { a kblcl \ k , l > 0 } are b oth context free. T he form er is generated by a giam m ai containing the rules: (1 8 -5 )
S -> B C B —* aBb B —> e C -> c C C -► e
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and the grammar for the lattei is similar T he intersection o f these tw o lan guages, however, is { a nbn cn |n > 0 }, which we proved above b y the Pum ping T heorem n ot to be a context free language R ecall in this connection what it means to say that a set is not closed under a certain operation. We have shown that the intersection o f two cfi’s is sometimes not a cfl It is not claimed that the result is never a cfl. Indeed, this could not be so, since the regular languages are necessarily cfl’s, and since the regular languages are closed under intersection, the result is regular, hence, context free. C o m p le m e n t a t io n : The cfl’s are not closed under com plem entation. Given two c fl’s L i and L% over some alphabet S , if their com plem ents L[ and L'2 (i e , £* — L\ and £* — L%, respectively) were context free, then so would be their union, L[ U L'2 The com plem ent o f this, in turn, ( L[ U L'%) ' , would also be context free, but this is equal b y D eM organ’s Laws to L\ fl L%, which is not necessarily context fr ee. Hence, the com plem ent o f a cfl is not necessarily a cfl I n t e r s e c t io n w it h a R e g u la r L a n g u a g e : A lthough the intersection o f tw o arbitrary cfl’s L\ and L 2 is not in general a cfl, it happens that if one o f the languages is restricted to being regular, then the intersection is always a cfl. A dem onstration o f this fact is somewhat involved and depends on constructing a non-determ inistic p d a accepting L\ fl L 2 out o f a p d a accepting L\ and a finite autom aton accepting L 2 (T he non-closure o f cfl’s under intersection implies that it is not in general possible to coalesce tw o non-determ inistic p d a ’s in this way.) The closure o f the cfl’s under intersection with a regular language can be a convenience in showing certain languages not to be context free. For exam ple, the language L = { x £ {a , b, c}* |x contains equal numbers o f a’s, b’s, and c ’s }, although not context free, resists application o f the P um ping T heorem However, if L is first intersected with the regular language a*b*c*, the result is { a nbncn | n > 0 }, which we have shown not to be context free. Now if L were a cfl, its intersection with a regular language would also be a cfl; hence, L is not context free.
18.5
Decidability questions for context free lan guages
C ontext-free languages differ from fal’s also in respect to which questions can be answered by algorithm. For the fa l’s we saw that there were algorithms
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W e shall see
that some o f these questions have algorithm ic solutions for the cfi’s and some do not. M e m b e r s h ip :
G iven an arbitrary cfg G and an arbitrary string x, is x
generated by G? One m ight propose an algorithm for answering this question o f the follow ing sort: Start producing derivations by G in som e system atic fashion, discarding any whose last lines are longer than x. This will be some finite num ber o f derivations, If x has not been generated by this p oin t, it is not going to be As matters stand, this algorithm might not be successful for two rea sons: first, the grammar m ay contain rules o f the form A —> e, which allows derivations to becom e shorter. Thus, we cannot be sure that we can stop exam ining derivations when their last lines reach the length o f x A deriva tion m ight produ ce longer strings which then shrink to p rod u ce x Second, because rules such as A —> B , B —> C , C —> A , etc, m ight be present in the gram m ar, derivations might continue indefinitely w ithout their final strings getting any longer
This subverts our claim that we need to exam ine only a
finite number o f derivations to see if any generate x. T he p roposed algorithm would work, however, if we could som ehow contrive to rem ove all rules o f b oth types from a cfg while leaving the generative pow er o f the grammar unchanged.. W e will now show that this can in fact be done. If there is a rule A —> e in the grammar, this rule can be dispensed w ith if we add m ore rules to the grammar in the follow ing way. W henever A appears on the right side o f a rule we add another rule identical to it except that the A on the right is deleted. For exam ple, given rules A —> e and B —> c A b B a , we would add the rule B —> cbBa. N ow what would have been accom plished by application o f the first tw o rales in sequence can be accom plished b y the last rule alone. The original rule is o f course preserved since there m ay be other rules expanding A. W e continue in this way for every rule containing an A on the right side, and repeat the process for every non-term inal which can be rewritten as e„
(N ote that if such a non-term inal appeared m ore
than once on a right side, e.g., B —> a A b A c , we would add rules B —> abAc, B —> aAbc, and B —> abc,) This process must eventually com e to an end since there are finitely m any rules to begin with, there are a finite num ber o f non-term inals, and a finite number o f rules aie added at each step. W h en we are done, we m ay remove all rules of' the form A —» e from the gr ammar since they are superfluous,. The one exception is the rule S —> e, if it is present, which must be retained in order to generate the em pty string as a mem ber o f
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T he presence o f this rule will not interfere with the workings
o f our algorithm , however, since if there is a derivation o f some non-em pty string x which involves one or m ore applications o f the rule S —* e, there will also be, after carrying out the procedure just outlined, a derivation o f x which does not involve any applications o f this rule Thus, we can produce an equivalent cfg in w hich derivations are essentially non-shrinking. W hat about rules o f the fo im A —> B I These can be rem oved in the follow ing way. P ick a rule o f the form A —> w, where w is som ething other than a single non-term inal symbol. (If there are no such rules, the grammar generates no term inal strings and the m em bership question is settled at on ce.)
N ow for each non-term inal C distinct from A, determine whether
C ^ A, i.e., w hether C can be rewritten in some finite num ber o f steps to give A This can be done by examining all sequences C, B i , B 2 , ■■•, Bn , A (where B\, B 2, are single non-teim inals) o f length no m ore than the number o f non-term inals in the grammar to see whether they are allowed by the gram m ai. The restriction on length o f derivations is possible because if there is such a derivation with repeated symbols C = > , . , = > Bi =>, . . . => Bi => B j => . . => A then there is also a shorter one with the section between repetitions rem oved: C = > . . . = > B{ => B j = > . . , = > A. Thus, the number o f non-term inals fixes an upper bound on the length o f such derivations, and we can effectively determine whether C ^ A. If so, then we add the rule C —> w to the grammar, and thus the derivation C ^ A ^ w can be replaced b y the derivation C ^ w directly. W e continue this process for all rules o f the form A —> w (w 0 Vjv) and all non-term inals distinct from A, W hen we have finished, all rules o f the form A —» B can be rem oved from the grammar w ithout affecting the terminal strings generated. Once all these steps have been carried ou t, the proposed algorithm for answering the m em bership question can be executed and is guaranteed to lead to an answer in a finite amount o f time. N o claim is m ade that this is particularly efficient way to answer the m em bership question, but we are not concerned with relative amounts o f com putational la b or here— only with showing that the question can be answered in some finite amount o f tim e by m echanical means. E m p t in e s s : Does an arbitrarily given cfg G generate any strings at all? There is an algorithm for answering this, the emptiness question, and it depends on the follow ing observation, If G generates any term inal strings, it generates some term inal string with a parse tree which has no non-terminals repeated along any path. Refer again to Fig. 18-2 which we used in proving
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the P um pin g Theorem . If a parse tree for some term inal string has a repeated non-term inal A along some path, the subtree rooted b y the upper A could be replaced by the subtree rooted by the lower A , and the result is also a parse tree for a terminal string generated by the grammar (cf. Fig, 18-3). Clearly all repeated occurrences o f non-term inals could be rem oved in this way. Thus, in order to see whether G generates any term inal strings, all we have to do is examine the finite num ber o f parse trees which contain no repeated elements along any path.
The exact num ber we need to look at
will depend on the number o f non-term inals in the grammar and the degree o f branching allowed by the rules o f G , but it will be finite I f no term inal string has appeared as the yield o f a tree by this poin t, none is ever going to appear. This answers the emptiness question,
Undecidable questions: M any problem s concerning context free languages have no algorithmic solution.
W e cannot provide dem onstrations o f these
facts here since they require results from Turing m achine theory, which we have not yet examined.
W e will sim ply list some o f the m ore im portant
undecidable questions: a. Given an arbitrary context free grammar G , there is no algorithm for determining: (i) whether L ( G ) = Vf> i e •>whether G generates all strings over the term inal alphabet (ii) whether the com plem ent o f L ( G ), i.e., V f — L ( G ) , is em pty, infi nite, regular, or context free.
b. Given tw o arbitrary context free grammars G i and G 2, there is n o algo rithm for determining: (i) whether I ( G 0 C L ( G 2) (ii) whether L ( G i ) = L ( G 2) (iii) whether L ( G i ) fl L ( G 2) = 0 (iv) whether L(G\) fl L ( G 2) is infinite, regular, or context free. R ecall that the lack o f a single algorithm for deciding every one o f an infinite class o f cases does not preclude the possibility that for certain context free grammars these questions might be answerable. In fact, for all context free gr ammars which happen to be regular, there are algorithm ic solutions to all o f the above questions, as we saw in Section 17,3,1.
A
18.6
r e
n a t u r a l
l a n g u a g e s
c o n t e x t
f r e e
5 03
?
A re natural languages context free?
In Section 17.3 2 we showed that English could not be a regular language. C ould it or any othei natural language be context fiee? This question has at tracted considerable attention in the years since Chom sky first outlined the hierarchical categorization o f form al languages (1963). The prevailing view has been that natural languages are not context free. A ttem pts to dem on strate this have usually centered on finding instances o f so-called “ crossserial” dependencies o f arbitrarily large size in some particular language. In a cross-serial dependency, items are linked in left-to-right order as shown in Fig
18-5.
®1
®2
' •
•
■
2/1
2/2
•
■
2 /n
Figure 18-5.
(C om pare this to the nested dependencies illustrated in Fig
17-17.)
A language such as { x x |x E {a , 6 }“ } exhibits cross-serial dependencies (for strings o f length 2 n in the ith and ( n + i) th symbols must m atch) and is not context free. Pullum and Gazdar (1982) review the various attem pts to establish that natural languages are n ot context free and find all o f them either form ally or em pirically flawed. In the latter category are instances in which claims that certain form s are ungram m atical are unjustified and m ore probably involve semantic or pragm atic anomaly. A com m on form al mistake was to assume that because one has found a subset o f a language which exhibits cross-serial dependency the language as a whole is thereby shown not to be context free. N ote that the n on -con text free language { x x \x £ {a , 6 }’ } is a subset o f the context free (indeed, regular) language {a , &}*, Recently, however, evidence has appeared for the non-con text freeness o f Swiss German which seems unassailable on either form al or em pirical grounds (Shieber, 1985). Swiss Germ an, like its m uch studied cousin D utch, allows cross-serial order in dependent clauses, Sentences like the follow ing are grammatical:
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Jan sait das m er em Hans es huus halfed aastriiche John said that we H an s-D at the h o u s e -A c c helped paint “ John said that we helped Hans paint the house ”
(1 8 -7 )
Jan sait das mer d ’ chind em Hans es huus lon d halfe aastriiche John said that we the children - A cc Hans - Dat the house - A cc let help paint “ John said that we let the children help Hans paint the house.”
The N P ’s and the V ’s o f which the N P ’s are objects occu r in cross-serial or der: in (18-7) d ’chind ( “ the children” ) is the o b je ct o f lond ( “ le t” ), em Hans is the ob ject o f halfe ( “help” ), and es huus ( “ the house” ) is the o b je ct o f aas triiche ( “ paint” ). Furtherm ore, the verbs mark their ob jects for case: Halfe requires dative case, while lond and aastriiche require accusative Sentences in which the case m arking does not follow this restriction are uniform ly re jected by native speakers as ungram m atical. (Since case m arking is unlikely to be accounted for semantically or pragm atically, this avoids the empirical trap m entioned above ) It also appears that there are n o limits other than perform ance constraints on the length o f such constructions in gram m atical sentences o f Swiss German, Shieber then intersects Swiss Germ an with the regular language:
(1 8 -8 )
R = Jan sait das mer (d ’ch in d)*(em H ans)* es huus haend wele (laa)* (halfe)* aastriiche John said that we (the children)* (H ans)* the house have wanted to (let)*(h elp )* paint
(Here the haend wele ( “have wanted to ” ) is present in order to put all the succeeding verbs in their infinitive forms, Schieber shows that this insertion does not affect grammaticality judgments,,) The result o f intersecting R and Swiss German is all sentences o f the follow ing form : (1 8 -9 )
L = Jan sait das mer (d ’ chind)71(em H ans)m es huus haend wele (laa)71 (halfe)771 aastriiche
where the num ber o f nouns in the accusative case m atches the num ber o f
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505
verbs requiring this case and similarly for the dative case, and all accusative case nouns (except the constant “ es huus” ) precede all dative case nouns and all accusative-case m arking verbs precede all dative-case marking verbs. The strings o f this sublanguage o f Swiss Germ an are o f the form wanbmx c ndmy , which can be shown to be n on-context free by the Pum ping Theorem . Since the con text free languages are closed under intersection with a regular lan guage, this dem onstrates fairly convincingly that Swiss German is n ot con text free N ote that the form al difficulty m entioned above has been avoided. The sublanguage o f Swiss German shown to be n on-context free is not merely a subset o f the original but a subset obtained by intersection with a regular language The latter operation preserves context freeness while the operation o f sim ply selecting a subset in general does not. A ttem pts to arrive at the corresponding results for D utch could n ot suc ceed because D utch does not have verbs with differing case-marking proper ties w hich can occu r in arbitrarily long cross-serial dependent clauses.
Exercises 1. C onstruct context free grammars generating each o f the follow ing lan guages., (a)
L i = anbman ( n , m > 1)
( b ) L 2 = anbnambm(n, m > 1) (c ) Lz = { x | x E {a , b}* and x contains tw ice as m any 6’s as a ’s }
( d ) Ir4 = { x x R |x £ {a , 6 }* }
( e ) L s = { x £ { a , b } x \x -- x R} 2. Show that for every context free grammar there is an equivalent gram m ar in which all productions are o f the form A —* B C or A —* a ( A , B , C in Vn , a in Vt )- Such a grammar is said to be in Chomsky Norm al F orm (C hom sky, 1959). 3. Show by means o f the P um ping Theorem that the follow ing languages are not context free, (a )
{ a " 2 |(ra > 1 ) }
( b ) { a 711n is prim e (i.e., divisible only by 1 and by itse lf)}
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4 . Given a language L we define the reversal o f L, denoted L R, as { x R I x £ L } , where x R is the reversal o f x.
Show that the context free
languages are closed under reversal, 5. Construct a deterministic pda which accepts the language ( ab)n (cd)n for all re > 1. (T h e parentheses are not part o f the language nor do they indicate optionality; they are used in the expression above only for grouping.)
6 . C onstruct a deterministic pda which accepts every string o f the form xc, where x is a string o f a ’s and 6’s o f length 0 or more in which the total num ber o f a’s is exactly equal to the total num ber o f 6’s 7. Construct a non-determ inistic p d a which accepts every string which is o f the form an ban or o f the form a2nban for all re > 1 8. Is the union o f the languages o f two determ insitc p d a ’s necessarily the language o f some deterministic pda? Justify your answer 9. Show by means o f the Pum ping Theorem for context free languages that a V c max(1J') is not context free, where m a x ( i j ') is the larger o f i and .j.
Chapter 19
Turing Machines, Recursively Enumerable Languages and Type 0 Grammars 19.1
Turing machines
W e have seen that a pushdow n autom aton can carry out com putations which are beyon d the capability o f a finite autom aton, which is perhaps the simplest sort o f m achine able to accept an infinite set o f strings. A t the other end o f the scale o f com putational pow er is the Turing m achine (after the English m athem atician, A. M„ Turing, who devised th em ), which can carry out any set o f operations which could reasonably be called a com putation. Like the previous classes o f autom ata, a Turing machine can be visualized as having a control b ox, which at any point is in one o f a finite num ber o f states, an input tape marked o ff into squares with one sym bol o f the input string being inscribed on each square, and a reading head which scans one square o f the input tape at a time. The Turing m achine, however, can write on its input tape as well as read from it, and it can m ove its reading head either to the left or to the right. A s before, a com putation is assumed to begin in a distinguished initial state with the reading head over the leftm ost sym bol o f the input string.. W e also assume that the tape extends infinitely to the left and right and that all tape squares not occu pied by sym bols o f the input string are filled by a special “ blank” sym bol 507
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The m oves o f a Turing m achine (henceforth, T M ) are directed b y a finite set o f quadruples o f the form (
The set o f states K is { g o ,g i } ; the alphabet S is { a , 6 , # } ; the initial state is go; the set o f instructions S is written w ith an arrow between left and right halves for clarity: (go, a) —* (gi ,6)
(go, b) —» (gi , a) (?o,#) -*■ (?1 , # ) (gi,a) -> (g0, R) (<7i > t>)
— *
( g o , R)
This m achine scans an input string from left to right, changing a ’s to 6’s and 6’ s to a ’s, until it encounters the first # . It then rewrites the # as changes state to gi and halts (since there is no instruction beginning (<7i ! # ) ) Since a # is sure to be found eventually, this T M has the property o f halting on all inputs. N ote, however, that if it had also contained the
instruction ( g i , # ) —> ( g i , i i ) , then it w ould com pute forever once it had reached the string o f # ’s to the right o f the input. The same result could also be achieved by the instruction ( g i , # ) —> ( g i , # ) , except that instead
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m a c h in e s
509
o f scanning endlessly to the right, the TA1 w ould stay on one tape square forever reading and writing # , (1 9 -2 )
Exam ple o f com putation:
W e have not designated the states o f a T M as final or non-final. It would be perfectly feasible to do so and to define acceptance o f an input string in terms o f halting in a final state. It will be slightly more convenient, however, to say that a string is accepted if, when it is given to the m achine in the standard starting configuration, it causes the T M to halt after some finite num ber o f m oves; otherwise, it is rejected (i.e., the T M never halts).
In
(19-3) we give a machine which accepts all strings in {a , &}“ which contain at least one a and which rejects, i.e., com putes forever, when given anything else in {a , &}*. (N ote that we are here concerned only with strings in {a , 6}* n ot { a , T M ’s are assumed always to be able to read and write the # sym bol, but we will ordinarily confine ourselves to strings over alphabets which do not contain # . A T M which accepts some language in {a , 6}* may give bizarre results when given a string not in this alphabet, but that d oesn ’t m atter. W e are only concerned with its behavior when given inputs from
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t h e r e le v a n t set
(1 9 -3 )
M = ( i f , S , s, 6 )\ K = {g0,gi}; S = {a , 6 , # } ; s = g0;
'(go,a) (go, 6) (go,#) (gi,a) . (gi»6)
-> (gi,J2) —* (?oi -K) -*■ (go,-K) -*■ (gi,J2) -*■ (gi.-K) ,
This m achine scans left to right and stays in state g0 so long as it sees b’s. Once it encounters an a, it changes to state gj and continues rightw aid in this state until the first # and then halts It if meets the first # in state g0) it scans right forever
1 9 .1 .1
F o r m a l d e fin itio n s
A T u r in g m a c h in e M is a quadruple ( K , ' S , s >6 ), where K is a finite set o f states, S is a finite set (the alphabet) containing # , s & K is
D e f i n i t i o n 19.1
the initial state, and 5 is a (partial) function from i f x S to K x ( S U { L , R } ) .
A situation o f a T M will be a quadruple o f the form ( x , q , a, y ), where q is the current state, a is the sym bol being scanned, and x and y are the strings to the left and right, respectively, o f the reading head up to the beginnings o f the infinite strings o f # ’s. This last provision is necessary to insure that a situation is uniquely specified. The T M in (19-2) is in situation (e, go, a,6a) at the beginning o f the com putation and in ( 6 a 6 , g i ,# , e ) when it halts, e f i n i t i o n 19 2 A situation o f a T M M = ( K , ' £ , s , S ) is any m em b er ( x , q , a , y ) o f £ * x i i f x S x S " such that x does n ot begin with # and y does n o t end with # . ■
D
W e om it the form al definition o f the produces-in-one-step relation, I on pairs o f situation since it is rather com plex when specified in full detail. N ote that one must allow for cases such as the follow ing: situation ( a b b , q , # , e ) , as shown in F ig
the T M is in
19-1, and executes the instruction
(g, #) — ^(g',
L ). The resulting situation is ( ab, q',b, e), as in Fig. 19-2, where the # originally being scanned has joined the infinite string o f # ’s to the right and has thus dropped out o f the form al specification o f the situation.
T
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511
m a c h in e s
In a s im ila r v e in , i f th e in s t r u c t io n h a d b e e n (g, # ) —> (g', R ), th e r e s u ltin g s it u a t io n w o u l d b e ( abb# , g', # , e ) w ith a #
ta k e n f r o m th e s tr in g o f # ’s t o
th e r ig h t a n d p la c e d u n d e r th e r e a d in g h e a d .
#
b
#
b
#
#
I
Figure 19-1,
#
#
a
b
b
#
#
i
Figure 19-2,
Given the produces-in-one-step relation, we define the produces relation, Ij^- as its reflexive transitive closure.
e f i n i t i o n 19.3 Given a T M M = { K , S , s, 5) and S i, a subset o f S which does n o t contain # , we say that M accepts a string x = a 1 o 2 . . an £ S j if f
D
(e, s, a i, 02 „. g, 6, y '), where y and y' are strings in b £ S , and there is no instruction in 5 beginning (q,b) (i . e ., M has halted). (In case x = e, the initial situation is (e, s , # , e ) J
■
D e f i n i t i o n 19,4 A T M M a c c e p t s a la n g u a g e L £ S j if f M accepts all strings in L and rejects (i.e., fails to halt on) all strings n o t in L. ■
N ote that we have defined acceptance so that it holds only o f strings and languages defined over alphabets not containing This is prim arily a technical convenience.
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We say that a language is T uiin g acceptable iff there is
som e T M which accepts it,
■
In virtue o f E xam ple (19-3) above, we can say that the language L = { x £ {a, b Y | x contains at least one a} is Turing acceptable
In fact, all the
regular languages and the deterministic and non-determ inistic p d a languages are Turing acceptable,
(It should not be to o difficult to im agine how one
would go about constructing a T M to m im ic the behavior o f a fa or a p d a ) Are there any languages, then, which are not Turing acceptable? There are, but it is not easy to exhibit one. W e will return to this im portant question below W e have seen that a given T M might not halt on certain inputs— indeed, we make use o f this property in characterizing rejection o f an input string. But this pr esents us with a problem
Suppose we have set a T M to com puting
on some input and it has n ot yet halted. Can we tell in general whether it is going to halt eventually or whether it is going to com pute forever? This is the renowned Halting P roblem for Turing machines, and we will show in a later section that there is, in general, no way to tell— at least if we m ean by “ a w a y to tell” some explicit procedure which can be com puted m echanically by, say, a Turing machine. A nother way to form ulate the Halting P roblem is this: C ould we take any T M which accepts a language L and convert it in to a m achine which halts on all inputs (over the relevant alphabet) and signals its acceptance or rejection o f the input by, say, the state it is in (or some special sym bol printed on the tape, etc,)? Let us, in fa ct, make the follow ing definition: D
e f in it io n
19.6 A T M M = ( K ,' S ,s ,S ) with som e designated set F C K
o f final states, decides a language L C SJ, (where S i C S and does not contain # ) iff for all x £ M halts in a final state i f x £ L and M halts in a non-final state i f x £ L. ■ e f i n i t i o n 19.7 W e say that a language is Turing decidable iff there is some T M which decides it, in the sense ju st defined, ■
D
It is n ot difficult to convert the T M o f Exam ple (19-3) above in to one which decides, rather than accepts, the language L = { x £ {a , 6}* | x contains at least one a } — simply rem ove the instruction (go, # ) —> (go,-K) and designate gi as the only final state. The question is whether such a conversion can always be carried out. I f so, then every Turing acceptable language is Turing
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m a c h in e s
decidable. and, provided that the conversion can be carried out “m echani cally” , i.e., algorithmically, the Halting P roblem is solvable
Since this turns
out not to be so, we will have to conclude that there are Turing acceptable languages which are not Turing decidable. Again, m ost o f the languages one ordinarily encounters as examples in form al language theory are Turing decidable, so one must look further afield to find one which is not N ote that the im plication in the opposite direction is easy to establish: every Turing decidable language is Turing acceptable. Given a T M which decides a language L, it is a simple m atter to convert it into a T M which com putes forever just in case the original m achine would have halted in a non-final state, (Just add instructions which rewrite sym bols as themselves while staying in the same state), Turing m achines are probably m ost often viewed n ot as language accep tors but as devices which com pute functions. The initial input string is the argument for the function, and the expression on the tape when the machine halts (if it halts) is taken to be the value o f the function at that argument For exam ple, the T M o f (19-1) com putes the function / : {a , b}* —> {a , &}* such that when / ( « ) = v, v is like u with a ’s and 6’ s interchanged. If the T M does not halt for a certain input, the function is not defined at that ar gum ent, Thus, T M ’s in general com pute partial functions, but a T M which halts for all inputs in some set A com putes a total function from A to its range, A T M which decides a language L C £ J , in the sense just defined, com putes the characteristic function o f L. A T M which accepts L com putes a different function— one which is defined for all strings in L and undefined for all other strings in £ j. B y coding natural numbers as strings— say, in binary or in a unary en codin g in which n a ’s represent the natural num ber n— we can let T M ’s serve a com puters o f functions from natural num bers to natural num bers. We can then ask whether functions such as f ( x ) = x 2 or f ( x ) -- x\ are com putable by Turing m achine (they are, in fact, as are m ost o f the functions ordinarily encountered). W e can generalize this approach to functions o f k arguments by letting the initial string given to the T M be, say, k blocks o f a ’s o f appro priate size separated by # ’s. The T M which com putes the addition function on natural num bers, for exam ple, w ould start with , ,, # # a " # a m# # . . . on its tape and end with . , „ # # a n+m# #
,,
Functions which are com putable by Turing m achine are called partial recursive functions (partial, because the T M m ay not halt for all arguments and thus m ay leave some values o f the function undefined).
The Turing
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com putable functions which happen to be total functions ought to be called “ total recursive functions,” but aren’t. T h e y ’re called simply recursive fu n c tions, A recursive function, thus, is a function which can be com pu ted by a Turing m achine which halts on all inputs in the dom ain o f the function A T M m ay also be regarded as a device for generating, rather than accepting or deciding, a set o f strings. Let a T M be given as input some encoding o f a natural num ber (say, n a ’s representing n) and let it com pute until it halts, if it does, T he contents o f its tape betw een the infinite strings o f $ ’s is some string over the alphabet o f the T M (w hich m ay contain more than a ’s, o f course), and this string is said to be generated by the TM , The set o f all such strings generated, given all the natural numbers as inputs, is said to be the set recursively enumerated by the TM , A set is said to be recursively enumerable (abbreviated r e )
if theie is some T M which
recursively enumerates it in the way just described It turns out that a set o f strings is r.e, just in case it is Turing acceptable. That is, if A is recursively enum erated by some T M T , there is a T M T 1 which accepts A , and conversely, (W e will om it the p r o o f o f this result.) Note also that a set which is recursively enum erated constitutes the range of' a partial recursive function from the natural numbers to the set o f all strings ovei the alphabet o f A. In fact it is inessential here that the domain be the natural numbers— any denumerably infinite set would do. Thus, we can say that a set is recursively enumer able if it is the r ange o f some partial recursive function. The follow ing three statements, therefore, are equivalent: (1 9 -4 )
(i) A is accepted by some T M (ii) A is the range o f some partial recursive function. (iii) A is recursively enum erated by some TM .
19.2
E q u iv a le n t fo rm u la tio n s o f T u rin g m ach in es
There are many ways in which Turing m achines can be defined which turn out to be equivalent in com putational power. The input tape can be stip ulated to be infinite in one direction only, or the m achine can be endowed with any finite num ber o f tapes which extend infinitely in one or b o th di rections, or the machine may have one m ultiple-track tape w ith m ultiple reading heads. It may even be regarded as operating on an ra-dimensional grid extending infinitely in all dim ensions, so lon g as ra is a finite n um ber.
U n r e str ic te d
grammars
and
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515
Even m aking a T M non-determ inistic does not change its capabilities in any essential way Suppose a non-determ inistic T M had k distinct moves allowed from any given situation, Then from the initial situation there could be at most k possible situations after one step, at m ost k 2 after two steps, ,,, at m ost kn after n steps o f the com putation. A deterministic T M could keep track o f all o f these (at some expenditure o f tape and tim e) I f there is a halted com pu tation by the non-determ inistic m achine, it occurs in some finite num ber o f steps r. The deterministic m achine will therefore discover this fact after having exam ined at m ost kT possible situations— again, a finite num ber— and can then halt, A language accepted by some non-determ inistic T M can therefore be accepted by a deterministic TM, W hat seems to be essential to the form ulation o f a Turing machine, therefore, is that it has a finite num ber o f states, a finite alphabet, a finite num ber o f instructions, and an unbounded amount o f com putational space available to it.
19.3
Unrestricted grammars and Turing machines
A n unrestricted (or T ype 0) grammar G — (Vt , Vj\t, S, R ) is one in which the only lim itation on the form o f the rules is that the left side contain at least one non-term inal symbol. Thus, letting upper case letters be non-terminals and lower case letters be terminals as usual, aAbb —> ba, aA bB —> e, and A —¥ b C aB would all be allowed rules in such a grammar. Rules such as ab —> 6a, b —> B A , or e aA would be excluded. N ote that because more than one sym bol m ay be replaced in T ype 0 rules, it is in general not possible to associate a phrase structure tree with a derivation in such a grammar. T yp e 0 grammars can generate languages which are not context free Here, for exam ple, is an unrestricted grammar generating { x E {a , 6, c}* | x contains equal num bers o f a ’s, 6’s, and c ’s } which, as we have seen, is not a cfl, (1 9 -5 )
G = ({ a , 6, c }, {S , A, B , C } , S, R ), where
R =
S S
-► S A B C _» e
A B -+ B A B A —> A B
A C —> C A C A -> A C
A —> B —>
B C —> C B C B —►B C
C ->
This grammar works by produ cin g strings o f the form ( A B C ) n then per m uting non-term inals freely by means o f the rules A B —> B A , etc. Finally,
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19
non-term inals are rewritten as the corresponding terminals. Here is a deriva tion of' cabbca (1 9 -6 )
S => S A B C => S A B C A B C => A B C A B C => A C B A B C C A B A B C => C A B B A C => C A B B C A = > .,..= > cabbca
=>
A nd here is a T ype 0 grammar generating the n on -con text free language { x x |x E {a , 6 }* }: (1 9 -7 )
G = ( { a , &}, { S , S ', A , B , # } , S ,R ) , where 5
A cl —> clA
#a
S' -> aAS'
A b -> bA
# b -*■ 6 #
S' -* bBS' S' —> e
B a —^ clB Bb -> bB
A # ->■ # a
a#
B # —> # 6 ##
e
Here is a derivation o f abaaba: (1 9 -8 )
S => => # a A S ' # => # a A b B S => f t a A b B a A S ' => # a A b B a A # => f t a b A B a A # => f t a b A a B A # => f i a b a A B A # => a # b a A B A # => a f t b a A B f t a = > .. ,= > a b a # # a b a => abaaba
This grammar works by generating sequences o f a A ’s and b B ’s betw een # ’s as endm arkeis and then letting the non-term inals m igrate to the right, where they can h op over the # and becom e terminals. The terminals in the left h a lf similarly h op over the left end marker, and when the tw o # ’s meet in the m iddle they are erased The languages generated by the T ype 0 grammars are exactly the lan guages accepted by Turing machines, i.e., the r e . sets. W e will not give detailed proofs o f this equivalence here but will sim ply suggest h ow the proofs are constructed. Given a T yp e 0 grammar G generating L ( G ) a T M M accepting L (G ) can be constructed as follows, M is non-determ inistic and has two tapes. Its input is given on the first tape where it is stored intact throughout the com putation. The instructions o f M essentially m im ic the rules o f G The initial sym bol S is placed on the second tape, and M proceeds to rewrite as G w ould
A fter the application o f each rule, M com pares the contents o f
the second tape with the input on tape 1. If they m atch, M halts, and thus accepts its input. I f they do not m atch, M continues applying rules o f G to
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H y po th e sis
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the string on tape 2, and if no rule is applicable, M cycles endlessly in some fashion. Clearly, if there is a derivation o f the input string b y G, there will be some com pu tation b y M which discovers this fact and thus M will halt and accept I f there is no such derivation, M com putes forevei, as required. The sim ulation in the reverse direction— making a T y p e 0 grammar m im ic a Turing m achine— depends essentially on the fact that a situation o f a T M can be regarded sim ply as a finite string o f symbols and that to get from one situation to the next, some substring o f these sym bols is rewritten as some other string. For exam ple, a T M instruction o f the form (g, a) —> (g', b) would correspond to the grammar rule qa —> q'b Thus, situ ation (aab, g, a, bb) becom es ( aab, q1, b, bb) in one move b y the machine, and, correspondingly, the string aabqabb is rewritten as aabq'bbb b y the grammar. Left-m ovin g and right- m oving T M instructions require somewhat more com plicated grammar iules to cover all possibilities The details are tedious and not to o instructive,. Now , given a T M M which accepts language L ( M ) , we first convert it to a machine M ' which behaves like M up to the point at which M w ould halt
M ', however, replaces all non-blank sym bols on its
tape by # ’s, then writes S, and halts in some designated state gj Thus, M ' accepts L ( M ) also but does so in such a way that it always halts in situation ( ^ 5 g i 7 *^5 ^ ) ,
W e now construct G so that it simulates the moves o f M ' in reverse. The initial sym bol o f G is S ', and it first rewrites S' as qxS Then, m im icking the moves o f M ' in reverse it can arrive at the string qQx, where go is the initial state o f M ' and x was the input accepted. N ow all G has to do is to erase go, thereby generating x. The only com plication here is that we d on ’t want to erase g0 unless it is part o f an initial situation o f M 1; i.e., go might be entered at other points in the com putation b y M ', and we d on ’t want to erase it in these cases This difficulty can be taken care o f by adding new states to M ' to insure that once it has left its initial state go it never enters it again in the course o f any com putation, W ith this repair, the grammar G generates exactly the strings accepted by M ' (and M ). Thus, we can add a fourth equivalent statement to those given in (19-4) above: (iv) A is generated b y some unrestricted grammar
19.4
Church’s Hypothesis
A n algorithm is a fixed, deterministic procedure which can be applied m e chanically to yield a result within a finite amount o f time.
For example,
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theie is an algorithm for finding the square root o f any positive num ber to any desired number o f decim al places, Algorithm s are norm ally designed to apply to a class o f problem s, not to a single p roblem T he algorithm for find ing square roots can b e applied in the same way to any positive num ber— we do not have to hunt for a new procedure for every case Some classes o f problem s do not have algorithm ic solutions. Ther e is no algorithm , for exam ple, for supplying proofs for theorems o f geometry. T o find a p roo f for a given theorem often requires some ingenuity, skill, and even luck, whereas algorithms, b y definition, dem and only simple clerical abilities The definition o f algorithm given above is not m athem atically precise, relying as it does on such intuitive notions as “ m echanical,” In the 1930’s, a num ber o f attem pts were made to find a precise, form al characterization o f the notion o f algorithm as applied to m athem atical problem s T he Turing m achine was the result o f one such attempt It is clear that a Turing machine satisfies our intuitive notion o f what an algorithm should be; a T M which com putes a fu n ction , for example, determines the value at any agrument in a fixed, deterministic, mechanical way and in a finite amount o f time. T he question then arises whether all things which we w ould intuitively call algorithms can be form ulated as Turing machines. T he conjecture that this is indeed the case has been given the name Church’s Thesis or Church’s Hypothesis (after the logician A lon zo Church). It is not a theorem , since it relates a m athem atical construct— the Turing m achine— to an intuitive, im precise notion— an algorithm. It is nonetheless widely believed to be correct. The evidence in its favor arises basically from the fact that all independent attem pts to characterize the notion of algorithm b y m athem aticians such as Kleene, P ost, M arkov, Church, and others turned out to be equivalent to the Turing machine, Rogers (1967) calls this the B asic Result in Recursive Function Theory: The classes o f partial functions (and hence total functions) o b tained b y the characterization o f Turing, Kleene, Church, P ost, M arkov, and certain others, are identical, i e , are ju st one class. Further support for Chur ch ’s Hypothesis com es from the fact that m od ifications and enrichments to the definition o f a Turing machine which keep intact our view o f it as a m echanical com puting device with a finite num ber o f states and instructions but with a potentially unlim ited am ount of' space for’ com putation, always produce an equivalent d e v ice ,
R
ec u rsive
versus
rec u r siv e ly e n u m erable
sets
519
These results suggest that the characterizations are not purely arbitrary but do in fact define a natural concept, i.e,, sets which are recognizable, or functions which are com putable, b y algorithm If we accept C hurch’s H ypothesis, then, we m ay add a fifth equivalent statement to the list in (19-4) above (v ) There is an algorithm for recognizing strings in A.
19.5
Recursive versus recursively enumerable sets
A Turing m achine which accepts a set A halts eventually whenever given a m em ber o f A as input but fails to halt w hen given a non-m em ber o f A. If we accept C hurch’s H ypothesis, this means that an algorithm m ay w ork in such a way that it yields an answer in a finite amount o f time for all members of’ a particular class but m ay yield no result for things not in the class. It m ay happen, however, that there exist algorithms for recognizing not only all m em bers o f A but also one for recognizing all m em bers o f S ’1—A , i e ., the com plem ent o f A W h en this is so, A is called a recursive set. T o state the definition form ally in terms o f Turing machines, a set A is recursive iff both A and A ' are recursively enum erable (= Turing acceptable). It now follows that the recursive sets are just the Turing decidable languages defined above. R ecall that a language is Turing decidable if there is some T M which halts on all inputs over the relevant alphabet and signals whether or not the input was in the language. A s we have seen, it is a simple m atter to convert a T M which decides L into one which accepts L (or in to one which accepts L ') b y causing it to com pute forever on negative outcom es. Conversely, if we were given tw o T M ’s, one accepting L and one accepting L ' , we could construct a T M for deciding L by having it simply alternate the instructions o f the tw o m achines on tw o copies o f the input One o f these will eventually halt, since b y assum ption b o th L and L' are Turing acceptable, and then our com posite m achine can signal whether the input was in I or I '
Thus it decides L.
A nalogous remarks could be made about com putation o f functions rather than recogn ition o f sets. T he functions which aie algorithm ically com putable are, b y C hurch’s H ypothesis, just the parital recursive functions. If the fu n ction is properly partial and not total, the algorithm will yield no value at arguments for which the function is not defined
If the function is total,
how ever, the algorithm yields a value at each argument As we have seen, a T M which decides a language in effect com putes the char acteristic function o f
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that language. The recursive sets, then, are just those w hich have recursive characteristic functions, The recursively enumerable sets have characteristic functions w hich are partial recursive functions. W e m ay summarize our statements about recursive sets as follows: (1 9 -9 )
The follow ing statements are equivalent: (i) A is a recursive set (ii) A is Turing decidable (iii) B oth A and A 1 are recursively enumerable (iv) A has a characteristic function which is (total) recursive.
W e do not yet officially know, o f course, whether there are actually any r.e, sets which are not recursive. W e turn our attention to this m atter in the next section.
19.6
The universal Turing machine
Since a Turing m achine is defined as a finite set o f quadruples together with a designated initial state (the set o f states and the alphabet are im plicit in the quadruples), it is possible to enumerate all possible Turing machines To be somewhat more explicit, we m ight code the states and alphabet symbols as sequences o f l ’s and separate them b y 0’s. A com plete cod in g for a Turing machine m ight, then, look something like this: W e m ight also agree on a fixed or der o f listing the quadruples so that each T M has a unique representation in this coding scheme, W e could now enu merate T M ’s b y listing them with machines w ith the smallest num ber o f quadruples first in increasing order according to their encodings interpreted as a binary num ber. Thus we have a on e-to-one correspondence between T M ’s and the natural numbers, (Incidentally, this fact shows us that there must be languages which are not Turing acceptable sim ply b y considering the cardinalities o f the sets involved, Given a finite alphabet A , there are Ho strings in A*. There are 2^° subsets o f A ", i.e., languages over the alphabet A B y C antor’ s T heorem , 2^° > Ho and since there are on ly Ho Turing m a chines, there is an uncountable infinity o f languages over any given alphabet w hich are not Turing acceptable.)
T
he u n iv e r s a l
T
u r in g
m a c h in e
521
O i l . .. 11101. . . 11011 ,. 111011.. . 11011110111. . 1 1 0 1 1 ... 1 etc, c o d in g fo r in i t ia l sta te
sym * b o l or L or R
sym bol
c o d i n g f o r fir s t q u a d r u p l e
|
c o d in g fo r secon d q u a d r u p l e , etc,,
Figure 19-3
Let us also assume that input strings are encoded into l ’s and 0 ’s in some fixed fashion, W e may also assume w ithout loss o f generality that whatever ou tpu t a T M leaves on its tape can also be encoded in to l ’s and 0 ’s. Let us denote b y E ( M ) an encoding o f a Turing m achine M and b y E ( x ) the en codin g o f an input string x. It now turns out that there is a T M U, the universal Turing machine, which can take as input E ( M ) E ( x ) and mimic the behavior o f M on x. That is, if M halts on x, U halts given E ( M ) E ( x ) and leaves on its tape an encoding o f whatever output M w ould have left when it halted; if M does not halt on x, then V does not halt on E ( M ) E ( x ) W e will not attem pt to give the construction o f U here, but it can be thought o f as a three-tape machine which keeps on its tapes ( l ) the encoded instructions o f M , (2) an encoded version o f the non-blank p ortion o f the tape M w ould have at each point in its com putation, and (3) an encoded representation o f the current state. U consults tape 3 for the current state, examines tape 2 to see which sym bol is under the reading head (o f M ) and then consults tape 1 to find the instruction beginning w ith that state and symbol. If none is found, M w ould have halted, and U halts. If one is found, U makes the appropriate changes to tape 2, changes the state on tape 3 and repeats the cycle.. T o dispel any possible air o f m ystery surrounding the universal Turing m achine, let us poin t out that sim ulating the moves o f any given T M on any given input tape falls under the class o f procedures w hich can be carried out in purely m echanical fashion.
(T h in k what w ould be involved if you
were asked to carry out this task yourself.) It is therefore executable b y a
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Turing machine. T he universal character o f the machine arises from the fact that all Turing machines and input tapes are given an encoding over a fixed alphabet (here, l ’s and 0 ’s, although any convenient alphabet w ould do). Thus U needs to be program m ed only to find instructions and carry them out on tapes all cod ed in the same way. A nother w ay o f form ulating what we have just said is that the language { E ( M ) E ( x ) | M accepts x } is Turing acceptable. This is a language over the alphabet { 1 , 0 } , and it is accepted b y the Turing machine U (actually, a slight variant o f U which first checks to see if the string which it has received as input is in fact o f the right form to be a T M encoding follow ed b y a string encoding; this again is an easily arranged m echanical procedure )
19.7
The Halting Problem for Turing machines
W e are now ready to address the problem m entioned above, namely, the problem o f deciding for an arbitrarily given T M and an arbitrarily given input string whether the T M will ever halt on that input, G iven our m ethod o f encoding T M ’s and input strings, it is easy to state the halting problem in terms o f Turing acceptable and Turing decidable languages. W e have just seen that the language (1 9 -1 0 )
L = { E ( M ) E ( x ) | M accepts x }
is a Turing acceptable language.
Thus, in order to determine whether M
halts given x, sim ply give the encoding o f M and the encoding o f x to (m odified) U I f M halts on x, U will also But now if M does not halt on x, U doesn’t halt either. W e want to know if there is some way to tell that M will not halt on x when that is the case. In other w ords, is L Turing decidable? Is there some T M which will halt and say yes if M halts on x, and will halt and say no when M does not halt on x? W e will show that this cannot be the case. Assum e L is Turing decidable b y some T M M l . Since L is decidable for all E ( M ) and E ( x ) , it will be decidable in the special case in which x happens to b e E ( M ) itself. It may seem strange to give a Turing machine its ow n encoding as an input tape, but since this encoding is just a lon g string o f l ’s and 0’s, there is nothing in principle to prevent us from doing so. T hat is, the follow ing language L\ is Turing decidable if L is:
T
(1 9 -1 1 )
he
H altin g P
roblem
for
T
u rin g
m a c h in e s
523
I i = { E ( M ) | M accepts E ( M ) }
It w ou ld in fact be decided b y a T M M\ which first encodes its input and copies it directly to the right of the original and then behaves like M l B ut now , since
is decidable, it follows that its com plem ent is decidable
and hence, Turing acceptable; i.e., there is a T M , call it M *, which accepts L[. (1 9 -1 2 )
L[ = { x £ { 0 ,1 } * | x is not a T M encoding, or else x is the encoding o f a T M T and T does not accept E ( T ) (i.e., x ) }
We now ask whether the encoding o f M " itself is in
that is, is M * a
m achine which does not accept its own encoding as input? First case: E ( M * ) £ L[. Then E ( M ><) is one o f the strings accepted by M " , b y the assum ption that M “ accepts L[. So M * accepts E (M ~ ). But because E ( M X) £ L [, it is the encoding o f a Turing machine which does not accept its ow n encoding, i e., M " does not accept E ( M “ ). C ontradiction Second case:
E (M ”) £ Lj
Then E ( M *) is not a string accepted by
M * , b y the assum ption that M “ accepts L[. So M “ does not accept E (M * ). Therefore, M " is a T M which does not accept its own encoding; therefore E ( M “ ) is a m em ber o f L [. Contradiction, Since E ( M *) must either be in L[ or not in and either assumption leads to a contradiction, we conclude that there is no such machine as thus, L[ is not Turing acceptable (our first exam ple o f a set o f this sort). But the Turing acceptability o f L[ was im plied b y its Turing decidability, w hich in turn was im plied by the Turing decidability o f L\ Therefore, we conclude that Li cannot be Turing decidable after all. Finally, Turing decidability was im plied b y the assumed Turing decidability o f L. Thus, we conclude that L is not Turing decidable, and the Halting P roblem for Turing m achines is not decidable b y Turing machine; hence, given C hurch’s H ypothesis, not decidable b y algorithm N ote that in the process o f proving the undecidability o f the Halting P roblem , we have exhibited a set which is not Turing acceptable (namely, L [ ) and sets w hich are Turing acceptable but not Turing decidable (namely, L and i i ) , W e can thus state the following: T
heorem
19.1
There are sets which a re n o t recursively enumerable,
524
T
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19,2
hapter
19
There are sets which are recursively enumerable but not
recursive. As we have seen, these are not ordinary garden variety sets.
■ T he latter
are exemplified by the set o f all encodings o f T M ’ s which accept their own encodings as input; the form er b y the set o f all Turing machines (in encoded form ) which do not accept their own encodings as input However, having established a footh old in this territory we can use these sets to discover others o f their class W e m ay also use the undecidability o f the T M Halting P roblem to prove that other problem s are undecidable as well, For example, we will show the follow ing problem for Turing machines to be undecidable: Problem: For an arbitrarily given T M M , does M halt given e, the em pty string, as input? W e first express the problem as a language: (1 9 -1 3 )
L 2 = {.E (M ) |M accepts e}
and ask whether there is a T M M 2 which decides this language. W e show that there is n ot, and the p r o o f technique is to show that if such a machine existed then it could be m odified to produce a m achine which decides the Halting P roblem Since the latter cannot exist, neither can M 2„ Suppose we have M 2, which b y hypothesis decides L 2, W e show how to use M 2 to construct a machine M , which decides L, where L is the language o f the Halting P roblem : (1 9 -1 4 )
L = { E ( M ) E ( x ) |M accepts x }
First o f all, for any given T M M and any given input string x, one can m odify M so that if it is started on the em pty tape it will first write x on it and then proceed as M w ould have, given x. Call this m odified machine M x . M x first checks to see if its input is the em pty tape, (If n ot, it runs forever in some fashion ) If so, it writes x (a finite string, so this is done b y some finite set o f instructions added to M ) and then positions its reading head for the start o f a com putation and executes the m oves o f M thereafter. Now if we assume that we have a machine M 2 w hich decides L 2, then it will work, in particular, if it is given the encodin g o f any m achine M x
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constructed in the way ju st described. ( M 2 works for any T M encoding; it will work i f that encoding happens to b e the encoding o f M x .) B ut notice that M x accepts e just in case M accepts x. That is, M x halts, given the em pty string as input, iff M halts given x. Therefore a machine which decides whether M x halts given e could be used in effect to decide whether M halts given x. This has been shown to be im possible so there is no such m achine as M 2 , and language L 2 is therefore not Turing decidable. A whole host o f problem s concerning Turing machines turn out to be undecidable: w hether an arbitrarily given T M ever enters a particular state, whether' it halts on any inputs at all, whether it halts on every input, whether it ever writes a particular sym bol on its tape, etc. These and other unde cidability results can, in turn, be used to establish undecidability results in other areas, For exam ple, the undecidability o f the T M Halting P rob lem can be used to establish the undecidability o f another problem called the Post C orrespondence Problem , This can then be used in showing that certain problem s concerning context free grammars and languages are undecidable. For exam ple, it is undecidable, given tw o arbitrary c fg ’s G i and G 2 whether L ( G i ) = L ( G 2), whether I ^ ) H L { G 2) = 0, whether L ( G i ) C L ( G 2), w hether L(G\) is inherently ambiguous, etc. One should note carefully that none o f these undecidability results im ply that for a particular T M , a particular cfg, etc. there is no way to determine whether it halts given the em pty string, whether it is inherently am biguous, etc. W e have seen examples o f T M ’ s, e.g., in (19-1), which can b e shown, quite easily in fact, to halt given the em pty string. W hat the undecidability result says is that there is no single, generally applicable algorithm which is guaranteed to w ork for every T M (or every arbitrarily given pair o f cfg ’s, etc.) It is also w orth noting that in view o f the correspondence shown above b e tween T M ’s and unrestricted grammars, the undecidability results for T M ’ s can be carxied over im m ediately to grammars. Thus, there is no algorithm for determ ining for' an arbitr arily given T yp e 0 grammar G whether G gener ates any strings, w hether G generates the em pty string, whether G generates all strings in S *, etc.
Exercises 1. C onstruct a Turing machine that accepts any tape written on the v o cabulary { 0 , 1 } and converts every contiguous string o f two or more l ’ s
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to 0’ s. Everything else is left unchanged. For exam ple, the input tape ••#01 011 0 1 1 1 0 1 # ■■ should end up as ■•■# 0 1 0 0 0 0 0 0 0 0 1 # • 2 . Construct a Turing machine w ith three states {<7o5<7i 5<72}: initial state go, that begins w ith an input tape consisting entirely o f blanks and halts with exactly three contiguous l ’s on the tape. 3 . Consider the follow ing Turing machine: M = {{ go , g i } , {a , b, # } , go, S) where (go, a)
6=
- > (go, R )
(g o ,# ) -*■ (gi,a) (qu a) —> ( g i , X ) , (?i, # ) ~ y (go,a)
( a ) W rite the first twelve situations o f the m achine M if it starts in the situation ( e , g o , a , # # a ) .
(b)
Describe verbally what machine M will continue to do after this m uch has b een done
( c ) W ill it ever halt? ( d ) W ill it use only a finite amount o f tape? ( e ) A ie there any squares o f the tape that it will scan only a finite num ber o f times?
(f) 4.
W h at will machine M do if started in another situation?
(a ) Make up a simple Turing machine which never halts n o matter what the initial tape sequence is. Give b o th the quadruples and a verbal description o f its behavior. Let the m achine be allowed to start scanning at any square but always start in state go-
(b)
Similarly, make up a Turing m achine w hich always halts eventu ally,,
5. Tell whether the follow ing functions are tota l or only partial A func tion is considered to be undefined if it w ould yield a value outside the set on which it is specified ( a ) A dd ition on the set o f all even integers.
(b)
A ddition on the set o f all prim e num bers,
(c ) Set union on the set {{0}, {1}, {2}, {0,1}}
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6 . Describe inform ally an algorithm for converting an integer in binary notation to decim al notation. 7. W rite a T yp e 0 grammar generating the language { a 2” j n > 0 }. 8 . Show that the follow ing problem for Turing machines is undecidable: For an arbitrarily given T M M , does M accept at least one string? (H int: Show that if a T M existed which decided the language { E ( M ) | M
accepts at least one string}, it could be m odified to produce a
machine deciding L 2 in (19-13).)
Chapter 20
Linear Bounded Automata, Context Sensitive Languages and Type 1 Grammars 20.1
Linear bounded autom ata
A linear bou nd ed autom aton (lba) is, in effect, a Turing m achine whose com putations are restricted to the amount o f tape on which the input is written, W e can imagine it as consisting o f a finite set o f states, a finite alphabet (including special right- and left- endmarkers [ and ]), a designated initial state, and a finite set o f instructions o f the same form as the quadru ples for Turing machines. W e assume, however, that the input to an lba is given betw een the designated endmarkers, i.e., as [uj] and that the lba has n o instructions which allow it to m ove past these endmarkers or to erase or replace them . Thus, the tape head can m ove on ly in the p ortion o f the tape originally occupied by w, although an equivalent form ulation o f lb a ’s sets the limit on usable tape not as equal to the length o f the input but rather as a linear function o f the length o f the input. (A linear function in a variable x is o f the form ax + b, where a and b are constants. P lottin g values o f ax + b for each value o f x on graph paper gives a straight line, whereas plots o f functions involving x 2 , x 3, etc. gives curves.) This is the source o f the name for these autom ata— the allowed com putational space is bou nded b y a linear function o f the length o f the input string. L b a ’s are thus placed between p d a ’s and T M ’s.
T hey can read and
write, and m ove left and right on the input tape like T M ’s, but T M ’s have 529
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a poten tially unlim ited amount o f tape available for com putation, A pda, on the other hand, uses a m axim um am ount o f tape for its pushdow n store which is a linear function of the length o f the input. (T h is is n ot obvious given the way we have form ulated p d a ’ s since a p d a could cycle endlessly on e-transitions adding m o ie and m ore symbols to its pushdow n store. It can be shown, however, that each p d a can be turned in to an equivalent pda which always reads to the end o f its input and which does not loop endlessly on e-transitions. Such a p d a uses at m ost an amount o f pushdow n storage which is a linear function o f the length o f its input.)
B ut p d a ’s lack the
pow er o f lb a ’s to m ove left and right and thus to have access to any portion o f the “m em ory” encoded on the tape during a com putation. L b a ’s com e in deterministic and non-determ inistic varieties, defined in ways analogous to the corresponding types o f Turing m achines, but it is not know n whether they are equivalent. T hat is, given language L accepted b y some non-determ inistic lba, we d on ’t know in general whether there is some determ inistic lba which also accepts L. This problem has remained open for m any years despite the efforts o f a g ood m any talented and energetic people to solve it.
2 0 .1 .1
L b a ’ s and con text sensitive gram m ars
R ecall that a context sensitive grammar (csg) is one in which every rule has the p roperty that the right side is at least as lon g as the left side. In particular, rules o f the form a —> e are excluded, and therefore the em pty string cannot be generated as part o f any context sensitive language
(A n
equivalent form ulation states that every rule is o f the form a A/3 —> aip/3, where A is a non-term inal, a,/3, and ip are strings over (Vt U V/v)"\ and ip ^ e„ See Section 16.4) E xcept for the m atter o f the em pty string, the languages accepted by non-determ inistic lb a ’ s are the languages generated b y csg’s. W e will sim p ly sketch here how this equivalence is established and refer the reader to H op croft and Ullman (1979, pp. 225-6) for details. Given a csg G , we can construct a non-determ inistic lb a M to simulate derivations b y G .
M is given an input string w em bedded betw een the
endmarkers; i.e., [w], and it sets up a work space to the right o f the input of length equal to the length o f w where it inserts the start sym bol S. M then non-determ inistically carries out a derivation from S according to the rules o f G, checking after each step to see if the contents o f the work space equal w. I f so, M halts and accepts; if not, M continues the derivation
C
o n t e x t
s e n s it iv e
l a n g u a g e s
an d
r e c u r s iv e
se t s
531
If there is a derivation o f w b y G, it will be found b y M , and, furtherm ore, it will be fou n d by some com putation which uses n o more than the allotted am ount o f work space Since rules o f G are “ non-shrinking,” n o line o f a derivation is shorter than the preceding one
Thus, n o derivation o f w by
G contains a line longer than w, and the derivation can b e carried out by M on an am ount o f tape which is a linear function o f (roughly, double) the length o f the input . T he construction in the other direction— sim ulating the moves o f an lba in reverse b y a csg— is similar to that for Turing machines and T yp e 0 gram mars. H owever, because o f the restriction that rules cannot make strings shorter, a problem arises in getting rid o f the boundary sym bols and the sym bol for the state o f the autom aton This can be accom plished b y coding sequences o f sym bols as com plex non-term inals; for details, see the section in H opcroft and Ullman (1979) referred to above. A nother m atter which must be attended to is that the grammar cannot generate e even if the lba accepts it; i.e., if L is accepted by a non-determ inistic lb a M , L — { e } is generated by the corresponding csg G.
20.2
Context sensitive languages and recursive sets
E very language generated b y a context sensitive grammar is a recursive set o f strings in the sense defined above in connection with our discussion o f recursive and recursively enumerable sets. That is, for every context sensitive language, there is a Turing machine which decides it. W e will show this— appealing to C hurch’s H ypothesis— b y sketching an algorithm for deciding any csl. It w ould not be very difficult to give this algorithm explicitly in the form o f a T M Given a csg G and a string x, we decide whether or not x £ L ( G ) as follows. Construct all derivations beginning with S and whose last lines are no longer than the length o f x. If any line in a derivation is repeated, the derivation can be disregarded, since, if there is a derivation o f x beginning with S, there is one with no repeated lines. The num ber o f derivations whose last lines are n o longer than the length o f x and which contain no repeated lines is a finite num ber. W e sim ply examine this finite list o f derivations to see if any end in x. I f none do, we can be sure that x is not generated b y G since derivations o f x cannot contain lines longer than x.
This follows
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20
from the fact that rules o f a csg never decrease the num ber o f sym bols in succeeding lines o f a derivation (N ote that the possibility o f such a decision procedure for T yp e 0 languages is precluded b y the possibility that rules in such grammar m ay decrease the number o f symbols. As a consequence, we cannot in general fix an upper b ou n d on the length o f a derivation o f a string o f given length by a T yp e 0 grammar
This reasoning does not prove
that n o decision procedure can be found for T yp e 0 languages— although in fact there is n one, as we have seen— only that we cannot arrive at one by enum erating derivations ) The context sensitive languages do not exhaust the recursive languages; there are recursive languages which are not csl’s. There is a sense in which this is trivially true since some recursive languages contain e, and n o csl’s do. B ut even if we consider the class o f all csl’s together with the class of all csl’s with e added (which are just the non-determ inistic lb a languages), there are still recursive languages which are not in this class. W e show this with respect to lb a ’s for simplicity, Like a T M , a non-determ inistic lba could b e encoded as a string over the alphabet {0 ,1 },, W e can then enumerate lb a ’s in the same fashion as we did for T M ’s— say, shortest encodings first and with encodings o f the same length b ein g arranged in increasing order when interpreted as a binary number. Thus, we have an enum eration o f lb a ’s:
(20-1)
M i , M2, , , , , M i , .
,
W e also construct an enum eration o f input strings (for exam ple b y coding them in the alphabet { 0 , 1 } and arranging them in the order o f increasing binary num ber o f the encoding), (2 0 -2 )
x 1 , x 2, ,, . , x i } ....
W e construct the language L in the follow ing way: (2 0 -3 )
L = { xi | and Xi is not accepted b y M ;}
T hat is, w is in L iff its encoding is not accepted b y the lb a at the corre sponding p osition in the enumeration, This language is not an lb a language since it is not accepted b y any o f the lb a ’ s in the enum eration. Given any M j , it cannot accept L because there is a string, nam ely x j , which is in L
iff it is not accepted b y M j . Thus, L differes from the languages accepted b y each o f the lb a ’s in at least one string
C
l o s u r e
and
d ec isio n
533
p r o p e r t ie s
L is nonetheless recursive There is a procedure for determining for any string w whether or n ot it is in £ E ncode w in the prescribed manner in to 0 ’s and l ’s, and find it in the enumeration, This can be done, since the encodings o f the x i ’s aie listed in order o f increasing length and within strings o f the same length in increasing order o f size as a binary number. H aving loca ted w in the enum eration, we get its in d ex (b y counting from the beginning) and find the corresponding lba in the enum eration o f lb a ’s W e then determine b y the algorithm referred to above whether the selected lba accepts w or not. If it does, w 0 L; if n ot, w £ L. Thus L is a recursive language which is n ot the language accepted b y any non-determ inistic lba.
20.3
Closure and decision properties
T he csl’s are closed under the operations o f union and intersection, but it is n ot know n whether they are closed undei com plem entaiton (this is essentially the p roblem o f determ ining whether deterministic and non-determ inistic lb a ’s are equivalent). T hey are closed under concatenation but not under Kleene star, inasmuch as L" must contain e, and no csl does.
The csl’s,
however, are closed under positive closure L + , defined as L" — {e } . T hey are also closed under intersection with a regular language. W e om it the demon strations o f these results here; the interested reader m ay consult H opcroft and Ullm an (1979, Chapter 11) and references therein for m ore details. A lth ou gh the mem bership question is decidable for csl’s, alm ost all the other standard question turn out to be undecidable: (i) Is L := 0? (ii) Is L ■= S '? (iii) Is Li (iv) Is Li C L 21 (v ) Is h
n L2
= 0?
534
C
h a p t e r
20
Exercises 1. Construct context sensitive grammars for each o f the follow ing lan guages. ( a ) anbncn (n > 1)
(b)
{ x £ {a , 6 ,c}* | x consists o f an equal num ber ( > 1) o f a ’s, 6’ s, and c ’s}
( c ) { x x |x £ {a , 6 }+ } (i.e.,, all non-em pty strings in {a , 6}* with iden tical left and right halves.)
Chapter 21
Languages Between Context Free and Context Sensitive
M ost investigators supposed from the time that the Chom sky hierarchy was first established that natural languages, considered as string sets, would fall somewhere betw een the context free and the context sensitive languages and, further, that they would lie in some sense “ close” to the context free class. On the one hand, the context sensitive languages seemed m uch to o inclusive, containing as they do species such as { a nbn'} (ra! is ra factorial, i.e ., 1 x 2 x 3 x . . . x » ) and { a n : n is p rim e }, which seem unlikely candidates for any sort o f linguistic m odel. On the other hand, a large part o f natural language syntax seems to b e handled quite nicely b y a C F G , and the aspects which seem to cause languages to fall outside the C FL class (as string sets) could be considered rather isolated and infrequent. A fter all, it took nearly thirty years to find one com pletely convincing exam ple o f a natural language which was not context free. Such considerations spawned work on the form al languages which would be, in A ravind Joshi’s phrase, “m ildly context sensitive.” In^th'e follow ing sections we will survey very briefly a few o f these whose form al properties have been m ost thoroughly investigated. 535
536
21.1
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h a p t e r
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Indexed grammars
A ccordin g to H opcroft and Ullman (1979, p.
389) “ o f the m any general
izations o f context free grammars that have been p roposed, a class called ‘ in dexed’ appears the m ost natural in that it arises in a wide variety o f con texts.” A n indexed grammar (IG ) differs from a context free gram m ar in that nonterm inal sym bols m ay carry a sequence o f indices chosen from an initially specified finite set, and its productions allow foi adding or rem ov ing these indices in the course o f a derivation.
W e write A [ i , j , k ] for the
nonterm inal sym bol A bearing the in dex sequence [t, j, k]
Indices can be
added or rem oved from the left end o f the sequence, and there is no upper bou nd on the length Thus, an index sequence is like a pushdow n stack attached to a nonterm inal. Further, when a rule o f the ordinary CF form , e.g,, A —> a B c d C , is applied, the index sequence on A , if any, is copied onto each o f the nonterm inals on the right, i.e ., B and C, Term inal sym bols never bear indices. If A is rewritten as a string containing no nonterm inals, e.g., A —> 6c, the indices on A sim ply disappear. P roductions m ay also be o f the form A[i] —> a, where i is an in d ex and a is any string, in which case the rule can only be applied to a nonterm inal A bearing index i at the left o f its index sequence, and if a contains any nonterm inals, then the sequence on A with i rem oved is copied on to each o f them. For exam ple, the rule A[i\ —> B a C could be applied to the indexed nonterm inal A [ i , j , k ] to produ ce B [j, k]aC[j, k]. The rule could not b e ap plied to A [j,k ] or A [ j , i ] , for exam ple, since neither in d ex sequence has i at the left. Thus, rules o f the ordinary CF form A —> a are accom panied by in dex copyin g, while rules o f the form A[i\ —> a aie “p o p and co p y ” rules. Finally, rules o f the form A —» B[i\ add in dex i at the left o f the sequence already present on A and cop y the result on to B (a “ push and co p y ” rule). Such rules have only a single nonterm inal cm the right because we d on ’t want to allow the possibility o f pushing different indices and copying onto different nonterminals; i.e., A —> 5 [z ]C [j] is not allowed. W e are follow ing here the form ulation in H opcroft and Ullman (1979, pp. 389-390) with slight notational changes. Different but equivalent form ulations can b e fou n d in A h o (1968), Salomaa (1973, pp. 257-259) and Gazdar (1985). Thus, the I G ’s properly include the C F G ’s, since a C F G can b e regarded as an IG in which all index sequences are empty. That the C F L ’ s are a p roper subset o f the index languages (IL ’s) is shown b y the follow ing IG generating the non-con text free language { a nbncn : n > 0 }:
I
G =
n d e x e d
537
g r a m m a r s
{ V t , V n , I , S , R ) , where
{ a , b , c , } ; Vn = { S , T , A, B , C } ; I = R =
{*>.?}> the set o f indices; {S -> T [j} A [i]^ a A A [j]^ e T —>T[i\
B[i\ —>bB
B [j\ ^ e
T —> A B C
C\i\-^cC
C\j\ —> e }
Fig. 21-1 is a derivation tree for a 262c2
5 T\j] T [i,j] T [i,i,j] A [i,i,j] a
B [i,i,j]
A [i,j] a
A [j]
b
C [i,i,j]
B [i,j] b
e
B [j] e
c
C [i,j} c
C [j] e
Figure 21-1.
N ote that it is necessary to use an index distinct from i as a “b o tto m o f stack” marker in this grammar. Because o f the convention that index sequences disappear when the right side o f a p rod u ction contains no nonter m inals, if we had only the rules A[i] —>a A and A[i\ —> a rewriting A, then A[i, i\ cou ld give rise either to a or aa. T he gram m ar containing the follow ing rules generates { x x : x £ {a , 6 }“ }:
5 38
C
S -+ T [k] T —>T[i\ T —>T\j]
h a p t e r
f [ i ] -> a F F [j]^ b F F [k ]^ e
21
G [i\ -^ a G G[j] —>bG G[k] —» e
T -^ F G A derivation tree for abbabb appears in Fig. 21-2.
Figure 2 1 -2 .
In the first exam ple, indices are used as counters to ensure that the num bers o f a ’s, b’s, and c ’s are all equal; in the second, the indices constrain the rewriting sequence so that the tw o halves o f the string are identical. The follow ing grammar (henceforth we give just the rules) com bines elements o f b o th techniques to generate { anbmcndJn : m , n > 0 }, which exhibits the kind o f cross-serial dependencies found in Swiss Germ an (Sec. 18.6 above):
In
d e x e d
g r a m m a r s
S-»T[Jfe]
A [i\ -^ a A
C\i\-^cC
T - » T [j]
A -^ B
C —>D
r - r
B \ j]^ b B
D [7 ] —> dD
f - * r '[ i]
B[Jfe]-»e
2?[Jfc] —►e
539
T '-^ A C Fig, 21-3 is a derivation tree for abbcdd.
Figure 21-3, It is worth noting here that alternative grammars are possible fo r the preceding exam ples, some o f which will p rod u ce very different derivation
C
5 40
21
hapter
trees. For exam ple, the follow ing grammar also generates { a nbncn : n > 0 } with the derivation tree for a 2b2c 2 in Fig. 21-4 S -> T [i] T —>T\i)
A [i]-^ a A A [j] —>e
B[i] —>bBc B [j] —>e
T —>A B
Figure 2 1-4.
T he context free languages can o f course be generated b y IL ’s w ithout indices, but it is also possible to use the pow er o f the indices to generate C F L ’s with right- (or left-) linear derivation trees. See, for exam ple, the follow ing IG , w ith an accom panying derivation in Fig. 21-5, w hich gener ates strings in anbn with a right-linear rather than a center-em bedded tree structure: S -^ A [k] T -^ A [i]
A —>aT A —* B
B [i]-^ bB B [k ]-^ e
I
n d e x e d
g r a m m a r s
541
5 A[k\ a
T[k] A[i, k] a
T [i,k ] A[i, i, k] k] b
B [i,k] b
B[i, k]
Figure 2 1-5.
A h o (1968) was in fact able to prove that the languages generated b y I G ’ s with entirely right-linear or left-linear tree structures were exactly the C F L ’s. Gazdar (1985) suggests that this fact m ay be o f linguistic significance, cit ing examples o f nested dependencies which appear to require a right- or left-linear rather than a center-em bedded constituent structure (e.g., nested com parative constructions and unbounded dependencies in the Scandinavian languages). The I L ’s have an equivalent characterization in terms o f autom ata, viz,, the nondeterm inistic “ nested” stack automata, A stack autom aton is like a P D A which, in addition to pop p in g and pushing sym bols at the top , can also enter its stack in “read-only” m ode, m oving up or dow n without writing or erasing any sym bols. A nested stack autom aton has the further capability
542
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21
o f creating stacks inside its stack, stacks inside these stacks, etc., but it can only em pty a stack after all stacks within it are empty. It can be shown that the I L ’s (w ithout e) are contained in the contextsensitive languages and that the containm ent is proper. T hat is, we have: (2 1 -1 )
C F L ’ s C IL ’s C C SL’ s
Exam ples o f C S L ’s which are not IL ’s aie {a nl : n > 1 } and { ( S u i) ^ : w £ { a , 6 }“ } (Hayashi, 1973). The language {w £ { a , b ,c}~ '■ w contains equal numbers o f a’s, b’s, and c ’s } is conjectured b y M arsh (1985) to be non-IL (we have already seen that it is CS), T he language { a nbn : n > 1 } is IL, however, and is generated by the follow ing grammar (m odified from Salom aa (1973)): S -^ T [k] T —>T\i\
A [i\ -^ a A A [k ]-^ a
Z [i]-^ bZ Z [k]-^b
B [i]-^ b B Z Z B [k]-^b
T —> A B Fig, 21-6 is the derivation tree for a3 b9. The I L ’s are closed under such operations as union, concatenation, inter section with a regular set, and Kleene closure, but they are not closed under intersection or com plem entation.
There exists an algorithm for deciding
whether the language generated b y an arbitrary IG is empty, G azdar and P ullum (1985, pp. 11) state that “ no phenom ena aie known which would lead one to believe that the NLs [natural languages] fall outside [the IL s’] purview... T he indexed languages thus p rovide us, at least for the m om ent, with a kind o f upper b ou n d for syntactic ph enom ena.” This is of course in the context o f the assum ption that languages are being regarded as string sets.
21.2
Tree adjoining grammars
Tree adjoining grammars (T A G ’s) were devised b y Joshi and his coworkers (Joshi, Levy, and Takahashi, 1975) as lim ited extensions o f C F G ’s
They
occu py, in terms o f weak generative capacity, a p osition betw een C F G ’ s and indexed grammars; i.e., (2 1 -2 )
C F L ’s C T A L ’ s C IL ’ s
T
r e e
a d jo in in g
543
g r a m m a r s
5 T[k\ T [ i t k} T [i,i,k ] A[i, i, k] a
A [i,k ] a
B [i,k ]
A [k ]
Z[i, k]
b B[k] Z[k] Z[k]
b
Z[i, k]
Z[k]
b Z[k]
Figure 21-6.
A T A G , unlike the grammars we have seen thus far, does not generate sentences by rewriting strings o f sym bols; rather, it begins with a finite set o f initial trees, which can be enlarged by recursively inserting at appropri ate positions one o f a finite num ber o f auxiliary trees. For exam ple, if the follow ing is an initial tree,
(2 1 -3 )
a
=
S
a
and /? is an auxiliary tree
b
544
C
21
h a p t e r
T c
S
T
a
then /? can b e “ adjoined” to a by 1) excising the subtree rooted b y T from a giving
S
A\ a
b
2 ) placing /? in the position just vacated giving
T c
S
T
a
3) reattaching the excised subtree at the position o f the T node in j3 giving
5
Tree
a d jo in in g
g r a m m a r s
545
In a T A G initial trees are always sentential in that they are rooted b y S and have a string o f terminals as their yield, A uxiliary trees can have any nonterm inal sym bol as root and must have that same sym bol occurring once in its yield (on its “frontier” ), all the rest being terminals. A lthough not part o f the form al definitions, the intent is that b o th initial and auxiliary trees be “m inim al” Initial trees should contain n o repeated non-term inals along any path , and auxiliaries should contain n o repetitions o f the nonterm inal labeling the root node and the n ode at the frontier (the “fo o t ” n od e) on the p ath connecting the two. G iven a tree 7 containing a nonterm inal node X , and an auxiliary tree /? with X as root (hence with X as the only nonterm inal on its frontier), we can represent the operation o f adjunction schem atically as in Fig. 21-7,
7
=
5
/?
=
X
Figure 21-7.
(T h e similarities betw een the generation o f trees b y a T A G and the de com position o f derivation trees em ployed in the p r o o f o f the Pum ping T he orem for C F L ’s is evident.) Formally; a T A G G = {I , A ) is com posed o f finite sets o f initial and auxiliary trees, I and A , respectively.
T he set o f trees generated b y G is
the set o f trees obtainable from any tree in I b y repeated (possibly zero) applications o f adjunction using trees from A. The language generated, L (G ) , is the set o f term inal strings on the trees generated by G N ot surprisingly, for every C F G there is a strongly equivalent T A G (one which generates exactly the same strings with the same tree structures). In fact, given any C F G there is a procedure for constructing a T A G which is strongly equivalent to it.
Strong equivalence implies weak equivalence o f
course, so the C F L !s are included in the T A L ’s. There are, however, T A G ’s
C
5 46
hapter
21
for which no strongly equivalent C FG exists (see Joshi, 1985, pp.
211-2,
for an exam ple), and, m oreover, there are T A L ’s which are not weakly CF, Since the T A L ’s are such a m odest extension o f the C F L ’s, it is difficult to exhibit a simple exam ple o f a non-C F T A L T he follow ing is from Joshi (op. cit , pp. 212-3): (2 1 -4 )
L i = { w c n : n > 0 and to is a string o f a ’s and b’s such that (i) the num ber o f a ’s = the num ber o f b’s = n , and (ii) for any initial substring o f w, the num ber o f a ’s is > the num ber o f 6’ s}
Li can be thought o f as m ade from the language {(a&)ncn} by displacing some o f the a ’s to the left. Thus, aabaabbbcccc is in L i but aababbbacccc is not. Li is not CF since its intersection with a'bxc* is { a nbncn} generated by the follow ing T A G : (2 1 -5 )
I: a =
B ut Li is
S e
A: 0 i = S / X a
/?2 = T / X a
T
A\ b S
5
/N c
b T
c
The derivation o f the tree for aabbcc appears in Fig. 21-8. {a n bncn : n > 0 } is an exam ple o f an indexed language which cannot be generated b y a T A G . I f there were a T A G generating this language, each o f its initial trees would have to contain an equal num ber o f a ’s, b’ s, and c ’s in Older (recall that all initial trees are sentential and therefore have yields which are in the language). Further, each auxiliary tree also would have to have equal numbers o f a ’s, b’s, and c ’s (because each adjunction o f an auxiliary tree also produces a sentential tree) with all a ’s preceding b’s, which precede all c ’s. B ut then adjunction cannot avoid m ixing the order o f the elements and thus some strings not in the language will be generated, Joshi (1985) shows that an extension o f the theory o f T A G ’s to include local context-dependence for adjunction o f auxiliary trees yields a class o f
T
r e e
a d jo in in g
a
T
547
g r a m m a r s
a
T
A\ b S
c
a
S b T
c
A\ b S
c
Figure 21-8.
grammars w hich can generate such languages as { anbncn} and { w w : w £ {a , &}*},, These grammars are also o f interest since with a suitable notion o f linking defined betw een pairs o f nodes in trees it appears that structures w ith cross-serial dependencies can be generated T he languages o f the ex tended T A G ’s still do not equal the full class o f indexed languages since languages such as { a 2 : n > 0 } are beyon d theii range. This is evident from the fact that in any T A G (con text restrictions are n ot relevant here) during an application o f adjunction a term inal string can be increased in length by only the num ber o f term inal symbols in an auxiliary tree. The result is that the grow th in length o f term inal strings for any T A L is represented b y a series o f linear progressions, none o f which grows as fast as the exponential grow th rate o f strings in a 2 . It has been shown, however, (Vijay-Shankar, 1987) that the T A L ’s are weakly equivalent to a restricted class o f indexed languages, viz., those generated by IG ’ s whose rules limit the inheritance and m anipulation o f in dex sequences to a single nonterm inal daughter on the right side o f the rule. Gazdar (1985) points out that such a restriction make I G ’s o f lim ited interest as natural language grammars since con ju n c tion constructions would seem to require the inheritance o f index sequences b y m ultiple daughter nodes. The T A L ’s are closed under union, concatenation, Kleene star, and in
548
C
h a p t e r
21
tersection with regular languages, but not under inter section or com plem en tation
21.3
Head grammars
P ollard (1984) proposed an extension o f C F G ’s in which each string con tains a distinguished sym bol called the head o f that string. In form al terms, a headed string can be thought o f as a pair consisting o f a string and a natural num ber (up to the length o f the string) indicating the position o f the head. T w o headed strings can be concatenated, in which case one o f the heads becom es the head o f the resulting string, and there are “ w rapping” operations as well in which a string is split at its head and a second string inserted A norm al form theorem due to R oach (1987) guarantees that Head Grammars (H G ’s) need only tw o concatenation operations (in which the left, resp. the right, head becom es head o f the result) and four wrapping opera tions (the second string is inserted either left or right o f the head of the first, and either the first or second head becom es the new head). Formally, a H G is a quadruple (Vn ,V t , S , P ) , where Vn and Vp are finite sets o f non-term inal and term inal sym bols, respectively; S , the initial sym bol, is a m em ber o f Vn ; and P is a set o f produ ction s o f the form
A —>£*1 or A —>f ( a i ....a n)
where A 6 Vn; o-i is either a nonterm inal or a headed string; and / is a concatenation or a wrapping operation o f the sort just m entioned The languages generated b y H G ’ s, Head Languages (H L ’s), are included in the T A L ’s (W eir, Vijay-Shankar, and Joshi, 1986). The inclusion in the reverse direction has not yet been dem onstrated owing to technical difficul ties arising from the n otion o f the ‘headed em pty string.’ This has led Joshi, et al„ to define a closely related class o f grammars, the so-called M odified Head Gram mars (M H G ’s) in which these difficulties are avoided and which are provably equivalent in weak generative capacity to T A G ’s Strings in M H G ’s have, instead o f heads, a designated position betw een sym bols where the string can be split and a second string inserted. A “ split string” is de n oted Wi | W2 , where wi and to2 are in V£, and in a M H G there are just three operations: tw o concatenations and one wrapping.
H
e a d
549
g r a m m a r s
Ci(u>l T ™2 l« l T «2 ) =
Wl T
C 2(w 1 f W2, U 1 f
U2)
=
W1W2Ui T u 2
W2, U 1 f
U2)
=
WiUi
W (w !
f
W2 U i U 2
f
U2W2
The form o f the produ ction s and the rest o f the form alism otherwise remain as in H G ’s except that the a» can be either a nonterm inal or a split string Here is an exam ple o f a M H G generating the language { a ngbnf c nh |n > 0 Vr = P =
{a , b, c , f , g , h } ; Vn = {S , S1 , S 2, S3} ; S —> W ( S i , f T) S - + W ( S 2, f T) Si -> C 2(a T, S s ) s2 - * g f h Sz —> W ( S i , b T c)
S z ^ W ( S 2, b U )
Here is a derivation o f aagbbfcch:
S =* W ( S 1, f r) =* W ( C 2(a T, 5 3), / T) =* W ( C 2(a T , W i , & T < = )),/T)=> W ( C 2(a T, W ( C 2(a T, S3),b T c), / T) => W ( C 2(a T, W { C 2{a t, W ( S 2, b f c )), b T c), / T) => P 7(C 2(a T, W"(C2(a T, W ( g T h, b f c )), b f c), / |) => P7(C 2(a T, W"(C2(a T,
^ (C '2(oT.Wr(aff6TcA,6Tc),/T)=> W^(C2(at,a?66T ccA),/T) => W (a a gbb f c c h , f T) =£• aagbbf | cc/i
The spht marker is ignored in the final string, so the grammar is taken as generating aagbbfcch T he H L ’s are included in the M H L ’s, but the converse has not been shown, again because o f the difficulty m entioned above concerning the headed em pty string,
The differences betw een H G ’s and M H G ’s, however, are
marginal, and the systems are equivalent in all im portant respects.
Thus,
M H G ’s, T A G ’s and I G ’s restricted to inheritance o f indices b y a single n on term inal daughter ( I G ( l ) ’s) form a stable class o f weakly equivalent gram mars in the “m ildly context sensitive” dom ain
5 50
C
21.4
21
h a p t e r
Categorial grammars
C ategorial grammars are based on the work o f A jdukiew icz (1935) on se m antic categories o f logical languages and were proposed has a system o f syntactic description foi natural languages b y B ai-H illel (1953), M on tagu e’s work in natural language semantics assumes a categorial syntax, and this in part has led to a ievival o f inteiest in such systems T he fundam ental idea in a categorial grammar (C G ) is that lexical items m ay be assigned not only to basic categories such as N and V , but also to com plex categories like S /V P , V /( N P /S ) , etc. Sequences o f categories can be reduced by a kind o f “ cancellation” rule o f the form A /B B =»• A That is, if x is o f category A /B and y is o f category B , then the sequence x y is o f category A. A “ backward cancellation” rule, B A \ B =£• A , is oidinarily included as well. T o take a natural language exam ple, M ary m ight be o f category S /V P and sleeps o f category V P ; then Mary sleeps would be o f category S. Further, if often were assigned category V P /V P , the sequence Mary often sleeps would be reduced to category S by the follow ing sequence o f steps: S /V P
V P /V P
VP
M ary
often
sleeps
=>
S /V P
VP
M ary
often sleeps
=>
S M ary often sleeps
T he same inform ation could be represented as a tree structure:
VP S /V P
V P /V P
VP
I
I
I
M ary
often
sleeps
N otice that in a C G the paucity o f syntactic rules is com pensated for by the com plexity o f the syntactic categories to which constituents belong. In some sense, C G ’s code their syntactic operations in to their lexicons One o f the advantages o f such a system lies in the close correspondence o f syntactic and semantic structures. Parallel to syntactic reduction o f categories X / Y
C
a t e g o r ia l
g r a m m a r s
551
and Y to X is the semantic operation o f applying a function (from Y -ty p e things to X -ty p e things) to an axgument (o f type Y ) to produce a value (o f type X ) It is for this reason that the reduction rules are often referred to as rules o f (forw ard and backw ard) Function A pplication, (See the discussion in Chapter 13 where this is applied to natural language semantics.) T he form al definitions o f what m ight be term ed a classical, bi-directional C G w ould be as follows: W e assume a finite alphabet V a o f category sym bols. T he set C o f all category sym bols is then given by: (1) For all x £ Va ,
x
£ C,
(2) I f x and y are in C , then so axe (x / y ) and (x \ y ) . (3) N othing else is in C. A C G then is a quadruple G = {V t , V a , S, F ) where Vt is a finite set o f term inal symbols Va is as above S is a designated mem ber o f V a F is a function from Vt to p(C)', i,e., a function assigning to each term inal sym bol a set o f categories from C . W e define the binary relation =s>, “reduces in one step t o ” , on pairs o f category sequences as follows: For any categories X , Y in C and any finite sequences o f categories a and /?, a X / Y Y /? => aX(3 a Y X \ Y j3 => a X j3 The relation =? “reduces to (in zero or more steps)” is the reflexive transitive closure o f =^. A string w £ V f is generated b y G iff w = wj ., . wn and there is some sequence o f categories Ci ... Cn such that F (w i ) = Ci for all 1 < i < n , and Ci . ,, Cn => S . In other words, w i . ... wn is generated iff there is some choice o f category assignments by F to the sym bols in wi . wn which reduces to S.
552
C
h a p t e r
21
E xam ple: G — { V t , V a , S, F ) where Vt = {a , &}; Fa = { 5 , B } ; F ( a ) = { S / B } ; F (b ) = { B , B \ S }
T he string a363 is generated as indicated by the derivation tree in Fig 21-9
Figure 21 -9 .
C G ’ s have been shown to be identical to C F G ’s in weak generative ca pacity (Bar-Hillel, Gaifman, and Shamir, 1960) T h e p ro o f is constructive; from any given (basic, bi-directional) C G , a weakly equivalent C F G can be constructed, and vice versa. M ore recent work in categorial grammar has focused on systems with expanded sets o f reduction rules. Some o f those m ost frequently encountered
(2 1 -6 )
1 X / Y Y j Z => X j Z
Function C om position
2, (X / Y ) \ Z => (X \ Z ) / Y
C om m utativity T yp e raising
3, X = > y / ( y \ x ) 4, X / Y => (X / Z )/ (Y / Z )
“ G each R u le”
These rules also have corresponding form s w ith / and \ interchanged; e.g.,
C a t e g o r ia l
grammars
553
Y \ Z X \ Y =£• X \ Z (B ackw ard Function C om position) A system containing all the rules in (21-6) (in b oth forward and back ward versions) together with the two standard rules o f Function A pplication was proposed by Lam bek (1958) and has com e be know n as the Lambek Calculus. Van B entham (1988) has shown that a particular variety o f this calculus (in which categories axe non-directional, i e , b o th X \ Y Y => X and Y X
\ Y => X
are valid) generates the perm utation closure o f the
C F L ’s The perm utation closure o f a language L is the set o f strings obtained by taking every string L and perm uting its sym bols in every possible w ay For exam ple, the perm utation closure o f { a n bn \n > 0 } is { x £ {a , 6 }“ | x contains equal numbers o f a ’ s and 6’s }
B oth these languages happen to
be C F , but the perm utation closure o f a CFL is not always CF. For ex am ple, { ( abc)n |n > 0 } is C F (indeed regular), yet its perm utation closure {a; £ {a , 6, c } “ |x contains equal num bers o f a ’s, 6’s, and c ’s} is not CF, Thus the languages o f the non-directional Lam bek calculus include some n on-C F languages and do not include all C F languages. W eir and Joshi (1988) re p ort that C G ’s with directional categories, forw ard and backwaxd Function A pplication , and a generalized form o f (forw ard and backward) Function C om position are weakly equivalent to T A G ’s, and thus also to M H G ’s and I G (l)’s A num ber o f C G systems based on the work o f Ades and Steedman (1982) and others have been investigated by Friedman and her co-workers (Friedm an et a l , 1982) These systems have unidirectional, parenthesisfree categories (parenthesization is assumed to the left) and a variety o f function application and function com position rules Some systems generate the C F L ’s, others a subset o f the C F L ’ s, and still others, depending on the particular inventory o f reduction rules, generate languages which are “m ildly context sensitive” . A t least all C S L ’s discovered thus fax in the languages o f these systems are generated b y I G ( l ) ’s, which as we saw above are weakly equivalent to T A G ’s and M H G ’s, and have CF-like growth functions. T he whole area o f C G ’s and their m athem atical properties is one cur rently under active investigation, and it would be premature to try to give a m ore thorough summary. A representative collection o f recent work in C G can be found in Oehrle, B ach, and W heeler (eds.) (1988).
Chapter 22
Transformational Grammars One o f the m ost im portant developm ents in the m athem atical study o f gram mars o f linguistic interest was the work o f Peters and R itchie (1973) and Ginsburg and P aitee (1969) on the so-called “ standard theory” transform a tional grammars o f the sort outlined in Chom sky (1965). The syntactic com ponent o f a transform ational grammar consists o f tw o parts: (1) a base com pon en t, which is a C F G or C SG , and (2 ) a transform a tional com pon en t, com posed o f an ordered set o f transform ational rules. The base generates an infinite class o f trees by recursion through the initial sym b o l S. E ach such tree serves as an input to the transform ational rules, each o f which defines a m apping from trees to trees. The transform ational rules are assumed to be linearly ordered, the output o f one form ing the input to the next. I f a given transform ational rule cannot apply to a particular tree, the tree is left unchanged and the next rule in the sequence is considered. In this version o f transform ational gram m ar, the transform ational rules are assumed to apply in accordance with the principle o f the transforma tional cycle This specifies that the entire sequence o f transform ational rules applies fast to the lowest S -rooted subtrees (those that properly contain n o S -rooted subtrees). A single pass through all the transform ational rules is said to constitute a cycle, Subsequent cycles take as their domains succes sively larger S -rooted subtrees, until finally on the last cycle the dom ain o f application o f the rules is the entire tree. The tr ee that is the final output o f the transform ational rules is called a surface structure. T ransform ational rules perform elementary operations o f (i) deletion, (ii) substitution, and (iii) adjunction, but for our purposes here it suffices to 555
Chapter
556
22
observe only that deletion is constrained b y a principle o f recoverability M ore specifically, any item deleted b y a transform ation must either' leave behind a copy in a specified place in the tree or else be a fixed lexical item m entioned in the specification o f the transform ation. N ot every phase marker generated b y the base need be m ap p ed into a well-form ed surface structure by the transform ational com ponent. For exam ple, a special bou ndary sym bol # , which is placed on b o th sides of every S by the rules o f the base, may be erased b y the application o f certain transform ational rules. I f a phrase marker that is the final output o f the transform ational com ponent contains any instance o f # , it is not a wellform ed surface structure, and the phrase marker that form ed the original input is said to have been filtered out by the transform ational rules. A transform ational grammar with this property is called a transform a tional grammar with filtering. On this view , then, a transform ational gr am mar is a device for pairing base trees with well-form ed surface structures. The language generated by the grammar is the set o f all strings that appear as a sequence o f leaves o f some well-form ed surface structur e tree Peters and R itchie showed that a transform ational gram m ar form alized in this way with a context-sensitive grammar as its base has the same weak generative capacity as the type 0 grammars. That is, the languages gen erated by context-sensitive based transform ational grammars are just the recursively enumerable sets. The proof, which we only outline here, proceeds by showing first, for any given context-sensitive based transform ational grammar G, how to construct a Turing machine that accepts all and only the term inal strings in L (G ), Thus, every transform ational language is a r e set, The converse is proved in the follow ing way. I f L is a r e set, then there is some type 0 gram m ar Go generating L. N ext, a type 1 grammar G\ is constructed from Go such that for every a; if a; £ L ( G n), xbm £ L ( G o ) for some integer m , and conversely. This is done by taking each rule in the type 0 grammar whose right side is shorter than its left side and adding a sufficient number o f 6’s (a special sym bol not occurring in L ) to make the rule “no-shrinking,” T hen rules are added which perm ute the 6’s with the other sym bols, allowing the 6’s to m igrate to the right side. Thus G\ generates all the strings in L (G o ) (equal to L) follow ed by some num ber o f occurrences o f 6 From G\ it is possible by using some results o f K u rod a (1964) to construct an equivalent type 1 grammar G [ in which every string xbm generated is assigned the sort o f tree structure shown in Fig,
22-1, where x = a i , , ., a n_ i a n , and
T
r a n sfo r m a t io n al
5 57
grammars
A , A', A " , A'" etc., are nonterminals distinct from S. G [ is then taken as the base o f a transform ational grammar whose only transform ational rule is to delete an occu rren ce o f b if it is the rightm ost term inal sym b ol o f the sentence being processed on that cycle. Since this is the deletion o f a fixed term inal sy m b ol in a fixed position , it m eets the con d ition on recoverabil ity o f deletions. T hus, the transform ational com p on en t takes as in pu t a tree o f the form given in Figure 22-1 and deletes all occu rren ces o f b (t o gether with the nonterm inals dom inating each b) to p rod u ce a tree having x = c*i. . . as it leaves. T his transform ational gram m ar, therefore, generates all and on ly the strings o f th e original recursively enum erable set
L. S
/
\
S 5
/ S
A '"
A
\
.
I A
\
b
I A
b
a n- 1
Figure 22-1.
Peters and R itchie were able to show that transform ational
gram
mars still generate all r.e. sets even if the base is made context free rather than context sensitive. W ith a context-free base, however, the filtering effect o f the transform ational com ponent must be used extensively; this feature is not needed at all for context-sensitive based transform ational grammars to generate the r.e. sets. These results raise serious difficulties for the standard theory. Since it is com m only supposed that natural languages are properly included in the recursive sets, which are, in turn, properly included in the r.e. sets, trans
C
558
hapter
22
form ational grammars, by this form ulation, are capable o f generating sets of strings that are not possible natur al languages. One o f the goals o f linguistic theory is to give a precise characterization o f the n otion ‘possible grammar o f a natural language, ’ and thus this version o f the theory o f tr ansform ational gram m ar is shown to fail to meet this goal by virtue o f being to o broad, i.e., allowing to o large a class o f grammars T he nonrecursiveness o f transform ational languages was shown b y P e ters and R itchie (1971) to arise solely from the fact that for any arbitrarily given transform ational grammar G, there is n o upper b ou n d on the number o f S -rooted subtrees that can occu r in the base phrase marker underlying x. This is a consequence o f the fact that the theory allows the transform a tional rules to pare dow n base phrase markers containing a large num ber of com ponent S trees to produce very short strings w ithout violating the con dition on recoverability o f deletion In order1for transform ational grammars to generate only recursive sets they would have to be restricted in such a way that for any given gr ammar and any given sentence there is a procedure for determining the m axim um num ber o f cycles that could be involved in the derivation o f the sentence b y the gram m ar T h is w ould be assured if transform ational gram m ars had, for exam ple, what P eters (1973) has term ed the Survivor P rop erty: “if
( 22- 1)
1. 2 , S ^ &\&2 •••
b
together with a small number o f transform ational rules, suffice to generate
T
r a n s f o r m a t io n a l
g r a m m a r s
559
any recursively enumerable language on the alphabet { a i , a2, ..,., a „ } . From this it is relatively straightforward to show that there are an infinite num ber o f such trivial “ universal bases.” This result still holds when the grammar is constrained to be ‘ descriptively adequate,’ in the sense that it give the intuitively correct results with respect to gram m aticality, ambiguity, and paraphrase. Thus, given the powerful nature o f the standard theory, certain propositions such as the Universal Base H ypothesis, whose truth value ought to depend on the facts o f natural language, turn out to be trivially true. Results such as these provide a particularly convincing dem onstration o f the im portance o f finding precise m athem atical form ulations o f the gram mars allowed by a linguistic theory, for it is only with the aid o f such for m ulations that the consequences o f the theory can be carefully investigated. W ith ou t them , we have little hope o f being able to prove a theory wrong, m uch less o f seeing wherein it fails and how it m ight possibly be corrected.
Appendix E-I
The Chomsky Hierarchy Figure E.I-1 shows the inclusion relations that hold am ong the form al lan guages considered in this section (this is essentially the Chom sky H ierarchy), Tables E.I-1 and E .I-2 are a summary o f closure and decision properties for these classes o f languages
561
562
A
p p e n d ix
E -I
* The context sensitive languages are the non-deterministic linear bound automaton languages—{ e } It is not known whether the deterministic linear bounded automaton languages are equal to the non-deterministic lba languages or are a proper subset.
Figure E .1—1: Inclusion relations o f the classes o f form al languages
T
he
C
H
h o m sk y
56 3
ie r a r c h y
Operation fl with a regu
com ple Class o f languages
U
n
m entation
•
*
lar language
Regular
yes
yes
yes
yes
yes
yes
D et, p d a
no
no
yes
no
no
yes
C ontext free
yes
no
yes
yes
yes
yes
no ?
yes
C ontext sensitive Recursive Recursively enu
no*
yes yes
yes yes
yes no
yes yes yes
yes yes
yes yes
yes
merable T The context sensitive languages do not contain e and therefore ale never equal to S*
Table E .I-1 : Summary o f closure properties o f classes o f form al languages
Question Class o f languages
x€ L{G)t
L{G) = 0?
L(G) = £*?
L(Gi)
£(
i(G i) = £(
C
L{Gi)n
2 0
L(G ) = ?
Regular'
yes
yes
yes
yes
yes
yes
Det. p d a
yes
yes
yes
no
7
no
C ontext free Context sensitive
yes yes
yes no
no yes}
no no
no no
no no
Recursive
yes no
no
no no
no no
ho
no
no
no
R ecursively merable
enu
no
^ This case is trivially decidable (the answer is always n o) because the context sensitive languages do not contain e.
Table E .I-2 : Summary o f decidability properties o f classes o f form al languages
Appendix E-II
Semantic Automata Since the earliest days o f m odern generative linguistics the theory o f au tom ata and their relation to form al grammars has been a cornerstone o f syntactic linguistic theories, but recently semantic applications have been developed b y Johan van B enthem for ‘quantifier au tom ata’ In this section the m ain idea is outlined and illustrated, but for proofs o f the advanced theorem s the interested reader is referred to van Benthem (1986), chapter 8. A generalized quantifier is interpreted as a functor D e A B which assigns to each dom ain E a binary relation betw een its subsets A and B (see Chapter 14), In the procedural view o f quantifiers the determiner D o f the N P takes as input a list o f mem bers o f A marked for their m em bership in B and either accepts or rejects the list. In its m athem atical form ulation a determiner D is presented with finite sequences o f 0 ’s and l ’s, where 0 stands for members o f A — B and 1 for mem bers o f A f ) B , respectively T he output o f D is either ‘y es’ or ‘n o ’, depending on whether or not D e A B is true for the sequence read Examples: E v e r y A B is recognized by the finite state m achine in Fig, E.II-1. E v e r y A B is true iff \A — B\ = 0, so the E v e r y autom aton should accept all and only those sequences consisting o f only l ’s. In all figures So is the initial state, and the square state always marks an accepting state, so in Fig, E.II-1 So is also the accepting state. For instance, for e v e r y m a n w a lk s the l ’s represent men who walk and the 0 ’s men who do not walk. T he autom aton accepts only l ’s, i.e., whenever A C A fl B. The negated quantifier n o t e v e r y A B is obtained by reversing the accepting and rejecting states o f the e v e r y autom aton. 565
A
566
Q
*^0
p p e n d ix
1
E -II
H
Figure E .II-1 : E v e r y autom aton
Figure E .II-2 : N o t e v e r y autom aton
T he n o t e v e r y autom aton accepts sequences that consist o f at least one A that is not in B. Reversal o f accepting and rejecting states in finite state autom aton for a determiner gives in general the autom aton for the (externally) negated determiner. T he autom aton for a t le a s t o n e ( s o m e ) , the dual o f e v e r y , should accept any sequence containing at least one 1, i e , one A that is also in B. It exchanges 1 and 0 in the initial state o f the e v e r y autom aton and reverses its accepting and rejecting states. To com plete the traditional Square o f O pposition the n o autom aton, the negation o f the at lea st (s o m e ) autom aton, reverses its accepting and rejecting states, and accepts only sequences consisting o f only 0’ s.
S
e m a n t ic
a u t o m a t a
567
Figure E ,II—3: A t le a st o n e ( s o m e ) autom aton
Figure E .II-4 : N o autom aton
C om p a rison to the tree o f num bers There is a clear relation betw een the finite state autom ata for these de terminers and their tree o f numbers representation (see Chapter 14), In a tree o f num bers a node ( x , y ) corresponds to x 0’s and y l ’s in the input for the autom aton, since x = \A — B\ and y = \A fl B\, The tree for e v e r y shows that only the right m ost pair o f numbers on each line is not in the quantifier. T he trees for a t le a st o n e ( s o m e ) and n o are related in exactly the same way. Thus we see that negation o f a quantifier reverses the tree pattern and switches accepting and rejecting states. From a given tree the corresponding autom aton can be derived by a reduction algorithm. Let us look at the example for a t lea st o n e first. N odes in the tree are first identified with states, connected by adding 0 ’s or l ’s, and accepting states are indicated by squares. This can be shown equivalent to the simpler autom aton in figure E II-3 b y the reductions in
568
A
p p e n d ix
E -II
Figure E ,II-5 : T he a t lea st o n e tree o f numbers turned in to an autom aton
Fig. E.II-6. T he procedure which, turns a tree into an autom aton is specified as: (i) look at the downward triangle generated by a node ( x , y ) (ii) turn it into a + / — pattern (iii) if it is identical to the triangle pattern o f the n ode above and to the right, it should correspond to the same state as that one (iv) i f it is identical to the node up and left, it is the same state as that one (v ) the m inim um number o f states should b e the num ber o f distinct triangle patterns in the tree This procedure m ay not be com pletely general for arbitrarily com plex de terminers, but it works well for the lim ited cases we consider here. Here is one more com plicated exam ple for e x a c t l y t w o A B , In Figure E .II-7 the square states are the accepting states again, and the b old face states represent the four states needed for the autom aton, corresponding to the four different triangle patterns in Fig. e x a c t l y t w o autom aton is shown in Fig. E.II-9-
E.II-8.
The
S
e m a n t ic
569
a u to m a ta
( 1 ) mark all accepting states with |+ |— all rejecting states with |— | ( 2 ) the configuration:
+
+
+
corresponds to the a c c e p t a n c e f o r e v e r a fte r-sta te ( 3 ) the configuration:
0/ \ 1
corresponds to the r e je c t i o n f o r e v e r a fte r-sta te ( 4 ) the configuration: +
O'
0/
corresponds to the a c c e p t a n c e w it h 0-state ( 5 ) the configuration: 0/
corresponds to the r e je c t i o n w i t h 0-state ( 6 ) the configuration: + o corresponds to the a c c e p t a n c e w it h 1-state ( 7 ) the configuration: Jx,y corresponds to the r e je c t i o n w i t h 1-state Figure E..H—6..
-
A
p pe n d ix
E -II
Figure E J I -7 : The tree/a u tom a ton for e x a c tly tw o
Se m a n t i c
0 ,2 :
autom ata
571
0,3:
+ +
/ A / /
+
Figure E .H -8.
H ig h er order quantifiers The determiner m o s t is not recognized by any finite state m achine; this follow s from the Pum ping Lem m a. T he language for m o s t is context free so it can be recognized by pushdow n automaton. T he idea is that the stack stores the values read, and that com plem entary pairs 0,1 or 1,0 o f the top stack sym bol and the sym bol being read are erased as they occu r. W hen the entire sequence has been read the stack should only contain l ’s. This is slightly liberal version o f a pushdow n autom aton accepting with a n on em pty stack, but it can be turned into a norm al pushdow n autom aton with som e additional encoding.
A p p en d ix E - I I
572
It turns out that higher order quantifiers m ay need context free languages, b u t, for instance, the higher order quantifier a n e v e n n u m b e r o f is rec ognized by a finite state machine, T he follow ing theorem characterizes the first-order definable determiners. T
heorem
E .H .l
('van Be n t h e m (1986)) Al l and only all first-order definable
q u a n tifie r s a re c omput abl e b y permutation-invariant and acyclic finite state
machines.
■
Perm utation-invariance means that any perm utation in the order o f l ’s and 0’s o f an accepted or rejected list is accepted or rejected, and an autom aton is a cyclic when it contains n o non- trival loops betw een states, A proportion al determiner like a t le a st t w o -t h ir d s is essentially context free A pushdow n autom aton for it keeps track o f two top stack positions, com paring them with the next sym bol read. I f it reads 1 with 1,0 or 0,1 on top tw o positions o f the stack it erases that top and continues. Likewise if it reads 0 with 1,1 on top. In the other cases the sym bol read is stored on top o f the stack. W hen the entire sequence is read, the autom aton checks if the stack contains only l ’s, and if so, it recognizes the sequence. M any other determiners can be recognized by finite state or pushdow n automata. It turns out that natural language determiners venture rarely, if at all, beyon d the context free realm. For the proofs and m ore substantial arguments the interested reader is referred to van B enthem (1986),
Exercises 1. Design finite state autom ata for: ( a ) at lea st t h r e e A B ( b ) a ll b u t o n e A B (c ) an even num ber o f A B ( d ) a lm o s t a ll A B (in the sense o f ‘ with at most finitely many ex ceptions’)
leview Exercises 1. Consider the follow ing grammar G: S —> N seems certain S
—► I am right
N
—> that S
( a ) List three members o f L ( G ) ( b ) A bbreviating so that £ = { s , c , i , a , r , t } , construct a push down autom aton which accepts L( G) . 2. For each language below , construct a non-determ inistic finite state autom aton which accepts all and only the specified strings. cases the input alphabet { 0 , 1 } .
In all
( a ) T he term inal language o f the follow ing grammar: s —¥ 0 A
s
—
A
—
A B
—¥ —¥
B
—¥
¥ ¥
IB 0A IB 1A 1
( b ) T he set o f all strings tw o or more sym bols long whose first and last symbols are identical, ( c ) The set whose regular expression is (01)*1"1. 3. F in d a regular expression for: ( a ) The language o f problem 2a above. ( b ) The intersection o f the languages o f 2b and 2c. 573
R
57 4
e v ie w
exercises
( c ) T he set o f all strings o f l ’s and 0’s in which every substring o f consecutive 1 ’s is of even length, e g. include: but not: 011001111000 0000
00100
00110111
1111
01
0111111
111
4. C onstruct a Turing m achine which operates with the alphabet { # , a, b} which when started in q0 will m ove to the right until it encounters an а, change the a in to an 6, then m ove left until it encounters a different б, change that b in to an # , and halt. I f there is no a to the right o f the initially scanned square, or if there is such an a but n o b to its left, the m achine will never halt.
Solutions to selected exercises PART A
CHAPTER 1
1. ( a ) t; ( b ) f; ( c ) f; ( d ) t; (e ) f; ( f ) t; ( g ) t; ( h ) f; ( i ) f; ( j ) t; (k ) f (1) t; ( m ) f; ( n ) t; (o) f; ( p ) f; ( q ) f; ( r ) t 2. (a ) yes; ( b ) no; ( c ) yes; ( d ) { { S } } 3 . (a ) Rule: 1. 5 £ A 2
If x £ A, x + 5 E A
Property: A = { x |x is positive and a; is a m ultiple o f 5 } ( b ) Propert y: B = { x |x + 3 is a positive m ultiple o f 10} or ( c ) Rul e
B = { x |x is a positive integer whose last digit is 7 } 1. 300 £ C
2 If x £ C and x < 400, x + 1 £ C ( d ) Rule: 1, 3 £ D 2 4 ED or
3. If x £ D , x + 4 £ D 1. 3 £ D 2. I f a; £ D and x is o d d , x + 1 £ D 3. If x £ D and x is even, x + 3 £ D
Property: D = { x |x is a positive m ultiple o f 4 or x + 1 is a positive m ultiple o f 4 }
So l u t io n s
( e ) Ride: 1, 0 E E 2. I f x £ E, then x + 2 £ E 3 If x £ E, then —x E E where n is a non-negative in teger}
( f ) Property: F = { x j x = ^
(c ) ( 0 } ;( d ) {{ 0 },0 } ;
(e) {0, { { a }} , {{&}},{ { a ; &}}, {0}, {0, { { « } } , {0, {6}},{0, {a,6}}, {{a}, { 6 } } , { { a } , { a , & } },{ {& } ,{ a , 6 } } , { 0 , { a } , { 6 } } , {0 , { a } , { a , b } } , { 0 , { 6 } , { a, &}}, { { a } , { 6 } , { a, b } } , {0 , { a } , {6 }, {a , 6>>} (a ) {a , 6, c, 2 }, ( b ) {a , b, c, 2 ,3 ,4 } , ( c ) { a , 6 , c , { c } } , ( d ) {a , 6, {a . 6}, { c , 2 } } , ( e ) { 6 , c } , ( f ) { a , 6 }, ( g ) { a, b} , ( h ) { c } , ( i ) 0, ( j ) 0, ( k ) 0, (1) {c , 2 ,3 , 4 } , ( m ) 0, ( n ) { 2 } , ( o ) { a , 6 , { c } } , ( p ) 0, ( q ) { { a , 6 }, {c , 2 } }
(a ) { a , b , c , 2 } , ( b ) { a , 6 , c , 2 } , ( c ) { a } , ( d ) { 2 } , ( e ) { 2 } , ( f ) { a , b , c , 2 , 3 ,4 , { c } } , ( g ) { 2 ,3 ,4 , { a, b}, {c , 2 } } , ( h ) { 2 ,3 ,4 , {a , 6 }, { c , 2 } } , ( i ) 0, (j) U , (k ) { 6 ,c ,2 } ,( l ) ( 2 } ,( m ) U , ( n ) U (a ) ( i ) { a, b, c , d} \ ( i i ) { c } ; ( i i i ) { a , b , c , d } ; ( i v ) 0; ( v ) { c, d} \ ( v i ) 0; (vii) {a,b}
( b ) ( i ) no; ( i i ) yes (b )
(b )
1 A n ( B - A) 2 A n ( 5 n i ') 3. ( B fl A' ) n A 4. B fl (A' fl A) 5 B fl ( A n A' ) 6. B fl 0 7. 0 .
C om m . Compl. Ident.
7.
(A U B) - ( A n B) (iu 5 )n (A n 5 )' (A u B )n(A 'u B ') ((iuB)n i')u((4u5)n5') { A n A' ) u { B n A' ) u ( A n B 1) u 0 u { B n A' ) u ( A n B' ) u 0 (.B n A' ) u ( A n B' )
8,
{ B - A) U ( A - B )
9.
(A - B ) U ( B - A)
1.
2. 3 4. 5 6.
(c )
C om pl. Comm. A ssoc.
(IU7)-
(X
n Y)
union and intersection.
=
Com pl. DeM. Distr, {B
(y U X) - (y n X )
n
B' )
Distr, (tw ice) C om pl, (tw ice) Ident, (tw ice) C om pl. (tw ice) C om m (tw ice)
by the com m utativity o f
P art A
C h ap ter 2
577
(d) (i) 0, (ii) A', (iii) A, (iv) B - A, (v) A U B ( A - B ) + ( B - A)
(e) 1 2
( ( A - B ) ' J ( B - A ) ) - ( ( A - B ) n ( B - A) )
3
{A + B ) - { { A - B ) n { B - A ) )
D ef of A + B
4.
(A + B ) - ( ( A n B ' ) n ( B n A ' ) )
C om pl
5 6.
(4 (A + B) — 0
7.
{A + B)
8.
(A + B ) (A + B)
9.
(f)
+ 5)-(Ani'nBn B' )
D ef of A + B
A ssoc., Comm. C om pl., Ident,
n 0' nU
C om pl C om pl Ident.
1.
(A + B ) C B
2. 3.
(A + B) U B = B ((iu B )-(A n 5 ))u B
4.
( ( 4 u B ) n ( A n 5 ) ') u 5 - 5
Compl,
5.
( ( 4 u f l ) n ( i 'u B ') ) u 5 = 5
DeM ,
= fl
Cons. Prin. Def. o f A + B
6.
( ( A u 5 ) u 5 ) n ( ( 4 'u £ ') u B ) = 5
Distr,
7.
(A
[ A U B ) fl ( A 1 U U) = B
Assoc, (tw ice) Idemp.,, Ident,
9. 10.
{ A U B ) fl U = B All B = B
IdentIdent
11.
A C B
Cons. Prin,
8.
U ( B U 5 ) ) fl {A! U ( B' U 5 ) ) = B
CHAPTER 2
1. (a ) ( i ) { ( 6 , 2 ) , ( 6 , 3 ) , ( c , 2 ) , ( c , 3 ) } ; ( i i) { ( 2 , 6 ) , ( 2 , c ) , ( 3 , 6 ) , < 3 ,c ) } ; ( i ii) { ( 6 , 6 ) , ( 6, c ), ( c , 6 ) , ( c , c ) } ; ( i v ) { ( 6 , 2 ) , ( 6 , 3 ) , ( c , 2 ) , ( c, 3 ) , ( 2 , 2 ) , ( 2 , 3 ) , ( 3 , 2 ) , ( 3 , 3 ) } ; (v )
0 (since
A
fl B
=
0);
( v i ) same as A x B
( b ) ( i ) T iu e; (ii) False; ( i ii) False, (c , c ) £ ( A x A) ; ( i v ) True; ( v ) True; ( v i ) True; ( v i i ) Time
(c) ( i ) dom(iZ) = A, ran(i?) = { 6 , 2 ,3 } ; ( i i) R 1 = { ( 6 , c>, ( 6 , 3 ) , (c ,6 > , ( c , c ) } , R~1= { ( 6 , 6 ) , ( 2 , 6 ) , (2 ,c > , ( 3 , c ) } ; ( i ii) No
So l u t io n s
578
2. In relations from A to B each, o f a, b, and c can be paired with 1, with 2, with b o th 1 and 2, or with neither, i e , in four possible ways, Therefore there 4 x 4 x 4 = 64 distinct relations. In functions from A to B each, o f a, 6, and c can be paired with 1 or 2, i e.., in two possible ways. Therefore, there are 2 x 2 x 2 =
8 distinct functions.
Six o f these are on to (only {( a , 1), (6 ,1 ), (c, 1 )} and { ( a , 2), (6 ,2 ), (c, 2 )} are not on to). Since none o f these are on e-to-one and on to, none have inverses that are functions There are 8 X 8 distinct relations from B to A o f which 3 x 3 = 9 are functions, N one are on to, six are one-to-one, and none have inverses that are functions 3 . ( a ) R 2 o R , = { { 1 ,2 } , (1 ,4 ), (1 ,3 ), (2 ,2 ), (2 ,4 ), (2 ,3 ), (3 ,4 ), (4 ,2 ), (4 ,3 ), { 4 ,4 } } , R 1 o R 2 = { { 3 ,4 } , {3 ,1 }, {1 ,1 }, {1 ,2 }, {1 ,4 }, {2 ,4 }, {2 ,3 }, {2 ,1 }, { 1 ,3 }} ( b ) R - ' o R , = { { 1 ,1 } , {1 ,2 }, {1 ,4 }, {2 ,1 }, {2 ,2 }, {2 ,4 }, {3 ,3 }, {3 ,4 },
{4 ,4 }, {4 ,3 }, {4 ,2 }, { 4 ,1 } }
CHAPTER 3
1. ( a ) (ii) irreflexive, non-sym m etric, non-transitive No t e : In considering transitivity, it can be misleading to consider only triples o f distinct elements, as this exam ple shows
For if a is the
brother o f b ( a Bb ) and bBc, and a, b, c are all distinct, a must indeed be the brother o f c. But consider a pair o f brothers: aBb and bBa\ now it is false that a is the brother o f a, as one would have to conclude if the relation were transitive. 2. ( a ) Irreflexive (n o utterance forms a minim al pair with itself), sym m et ric, nontransitive (e g ,, {c a t , b a t } and {b a t, b a g } are m inim al pairs but not { cat, b a g }), and nonconnected; ( d ) Irreflexive or reflexive (depending on how the term ‘ allophone’ is
interpreted), sym m etric, transitive (if “phonem ic overlap” is excluded, otherwise nontransitive) and nonconnected. I f one takes the view that it is reflexive, sym m etric, and transitive, then A is an equivalence rela tion that partitions the set o f English phones in to equivalence classes corresponding to the ( “ taxon om ic” ) phonem es o f English; (e ) Reflexive, sym m etric, transitive, and noncon n ected (in general). Each equivalence class contains all the sets that have the same num bers o f members;
P art A
C h ap ter 4
579
3. (a ) R 1 and R j"1: reflexive, antisym m etric, nontransitive, nonconnected; R [ : irreflexive, nonsym m etric, nontransitive, noncon n ected; R 2 and R 2 X: irreflexive, asym m etric, transitive, connected; R '2 : reflexive, anti sym m etric, transitive, connected; R& and R ^ 1: nonreflexive, sym m et ric, nontransitive, nonconnected; # 3 : nonreflexive, sym m etric, nontransitive, nonconnected; R 4 and R ^ 1: reflexive, sym m etric, transitive, n onconnected; R'4: irreflexive, sym m etric, intransitive, nonconnected; R 4 ( = R 4 1) is an equivalence relation. The partition induced in A is { { 1 ,3 }, {2 ,4 }} (b ) { ( 1 ,1 ) , (2 ,2 ), (3 ,3 ), (4 ,4 ), (2 ,3 ), ( 3 ,2 ) } ( c ) 15 4. T he fallacy lies in ignoring the if-clause in the definition o f symm etry: i f aRb, then bRa The “p r o o f” takes aRb for granted, so a counterex ample can be constructed by finding a relation R which is symm etric and transitive on a certain set S where for some a, a does not bear the relation R to any mem ber o f S Such an exam ple is the follow ing: Let S be the set o f all humans and let R be defined by ‘aRb if and only i f a and b have the same parents and those parents have at least two children’
Then an only child does n ot bear R to anyone, including
himself, (A simpler R might be ‘has the same oldest brother as’ , but it could be ob jected that the relation is simply not defined, rather than failing to hold among people who have n o older brothers ) 5. (a ) R = { { 1 ,1 } , (2 ,2 ), (3 ,3 ), (5 ,5 ), ( 6 , 6 ), (1 0 ,1 0 ), (1 5 ,1 5 ), (3 0 ,3 0 ), (1 ,2 ), (1 ,3 ), (1 ,5 ), (1 ,6 ), (1 ,1 0 ), (1 ,1 5 ), (1 ,3 0 ), (2 ,6 ), (2 ,1 0 ), (2 ,3 0 ), (3 ,6 ), (3 ,1 5 ), (3 ,3 0 ), (5 ,1 0 ), (5 ,1 5 ), (5 ,3 0 ), (6 ,3 0 ), (1 0 ,3 0 ), (1 5 ,3 0 )}, which, is reflexive, antisym m etric, transitive, and nonconnected. ( b ) 1 is m inim al and least; 30 is m axim al and greatest.
CHAPTER 4
1. There is on e-to-one correspondence o f every set with itself— for exam ple, the identity function.. Thus, the relation o f equivalence o f sets is reflexive I f / is a on e-to-one correspondence from A to B, then f ~ 1 is a on e-to-one correspondence from B to A. Therefore, the relation is sym m etric I f / and g are one-to one correspondences from A to B and
580
So l u t io n s
from B to C, respectively, then g o / is a on e-to-one correspondence from A to C (This can b e easily shown by an indirect p roo f,) Thus, the relation is transitive 2. T he set can be denoted { 101, 102, 103, 104 , . ., } 4 . T he cardinality is No.
A 1-1 correspondence betw een the set o f all
sentences o f English and the natural numbers could be established as follows: First, arrange all the sentences into groups according to the number o f sym bols in their written form , and order these groups lin early starting w ith the group o f the shortest sentences. W ith in each group, arrange the sentences alphabetically (using some arbitrary con vention for the punctuation marks and space).
This procedure puts
all the sentences in to a single linear order, and thus establishes a 1-1 correspondence betw een the sentences and the natural numbers 5. T he hotelkeeper uses his in tercom to ask each guest to m ove in to the room whose number is twice the number o f his present ro o m That leaves all the o d d num bered room s em pty, so each fo o tb a ll player can double the num ber on his shirt and subtract one to find his room number, 6. (a ) The turtles can be num bered as follows: 0 1 3
2 4
5
Since the turtles can be effectively listed, so can the corresponding m o notheistic sects, and the cardinality o f the set o f such sets is therefore No! ( b ) Taking the preceding enum eration, it is clear that each sect corre sponds to a subset o f the set o f all natural numbers; the atheistic sect corresponds to 0, the m onotheistic sets to singleton sets. T he set o f all sects therefore can be put in on e-to-one correspondence with V ( N ) and thus has cardinality 2Xo „ 7. ( a ) Let A = { 1 , 2 , 3 , . , , } and B = {0 } ( b ) Let A = { 0 , 1 , 2 , ....} and B = { a, b} .
The set { { 0 , a ) , ( 0 , 6 ) ,
P a r t A R e v ie w P r o b le m s
{1 , a) , {1 , b ) , ( 2 , a ) , (2 , 6 ), . . . } is m apped 1-1 on to {0 ,1 ,2 ,3 ,..
581
} by
f ( n , a) = 2n, f ( n , b) —2 n + 1. ( c ) Let A — {0,2,4,6, , } and B = { l , 3,5,7,.. . }. ( d ) Let A = {0,1,2,3,. . } and let B be the set o f “ prim ed” integers {O', 1', 2', 3', } disjoint from A A x B is equivalent to A x A, which has cardinality No.
R E V IE W P R O B L E M S , P A R T A
1. ( a ) A 4; ( b ) 0; ( c ) A 5; ( d ) A 4
2. Sample answers: ( a ) Let aRb be a + b = 5 or |a — b |= 1 or a x b = 24 or ‘ a + b is o d d ’. ( b ) Let aRb be a > b or ‘ a — b is p ositive’ .
3. ( a ) A ntisym m etric, transitive, reflexive ( b ) Yes. I f n £ N , n = {0,1,2,3,. , . ,n — 1 } . If x £ n, x is a natural num ber smaller than x. Then since all the members o f x are natural numbers smaller than x, they are also smaller than n and therefore members o f n. Hence a; is a subset o f n. It is a proper subset, i.e , not equal to n, because x was given as a mem ber o f n, and the members o f n do not include n itself.
5. (a ) The set o f all mem bers could be represented graphically in a family tree diagram:
where left-to-right order o f branches from a single n ode represents oldest-to-youngest order among brothers.
T hen the nodes can be ef-
582
So l u t io n s
fectively enum erated proceeding to p -to -b o tto m and left-to-right: 0
(b) Assum e that the set o f all clubs is denumerably infinite, so that the clubs cou ld be listed as Co, Ci , C 2, C 3 , club C “ by the m em bership requirement:
•. T hen define a new
m an; £ C ‘ if and only if maiij $ C r Since C* is thereby made distinct from every club in the putative listing, the listing could not have been com plete, i.e , no com plete effective enum eration is possible. 6. (asterisked entries are discussed below )
(a) (b) (c) (d) (e) (f) (g) (h)
PO
SO
WO
yes yes
yes yes
no no
yes
yes
no*
no
no* no
yes yes
yes no*
yes no
yes yes
yes
yes* no*
yes
( c ) It is easy to be m isled into thinking that the negative rational numbers with 0 are well-ordered by the relation > , because the whole set does have a first elem ent, namely 0, But well-ordering requires that every subset have a first element, and there are in this case infinitely many subsets that do n ot. Consider, for exam ple, the subset consisting o f all the negative rational numbers less than —1, There is n o first (i.e. largest) num ber in that set, since for any rational num ber x which is
P
art
A R
e v ie w
P
583
roblems
less than —1 it is possible to find another rational num ber y which is larger than x but still smaller than —1;
(d)
The set is n ot even partially ordered, because tw o distinct strings
m ay have the same length— i.e., antisym mnetry does not hold; ( f ) The ordering looks like this: 0,2,4,6,8,
..
1,3,5,7,9,.
with no relation holding betw een any evens and any odds. This sort o f ordering satisfies the partial ordering definition, but fails to meet the linear-ordering requirement that R hold in some direction betw een every pair o f elements. ( g ) T he ordering looks like this: 0 ,2 ,4 ,6 ,8 ,1 0 ,, , ,,1 ,3 ,5 , 7,9,.
.
This is a well-or dering; every subset has a first elem ent, which will be the smallest even num ber in the subset, if there axe any, and otherwise the smallest o d d num ber in the subset; ( h ) The ordering looks like this: ,9,7,5,3,1,0,2,4,6,8,,,,.
This is not a well-ordering, since any subset which includes infinitely m any od d num bers will lack a first member. 7 . I f |A | >
| B | and |B | > |C |, there are functions f : A —i B and g : B —> C which are on to B and C, respectively. W e prove that g o / is
an on to fu n ction (from A to C). Assum e that it is not on to. Then there is some z E C such that for no x £ A, (g o f ) ( x ) = z, i.e., g ( f ( x ) ) = z. But g is on to C , and thus there is some y E B such that g( y) = 2 , Thus it m ust be that there is no x £ A such that / ( x ) = y. But this contradicts the assum ption that / is onto B. Therefore, g o f i s onto C, and |A| > |C |
So l u t io n s
PART B
CHAPTER 6
(O ther proposition letters m ay be used, other interpretations m ay be argued for, and other equivalent sym bolizations are possible for any given interpretation.) ( a ) j = John is in that room , m = M ary is in that ro o m , j V m ( b ) a = the fire was set b y an arsonist, e = there was an accidental explosion in the boiler room , (a V e) & ~ (a & e ), or equivalently, a <-+ ~ e ( c ) r = it rains, p = it pours: ( 1 ) I f the statement means that every rainstorm is a big one, then “ when” must be translated as “if and only if” , since it can’t pour without raining: 7* <-+ p (2 ) In the famous salt advertisement, the second “it” refers to salt, which, presum ably can also pour when it isn ’t raining, 7* —> p ( d ) s = Sam wants a dog, a = A lice prefers cats, s & a, (T h e difference betw een “b u t” and “ and” relates to the content o f the conjoined sentences, not their truth values) ( e ) I = Steve com es hom e late, s = Steve has had some supper, 7* = we will reheat the stew: ( 1 ) I f one interprets the “if” as “if and only if” , then the p rop o sition can be sym bolized as (/& ~ s ) <-+ r ( 2 ) Under the interpretation that the stew m ay be reheated in any case, then the sym bolic form is ( /& ~ s ) —> r ( f ) c = Clarence is well educated, r = Clarence can read Chuvash, c —> 7* ( g ) c = this cat goes, i = I go, c V i, or equivalently ~ c —> i (T he statement might also be taken to m ean ~ c <-+ i.) (h ) m = Marsha goes out with John, b = John shaves o ff his beard, d = John stops drinking, m —> (b V d) (T h e statem ent m ight also be taken to m ean m <-+ (b & d).) ( i ) n — negotiations com m ence, b = Barataria ceases all acts o f ag gression against T itipu, n —> b
P
2.
art
B
Chapter 6
585
( a ) j = John is going to the m ovies, b = Bill is going to the movies, t = T om is going to the m ovies, j & 6 & ( b ) s = Susan likes squash, t = Susan likes turnips, ~ (s V t) or equivalently, ~ s & ~ t ( c ) p = Peter is going to the party, / = Fred is going to the party, i — I am going to the party, ~ {p V / ) t or equivalently, ( ~ p & ~ / ) —> ~ i ( d ) I = M ary has gotten lost, a = M ary has had an accident, h — M ary will be here in five m inutes, (~ I & ~ a) —> h or equivalently, ~ (I V a) —> h ( e ) b = a bear frightened the b oy s, w = a w o lf frightened the boys, by w ( f ) p = a party would have am used the children, s — a softball game would have amused the children, p S z s . (Perhaps p V s could be argued for, but under the natural interpretation that either one would have amused them , then one is saying b o th that a party would have amused them and that a softball game w ould have amused them .)
3 . ( b ) False; ( d ) True; (f)F a lse 4. ( a ) and ( b ) axe logically equivalent. 5. ( b ) p : 1, q : 0, or p : 0, q : 1; ( d ) p : 1, q : 0, r : 0 or 1, s : 1; ( e ) p : 1, q : 0, r : 1, s : 1, or p : 0, q : 1, r : 1, s : 1 6. ( a ) tautology; ( b ) contingent; ( c ) tautology; ( d ) contradiction; ( e ) con tingent 7. (a ) p —> q can be defined as ~ (p & ~ q) ( b ) p & q can be defined as ~ (~ p V ~
8. (a )
1.
V { p&q)
2. 3.
(~ p V p )& (~ j> V {) (pV p)k( ~pVq)
Distr. Com m .
4.
T k(-pVq)
Com pl.
5.
(~ p V ? )fc T
Comm.
6. ~ p V { Ident. ( b ) ~ p ; ( c ) F; ( d ) ~ p ; ( e ) T
So l u t io n s
586
(d)
1. p - > q 2 . q —> r 3.
3
~ r
4
4.
2,3, M T 1,4, M .T.
^p
5
P
5
q
6 7.
r^>7»
8. (f)
g * — r 3 ~ r —>5
1.
2
£ P
—
»
1. p V ( q & r ) 2, ~ t 3. 4. 5. 6. 7 8. 9. 10. 11
( p V q) —> (s V t) (p V q) & (p V r)
1, Distr.
pV { s Vt
5, Simpl,
s p Vr
2,7, D.S 5, Simpl,
r r&s
4,9, D.S. 8, 10, Conj.
1.
~ p —> 5
2.
r -* (« V i) s —¥'“•w7* P --»'■'Wt
3. 4. 5.
r
6. 7.
sV f ~ s t
8. 9. 10. 11.
3 r -> 3
3,6, M .P.
A uxiliary Premise 2,5, M.P. 3,5, M .P. 6,7, D.S. 4,8, M T , 1,9, M .P. 5-10, C.P.
5
A uxiliary Premise 1,4, M.P. 2,5, M .T . 3,6, M.P. 4-7, C.P.
P
0)
1 2.
p —> ( q & r ) q —> s
3. 4
r —> t
5.
6. 7
8.
(s u
9 s r
11. 12
t s &zt
13. 14.
~ u
1,6, M.P, 7, Simpl 2,8, M .P 7, Simpl. 3,10, M .P 9,11, Conj 4,12, M P
u& ~ u
5,13, Conj, 6-14, Indirect P ro o f
1. 2
P ( p t q ) V { p t r)
3. 4. 5
(P V 9) — py q rv ^
1, A dd n 3,4, M.P,
6.
p & (9 V r)
2, Distr.
7.
9 Vr
6 , Simpl,
8.
9 pkq
5,7, D.S
(p &zq) V ( ~ p & ~ 9 )
9, Addn, 10, Bicond.
9. 10. 11.
P
587
A uxiliary Premise
15. ( 1)
B Chapter 6
u
P q&r
9, 10.
art
1,8, Conj, 9
10. (a ) valid, B — the butler killed the baron, K = the cook killed the baron, C = the chauffeur killed the baron, S — the stew was poisoned, Q = there was a bom b in the car: 1. B V K
V C
2. ( K —> S ) & ((7 —►Q) 3. ~ S & ~ 5 4. ~ B
3, Simpl
5 K y C
1 ,4 , D.S.
6. K - * S
2, Simpl.
7 ~ S
3, Simpl.
8. ~ K
6,7, M .T . 5,8, D.S
9 C
So l u t io n s
588
( b ) Valid; ( c ) Invalid (recall that lp only if q ’ is represented logically as p —> 5 ); ( d ) Valid; ( e ) Invalid (let it be false that the segment is voiceless and let all the other elem entary propositions be true).
11.
( a ) R is an equivalence relation since ( 1 ) it is reflexive: x <-+ x is always true ( 2 ) it is sym m etric: whenever x <-+ y is true, y <-+ x is true (since they have the same truth tables) (3) it is transitive: if x <-+ y is true; y is true; otherwise, if x
z is true, then x <-+ z
z, then for some assignment o f truth
values to elementary statements there would be some y which is b o th true and false. (b ) { p , ( p V p ) , ( p & ( j V ~ ? ) ) , ( p V ( ? & ~ ? ) ) } {(P V ? ) , ( ~ 3 -> P ) , ( ~ P -> ? ) } {(pV ~ p ) , ( p - > ( ? - > p ) ) , ( p V (?V ~ ? ) ) }
FALSE
TRUE
pk
p —> (q —> r)
1.
~ (~f> V r)
2.
p \/
3. 4.
r
P
5.
p
6.
61
q —> r
6.2
6.1
6.2
P
7.
7.n
r
7-12
7.11
7-12
8.
8121
8-122
8.121
9.
9-122
P
8.122 ~ q 9.122
A ll the subtableaux close, so the tableau closes, and the argu ment is shown to be valid.
P
art
B
1,
TRUE { ( p q ) & {s V t)) (t —> q)
2.
(p -> q)
3
(svt)
Chapter 7
589
FALSE ((P
?)V ~ ( s -> q))
4.
(.P ->• ?)
5
~ ( s -> q)
This tableau remains open. 13.
( a ) ( i ) KKA.pqA.qrA.ps, ( i i) CKNpCNpgg, (iii) C Ap q K Er s p ( b ) ( i ) p V ((~J>& ~ ? ) -> ( p t ( q <-+ r ))) ( i i) ( ~ (( ( ( ( p <-» 2) <-+ r) & s ) V p) & q) V r) & s ( i ii) ~ ((((;> <-+ ? ) f c r ) V s ) - » i)
( c) NApg o KNpNg, NKpg o ANpNg
CHAPTER 7
1.
( a ) (\/x) ( B( x) V W^(i®)) : B ( x ) - ‘ x is b la ck ’ , W ( x ) - ‘ x is w hite’ ( c ) (\/x)(D(x) —> Q ( x ) ) : D ( x ) - lx is a d o g ’ , Q ( x ) - ‘ x is a qu adruped’ ( e ) ( \/x) ( 3y) L( x, y) : L ( x , y ) - lx loves j/’ ( g ) ( 3x) ( \/y) L( x, y) : L ( x , y ) - lx loves 3/ ’ ( i ) ( ' i x ) ( L( x , x) —> (x = j ) ) : L ( x , y ) - ‘ x loves y\ j - ’Joh n ’ ( k ) ( \f x ) ( ( W( x ) & L(g, x) ) - ( K ( g , x) V M ( g , x) ) ) : W ( x ) - lx is a w om an’ , L ( x , y ) - ‘ x loves y\ g - ‘ y ou ’ , K ( x , y ) - lx kisses y ’ , M ( x , y) - lx loses y ’ ( m ) ( \/x)( (P(x) & M ( x , h ) ) —►L ( x , h ) ) : P ( x ) - lx is a person’ , h - ‘New Y o rk ’, L ( x , y ) - ‘ x loves y\ M ( x , y ) - lx Hves in y ’ ( o ) ( \ f x ) ( ~ L ( x , h ) —> ~ N ( x , h ) ) : L ( x , y ) - lx loves y\ h - ‘New Y ork ’, N ( x , y ) - ‘ x knows y ’ ( q ) ( \ f x ) ( \ / y ) ( ( F ( y , x ) &G{ x , y , g )
(3z)(H(z,y) k l { g , z , x ) ) )
F ( x , y) - ‘ x is a finger o f y\ G ( x , y , z) - lx gives y to z\ H ( x , y ) - ‘ x is the whole hand o f which y is a finger’, I ( x , y , z ) - lx takes y from z\ g - ‘h e’
So l u t io n s
590
( s ) ((3a;)iV(a;) —►(\/x)A(x)) : N ( x ) - lx is n oisy’ , A { x ) - lx is annoyed’ (t) ~ ( 3 x ) N ( x ) k A(j) : N ( x ) - lx m ade n oise’ , A { x ) - ‘ x was annoyed’ , j - ‘Joh n ’ (v ) ( Vx) ( A( x) B(x)) : A { x ) - ‘ x causes bad accidents’ , B ( x ) - lx is a drank driver under 18’ ( x ) (\/x)(A(x) - * B { x ) ) : A ( x ) - lx is drunk’ , B { x ) - ‘for x to drive is risky’ 2.
(a ) ~ { 3 x ) ( P ( x ) k ( \ f y ) ( Q ( y ) -> A ( x , y ) ) ) ( b ) ( Vx ( Q( x ) - > ( 3 y ) ( P ( y ) & A ( y , x ) ) ) ( d ) (Bx ) ( P ( x ) k ( V y ) ( Q ( y ) A ( x , y ) ) ) , although ‘ some p e o p le ’ migh also be taken to m ean not ‘ at least one person ’ but ‘ at least two persons’ . ( e ) L ( x , y ) - lx likes y \ (V a;)(£(a;,M ary)) <-+ ~ I ( x , M a ry )) ( g ) T ( x , y ) - lx attem pted y\ (\/x){ {P{x) k { 3 y ) { Q{ y ) k A ( x , y ) ) ) -> ( 3 z ) ( Q ( z ) k T ( x , z ) ) )
3. ( a ) x in P { x ) bou nd , last x and the y free; ( b ) x free, y and z bound; ( c ) everything bou nd; ( d ) first x bou n d , everything else free; ( e ) ev erything bound 4.
( a ) ( l ) Yes. A pplyin g (7-24) to (V z) in the translation gives (V z )((V * )(B » )F (y , x) k ( 0 ( z ) -
J (z ))).
Then (Va;) and (By) can be m oved outside b y the follow ing se quence o f equivalences. N ote first that P k Q O Thus, ( \ / x ) P ( x ) k Q & ~{ ( \/x ) P( x ) - » ~ Q ) o
{P
Q) .
~ (3 a ;)(P (a ;) -+ ~ Q ) (by Law 11) o (Va;) ~ ( P ( x) —> ~ Q ) (by Law 1 '") O {\fx){P{x)kQ). T he steps for (3 y) are similar. ( 2 ) N o. A pplyin g (7-24) in reverse to (Vz) in (2) gives (B y )(V * )(JP (» , * ) k (V z )(0 ( z ) -
J (z ) ))
which, says ‘ There is something which is everything’ s father, and all od d numbers axe integers’ . ( 3 ) Yes. A pplyin g Law 6 to (1),
Part B
(b ) (1 ) No
Chapter 7
(1) is equivalent by Law 9 to B ( a ) —> (Va;)
591
(M(x) -
H { x ) ) , which is equivalent by the steps n oted in the answer to ( a ,l) to B{ a ) —> (\/x) ( M( x) k ~ H ( x ) ) , which says ‘I f A dam is a bachelor, then everything is a man and a non-husband’ , or loosely, ‘I f A d am is a bachelor, then everyone is a bachelor’ , (2 ) Yes. A pply Law 10 to (2 ), then Law 1. (3 ) No. ‘I t ’s not the case that if A dam is a bachelor then all m en are husbands.’ (N.B. This is equivalent to: ‘A d a m is a bachelor and not all m en are husbands. ’ ) (4 ) Yes., ( c ) ( l ) Yes (by equivalences concerning — k , and ~ ). (2 ) Yes, (by Law 12)
5.
(a ) (\/x)(I(x) -> ( 3 y ) ( I ( y ) k L ( y , x) ) ) , ( \f x) ( 3y) ( I ( x) -> ( I ( y ) k L ( y , x) ) ) , ~ ( 3 x ) ( I ( x ) k ~(3y)(L(y,x)) (and m any others; such listings are never exhaustive) ( b ) ( ( V x ) ( P ( x ) -> 0 ( x ) ) V ( 3 y ) ( I ( y ) k ~ 0 ( j / ) ) ) , { 3y ) ( ( \f x ) { P{ x) -> 0 ( x ) ) V ( I ( y ) k ~ 0 ( j / ) ) ) , B y various applications o f com m utativity and Laws 9-12: ('\/x) ( 3y) { { P( x) -> 0 { x ) ) V { I { y) k ~ 0 ( y ) ) ) , ( 3 j/)( V z )( (P (z ) -> 0 { x ) ) V ( I ( y) k ~ 0 { y ) ) ) (N ote that this is one case in which the order o f universal and existential quantifiers is im material; the reason is that there axe no tw o-place predicates relating x and y ) ( c ) (( 3 x ) ( P ( x ) k ~ 0 ( x ) ) - ( V y ) ( ( P ( y ) & < ? (» ,7 )) - 0 ( y ) ) ) . It is natural to try an alphabetic change o f variable and a reinterpre tation o f —> in terms o f ~ and V, in hopes o f leading to an ap plication o f Law 3. However, once we get to: ((Va;) ~ (P ( x ) k ~ 0 ( x ) ) V (V a;)((P(a;) k G( x , 7)) —> 0 ( x ) ) ) , we see that only Law 4, which is not an equivalence, will apply. W ith tw o variables, we can apply Laws 9 and 12 to get: (\/x)( Vy) ((P(x) k ((P(y)kG(y,7))-*0(y))) (d ) (Vx) ( f f ( x) -» M { x ) ) -» M( s ) , (3 x ) \ { H{ x ) - » M { x ) ) - » M( s ) ) , ~ A f ( s ) —> { 3 x ) { H { x ) k ~ A f(a ;)), etc..
0 ( x) ) -
So l u t i o n s
592
({{3x)A{x)k(3x)B(x)) 1. 2. 3, 4, 5. 6, 7, 8,
(b)
7 . (a )
C7(a))
( ( ( B y ) ^ ) ^ ) ^ ) ) - ^ ) ) (~ ((3 2 /)A (y ) -> ~ ( 3 z ) £ ( z ) ) -> C'(as)) ( ~ ( V 2/) (A (y ) -> ~ { 3 z ) B { z ) ) -> C ( x ) ( ~ ( \ f y ) ( A{ y ) -> (Vz) ~ £ ( z ) ) -> (7(a;)) (~ (V y )(V z )(A (y ) -> ~ 5 ( z ) ) -> (7(a;)) ((B 2/ ) ( B z ) ~ ( A ( 2/ ) - , ~ 5 ( z ) ) - , ( 7 ( S)) (V y )(V z )(~ (A (y ) -> ~ B ( z ) ) -> C (a;)) (\/y)(\/z)((A(y)tB(z))^C(x))
(Va;)yl(a;)
alph, variant put in terms o f ~, Law 12 Law 1 Law 9 Law 1 (2x) Law 12 (2 x)
<-+(3 x ) B ( x )
1.
~ (((Vas)yl(a;) -> (By ) B ( y ) ) -> ~ ( ( 3 w ) B ( w ) -> (\/z)A(z))) alph, variants and in terms o f —>
2.
~ ( ( 3 x ) ( 3 y ) ( A ( x ) -> B ( y ) ) -> ~ ( V w ) (V z ) (£ (w ) -> A ( z ) ) ) Laws 9, 10, 11, 12
3.
~ ( ( 3 x ) ( 3 y ) ( A ( x ) -> B ( y ) ) -> (3 w )(3 z ) ~ ( 5 ( w ) -> A ( z ) ) ) Law 1
4.
~ (\fx) ( \/y) ( 3w) ( 3z) ( ( A( x) -> B ( y ) ) -> ~ ( 5 ( w ) -> A ( z ) ) )
5.
Laws 10 and 12 (3x)(3y)(\fw)(\/z) ~ ( ( A ( x) -> B ( y ) ) -> ~ ( 5 ( w ) -> A (z ))) Law 1
6.
(3 z)(3 2 /)(V w )(V z)((A (a ;) -> B ( y ) ) & ( B ( w ) -> A { z ) ) )
l„ ~ ( 3 x ) ( P ( x ) k Q ( x ) ) 2. ( B s ) ( P ( s ) & ^ ( s ) ) 3.P(w)SzR(w) 4. (Va;) ~ ( P ( x ) & Q ( x ) ) 5„ ~ ( P ( w ) & Q { w ) )
2, E.I. 1, Quant. Neg, 4, U.I.
6, ~ P ( w ) V ~< 3(w )
5, D eM .
7. ^ ( w )
3, Simpl.
8 „~ < 3 (w ) 9. .ffi(w)
6,7, D.S.. 3, Simpl,
10. R( w) & ~<3(w )
8,9, Conj.
11, (3zO(.K(zO& ~<5(a=)) 10, E.G.
P art B
Chapter 7
593
( e ) 1 (\/x)( P( x) -> Q ( x ) )
2 R (a ) 3 P(a)
(f)
4, P { a ) -*■ <3(o)
1, U.I
5. Q{ a ) 6 R{ a) &z Q{ a)
3,4, M .P 2,5, Conj
7 (3 x){R{x)kQ{x))
6, E.G.
1 . (V x )((P (x ) V Q( x ) ) -> J2(*)) 2 . (Va;)((.ffi(a;) V S(a;)) —> n * ) ) 3. P(t>) 4. (p (« ) v $ ( « ) ) 5. 6 7, 8.
P («) V Q ( v ) (JZ(t>) V 5(w )) -» 2 » i2(«) V S{ v)
9. T( v ) 10. p ( v ) - » r(w ) 11.
( v * ) ( Jp ( SB) -
Auxiliary Premise 1 ,I.U . 3, Addn. 4,5, M .P. 2 , U.I. 6 , A ddn. 7,8, M .P. 3-10, C.P. 10, U.G.
r (* ))
8 . ( c ) D { x ) — x is a duck, 0 { x ) — x is an officer, P { x ) = x is (one o f) m y poultry, W { x ) = x waltzes 1. (V z)(Z > (x) -*■ ~ W ( x ) ) 2. ( V z ) ( 0 ( z ) - » W ( x ) ) 3, (\f x) ( P( x) —> D ( x ) ) 4. P { v ) —> -£>(«)
3, U.I.
5. -D('w) —> ~ ^ ( v ) 6. p ( v ) - + ^ w { v )
1, U.I. 4,5, H.S.
7. 0 ( v ) —> W ( v ) 8. ~ W ’r(v) —» ~ 0 ( v )
2, U.I. 7, Cond.
9. -P(^) —> ~ 0 ( v ) 10. (V *)(P (a !) - » ~ 0 ( * ) )
6,8, H.S. 9, U.G.
594
9.
S o lu t io n s
(a ) 1. 2.
TRUE
FALSE
~{3x)F{x)
(V s) ~ F ( x ) (3 x ) F ( x )
3. 4.
~ ^ (a ) F( a)
5.
(b )
D — { a , . .}
F(a)
TRUE
FALSE
1
(V a;)(B y)^(a;,y)
(B !/)(V z )J2(z ,!/)
2
(B y)J2(o,y)
3 4,
i2(a, a)
(Va:)i2( k , a) i2(a, a)
5. 6 7,
£> = { a , 6 }
i2(6,a) (3 y ) R { b , y ) R(b, a)
9. 10
R(b, b) i?.(a, 6)
So a counterexam ple consists o f a m od el with a universe o f dis course D = {a , 6} and an extension R = {( a , a ), (6, 6 }}
(c ) 1.
TRUE ( 3y) ( \/x) R( y, x)
2. 3.
(\f x ) R( a , x ) R(a,a)
FALSE (V z)(3 y ) R { y , x )
4.
(3 y ) R { y , a )
5.
R ( a , a)
6.
( 3y ) R( y , b ) R( a, b)
7.
R( b, b) 9.
i2(a, 6)
D = { a , ., .}
P
1 0 . (a )
art
B
Chapter 7
595
1.
Fi C F2 & F2 C Fs
A uxiliary Premise
2
V2 C Fs F2 C Fs & F 2 ^ Fs
2, D e f o f C
3 4, 5 6 7. 8 9. 10. 11. 12 13. 14. 15. 16. 17. 18, 19. 20, 21.
F2 C Fs Fi c f 2 Fi C F2 & F2 C Fs Fi C Fs
1, Simpl. 3, Simpl, 1, Simpl. 4,5, Conj 6, (7-54) 1, A uxiliary Premise
F = Fs Fi C Fs & Fs C Fi
8, (7-52)
Fs C Fi
9, Simpl,
Fs C Fi & Fi C F2
5,10, Conj. 11, (7-54)
Vz Q V2
F2 C Fs & Fs C F2 F2 = Fs F2 ^ Fs F2 = Fs & F2 ^ Fs Fi ^ Fs Fi C Fs k Fi ^ Fs Fi C Vz (Fi C V 2 i z V 2 C Vs ) Fi C V2 ( V X , Y , Z ) ( ( X C Y & Y C Z) X C Z)
11. (a ) A ssum e x £ ( A — B)
4,12 Conj, 13, (7-52) 3, Simpl. 14,15, Conj, 8-16, Indirect P r o o f 7,17, Conj. 18, Def. o f C 1-19, C onditional P ro o f 20, U.G.
T h en x € A and x $ B, from which it follows
that x £ A. Therefore, if a; £ ( A —B ) , then x £ A. Thus, ( A —B ) C A. ( b ) If A = B , then ( A - A ) u ( A - A ) = 0U0 = 0.. I f ( A - B ) \ j { B - A ) = 0, then b o th A — B = 0 and B — A — 0 (otherwise the union could not equal 0). A - B = 0 means ~(3a;)(a; £ A & a ; ^ 5 ) , which b y Quant. Neg., D eM ., and Cond. is equivalent to ( Vx ) ( x £ A —> x £ B ), i.e.., A C B , Similarly, B — A = 0 is equivalent to 5 C A. Thus, A — B, ( e ) I f a; £ p ( A ) U p ( B ) then x £ p { A ) and x £ p ( B ) . x £ p ( A ) iff x C A and x £ p ( B ) iff x C I?. I f x C A and x C B, then a; C A fl B, and thus x £ p ( A fl B). The converse is proved by taking these steps in the opposite order.
So lu t io n s
596
CHAPTER 8
1. Assum e that atomi c stat ement is defined as in (8-7). Base: Every atom ic statement is a P w ff (P olish well-form ed form ula). Recursion: For all a and /3, if a and f3 are P w fTs, then so axe (a) N a , (b ) A a/3, (c ) K a/3, (d ) C af3, (e) Ea/3 Restriction: N othing else is a PwfF,
2. / ( 0 ) = 2 , f ( n ) = ( f ( n — l ) ) 2; m
= 2
/ ( I ) = ( / ( 0 ) ) 2 = 22 - 4 / ( 2 ) — ( / ( I ) ) 2 = 4 2 = 16 / ( 3 ) — ( / ( 2 ) ) 2 = 162 = 256 / ( 4 ) = ( / ( 3 ) ) 2 = 25 6 2 = 6 5 ,5 3 6
4 . T he power set o f the set with zero m em bers, 0, is { 0 } , which has one m em ber. Since 2° = 1, this establishes the base. T o prove the in du ction step, let Ak+1, a set with k + 1 m em bers, be form ed from Ak, a set with k m em bers, by the addition o f some element x not in Ak\ i.e., Ak+i = ylfcU{a;},. Let B \, B 2, . , B 2k be the 2fc mem bers o i T ( A k ) . V { A k + i ) contains all these sets plus the sets form ed by taking B 2k l i { x } . This makes an additional 2fc set. Thus, V ( A k +1) has 2(2fc) = 2fc+1 mem bers. The desired result now follow s by M athem atical Induction.
6. The in du ction fails in going from 1 to 2 as the reader can verify by letting n take on the value 1 in the in du ction step.
P art B Chapter 8
1. 2.
597
P-*{<1-*P)
(A l)
p)-*{p-*q))-+ ( ^ p - + ( ( ~ q - > ^p)-+(p-+q)))
3.
(Subst. ( ~ q—» ^ f q ) - * ( q - +p ) [ ^ p —¥
4
(~ g -> ^ p ) - y ( p - y q )
o
r
1, (R 2) p and ^ p for q) (A 3 ) 3, (R 2)
(Subst p for q and q for p) 5 6
7.
~ j> - > ( (~ ? -+ ~ p ) - > ( p - > q ) ) ( p - > ( q - i.r ) ) - ^( ( p - >q ) - > ( p - >r ) ) p )_^ (p _^ 5)))_^ ^'"ssp--^ ^
2,4, (R I ) A2
((~ p -> (~ g -> ~ p ))-» (~ p -» (p -+ ? ))) 6, (R 2) (Subst for p, ( ~ 9—> ~ p ) for 5 , and ( p—>q) for r) 9. 10
(~ .p -> (~ ? - » ~ p ) ) - » ( ~ P - » ( p - » ? ) ) ~ p —> (~ g —> ~j>)
5,7, (R I ) 1, (R 2)
(Subst. ^p-^(p-^q)
8,9, (R I )
for p and
for q)
8 . The alphabet is (p, ', N, A , K , C, E ); the axiom s axe the elementary propositions as defined in (8-7); the productions are: x —> Na; x, y —> Aazy x,y x,y
—> K x y -> Cazy
x , y —> Eazy where x and y are any strings on the alphabet, A n equivalent semiThue system has a basic alphabet as above, an auxiliary alphabet {Q> -K}> and axiom set {<3 } , and productions: Q —¥ N Q Q —¥ A.Q Q Q Q Q Q R R
—¥ K Q Q — ¥ CQQ — ¥ EQQ —¥ R —¥ —
R'
¥ P
9. A = { J , K } , B = { a } , S = { J , K } - ,
So l u t io n s
5 98
P :
aJ/3 aJ/3 aK/3
—> aJaa/3 —> a/3 —> aKaaa/3
aK/3 —> a/3 where a,/3 are any strings on { A
U
B)*.
11. I f the axiom had not been independent, it would be provable from the remaining axiom s
But then its deletion from the set o f axiom s would
have no effect on the total system: anything that could be proved before could still be proved I f that is not the case, the axiom must have been independent. 1 2 . ( c ) “ 0” is interpreted as + 1 ; “is a natural num ber” as “is a power o f —2” ; “ successor o f x ” as “ —2 times x ” Thus the sequence is viewed as having the form : ( —2 )°, ( —2 )1, ( —2 )2, ( —2)3, . . . 1 3 . ( a ) In this m odel, the “ lines” axe not the sides or diagonals o f the rectangle, but rather pairs o f vertices, by A xio m 1, a
b
( b ) Consider an arbitrary point p, By A xiom 2 there is at least one other poin t; call it q. B y A xio m 3, there is a line containing p and q; call it L\, B y A xiom 4, there exists a point not in L\\ call it r. A x iom 3 again, there is a line containing p and r; call it in at least tw o lines, Li , and £2
By
■ Thus p is
( c ) Assum e the em pty set is one o f the lines.. Consider any poin t p: by ( b ) above, p is in at least tw o distinct lines Li and L 2 But then A xiom 5 is violated, because p is a point not in the em pty line, but L\ and L 2 are b o th lines containing p and disjoint from the above em pty line
( d ) This was proved in the course o f proving ( b ) above.
P art B R
e v ie w
P
roblems
599
( e ) N o, because A xiom s 2 and 3 together contradict the new axiom,
R E V IE W P R O B L E M S, PART B
1 . I f P <-+ Q is true, then either P and Q are b o th true or P and Q are b oth false. In either case P V ~ Q is true.
2 . Other valid derivations are possible in all the follow ing examples: 1, p
(a)
2 q 3. p & q
1,2, Conj.
4 . (p & q) V r
3, A d d n
( b ) 1 . p <-+ q 2 (p —> q) & (q
(c )
—> p) 1 , B icond.
3, (q —i►p ) 8z { p —> q) 4 . q —> p
2, Comm, 3, Simpl,
5
4, Cond.
V ?
1 . p —> (5 —> ~ ~ r ) 2.
~r
2, Simpl.
3. p 4. 9 —»
r
5 . q —> r
( d)
3,1, M .P.
6. ~ r & p 7. ~ r
4, Com pl. 2, Comm, 6, Simpl.
8. ~ 9
7,5, M ,T .
l,p V { 2
V r
3 ~ 9 4. 9 V p 5. p
1, Comm, 4,3 D.S,
6 . p —> r
2, Cond.
7. r
5,6, M .P.
S o l u t io n s
1, p&z (q —> ( r V
~ ~ s))
2 q 3. p 4. q —> ( r V
1, Simpl. 1, Comm. Simpl (2 steps)
~ ~ s)
5. r V ~ ~ s
2,4, M .P
6. ~ ~ s V r 7. s V r
5, Com m . 6, C om pl
8. p & ( s V r )
3,7, Conj.
1. ~(j> V ~ ? ) 2, r V p
3 r'"' 4.
q
1, D e M 3, Simpl
5. p V r 6. r
2, Com m . 5,4, D.S
7. ~
3, Comm. Simpl. (2 steps)
8. q 9. q & r
7, Compl. 8,6, Conj.
1. p -4 q 2, p -+ (5 -+ r) 3.. q —> (r —> s) 4. P 5. 9 6. q —t r 7. r 8.
r —> s
9. s 10. p —> s
Auxiliary premise 4,1, M.P. 4,2, M .P. 5,6, M.P, 5,3, M .P. 7,8, M,P„ 4-9, C onditional p ro o f
P
1
B R
e v ie w
P
r —> (p V s) 1
2.
3. ~ s 4, 5, ~ p
Auxiliary premise 4, Simpl. ~s
6. 7.
5,3, Simpl.
~ ( p V s) '•vT
8,
6, DeM . 7,1, M .T . 4, Com m . Simpl,
9. 10. 11 12. 13. 14.
~ 5& ~ t
3,9, Conj
~ ( s V £)
10, D eM 11,2, M T.
^ T Si ~ £ J? & ~ i) —» ( ~ r &
~g)
( i ) 1. p -> q 2, ~ q & r 3 4.
roblems
<
(h )
art
Auxiliary premise
P
5
3,1, M P . 2, Simpl.
6,
4,5, C onj.
7. ~ p
3-6, Indirect P ro o f
(j) 1 2. r &
P V q
3
~ q
Auxiliary premise
4.
p
1,3, D.S.
5.
&r
2, C om m .
6.
5, Simpl,
7.
4, 6, Conj.
8. 5
3-7, Indirect P ro o f
8,12, A dju n ction 4-13 C o n d itio n a l!
6 02
S ol u t io n s
(k )
1. p ( ~ 5 —> 7*) 2. ~ r & ~ (s —> q) 3. 4. 5 6. 7. 8. 9. 10. 11. 12.
P (P -> ( ~ ? r )) & ((~
4, Simpl. 3,5, M P. 2, Comm. Simpl.
9,6, M .P. 2, Simpl.
rv ^
r k ~r
10,11, Conj 3-12, Indirect P ro o f
1, r fe(j) V 5 ) 2. ~ ( p k r)
Premise Premise
3. ~ ( q k r )
Premise
4. r
1, Simpl
V ~
5.
1, Bicond.
7, C ond., Compl.,, D eM 8, C om m . Simpl.
13.
3.
Auxiliary premise
v
6. ~ r V
2, DeM . 3, Comm.
7. r —>
6, C ond
8.
4,7, M .P, analogous to 5 -8
9—12. ~ g
8,12, Conj.
13. 14. ~ (p V 15.
P V q
5)
13, DeM 1, C om m ., Simpl.
Since 14 is the negation o f 15 and b oth are derived from the premises, the premises axe inconsistent. 6. (a ) ( y x ) { h x —> qx) k ( 3y) ( qy k ~ h y ) or (\/x)(hx —> qx) k ( 3 x ) ( q x k ~ h x ) or (\/x)(hx —> qx) k (3a;) ~ (qx —> h x ) or (\/x)(hx —> 5 a;) & ~ ( x) ( qx —> hx) (or yet others); ( b ) Let I x y : x is identical with y , P x y : a; is a phonem ic transcription o f y , C/®: a; is an utterance, (Vx)(\/y)[(Ux k U y k
/a ;y ) ->
(Vz)(Vw)((.Pza; & P w y ) —> ~ Jzm ))] or (Va;)(Vy)(Vz)(Vw)((£/'a;&: [fy fe ~ I x y k P z x k P w y ) —> ~ I z w )
P art C Ch apter 9
603
( o r a n y o t h e r l o g i c a l l y e q u iv a le n t e x p r e s s i o n ) ;
( b ) D ( x ) = x is a cab driver, H { x ) — x is a head waiter, S ( x ) = x is surly, C { x ) = x is churlish
1. ( V x ) ( ( D ( x ) V H ( x ) ) —> ( S( x ) & C ( x ) ) 2 D{ v) 3 4,
D{ v) V H( v) ( D { v ) V S { v ) ) -4 ( S ( v ) k C ( v ) )
5.
S{v)kC{v)
Aux
Premise
2, A ddn , 1, U.I. 3,4, M .P. 5^ Simpl.
6, 7
S( v) D ( v ) —> S( v) S( x ) ) 8, (\/x)(D(x)
2-6, Cond. P r o o f 7, U.G.
PART C
CHAPTER 9
(a ) T he universal set U, the union o f all the sets in the collection, is the (tw o-sided) identity. ( b ) I f U is a m em ber o f the collection, then it is the only element with an inverse; viz., itself. ( a ) (j> is the (tw o-sided) identity element for union.
element with an inverse; viz., itself. ( b ) ^ is also the (tw o-sided) identity element for sym m etric difference (for all A, A +
604
S o l u t io n s
C H A P T E R 10
1. Consider the group operation table, or “ m ultiplication table,” for ad dition m od u lo 4: + m od 4
0
1
2
3
0 1
0 1
1 2
2
3
3
2
2
3
3
3 0
0 1
0 1 2
Closure is shown by the fact that every ceL1 o f the table if filled with an element o f the set {0 ,1 , 2 ,3 }, A ssociativity must be verified by exhaustion (or else by a general argument, which in this case would take even longer, since the best general argument would go by way of proving associativity for addition over all the integers) T he fact that 0 is the unique identity element is evident from the fact that the 0colum n and the 0-row m atch the outside colum n and row respectively, and no other rows or columns do. T he fact that 1 and 3 are inverses o f each other and that 2 and, o f course, 0 are each their own inverses can be seen from the position o f the 0 ’s in the table. 2. (a ) N o; e.g.
7o 3 =
10 (m od 11), and 10 is not in the set, so the
closure property fails. ( b ) Yes, if it ’s associative, which in fact it is, ( c ) N o, a is the identity element; c is its own inverse, but has d sis an additional “right-inverse,” i.e. c o d = a, which violates the uniqueness condition on inverses (C orollary 10.2); in addition, d has n o “ rightinverse,” which violates part o f axiom G4. A lso, the operation is not associative. ( d ) Yes, (R em em ber to check associativity.) ( e ) No.
T he identity element is 0, but n o sets other than 0 have
inverses. ( f ) No. ( g ) Yes. ( h ) No. 3 . (a ) T he convention for m ultiplication tables is that the entry in the a.th row and the b tfc colum n shows a o b, not b o a . This makes a difference
P a r t C C h a p t e r 10
605
for non-com m utative operations like this one. b a o b
a
I
R
I
I
R
R' R'
R
R
R'
R"
R’ R"
R' R" H V D
R" I
I R
D' D
V H
H
D'
V
D' D
H V D D'
R"
H
V
D
D'
R" I
H D
V D'
D
D'
V
R
H D
D'
H D
R' D
V D' I
R"
D'
R'
R' I
R
V R R"
V H
R R"
R" R
I R'
R' I
H
( b ) < {/, R, R', R " } , o ), { { I , R ', S , V } , o } , { { / , R' , D, D ' } , o); ( c ) T he sets are { / , £ ' } , { I , H } , { / , F } , { / , £ > } , { I , D ’ }, The opera tion tables all look alike:
I X
I
X
I X
X I
( d ) { { / , R', S , V } , o) and { { / , R', D, D ' } , o) It turns out that any cor respondence which m aps I on to I is an isom orphism in this case, e g,, I —> I , R —> R', H -> D, V - » D' , ax I - * I, R' - * D, H —> D 1, V —> R', etc. ( e ) For the subgroup { { / , R, R ' , R " } , o ), the only non-trivial autom or phism is I —» / , R —> R " , R! —» R 1, R " —» R For the other two subgroups, any o f the n on-I elements can correspond to any other, so there are actually five different non-trivial autom orphism s for each o f them, o f which the follow ing is one exam ple for { { / , R', H, V } , o): I
I , R'
V , H —> R ' , V —> H.
( f ) For the subgroup { { / , R, R ' , R " } , o) and any o f the subgroups { { / , X } , o), the only possible hom om orphism is I —> J, R! —> I, R —> X , R " -> X . It m ay be useful to see why some other correspondences do not give hom om orphism s. Consider, for instance, the correspondence / ( / ) = / , f { R ) = R' , f ( R ' ) = R', f { R " ) — R' - This is not a hom om orphism because, for instance, R o R' = R " but f { R ) o f ( R ' ) = R' o R' = I, and I 7^ f ( R " ) -
For either o f the other tw o subgroups, the one with
{ / , R ' , H, V } or the one w ith { I , R ' , D , D ' } , there are several possible
S o l u t io n s
606
hom om orphism s with any o f the { / , X } subgroups
T hey all have the
follow ing form : Let I and any other one element correspond to / , and let the other tw o elements correspond to X , e.g.., for { { / , R', H, V } , o) and ( { I , D } . o), one possible hom om orphism is I
—
I , R'
—> / ,
H -4 D , V —> D. 4 . It must be a subset because the original group had to contain an iden tity element; by definition the identity elem ent is included, associativ ity is autom atic for subsets o f groups, and since the identity element is its own inverse, all the group axioms are satisfied 5. (a ) N o T he second and third group axiom s are satisfied, but 2, 3, and 4 lack inverses, and 2 o 3 = 0, which is not a mem ber1 o f the set ( b ) M ultiplication is associative, the set is closed under it, 1 is the identity element, 2 _1 = 4, 3_1 = 5, 6_1 = 6. ( c ) { { l } , X m od 7), {{1 , 6 }, X m od 7), {{1 , 2 ,4 } , X m o d 7). ( d ) { { 0 ,1 ,2 , 3 ,4 , 5 }, + m o d 6} Correspondence:
0- *l 1 — 3 2 — 2
0 —1 or
1 - 5 2 — 4 (n o others)
3 - 6 4 — 4
3 — 6 4 — 2
5 — 5
5 — 3
( e ) C ondition: n must be a prime number.
6. S must be a group to be a subgroup o f S'. T he only further condition to b e m et is that the set o f S be a subset o f the set o f S ", and that follows from the transitivity o f “ is a subset o f,” plus the fact that the subset relation must hold betw een the sets o f S and S' and betw een those o f S' and S " 7. (a ) Exam ples: the set o f all positive rationals o f the form l/n, or the set o f all positive integers, or the set o f all rational numbers equal to or greater than 1. ( b ) It is a semigroup and a m on oid, but not a group because 0 has no inverse. 8. Sym m etric difference can easily be shown to be com m utative and asso ciative either by set-theoretic equalities or by Venn diagrams. It is not
P a r t C C h a p t e r 10
607
idem potent since A + A = 0. 0 is the tw o-sided identity element, and every set is its own inverse, ( V ( A ) , + } is an A belian group o f order 4
10. (a )
1.
a + b = a + c
Premise
2,
—a + (a + 6) = —a + (a + c)
B y D2, a has an additive in verse —a and addition is well-
3,
( —a + a) + b = ( —a + a) + c
B y D2, + is associative
4,
0 + 6 = 0+ c
defined and unique
Def, o f inverse, 0 is the addi tive identity
5 (b )
b —c
Def. o f additive identity
1,
o+0 = o
Def. o f identity
2.
o o ( f l +0 ) = floo
Uniqueness o f m ultiplication (i.e. if a + 0 and a are the same element, then there is a single element which is that element m ultiplied by a
3. 4.
ao(a+0) = aoa+ao0 ooo+ooO =aoa
D istributive law 2,3, Transitivity o f =
5.
—(a o a) + (a o a + a o 0) = —(a o a) + a o a
Existence
o f add.
inverse,
Uniqueness o f addition 6.
( — ( a
o a )
+
a
o a )
+
a
o O
=
—(a o a ) + a o a 7. 8.
0 + ao0 = 0 ao0 = 0
The p r o o f that 0 o a = 0 is similar, from a o 0 = 0.
A ssociativity o f + Def. o f inverse Def, o f identity 0 o a = 0 also follows directly
So lu t io n s
608
(c)
( l ) A ssociativity o f + (2) D istributive law (3) D e f o f inverse (4) B y the theorem o f problem b . (5) D istributive law (6) D e f o f inverse (7) B y the theorem o f problem b . (8) From 4, 7, by sym m etry and transitivity o f :
1 1 . Given: a < b and b < c T hen a + b < b + c, b y the addition law. Since -b = -6 ,
13.
(a + b) + ( —6) < (6 + c) + ( —6)
b y the addition law
a + (b + ( —6)) < (( —6) + b) + c a+ 0 < 0+ c a < c
by assoc, and com m , b y def. o f inverse by def. o f 0
( a ) T o show: every positive integer n has the property that for all m , am o a " = am+n ( i ) To show that 1 has that property: am o a1
=
am o a
(b y def.)
=
am+1
(by def.)
(ii) To show that if k has the property, then k + 1 must have it: am o ak
— am+k
a
■= a
oa
o ar o a)
conditional premise b y def
= ( am o ak ) o a by assoc, o f o in integral domains _ am+k 0 a con(iitional premise _ 0(m+fc)+1 b y def. _ am+(k+ 1)
ky assoc Qf _)_ in ordinary arithme
From ( i ) and ( i i) the conclusion follow s b y the principle o f m ath em atical induction, ( b ) We will use induction on n ( i ) for n = 1:
(a,771) 1
P ( n ): for all m , ( am)n = ( an )m
=
am
(b y def.)
=
(a,1)” 1
(since a = a1, by def.)
(ii) T o show that P ( k ) —> P ( k + 1):
P a r t C C h a p t e r 11
609
( am) k
=
( ak)m
conditional premise
(a
= =
( am)k o a m ( ak)m o am
by def. by the conditional premise
=
(
) k+1
ak
o
a)m
an
o
bn
(a
=
o
b)n
(theorem proved in text) =
(a k+1)m
by def.
Since P ( l ) holds and P ( k ) —> P ( k + 1), it follow s from the prin ciple o f m athem atical induction that P ( n ) holds for all n.
C H A P T E R 11
{a,b},
11-1 and 11-2 are lattices; in 11-3 there is n o lub for gib for {c ,
nor any
d}.
( i ) (a ), (b ), and (c) are posets; (d ) and (e) are not transitive. ( i i) (a) is a semilattice; (b ) and (c) are not lattices. Let the poset A = ( A, > } be a meet semilattice. Set a
A b=
in f{a , 6}.
Then the algebra A a = { A, A) is a semilattice. ( a ) For all a, a A a = a (idem p oten t), since in f{a , a } = a. ( b ) For ah ( c ) Let
d< d <
d=
a,b, a/\b = b/\a (com m u tative),
since in f{o , 6} = in f{6, a},
in f{a ,in f{6 , c } } and e = in f{in f{a , 6}, c }
Then
d
< a and
in f{6, c } Therefore, d < b and d < c. Since d < a and d < in f{o ,6 }„ A nd since d < c, d < i n f { in f { a ,6 } , c }; i.e., d <
Similarly, e < (associativity).
d;
thus,
d
Suppose ( D l ) holds. Then a V (6 A c) = (a V (a A c)) V
= e.
Hence,
aA
(6 A c) =
(b A c)
(L4)
=
a V ((a A c) V (6 A c ))
(-^2)
=
a
=
= =
=
(a A
b, e.
6) A c
V ((c A a) V (c A b)) ( L 1) a V (c A (a V b)) (Dl) a V ((a V b) A c) (£ 1 ) (a A (aV b)) V ((aV b) A c) (LA) ((a V 6) A a) V ((a V b) A c) (£ 1 )
(a V 6) A (a V c) (-^1) So ( D 2) also holds A similar p r o o f shows that if ( D 2) holds, so does
=
So l u t io n s
610
a*
7. By Theorem 11-8,
has a unique complement in a complemented
a" Aa
distributive lattice. Since of a*, i.e ,,
a
= 0 and o*V o = 1, a is the complement
= (a*)*.
8 . Interchange A and V, 0 and 1 in the proof of Th, 11-11.
a“
9. Th. 11.8:
=
a* A 1
=
a" A (a V 6*)
=
(o’” A
=
OV(a*A
d
11,9:
a) V (o x A
D
1
C2 def, of 0 and V
=
dAc d A (6
=
(
=
d A d'
=
6*)
6*)
o* A 6*
= Th
def, of 1 and A
def o f com pl, ( C l )
R C 2, def of V and A V
£C 2 D1
R C 1, def of V
C H A P T E R 12
If o A
b=
0
then
a*
=
a* V (a A 6) (a* V a) A (a* V 6) 1 A (a* V 6) a“ V 6 ; hence
If a V
b=
1
then
a*
=
a* > b
a* A (a V 6) (a* A a) V (a* A 6 ) 0 V (o’* A b) a* A 6 ; hence
So
a* < b
b = a"
2. (Idempotent Law) For all a £ 2?, a U a = a and o fl
Proof: 1. a = a U
0 (o flo *)
a = a U
3.
a = ( a l l a) Cl (a U a x) B 3
4. 5.
a = aU a a fl
a= a
= a
54
2.
a = (a U a) fl 1
a
55
55 BA can be proved similarly
Part C C
hapter
12
6 11
3. ( i ) I f a — 0, the set o f join-irreducible elements S(a) = { x
jx
< o } = 0,
and 0 is the lub o f the em pty set, ( i i ) Let P ( n ) be the statem ent that every element is the join o f some join-irreducible elements if the num ber o f elements x < a in the finite B oolean lattice L is n. P ( n ) is t r iv ia lly true i f a is jo in -irred u cib le. I f a is n ot join-irreducible and not 0, then a — x V y, where x < a and y < a. So n ( x ) < n ( a ) and n{ y) < n(a). B y induction on n , it follows that x and y are joins o f join-irreducible elements: x = V s (a) n { x ) and V - V 5 (a) n ( y ) ’ so a = V n ( x ) V V n(y). 4 . Suppose a < b V c, then a = (b V c) A a
= =
(6 A a) V (c A a ) (d is trib u tiv ity ) (6 A a) or (c A a) since a is join-irreducible
= So a < b or a < c,
5. ( i ) I f L has 1, then a is its own relative pseudo-com plem ent, since by definition for all x £ L, x < a => a iff x A a < a, so a < a => a, ( i ii) For &]1 x E L, x < a = > b i f f a f \ x < b ; since a A b < b we infer b < a=> b (provided a => b exists) 6. Since the lattice if finite, it has 0, Given a, b, define C - { c \ a h c < b } = { c 1, c 2 lc3 l , , , c k} Let c0 be the lub o f C, so c0 = ci V c2 V , , , V c*. Then, if a A c < 6, also c < CoConversely, i f c < Co
aAc
<
a A c0
=
o A (c i V c2 V ,,
=
(a A Ci) V ....V (a A Cfc) (d is trib u tiv ity )
V ft)
But a A c; < 6 for each i, 1 < i < k. Therefore, o A c0 < 6 - Hence o A c < b. Thus Co satisfies the conditions on o => b So the lattice is a H eyting lattice
S o l u t io n s
61 2
R E V IE W E X E R C IS E S , P A R T C
2. For instance, P = ({ 0 ,1 , 2 ,3 ,4 } , + m o d 5) 3. Care must be taken in this problem to rem em ber that x + (y — 3) is defined as a single operation on two elements x and y Thus to check associativity it must be verified that ((* + (y ~ 3)) + (z - 3 )) = (x + (y + (2 - 3)) - 3)) which turns out to be true because b o th sides can be reduced to (x + + y + z) — 6. T he identity element is 3 and the inverse o f x is 6 — x . 4. (a ) The only finite one is ( { 0 } , + }
A ll the others consist o f all the muti-
ples o f any one integer, e.g. { 0 ,2 , —2 ,4 , —4, etc,
.}, {0 ,1 0 , —10, 20, —20,. ...}
( b ) ( { 0 } , + } certainly is not. A ll o f the others are. To set up an iso m orphism betw een the group with all integers and the group with all multiples o f a, let n —> an.
This correspondence is clearly one-one,
and preserves addition, since an\ + an 2 = a{n~i + 712) 5. There are only two:
0 —* I 1 ->■ R 2
0 —* I 1 —> R "
R'
3 —> R "
2 -> R 1 3 -> R
6. A non-A belian group o f order 6 7. (a ) E very string is a conjugate o f itself since x = x ^ e = e~^ x
Conju-
gacy is sym m etric by the definition. To prove transitivity, let x and y be conjugate and also y and 2 . Then for some t , u , v , w , x = t ^ u , y = u ~ ' t , y = v ^ 'w , and 2 = w ^ v . Case 1: let u = v and t — w. Then x = t ^ u = w ^ v = 2 ; thus x and z are conjugate because they are identical. Case 2: let u b e shorter than v, that is, there is some r such that u ~ ' r — v.
Since y = v ^ w = ■uT'r'^'w = u~' t , it follow s that
r ^ w = t. Therefore, x = t ^ 'u =
and z = w ^ v =
r\
thus x and 2 are conjugate. Case 3: let u be longer than v; that is, for some s , v ^ s = u. Since y = v T 't — v ^ ' s ^ t = v ^ w , it follow s that
P a r t C R ev iew P r o b le m s
613
Therefore, x — t ^ u = t ^ v ^ s , and 2 = w ^ v = s ^ t ^ v ,
w — s~ t
thus, x and 2 are conjugate. This exhausts the possible cases. This relation partitions A x in to equivalence classes, each class containing all the strings that are conjugates o f each other A string o f length n m ay be the only string in its equivalence class (o o o o , for exam ple, is conjugate only with itself) or there m ay be as m any as n strings in the class (abca for exam ple, is conjugate with itself and with bcaa, caab, and aabc). ( b ) Let x and y be conjugates Therefore, x = uv and y = vu, for u and v. T he string u is a string such that x ^ 'u = u ^ y , since x
/ “" “V
u = u
/ “" N
v
r'—V
u = u
/ “"N
y,
8. (1 ) ( a ) all; ( b ) all but G4; ( c ) Yes (2 ) (a ) none: G l n ot, because 2 — 4
a non-neg. integer, G2 n ot,
because (9 — 5) — 4 = 0, 9 — (5 — 4) = 8, G3 n ot, because there is no e such that for all x, b o th x — e — x and e — x = x, G4 n ot, because G3 n ot; ( b ) all but G4; ( c ) no; D1 n ot, D2 not (3 ) ( a ) all; ( b ) all but G4 (5 has no inverse); ( c ) no: D 4 n ot, because 5 2 = 10 (m od 25) and 5 ■7 = 10 (m od 25), but 2 ^ 7 (m od 25) (4 ) (a ) only G2: G l n ot, because 1 + 1 0 = 0 (m od 11), and 0 ^ A; G3 n ot, because 0 ^ A; G4 n ot, because G3 not; ( b ) all; ( c ) no: D1 n ot, D2 not (5 ) ( a ) only G2; ( b ) all; ( c ) no: D1 not, D2 not (6 ) ( a ) only G2: G l n ot, because 2 /3 + 2 /3 = 4 /3 , 4 /3 ^ A, G3 n ot, because 0 ^ A, G4 n ot, because G3 not; ( b ) only G l and G2; ( c ) no: D1 n ot, D2 n ot, D3 not ( 7 ) (a ) all; ( b ) G3 n ot, because for no x does 3 0 x — 3, G4 not — only G l and G2; ( c ) D3 n ot, D 4 n ot, because 3 0 4 = 3 0 5 = 0, but 4^5
9. ( a )
(o + b) - (c + d)
= =
[(a + b) ■c] + [(a + 6)
D3 (com m ut,
= =
(c a + c ■b) + (d a + d b) (o c + 6 c ) + ( o d + b d)
D5 D3
( b ) - 0 + 0 = —0 because x + 0 = x for all x (G 3) —0 + 0 = 0 because —x + x = 0 for all x (G 4) hence - 0 = 0 because —0 + 0 must have a single value ( G l )
S o l u t io n s
614
(c)
a 6=0 a ^ 0
3
o 0= 0
auxiliary premise P roblem 10(b) in Chapter 10
5.
6= 0 a ^ 0 D 6= 0
6. 7.
o
1, 2,
—a H— ( —a) = 0 —a + a = 0
Inverse law Inverse law
3,
—( —cl) = a
1,2, problem 10(a) in Chapter 10
4.
(d)
premise
1 2.
1,3 cancellation law (D 4) cond, p r o o f
5, conditional law neg. = 0 V 6 = double 0
~o^ 0V 6 = 0
1 0. (a )Y es group ( b ) No, - 2
|y| = \x ■y\ and we can show the new system is also a ->■ 2, - 4 -> 4, 8 -> - 8 but - 2
- 4 = 8 while 2 4 ^
( c ) No. 2 —> 4, 3 —» 6, 6 —> 12 but 2 3 = 6 while 4 •6 ^ 12. ( d ) Yes, 1/x ■1/y = 1/x ■y and the new system is a group ( e ) Yes. x 2 y2 = (x
y ) 2 and the new system is a group,
11.* (a ) R e fle x : for all a, a -1
a = e & e £ 5;
S y m : given a 1 ■b £ 5 , does it follow that 6 1 a £ 5 ? (6_1 ■a) ■(a -1
6)
=
6-1 ■(a ■a ~ 1) - b
=
6-1
e
=
6- 1
6
= Since (6-1 •a) ■(a -1
b) =
e e, 6-1 ■a is the inverse o f a-1 6 Therefore
6_1 a must also be in S. T r a n s : given a -1 ■6 £ 5 , 6-1 6 ) ■( 6 -
£ 5
b
= =
a o'
c £ 5 then (6
6-
e ■c
= a ■c so a c £ 5 It is an equivalence relation. One o f the equivalence classes is {as |x £ 5 } The others can vary from case to case. For exam ple, for G = the symmetries o f the square and S = { I , H } , the equivalence classes are:
Ei
={I,H}
2 E3 Ea
= { # ',n = {D,R} ={D',R"}
e
P a r t C R ev iew
P ro b le m s
615
But for the same G and S = { I , H, V, R' } : E 1 = { I , H , V , R 1} E2
(b)
={D,D',R,R"}
R e fle x : a + ( —a) = 0 and 0 is even, S y m : if a + ( —6) = c and c is even, then b + ( —a ) = —c which is also even, T r a n s : if a + ( —6) is even and b + ( —c) is even, then a + ( —c) is even because it is the sum o f two even num bers, a + ( —6) and b + ( —c), Equivalence classes: Ei = { x |x is od d }
(because the difference o f any two od d numbers is even)
E 2 = { * |* is even }; ( c ) R e fle x : o + ( —a) = 0 not odd: no, S y m : if a + ( —6) = c is od d , so is b + ( —a) = —c, T r a n s : if a + ( —6) is odd and 6 + ( —c) is o d d , o + ( —c) is not od d , a + ( —c) = (a + ( —6)) + (6 + ( —c)) = od d + od d = even: No not an equivalence relation;
(a)
I R 120° R 240° Di flip D 2 flip D 3 flip
axis 1
clockwise clockwise in axis 1 in axis 2 in axis 3
Sol u t io n s
6 16
y i
I I
R R
R R'
R R'
R' I
Di d2 D3
Di Dz d 2 Di Dz d2
y
R' R' I R
d2 Dz Di
Di Di d2 Dz
d2 d2 Dz Di
Dz Dz Di d2
I
R' I
R R'
R
I
R R'
( b ) { I , R, R ' } , { I , D i } , { I , D 2} , { I , D 3} , { I } I
I
'
R
R R' Di d 2 Dz
>-+T
R
D
D D
J
14. ( a ) a < b i S a = > b = l . ( c ) b A a < a A 6, and c < a = > 6 i f f a A c < 6 , so a < 6 => ( a A 6)
PART D
C H A P T E R 13
1. (a ) tree (ii)
( b ) interpretation (ii)
( ~ ( p & g ) V p), 2.3
~ { p k q ) , 2.1 { P k q ) , 22 X \
p
q
P
1, 2.3
P a rt D C h ap ter
( b ) tree (ii)
13
617
( e ) interpretation (ii)
1
C onn (N eg (C on n (p , q )) ), p, 2 V
Neg (C on n ~
(p,q)),
2
P
Conn (p, q), 1
A\ & P
(1 ,1 } -> 1
|
( 0 ,1 ) -* 1
0
(0 , 0 )5
(0 ,1 )
< 1 ,0 }-»1
k
0 l -> 0
0-+1 (M H l
(1 ,1 )
( l ,0 > -» 0
(0 ,l)-» 0 ( 0 ,0 ) —>0 ( a ) A possible m odel making all four formulas true is: M = ( D , F ) where D = {a , 6} F ( P ) = {a } F ( R ) = { { a , a), {b , a) } with assignment given g ( x ) = a and g ( y ) = b N ote that although (iii) is logically equivalent to (iv ), the inter pretation process is quite different. ( b ) (i) E very student kissed someone (ii) There is som eone w h om every student kissed. (iii) & (iv) I f som eon e/anyon e is a student then everyone kissed him , i.e., Everyone kissed every student. Take a very simple m od el where your choice o f statement for 7 is true The entire statem ent-schem a could be instantiated b y e.g., ( ( ~ ( p—*q)) V r ) , but the choice o f statement constants is yours. ( b ) t, F A L SE , ( d ) t, T R U E (convert 5 / x, 8 / y), ( e ) t, F A L SE (convert 5 / y, 8 / x) , ( g ) t, T R U E
61 8
S o l u t io n s
C H A P T E R 14
Conservativity: only a and b m atter to D A B . E xtension: E and E 1 are irrelevant to D A B . Quantity: only the cardinalities a , b , c , e m atter to D A B , Variety: D ( A ) $_ p ( E ) - 0. C um ulative effect: D { A ) is com pletely specified b y the set o f all pairs ( a, b), the cardinalities o f A and B. 2. ( a )
0,3
(b)
0,3
(c) 0,1
1,0 2, 0
1,1 2,1
3,0
(d)
0,2 1,2
0,3
00 10 20
01 11
21
02 12
L3
Part D C
hapter
1,0 2,0 3,0
14
619
0 1
'
11
.
-/ |2 1
0 2 12
0
( g ) Some p art(s) o f the tree must be shaded ( h ) A n y node to the right o f a n ode in D ( A ) is itself in D ( A) . ( i ) If {a, b) belongs to D ( A ) then (0 ,6 ) belongs to D( A) „ ri ght several at m ost 3 none at least n
T + +
left
i
T
I
+
+H -
+ -
+ +
+
-
-
+ -
-
-
-
some these neither
+ + -
every
+
+ -
-
+
aH each
+ +
-
-
+ +
infinitely many a finite num ber o f m ost
+ -
-
+ -
+ -
+ -
+ many + See section 5 for most and many.
-
4. ( a ) ( i ) Some men walk fast —» some men walk. Several students failed the final test —» several students failed a test. Infinitely m any numbers are even primes —►infinitely many numbers are prime. (ii) Some men sing, dance and laugh —» some men sing and laugh A t least five students got A ’s on their m idterm s and finals —*
S o l u t io n s
620
at least five students got A ’s on their finals.
(b) \E\ > 2, D ( A )
= { X C E\A n X ^ 0 and A - X ± 0} D ( A ) means ‘ some but not all A ’s ’ . I f A C B , D ( A ) C D ( B ) , so D
is left m onotone increasing
B ut D is not right m on oton e increasing
since E 0. D ( A ) for all A. 5. M od el should show m apping 7T : A —> A' where at least one a £ A fl X , 7r(o) 0 A fl X , if X is the set o f entities owned by John. 6. Suppose Q e meets Variation, i.e. X £ Q e ’ and ~ Y X , Y C E ' . ~ Q e —def { % S E\Z 0 Q e } For Q similarly,
£ Q e ’ for some
B y Extension ~ Z £ Q E’ .
7. (a ) W e know that if Q is m onotone increasing, Q ~ and ~ Q are m onotone decreasing, and if Q is m onotone decreasing, Q ~ and ~ Q are m onotone increasing Since Q~ negation that if Q is increasing, Q ~ is. (d ) If Q = Q ~ , X
£ Q +-> ( E - X )
) it follows b y double 0 Q.
So if for some Y
C
E , ( E - Y ) E Q then Y 0 Q and if Y 0 Q then { E - Y ) e Q 8 . Since D is definite, D { A ) is the principal filter generated b y some B C E D A E b y definition o f principal filter, so by theorem 14,2 D A A N ote that D A B - + D A A is the relational definition o f positive strength, also called the property o f quasi-reflexivity 9.
(a )
(i ) D A B <-> D { A f ) B ) E \ D A B <-> D A ( A fl B ) (conservativity)
<-» D ( A fl (A fl B ))
( A fl B ) (intersective) D (A fl B ) ( A fl B ) ( A fl B = A fl (A fl B )) <-» D (A fl B ) E (T heorem 14 2) □ (ii) D A B ^ DBA-. D B A <-> D (A n B ) E (b y 9 .( a ) ( i ) ) and D A B +-> D ( A Cl B ) E (by ( a ) ( i ) ) so D B A <-» D A B . □ ( b ) Some students walk —» some walkers are students —> some walking students are individuals. (There exist walking students,) 10. ( a ) D A B & D B C , so A C B and B C C by theorem 14.3, so A C C B y Conservativity D A (A fl B ) , so D A A , since A fl B = A. But then also D A (A fl C ), since A fl C = A . So b y Conservativity D A C □
(b) I f D is connected, ~ D is antisym m etric b y definition o f connect edness. W ith the result o f ( a ) , ~ D is transitive. connected by definition o f transitivity. □
So D is alm ost-
P a r t D C h a p t e r 15
6 21
( c ) Suppose D E A B and D transitive. B y Conservativity D EA ( A P \ B ) I Take E' D E and A' C E' such that \A'\ = |A| and A fl A' = A fl B. Clearly D EA ( A fl A'), and with Extension D E’ A ( A fl A'). W ith Con servativity D e ’ A A ' Take 7T a perm utation o f E' which leaves A n A' and E 1— ( A U A') intact, but interchanges A — A' and A' — A. Quantity gives D E’ 'k{A)'k(Ai), i.e, D E’ A'A. Since D is transitive, D E' A A ' and D e ' A ' A im ply D E’ A A , and Extension gives D E AA. □
C H A P T E R 15
1. g " ( x ) =
3. (a)
o «o•O *1
J
i0T
reflexivity f true on io , i i , i 2
(b) ^ ....- . t0
ij
*o
H
----- .»i2
transitivity V true on i 0 , h , i 2
(c) sym m etry ip true on io and ij
622
S o l u t io n s
PART E
C H A P T E R 17
1. (a ) (ii), ( iii), (iv ); ( b ) The set o f all strings containing a total o f n l ’s, where n = 2 (m odu lo 3) 2. ( c )
3 . (a )
0
0
P a r t E C h a p t e r 17
623
S o l u t io n s
624
4. (c)
5. (a)
(b)
6 . (a ) D uplicate the states o f the original machine, and duplicate all the transitions except those leading to So- A d d a new state S'0. W herever the original m achine had a transition from a state 5; in to So, labelled a, let the new autom aton have a transition from Si in to S'0, labelled a. I f the original autom aton had a loop transition on So, add a loop transition on Sq instead. For every other transition from So to a state
Part E C
Sj,
add a transition from
17
hapter
S'0 to Sj,
625
with the same label. I f 5 0 was
a final state, make Sq a final state as well. The resulting autom aton starts in So, but for the rest o f the com putation, S '0 plays the role previously played b y So
7. (a )
So
(b )
(c)
( d ) A determ inistic equivalent o f C is: 0
0
0
T he com plem ent construction then gives us D :
S o l u t io n s
626
D accepts any string containing a positive even num ber o f l ’s.
(b )
9. ( a ) the (m an is U men are) here ( b ) the old* (m an is U men are) here ( c ) the old* (m an is U men are) here (and the old* (m an is U men
P a r t E C h a p t e r 17
627
are) here)*
(d) O’ KTIO' ( e ) (0 U 1)“ 101(0 U 1)* or (corresponding to the autom aton given as the answer to the original problem ) (0 “ 11“ 0 0 )’‘ 0''11“ 01(0 U l)*
\
)i
So l u t io n s
628
11. ( a ) S —>aa ( b ) S —♦ e ( c ) 5 —♦aA S —>ab S aS S —+bS S —*ba S ->bb
(d ) S-* a
S —>a 5 —>65 S ->b A ->bA
C -*aC
A ->aB A —» a
C -*bD C —>6
S ->bA
B
D
B —>b (f) S - * b A S —♦al?
12. ( a ) (a ,g 0,3 i )
A —* aS A —* a
B —+bA B —♦ aC
S ->bC
A->bB
C —> aB C->bS C ->b
aD D —>a
A-+bS j4 —♦aC
C —>bB C —» aE
B -*bC B —>a D
D ->bE E —+bD
B —> a
E —>b
( c ) (&,9o,9o)
( e ) (a ,9 o ,9 i)
(a, 33,32)
( a , 9o, 9 i)
( ffl, 9 i , 9 o )
( 6 , 93, 9o)
(6 , 9 i ,9 2 )
( 6 , 9o> 9 3 )
(6 ,9 2 , 9 i)
(6 , 9 2 , 9 2 )
( 0 , 92, 93)
to’ to '-v'
**3 II
O '-v'
( a ,3 i, 32) (6,91,91)
to '-v'
(a, 3 i , g 2 ) ( 6 ,5 1 ,9 2 )
**3 II
9 o ,9 i )
( e ) 5 —»e S —» a A
**3 II
(b,
B —*b B bB
A —> bA
S -*b S —» aC
bB
A -*a A —> aB
C H A P T E R 18
1. (a ) S - * a S a ( b ) 5 —> (c) S-*aBB A -*aS S —* a B a A —+aAb S —* b A B A —* b B A A B ->bB B —*b
A —*ab
S - > b B A A —>b A B A S -> e A - > b A A B
B -*bS B —* a B B B
P a r t E C h a p t e r 18
629
2. A ny rules which are already o f the form A - > a or A —>B C can be left unchanged R em ove all rules o f the form A —♦ B b y the procedure ou t lined in Sec. 18.5, For each rule o f the form A —» a.\a2 . a n , where n > 2 and one oi m ore o f the a t is a term inal sym bol, replace each occurrence o f such an a ; b y a new nonterm inal Ai that occurs nowhere else and add the rule Ai —♦a* (w hich is o f the allowed form ).
The
right sides o f all rules now consist either o f a single term inal sym bol or else a string o f two or m ore nonterminals. E ach remaining rule o f the form A —» Bi B 2 ■■■B n ( n > 3) is now replaced b y the rules A —♦ Bi C\, Cl -+ B 2c 2, .... , C n - i -*■ B n - x B n , where C l t C2,. . , Cn_ i are new non terminals that occur nowhere else in the grammar. A ll rules are now o f the required form. Clearly for every derivation o f a term inal string in the original grammar there is a derivation o f that string in the new grammar and conversely.
3. ( b ) I f L is context free, then for some sufficiently long string w £ L, there exist u , v , x , y , z such that w = u v x y z , v and y not b o th e, and uvlxy'lz £ L for all i > 0, Let u — a?, v = aq, x = ar , y = a‘ , z = at, and call b = p + r + t and c = q + s. Then b + ic must be prim e for all i > 0. W hen i = be2, b + ic = b + (bc2)c = b + be3 — 6(1 + c3) = 6(1 + c ) ( l — c + c2). Since c > 1, (c + l ) > 2 and thus 6 + 6c3 is divisible by 2 or some large integer This establishes that 6 + 6c3 cou ld not be prim e unless it happened to equal 2 However, in this case 6 = 1 and c = 1, and it is obviou sly false that 1 + i ■ 1 is prim e for all integral values o f i. Therefore an ( n prim e) is not type 2.
4 . If L is generated b y a cfg G, construct G' b y reversing the right sides o f all rules o f G. G' generates LR
5. M = { K , - £ , T , A , s , F ) K = {q0,qi} £ ={a,b,c,d} T - {A } s =q0 F = {li}
A = { ( q 0, a b , e ) ~ ^ ( q 0, A ) (q0 , c d , A ) - * ( q 1 , e ) (q 1 , c d , A ) ~ ^ ( q 1 , e ) }
So l u t io n s
630
6
.
A = { { q 0 , e , e ) - * { q 1,$) ( qi , a, $) — (g i, j4$) ( g i, 6, $) —» (q1: B$) (q 1 , a , A ) - > ( q 1 , A A ) (
(g i,6 ,
( ? i ,c , $ ) - * ( g i ,e ) }
^ = { 9i} 7. The m achine will store the num ber o f a ’s preceding the 6.
Then it
will make a choice o f whether to start checking o ff 1 or 2 a ’s from the storage tape with each new a o f the input tape, Once m ade, the choice must stay consistent for the rest o f the com putation, A = {(g o, a, e ) ~* (g0> A) ( g 0 , 6, e ) ~ * { q 1 , e) ( g 0> 6, e ) —> ( g 2 , e )
(qu a >A ) ^ ( q 1 , e ) (q2 ,aa, A ) —> (q2, e )}
^ = {91,22} 8. No, P roblem 7 provides a counterexam ple, 9. C hoose akbkck in L. Then i f L is context free, there exist u , v , x , y , 2 , (\vy\ > 1) such that u v x y z = akbkck and uvn x y nz £ L for all n > 0, But if so, then neither v nor y can contain more than one typ e o f letter since “pum ping” w ould produce a string with 6’s preceding a ’s, or c’s preceding b’s, Therefore, the only remaining possibilities are: 1) b o th v and y in a*, 2) b oth v and y in 6” , 3) b o th v and y in cx, 4) v in a* and y in 6*, 5) v in ax and y in c*, or 6) v in 6* and y in c“ , provided, o f course, that n ot b o th v and y are empty. But in cases 1-4 “pum ping” v and y produces a string not in L since the a ’s, b’s, and c ’s will not b e in the required p rop ortion o f i a’s, j b's, and m a x ( i ,i ) c ’s. In case 5, if y — e, then it is like case 1; if v = e, then it is like case 3; if neither v nor y is em pty then n = 0 ( “pu m pin g” zero tim es) produces
Part E C
a string not in L.
hapter
Case 6 is similar.
19
631
Therefore, no such u , v , x , y , z
exist, and L is not context free.
C H A P T E R 19
1.
2
£ = { ( g o , 0 ) - * ( g o, £ )
(go, i)->
* = {(8b ,# )-> («i,l)
(gi,i)-*(gi ,R) (3i,#)-»(?2,1) (?2>i)->(32,^)
( q i , 0 ) - > ( q o, R )
(gi>i) (g2 ,l ) (32,l)-»(33,0)
( ? 2 ,# ) - » (go,l)}
(ffs, o) —+(34, jR)
« = go
( ? 4 , 1) - » ( ? 3 , 0 ) (qi, 0)
(q0, R) }
s — qo 3 . (a )
qo So a a
a
# ## #
a
( b ) T he m achine moves back and forth
So
# a
# #
a a
along the tape; each time it encounters an # it changes it to a and reverses direction;
Si Si a
a
# a a
a
#
# #
a a
a a
# #
a a
So
# a
a a
a
a a
Si a
a
Si So a
a a
a a
a
a
a
a
Si
a a a So a a Si a
Si a a
a
( c ) N o; ( d ) N o; ( e ) N o; ( f ) Same as ( b )
4. (a ) M = ( { g o , g i } , { a , # } , <5,go) £ = { ( g o , # ) - * (go,-ft)
(go,a)— *(?i> R) (ffi, #) —+ (qi>R) (?i,a)-> (qi,R)}
(b ) M — ({go,gi},{a,#},<^,go)
5=0
(
S o l u t io n s
632
5. (a ) T otal, since the sum o f any two even numbers is even. ( b ) Partial, There ar e some prim e numbers whose sum is prim e (2 + 5 = 7,2 + 11 = 13), but unless one o f the prim es is 2, b o th primes will be od d , and their sum will then be even, and hence divisible b y 2, i.e. not prim e ( c ) Partial 6 . Reading the digits from right to left, m ultiply the first digit b y 2 ° (= 1), the second b y 21, and so on; in general the nth digit is to b y m ultiplied by 271. Then add the resulting numbers to get the result, e g. 101 ==> (1 x 2°) + (0 x 21) + (1 x 22) = 1 + 0 + 4 = 5. 7. G = ( { [ ,] , A, D , S, a } , { a } , R, S) R = {S-+[A], [-* [£ , D A -* AAD, [-* e , ] e > A —» a} 8. Given arbitrary M , m od ify it so that it first erases its input tape and then proceeds as M would (on the resulting em pty ta p e). Call this m achine M '
Then M ' accepts its input (in fact, all inputs) iff
M accepts e, Thus, a T M which decided whether an arbitrary T M accepts at least one input could be applied to M ' to decide, in effect, whether M accepts e. Since the latter is im possible, the p roblem is undecidable.
C H A P T E R 20
1. (a )
S —>a S B c S —» a Bc c B —>B e aB —» ab
bB —>bb ( b ) R eplace the rule 5 —» e b y 5 —» A B C in the gram m ar o f (19-5)
A p p e n d ix
5 -+ aa S — bb S — a AS' S-
A a —>aA A b - > bA B a —♦aB Bb —>bB
bBS'
E II
633
A ' A -+ aA" A' B — aB B ' A — bA' B ' B - bB'
S' a AS' S' — bBS' S' — a A "
A A - > aA' A B - » aB' B A - > bA!
aa A'A" A ' B " — ab B ' A " — ba
S'
B B - » bB'
B'B"
b B"
bb
A P P E N D I X E II
1.
(c )
Q
1 So
(
A n even number o f autom aton. N ote that there is a non-trivial loop betw een So and Si, so this autom aton is cyclic
(d)
Almost all automaton. N ote that this autom aton is not perm utation invariant
S o l u t io n s
634
R E V IE W P R O B L E M S , P A R T E
1. (a ) I am right; That I am right seems certain; T hat that that I am right seems certain seems certain seems certain. ( b ) For example: K = {go,gi},£ =
{s ,c ,i,
=
( 0 , - F = {go}
A = {(g0,£, e)-*(go,*) (go, i a r , e) —>(g: , e) (?i> s c , t ) —> (g1; e )} 2. ( b )
0,1
0,1
(c) 1
J '
3 . (a ) (0 * 1 1 )* 0 1 1 or 0*11(0*11)*; ( b ) 111* or 1*11 or 11*1; ( c ) (0*(11)*0*)*
Part E R
e v ie w
P
roblem s
4. K = {go, 2 1, 22, £3}, £ = {a, b, # } , s = 2o
6 = {(go, b) -*■ (g0,-£) (go, # ) -*■ (go, R ) (go, a ) - * (gi,6)
{ < h , b ) - > ( q 2>L ) {q2, a ) - * { q 2, L )
(g2 ,#)-(g2,i) (g2 ,&)— +(gs, *)}
635
Bibliography PART A Introductory textbook: Halmos, P, R : 1960, Naive Set Theory, Princeton, N. J , Van Nostrand Set Theory, Paradoxes and Foundations: Anderson, A. R. (e d ): 1964, Minds and Machines, Englewood Cliffs, N, I , Prentice-Hall Barwise, J, and J. Etchemendy: 1987, The Liar An Essay on Truth and Circularity, New York, Oxford University Press, Cohen, P.: 1966, Set Theory and the Continuum Hypothesis, New York, Benjamin. Copi, I, M,: 1971, The Theory of Logical Type, London, Routledge & Kegan Paul., Dauben, J W .: 1979, Georg Cantor Harvard University Press,
His Mathematics and Philosophy o f the Infinite, Cambridge, M,A.,
Fraenkel, A . and Y, Bar-Hillel: 1973, Foundations of Set Theory 2nd edition, Amsterdam, North-Holland. Godel, K.: 1940, The Consistency of the Axiom o f Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory, Princeton University Press Hofstadter, D R : 1979, GSdel, Escher, Bach An Eternal Golden Braid, New York, Vintage Books Kunen, K: 1980, Set T h eory An Introduction to Independence Proofs, Amsterdam, North-Holland. Martin, R, L : 1984, Recent Essays on Truth and the Liar Paradox, New York, Oxford University Press Moore, G H : 1982, Zermelo’ s Axiom of York/Heidelberg/Berlin, Springer Verlag,
Choice
Its
origins,
development
and influence,
New
Quine, W. van Orman: 1963, Set Theory and Its Logic, Cambridge, M, A >Harvard University Press. Quine, W. van Orman: 1966, The Ways of Paradox, New York, Random House, Russell, B : 1919, Introduction to Mathematical Philosophy, London, Allen and Unwin, Smullyan, R,: 1978, What Is the Name o f This Book?, Englewood Cliffs, N J , Prentice-Hall, Zadeh, L A : 1987, Fuzzy Sets and Applications: Selected Papers, New York, Wiley. PART B Introductory textbooks: Copi, I M: 1965, Symbolic Logic 2nd edition. New Yoik, Macmillan, Jeffrey, R C : 1967, Formal Logic Its Scope and Limits, New York, McGraw-Hill, Kalish, D and R Montague: 1964, Logic, New York, Harcourt, Brace and World Mates, B : 1972, Elementary Logic, 2nd edition, New York, Oxford University Press. Thomason, R H : 1970, Symbolic Logic, New York, Macmillan. An innovative method for learning predicate logic on a Macintosh computer: J Barwise and J. Etchemendy, Tarski’ s World. Available from Kinko’ s Courseware 1987,, Mathematical logic, axiomatization and metatheory: Barwise, J (ed ): 19 7 7, Handbook of Mathematical Logic, Amsterdam, North-Holland, Bell, J, L : 1978, Boolean-valued Models and Independence P roof s in Set Theory, Oxford, Clarendon Press Bell, J,, L and M, Machover: 19 77, A Cour se in Mathematical Logic, Amsterdam, North-Holland.. Beth, E,: 1962, Formal Methods, An introduction to symbolic logic and to the study o f effective operations in arithmetic and logic, Dordrecht, Reidel. Beth, E„: 1970, Aspects o f Modern Logic, Dordrecht, Reidel Boolos, G„ and R, Jeffrey: 1980, Computability and Logic, 2nd edition, Cambridge, England, Cambridge University Press, Chang, C C , and Keisler, H J,: 1973, Model Theory, New York, American Elsevier and Amsterdam, NorthHolland, Davis, M : 1965, The Undecidable, Hewlett, N.Y., Raven Press,
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Davis, M and E J, Weyukei: 1983, Computability, Complexity and Languages Fundamentals o f Theoretical Computer Science, New York/London, Academic Press Gamut, L, T F : Logica, Taal en Betekenis, vol 1 and 2 (in Dutch), Spectrum, Utrecht (English translation to appear with the University of Chicago Press), G5del, K: 1962, On Formally Undecidable Propositions, New York, Basic Books Kleene,S G : 1967, Mathematical Logic, New York, W iley Kneale, W. and M Kneale: 1962, The Development of Logic, Oxford, Clarendon Press Kreisel, G , and Krivine, I L„: 1967, Elements of Mathematical Logic:. M odel Theory, Amsterdam, NorthHolland Landman, F.: 1986, Towards a Theory of Information The status o f partial objects in semantics, Dordrecht, Foris. Malitz, J.; 1979, Introduction to Mathematical Logic Set-theory, Computable Functions, M odel theory, N.Y., Springer Verlag Peano, G: 1973, Selected Works of Giuseppe Peano, ed and transl, by H C. Kennedy, Toronto, University o f Toronto Press, Quine, W, van Orman: 19 72, Methods o f Logic, 3rd edition, N Y , Holt, Rinehart and Winston, Inc Rasiowa, H, and R Sikorski: 1970, The Mathematics of Metamathematics, 3rd edition, Warszawa, Panstwowe Wydawn Robinson, A ,: 1965, Introduction to Model Theory and to the Metamathematics o f Algebra, 2nd edition, Amsterdam, North-Holland, Shoenfield, J. R : 1967, Mathematical Logic, Reading, M, A , Addison-Wesley, Smullyan, R : 1961, Theory of Formal Systems, Princeton, N J , Princeton University Press Tarski, A. M ,: 1956, Logic, Semantic s, Mathematic s, Oxford, Oxford University Press Tarski, A. M. and R M Robinson: 1953, Undecidable Theories, Amsterdam, North-Holland. Tarski, A M and J, C, C McKinsey: 1948, A Decision Method for Elementary Algebra and Geometry, 2nd edition, Berkeley, U G Press. PART C Abbott, J, G : 1969, Sets, Lattices and Boolean Algebras, Boston, Allyn & Bacon B iik h o ff,G : 1961, Lattice Theory, Providence, R I , A M S, Colloquium, vol., 25 Birkhoff, G, and S MacLane: 1977, A Survey o f Modern Algebra, New York, Macmillan Gill, A : 1976, Applied Algebra for the Computer Sciences, Englewood Cliffs, N J , Prentice-Hail. Gr&tzer, G .: 1968, Univer sal Algebra, New York, Van Nostrand GiMtzer, G : \91 \, Lattice-theory First concepts and distributive lattices, San Francisco, Freeman and Co MacLane, S : 1971, Categories for the Working Mathematician, Berlin, Springer MacLane, S, and G. Birkhoff: 1979, Algebra, 2nd edition, New York, Macmillan Scott, D, S.: 1972, ‘ Continuous lattices’ , in: Toposes, Algebraic Geometry and Logic, F W, Lawvere ( e d ), 97-136, Berlin, Springer, PA R T D Ajduciewicz, K ,: 1935, Die syntakti sche KonnexitQl, Studia Philosophica 1 Asher, N, and D Bonevac: 1985, ‘ How extensional is extensional perception?’ , Linguistics and Philosophy 8.2, 203-228 Barendregt, H : 1984, The Lambda Calculus Its syntax and semantics, North-Holland, Amsterdam/New York/Oxford, Barwise, J..: 1981: ‘ Scenes and other situations’ , The Journal o f Philosophy 78, 369-397, Barwise, J„ and R, Cooper: 1981, ‘ Generalized quantifiers and natural language’ , Linguistics and Philosophy 4, Baiwise, J. and J, Perry: 1983, Situations and Attitudes, Cambridge, Bradford Books, B&uerle, R., U, Egli and A, von Stechow (ed s): 1979, Semantics from Different Points o f View, Berlin, Springer. vanBenthem, J : 1983, The Logic of Time, Dordrecht, Reidel, van Benthem, J : 1985, A Manual o f Intensional Logic, Chicago, the University o f Chicago Press, vanBenthem, I : 1986, Essays in Logical Semantics, Dordrecht, Reidel
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Keenan, R : 1987, ‘ A semantic definition o f “ indefinite NP’ ” , in Reuland and ter Meulen (eds ), 286-317, Ladusaw, W.: 1979, Polarity Sensitivity as Inherent Scope Relations, P hD Diss University o f Texas, Austin. Ladusaw, W ,: 1982, ‘ Semantic constraints on the English partitive construction’ , in D Flickinger et al (eds,), Proceedings of the First West Coast Conference on Formal Linguistics, Dept o f Linguistics, Stanford University, 231-242 Lewis, D, K : 1973, Counterfactuals, Oxford, Basil Blackwell Lindstrom, P : 1966, ‘ First order predicate logic with generalized quantifiers’ , Theoria 32, 186-195. Link, G Link, G
1979, Montague Grammatik I Die Logische Grundlagen, Wilhelm Fink, Mtinchen 1983, ‘ The logical analysis of plurals and mass terms, a lattice-theoretic approach’ , in R, Bauerle et al,
(eds,) (1983), Meaning, Use and Interpretation o f Language, Berlin, de Gruyter, 302-323 Link, G.: (1984), ‘ Hydras. On the logic of relative constructions with multiple heads’ , in F Landman and F„ Veltman (e d s ) (1984), Varieties o f Formal Semantics, Dordrecht, Foris, 245-258,, Lo Cascio, V and C . Vet (e d s ): 1986, Temporal Structure in Sentence and Discour se, Dordrecht, Foris. Loux, M. J., (ed ): 1979, The Possible and the Actual, Ithaca, N Y , Cornell University Press May, R : 19S5, Logical Form Its structure and derivation, MIT Press., ter Meulen, A. (ed,): 1983, Studies in Modeltheoretic Semantics, Dordrecht, Foris Montague, R : 1974, Formal Philosophy, R H Thomason (ed ), New Haven, Yale University Press, Mostowski, A,.: 1957: ‘ On a generalization of quantifiers’ , Fundamenta Mathematicae 4 4 ,1 2 -3 6 Paitee, B„ Hall: 1975, ‘ Deletion and variable binding’ , Linguistic Inquiry 6 , 203-300 Paitee, B, H„: 1979, ‘ Semantics - mathematics or psychology?’ in Semantics p o m Different Points o f View, R, Bauerle et al (e d s ), Berlin, Springer, 1-14, Paitee, B. H,,: 1984, ‘ Compositionality’ , in F. Landman and F, Veltman (eds,), Varieties o f Formal Semantics, Dordrecht, Foris, 281-312, Quine, W„ van Orman: 1953, From a Logical Point o f View, 2nd edition (1961), Cambridge, Harvard University Press. Quine, W, van Orman: 1956, ‘ Quantifiers and propositional attitudes’ , The Journal o f Philosophy 53,177-187, Reichenbach, H.: 1947, Elements o f Symbolic Logic, New York, MacMillan Reuland, E. and A ter Meulen (e d s ): 1987, The Representation of Jndefiniteness, Current Studies in Linguistics, Cambridge, MIT Press. Scott, D ,: 1969, Models for the lambda-calculus, ms (unpublished) 53 pp Scott, D.: 1980, ‘ Lambda-calculus: some models some philosophy’ , in Barwise, I et al (eds.), The Kleene Symposium, North-Holland, Amsterdam/New York Soames, S.: 1985, ‘ Lost innocence’ , Linguistics and Philosophy 8.1,59-71. Turing, A : 1937, ‘ Computability and lambda-definability’ , Journal o f Symbolic Logic 2 ,1 5 3-1 6 3 Vendler, Z : 1968, Linguistics in Philosophy, Ithaca, Cornell University Press Westerstahl, D : 1984, ‘ Some results on quantifiers’ , Notre Dame Journal of Formal Logic 25,152-170. Westerstahl, D : 1985a, ‘ Logical constants in quantifier languages’ , Linguistics and Philosophy 8 ,3 8 7—413. Westerstahl, D.: 1985b, ‘ Determiners and context sets’ , in van Benthem and ter Meulen (eds ): (1985), 45-71. Zwarts, F : 1983, ‘ Determiners: arelational perspective’ , in ter Meulen (ed,) (1983), 37-62,, P A R TE Textbooks: Arbib, M. A : 1969, Theories o f Abstract Automata, Englewood Cliffs, New Jersey, Prentice-Hall, Brainerd, W . S. and L H.. Landweber: 1974, Theory o f Computation, New York, John W iley & Sons, Cohen, D, I, A ,: 1986, Introduction to Computer Theory, New York, John W iley & Sons, Gill, A : 1963, Introduction to the Theory o f Finite-State Machines, New York, McGraw-Hill. Ginsburg, S : 1962, An Introduction to Mathematical Machine Theory, Reading, Massachusetts, AddisonWesley Publ, Co, Ginsburg, S.: 1966, The Mathematical Theory of Context-Free Languages, New York, McGraw-Hill. Gross, M, and A, Lentin: 1970, Introduction to Formal Grammars, New York, Heidelberg, and Berlin, Springer-Verlag Harrison, M, A ,: 1965, Introduction to Switching and AutomataTheory, New York, McGraw-Hill B ook Co. Hopcroft, J. E and J D, Ullman: 1979, Introduction to Automata Theory, Languages, and Computation, Reading, Massachusetts, Addison-W esley Publ Co
Lewis, H, R and C. H, Papadimitriou: 1981, Elements of the Theory of Computation, Englewood Cliffs, New Jersey, Prentice-Hall, Minsky, M, L : 1967, Computation ■Finite and Infinite Machines, Englewood Cliffs, New Jersey, Prentice-Hall. Nelson, R J : 1968, Introduction to Automata, New York, John W iley & Sons.. Rogers, H , Jr.: 1967, Theory of Recursive Functions and Effective Computability, New York, McGraw-Hill Salomaa, A,,: 1969, Theory o f Automata, Oxford, Pergamon Press Salomaa, A.: 1973, Formal Languages, New York, Academic Press. Collections: Davis, M, (ed ): 1965, The Undecidable Basic Paper s on Undecidable Propositions, Unsolvable Problems and Computable Functions, Hewlett, New York, Raven Press. Dowty, D R., L, Karttunen, and A M Zwicky (ed s): 1985, Natural Language Processing Theoretical, Computational and Psychological Perspectives, New York, Cambridge Univ Press Luce, R, D , R, R Bush, and E Galanter (eds ): 1963, Handbook o f Mathematical Psychology, Vol, 2, New York, John W iley & Sons. Luce, R. D , R R. Bush, and E, Galantei (eds ): 1965, Readings in Mathematical Psychology, Vol 2, New York, John W iley & Sons, Manas ter-Ramer, A (ed ): 1987, Mathematics o f Language, Amsterdam, John Benjamins, Moore, E F. (ed.): 1964, Sequential Machines Selected Papers, Reading, Massachusetts, Addison-Wesley Publ, Co. Oehrle, R. T ,, E Bach, and D, Wheeler (e d s ): 1988, Categorial Grammar s and Natural Language Structures, Dordrecht, Reidel. Shannon, C E, and J. McCarthy (eds ): 1956, Automata Studies, Princeton, New Jersey, Princeton Univ. Press.. Savitch, W J, E. Bach, W Marsh, and G, Safran-Naveh (eds,): 1987, The Format Complexity o f Natural Languages, Dordrecht, Reidel,, Other books and articles: Ades, A, E and M, J Steedman: 1982, ‘ On the order of words’ , Linguistics and Philosophy 4,517-558, Aho, A. V.: 1968, ‘ Indexed grammars - an extension o f context-free grammars’ , Journal o f the Association for Computing Machinery No,, 15, No., 4,647-671 Ajdukiewicz, K.: 1935, ‘Die syntaktische Konnexitat’ , Studia Philosophica 1, 1-27 Storis McCall (ed.) (1967), Polish Logic, Oxford, Oxford Univ, Press
English translation in
Bar-Hillel, Y.: 1953, ‘ A quasi-arithmetical notation for syntactic description’ , Language 29,47-58, Bar-Hillel, Y„, C Gaifman, and E. Shamir: 1960, ‘ On categorial and phrase structure grammars’ , Bulletin o f the Research Council of Israel 9, 1-16 Reprinted in Bar-Hillel, Y : 1964, Language and Information : Selected Essays on their Theory and Application, Reading, Massachusetts, Addison-Wesley Publ Co. van Benthem, J.: 1986, Essays in Logical Semantic s, Dordrecht, Reidel., van Benthem, J.: 1988, ‘ The Lambek calculus’ , in Oehrle, Bach, and Wheeler (eds.), pp. 35-68. Chomsky, N.: 1956, ‘ Three models for the descriprion o f language’ , IRE Transactions on Information Theory 2, No. 3, 113—124 A corrected version appears in Luce, Bush, and Galanter (eds): 1965. Chomsky, N : 1957, Syntactic Structures, The Hague, Mouton & Co, Chomsky, N : 1959, ‘ On certain formal properties of grammars’ , Information and Control 2, No, 2, 137-167. Chomsky, N : 1963, ‘ Formal properties of grammars’ , in Luce, Bush, and Galanter (eds,,), pp, 323-418.. Chomsky, N : 1965, Aspects of the Theory of Syntax, Cambridge, Massachusetts, MIT Press. Chomsky, N, and G. A, Miller: 1958, ‘ Finite-state languages’ , Information and Control 1, 91-112 Reprinted in Luce, Bush and Galanter' (eds.,): 1965. Chomsky, N, and G A, Miller: 1963, ‘ Introduction to the formal analysis of natural languages’ , in Luce, Bush and Galanter (eds ) (196,3), pp, 269-321 Davis, M ,: 1958, Computability and Unsolvability, New York, McGraw-Hill,, Friedman, J , D, Dai, and W, Wong: 1986, ‘ The weak generative capacity o f parenthesis-free categorial grammars’ , proceedings o f the 11th International Conference on Computational Linguistics, Friedman, J„ and R, Venkatesan: 1986, ‘ Categorial and non-categorial language’ , proceedings o f the 24th Meeting o f the Association for Computational Linguistics. Gazdar, G„: 1985, ‘ Applicability of indexed grammars to natural languages’ , Report No,, CSLI-85-34, Center for the Study o f Language and Information, Stanford University,
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Gazdai, G. and G K Pullum: 1985, ‘ Computationally relevant properties o f natural languages and their grammars’ , New Generation Computing 3, 273-306; also appeared as Report CSLI-85-24, Center for the Study o f Language and Information, Stanford University Ginsburg, S.: 1975, Algebraic and Automata-Iheoretic Properties o f Formal Languages, Amsterdam, NoithHolland, Ginsburg, S. and B, H Partee: 1969, ‘ A mathematical model of transformational grammar’ , Information and Control 15, 297-334 Ginzburg, A ,: 1968, Algebraic Theory o f Automata, New York, Academic Press Hayashi, T.:1973, ‘ On derivation trees o f indexed grammars - an extension o f the uvwxy theorem’ , publ, o f the Research Institute for Mathematical Sciences, Kyoto University, Vol, 9, pp, 61-92, Joshi, A K : 1985, ‘ How much context-sensitivity is necessary for characterizing structural descriptions - tree adjoining grammars’ , in Dowty, Karttunen, and Zwicky (ed s). Joshi, A, K,, L Levy, and M. Takahashi: 1975, ‘ Tree adjunct grammars’ , Journal o f the Computer and System Sciences 10, No, 1,136-163. Kuroda, S. Y : 1964, ‘ Classes of languages and linear bounded automata’ , Information and Control 7,207-223,, Lambek, J,: 1958, ‘ The mathematics of sentence structure’ , American Mathematical Monthly 6 5,1 54 -1 70 Langacker, R W,: 1969, ‘ On pronominalization and the chain of command’ , in D„ A., Reibel and S. A, Schane (eds.), Modern Studies in English Readings in Transformational Grammar, Englewood Cliffs, New Jersey, Prentice-Hall. Marsh, W. E.: 1985, ‘ Some conjectures on indexed languages’ , paper presented to the Association for Symbolic L ogic Meeting, Stanford Univ., July 15-19 Peters, P, S„, Jr: 1973, ‘ On restricting deletion transformations’ , in Gross, Halle, and Schutzenberger, The Formal Analysis o f Natural Languages, The Hague, Mouton. Peters, P, S , Jr, and R. W.. Ritchie: 1973, ‘ On the generative power of transformational grammars’ , Information Sciences 6,49-83, Pollard, C : 1984, Generalized Phrase Structure Grammars, Head Grammars, and Natural Language, Ph D Dissertation, Stanford University, Pullum, G, K, and G.. Gazdar: 1982, ‘ Natural languages and context-free languages’ , Linguistics and Philosophy 4, 471-504. Reprinted in Savitch, Bach, Marsh, and Safran-Naveh (eds ). Rabin, M, O and D. Scott: 1959, ‘ Finite automata and their decision problems’ , IBM Journal o f Research and Development 3, No, 2,114-125. Roach: 1987, ‘ Formal properties of head grammars’ , in A Manaster-Ramer (ed,) Shieber, S M.: 1985, ‘ Evidence against the context-freeness of natural language’ , Linguistics and Philosophy 8, 333-343, Reprinted in Savitch, Bach, Marsh, and Safran-Naveh (eds,), Vijay-Shankar, K ,: 198 7, A Study o f Tree Adjoining Grammars, Ph.D. Dissertation, Univeisity o f Pennsylvania, Wasow, T\: 1978, ‘ On constraining the class of transformational languages’ , Synthese 39, 81-104, Reprinted in Savitch, Bach, Marsh, and Safran-Naveh (eds ), Weir, D. J, and A K Joshi: 1988, ‘ Combinatory categorial grammars: generative power and relationship to linear context-free rewriting systems’ , proceedings of the 26th Meeting of the Association for Computa tional Linguistics, Weir, D. J,,, K. Vijay-Shanker, and A, K, Joshi: 1986, ‘ The relationship between tree adjoining grammars and head grammars’ , proceedings of the 24th Meeting o f the Association for Computational Linguistics,
General collections and r eaders: Bernacerraf, P, and H, Putnam (eds,): 1964, Philosophy o f Mathematics Selected Readings, Englewood Cliffs, N, J., Prentice-Hall, Copi, I, M, and J A, Gould: 1967, Contemporary Readings in Logical Theory, New York, Macmillan van Heijenoort, J : 1967, From Frege to Gddel, a sourcebook in Mathematical Logic 1879-1931, Cambridge, M.A,, Harvard University Press. Kline, M ,: 1972, Mathematical Thought from Ancient to Modern Times, New York/Oxford, Oxford University Press. Martinich, A P, (ed.): 1985, The Philosophy of Language, New York/Oxford, Oxford University Press.
Index Abelian group 257,266,312,607 monoid 264—266,434 Absorption 283,293 law 282, 292 Abstract collection 3 concept 7 relation 28 Abstraction, lambda 338-371 Acceptance for ever after state 569 Acceptor 454 Accessibility 414-419,424,429 Ackermann, W. 220-224 Acoustic phonetics vi Action, collective v distributive v Acyclic automaton 572 Adamczyk, K. vii Addition 20,21, 31,73,76-80, 83,215,413, 593,599 law 274, 603, 608 Ades, A. 553 Adjective, autological 26 heterological 26 Adjunction 545-547,555 Agentless passive 363, 367 Aho, A. 536, 541 Ajdukiewicz, K. 550 Algebra 20, 215, 247, 249-256, 257-276, 277-296, 297-314, 334, 335,369,388, 414,415,609 Algorithm 517-519, 523, 527, 531, 533, 542 Allophone 52,578 Almost connectedness 391 Alphabet 66, 71, 188-190, 235, 254,433, 435-437, 451,453,454,456,458,465, 472,476,478,484,485,487,491,492, 494, 502, 508-515,519-522, 529, 532, 558,573,574 Alphabetic variant 150-153 Alternation 135 Analogue, arithmetic 21 Analysis 3,373-375,388, 391, 295,399,401, 403,408,427 Anaphora 389, 419 Andrews, A. vii Anti-euclideanness 391 Anti-symmetric relation 41 Antipersistent 381 Antisymmetric relation 44,47,48 Antisymmetry 44,45,47,48, 207, 208,209,
210, 212, 213, 277, 278, 282,391-394, 398,401,440,444,579,581,583, 620 Application 351 Argument 31,34,65, 97,115-120,134,135, 137,155, 159,161,162,164-169,175, 178,179, 233,303,339-343,348,351, 362,363, 366,373, 376,380,381,398, 421,444, 513,518,551,588,604 Aristotelian syllogistic logic 333 Aristotle 333 Arithmetic analogue 21 Arithmetic, language of ordinary 93 Aronszajn, M. vii Asher, N. 411 Assertability condition 242 Assignment 346 Associative operation 73, 214, 215, 251, 257, 261, 263, 265,274, 284, 293,434,576, 577 Associative law 18, 19, 21, 22, 112,114, 120, 199, 281, 292, 297,421 Associativity 270,273,604,606,607,608, 609, 612 Asymmetric relation 40,42,43,44,45,47, 49, 85 Asymmetry 208,209, 212,213,349,366, 391-393, 396,424,444,579 ‘At least one’ automaton 566, 567 ‘At least one’ tree 568 Atom 298,303 Atomic formula 140,152,308,325 sentence 100 statement 99, 100,106,107,110,113,115, 125-132, 155,184, 185,191, 220, 221, 235, 254,307,319, 321,322,368 Autological adjective 26 Automatagu, 94,433-574,624,625,627, 633 Automorphism 253,274,378, 605 Auxiliary alphabet 188-190,236, 596,597 Auxiliary premise 120-123,161, 162,173, 174,586, 587,593, 595, 600-603, 614 Auxiliary proof 172 Auxiliary symbol 6,140 Auxiliary tree 543,544,546 Axiom 90-94, 185-246, 249,250, 257,260, 266, 268, 269,299,312,313,314,348, 349,437,596, 597,599,604,606 of asymmetry 210 of choice 219,232,288 of extension 171 643
644
In dex
of iireflexivity 210 of regularity 218 of separation 219 of transitivity 210 schemata 191, 225-227 Axiomatic system vi, 185-188,191,192, 194, 211,235, 236 Axiomatization 8, 89, 90, 92, 181, 195, 201, 208, 213, 217, 219,223,225, 226, 228, 230, 234, 349 Bach, E. 362,363, 364,553 Backward function composition 553 Bar-Hillel, Y, 550,552 Barendregt, H. 351 Barwise, J. 373,384,385, 387,389, 397,411 Basic alphabet 188, 189 Basic clause 320, 321 Basic concepts of logic and formal systems 89 Basic concepts of set theory 3 Basic result m recursive function theory 518 Behavioral science vii Bell, J 231, 301,303 Bernays, P. 224 Beth, E. 123,124,128,135,165,168,169, 179, 229, 230, 234, 246 Beth tableaux 123, 135,165,168,169, 179, 229,230, 234, 246 construction rules for 128 Biconditional 99,105,113,135,171,174 laws 112 proof 599,602 statement 110 Bigelow, J. 405 Bijection 253,285,286,378 Binary connective 100 notation 66, 69 operation 266, 267,282, 297,312 product 191 relation 30, 39,40,47,174,175,180,205, 206, 208, 374,391,459,460 Binding 150, 323, 342,373 Biikhoff, 52, 250, 291, 292, 302 Bolyai, J. 90,91 Bonevac, D. 411 Boole, G.E. v, 21,215,297, 298, 300-304, 309, 314, 352, 354, 388,478,611 Boolean algebra v, 21,215,297,298, 301-303,388,478 laws 300 phrasal conjunction 354 Boolean lattice 611 Bound, upper 71
of subjects
Bound variable 141,142,144,177 Bounded function 70 lattice 290 Branch 440,497, 581 Bresnan, J. 362,364 Brolio, J vii Brouwer, L.E.J. 305, 306 Brouwerian lattice 305 Calculus 82,95,135,142,144, 147, 160, 235, 317, 339, 341, 343, 348, 349, 350, 351, 367, 553 Cancellation law 266 Cantor, G, 7, 62,64,73, 82, 300, 351,520 Cantor ’s set theory 7 Theorem 62,64,73, 300, 351,520 Cardinal number 56, 59, 62, 63, 65,73 Cardinality 9,11,58,60,64-72, 300,379, 390,395, 396, 398, 399,435, 580,581, 618 equivalent set and 55 set of 82 set-theoretic identity and 8 Carnap, R. 366 Cartesian product 27-30, 60,62,464 Categorial grammar 550, 552 language vi Categorical system 206, 207 Category 318,320,323, 324,339,351,354, 359,364, 368, 373, 375, 376, 386,440, 443, 503,550, 551, 553 theory 250,254, 281 Cell 46,47 Center embedded constituent structure 541 Choice-function 232 Chomsky, N, 94,436,451,452,453,463, 481, 503,505,535, 555,561 Chomsky hierarchy 436,451,452,535, 561 Chomsky normal form 505 Chomsky’ s program 94 Chronological order 86 Church, A. 317, 338,350, 352,517,518,519, 523,531 Church’ s hypothesis 517-519,523,531 Church-Rosser theorem 350 Circularity 391,392, 394 Circumlocution 101 Class 301, 306, 311,313,379,383,435,464, 578,613-615 context free vi context sensitive vi equivalence 45-47,56,58,59,135,209 of finite sets 59
In dex
Clause 320, 321, 351,355,357,373,415,505 Clause mate 445,446 Clitic 351 Closed formula 142, 150 Closure 249, 250, 263,335,462,464,492, 499, 511, 533, 542,551, 553,561, 563, 604 Coextension 417 Cohen, P.J. 65,219 Collection, abstract 3 Commutative 73, 224,251, 252,257,262, 263, 274, 284, 293,434, 576,577, 609, 613 Commutative group 311 Commutative law 18, 20, 22, 112, 114, 120, 121, 281, 292,297,421, 599,600,601, 602 operation 264 Commutativity 270,552 Complement 20,218,293,294,364,405, 410,413,418,478,485,576,577,610, 625 definition of 19 inverse of 36 inverse relation and 52 law 18,19, 22,112,114,120,123,156, 421 of arbitrary set, difference and 14 of relation 15, 29,33,44 o f set 16,17 Complementary distribution 52 relation 36 Complementation 26,298,300,477,498, 499, 542, 547,548, 563 laws 298 Complements, properties of inverses and 44, 45 Completeness 202, 203 proof 227, 228 theorem 308 Complex statement 113,115,126,127,145 Component, semantic 93,95 syntactic 95 Composite 37 Composition 33—35,258,285 of operations 263 Compositionality 317-338, 339,340, 351-357, 360,368, 369, 387,407-409 Compositionality principle 254,337 Computability 350 Computation, mathematics of vi Computational linguistics vi Computing function 454
o f sub jects
645
Concatenation 21.3—215, 318, 324,434-436, 464—468,471,477,498, 53.3,547, 548 Concept, abstract 7 of equality 78 of infinity 55 of logic and formal systems, basic 89 of set theory 45 Conclusion (of deduction) 115-126, 129,155, 162-168,183,421 Condition 398,442-444,450 Conditional99,104-106,108, 111, 115,119, 124,129,135,146,147, 162,166, 173, 239,242,243,246,307-309,333,473 connective 124 law 112-114,120,154,421,614 premise 270, 608,609 proof 120-123,133,160,161,171, 172, 174,198, 586, 59.3,599, 600,601, 602, 603,614 sentence 104 statement 127 Conjunction 95, 99,102,103,118,120,122, 125,129,135,149,156,159,161, 184, 199, 239, 246, 251,33.3,350, 351, 352, 354,357, 364, 382,383,407,420, 421, 547,587, 592, 593,595, 599,600-602 Connected 45,51 relation 42-44 Connectedness 39,42,44, 210, 211, 268, 391-393,401, 579,620 Connective 100-107, 111, 115, 124-126, 130-1.32,135,139,140, 145,15.3, 159, 166, 169, 220,22.3, 225, 236, 239, 240, 243, 250, 318, 320, 324, 368,420 binary 100 sentential 95,97,100 Connotation, temporal 102 Conover, L. vii Consequence and laws, logical equivalence, logical 110 Consequence, logical 111,116 Conservative 387,388 Conservativity 376, .377, .379,380,383, .387, 388,392, 394-398,618, 620,621 Consistency 202,203, 221, 22.3 principle 18,20, 22,577 Constancy .377, .378 Constant 138, 140,141,143,155, 156, 157, 158,160, 163,167, 168, 169,195, 225, 228,307, .308,319,419,496,617 Constituent 101,114, 1.38,360,439,440,441 sentence 102 statement 106,110,124
040
In d e x
structure tree 439,442,444,449 Constraint on coordination 382 Construction, intensional vi Construction rules for Beth tableaux 128 Construction, set-theoretic 79 Construction, tableau 130 Context 396-399,404,406,408,409,413, 418,420,421,425,427-429 dependence 395,398,546 free 449,450 class vi grammar 492,494,496,497, 500, 501, 502, 505, 525,535,536, 542, 544-546, 548, 552,555 language 494,496-500,505,506,525, 535,540-542, 545,553,562,563, 572 sensitive 449-451 class vi grammar 530-534, 556 language 529-535,542,553,555,562, 563 Contingent 108-110,132,144, 584 Continuity 383 Continuous structures v Continuum hypothesis 65, 66 Contradiction 107,110,113, 122, 123, 125, 126,132,143,144,228,242, 269,301, 385, 388,389,393,407,421, 523,585 Convention 345, 353,368,370 Convergence, definition 82 Converse 40 Conversion 342,349,350,353,357,359,360, 361 Cooper, R. 334,338, 373,384,385,387,389, 397 Coordinate, first 30 projection onto first 28, 29 projection onto second 28, 29 system 31 Copi, 1,202 Correspondence 31,46, 63,311,313,318, 319,336,579,580, 605, 606, 612 one to one 32-36,55-62, 67,69,76, 86, 198, 200,204,205, 253, 271 Count, frequency vi Countably infinite set 59 Cresswell, M. 352,358,413 Cross-serial dependency 503,538,547 Curry, H. 206 Cycle 555, 558 Dagger 239 Dahl, 0. 361
of subjects
Daughter 440,441, 547, 549 De dicto interpretation 409,410,412 De Jong, F. 398 De re interpretation 409 Decidability 229,563 Declarative sentence 95,100,1-2 Decreasing 381,382, 389, 399 determiner 383 simple quantifier 384 Dedekind, R. 82, 194,195,291 Dedekind infinite 195 Deduction 246 natural 115,154,170 Definability 374 Definition 234, 236, 270, 302,325,327, 339, 346, 350, 366, 367, 374, 383, 384, 387, 388,389,399,417,421,435,438,443, 445,446,447,458,459,460,462,464, 465,491, 492, 510,512,577, 594,607, 608, 609, 610 of complement 19 of composition 34 of convergence 82 of intersection 17,19,20 of inverse 174,175 of subset 10,17, 20 of union 17,19,20 recursive 181-186,188,194,196 Deletion 360-362, 555, 556-558, 597, 598 Demonstrative 390 DeMorgan, A. 18,20-22,112,136,149,154, 173, 200, 294, 296,421, 576, 577, 592, 595.600-602 identity 294, 296 DeMorgan’s laws 18,20-22,112,136, 149, 154,173, 200,421,499, 576, 577, 592, 595.600-602 Denotation 348 Dense relation 51 Denumerability of set 58,59 Denumerably infinite set 59,60,71 Deontic logic 413 Dependence 395, 546 law of quantifier 150 Dependency 398,481,503,538, 541 Derivation 119,185-189,190,192,193,225, 227,318, 322, 327, 328, 331, 337,345, 359, 362,365,437,438,449,474-477, 493,494,496,497, 500,501, 516,517, 530,531,532, 536, 546, 549,558,599, 629 schema 114 tree 367-370,370,537-539, 540, 542, 552
In d e x
of subjects
Descartes, R. (Cartesian) 27-30,60, 62,464 Designator 419,421,429 Determiner 95,373,374-401,439,440,449, 565,567, 571, 572 Determinism 490,506,508, 512,515, 530, 533, 625 Deterministic finite automaton 457-463,477 language 562 pushdown automaton language 562, 563 Diagonal argument 65 Diagram 278-280,441,442, 457,459,461, 476,479,482,484, 581, 606 o f relation 43,440 Venn 5, 12-15,17, 19, 20, 25 Difference and complement of arbitrary set 14-16 symmetric 25 Discontinuous connective 104 Discourse, domain of 16,142,147,149,158 universe of 16,25, 145,147,149,155,157, 166,168,169,176 Discrete entities v mathematics v, 82 Disjoint set 73 Disjunction 99, 103, 107,126-129, 149, 220, 223, 227, 240, 251, 308, 350,352, 354, 382,413 inclusive 12 Disjunctive syllogism 118,120-122,173, 586, 587, 593, 599-601 Distinct element 42,262 pair of integers 43 set 9 Distribution, complementary 52 laws of quantifier 149 Distributive 251, 309,390, 576, 577 lattice 297,302, 305, 308 Distributive laws 18,19,21,22, 25,112,198, 199, 266, 275, 290, 296, 297,421 Distributivity 585,586, 607, 608, 611 Division 31, 80, 81 Domain 29,30, 34, 36,45, 168, 169,170, 200, 208, 212, 219, 230-232, 242, 266-272, 275-277, 286,312,313,325, 331, 340,346, 347, 351, 360,373, 374-379, 383, 387, 390,392, 394, 395, 398-400,415, 419,421,423,426,443, 514, 549, 555 of discourse 16, 142, 147,149,158 of function 31 subset of 32 Dominance relation 440-445 Double indexing interpretation 428
647
Double negation 614, 620 law o f 148, 154 Dowty, D. vii, 320, 334,352,362,364,365, 415,425 Dual 384,385 atom 298 Duality 277, 278, 280, 282, 283,385,401 Effective enumeration 230 Effectively listing 62 Element 49, 51,62, 63, 67,78, 80, 86,168, 169,203-205,207, 212, 215, 219, 232, 249, 251-253,255, 257, 258, 261-269, 274, 276-280, 283, 288-290,292, 295, 297, 298, 300, 302, 303- 306, 309, 311, 328, 329, 366, 377, 380,390,391,398, 399,434,441,446, 538, 546, 578,582, 583,603-607,611,612 distinct 42 finite 76 first and second 29 greatest 50,51,53 least 50, 51, 53 maximal 50, 51,53 minimal 50, 51,53 of set 3 Elementary statement 137 Emptiness 501 question 479 Empty list notation 9 set 4, 9,75, 218,237 axiom 218 string 58, 214, 215,434 Enderton, H. 231 Entity 346, 347, 374,380,398,415,419, 620 Enumeration 230 Epistemology 409-414,418 Equal relation 76 set 55 Equality 79, 81 set-theoretic 17,18,25, 26 Equation 34 Equivalence 118,148,177, 313, 332,337, 342,350, 353,367, 376,377, 392,394, 401,417,418,420,421-423,462,463, 477,490,492, 514, 516, 530, 578,579, 585, 588, 590,591,613-615 class 45-47,56, 58,59, 135, 209,302, 306, 311 logical 111, 113,120,121,172 consequence and laws 110 relation 46,47, 52, 56,71,79,135, 205, 272,301,311
0 48
INDEX OF SUBJECTS
and partitions 45 Equivalent infinite set 58 set 56,57, 62 and cardinality 55 statement 120,123,132 Etchemendy, J, vii Euclid 89-91,195, 200, 216,394 Euclidean axiom 90, 91, 200 axiomatization 195 Euclideanness 391 392 Evaluation 331,416-419,423 Even integers, positive 56 ‘Every AB’ automaton 565 ‘Every’ automaton 566 ‘Exactly two AB’ automaton 568 ‘two’ automaton 570 ‘two’ tree 570 Exclusive disjunction 103 Exclusivity condition 442-444 Exercises 23,36,51,71, 85,1.30, 175,234, 255,273,295,309, .367,400,429,482, 505,525, 534, 572 Existential generalization 154,157,159,161, 162,164,419,592 instantiation 154,157,159,162-165,592 quantifier 138,141, 149,150, 225 Existentiality 407,408,416,421,424,426 Expression 21,348,352,466,485,513,573 Extended axiomatic system 188-193 Extension 377, 379,380,38.3,385,392,394, 395-398,414,415,417-419,421-424, 426,428, 618, 620,621 axiom o f 171 to the set of all integers 78 to the set of all rational numbers 80 to the set of all real numbers 82 Extensional equivalence 417 Extensionality 218,413 Factorization 497 Fallacy 117 False conditional 146 Faltz, L. 354, 379,390 Filter 287-289, 298,299, 301, 305-308,389, .390,398, .399,620 Filtering effect 555 557 Final state 455,457,458,469,475,476,479, 488, 625 Finite alphabet 66,71 automaton 455-457, 460-462,464,467, 469,473,475-480,483,486,487,490, 507, 508 language 464,466-468,470-472,476,
477,479,481,494,497,498 cardinals 6.3 element 76 linearly ordered set 51 numbers 72 power 73 set 6,9,56, 58,70, 86, 212, 266,458 class of 59 state 485 string 66, 71,86 First and second coordinates, projection onto 28 elements 29 members 42,44 coordinate .30 projection onto 29 member 27 order definability 374 logic 93 predicate calculus 95 quantifier law 148 Form, argument 97 prenex normal 151 quantifier laws and prenex normal 148 Formal and informal proofs 170 definition 545 grammar 433,435,4.38,449,47.3,481,565 language vi, 95,115,155, 250, 277,306, .317,318, 322- .343, .374, 385, 425,437, 481,503,513, 561-563 natural languages and 93 logic v, 96 proof 173, 180 semantics 95 system vii, 91,92, 94, 181, 202, 205, 210, 212, 220, 277,304, .33.3 basic concept of logic and 89 theory 348 Formalization v, vii, 89,91, 92 Formula 140-142,144,145, 148,150-154, 156, 160,166, 167,169, 176,177, 183-185,188,190,191, 208, 225, 228, 2.35, 2.36, 246, .308, 309,318, .320-332, 336-338, 342, 350, 353, .369, 373,414, 415,417,429, 596 well-formed 99,100-102,114,121,122 Fortran 93 Found 440,442 Foundation 348 Frame 306-308 Frankel, A. 217-219,348 Frege, G. 317,318,323,3.33,382,403^05, 408,414,415,428
Index
principle of 317,333 Frequency count vi Friedman, B. 553 Function 30-32, 36,37, 55,138,142,144, 145, 163, 196, 219, 231-234, 241, 249, 253,271, 307, 312,318, 326, 329,331, 335, 336, 339-343, 346,348, 349, 351, 352, 355, 356,362-364, 368,374,388, 414,415,417,421,422,424,426,428, 434,443,444,454, 458,459,462, 510, 513, 514, 518, 520,526, 529-531,551, 578, 579, 584 bounded 70 composition 33, 251, 552 graph of 31 identity 34, 35 infinite vs unbounded 70 into 32,34 inverse of 33 many to one 32 mapping 62 negative 59 one to one 32, 34, 56,62 onto 32, 34 partial 31, 32 positive 59 quaternary 36 range of 32, 34 relation and 27 ternary 36 value o f 34 Fuzzy set 6 Godel, K 219, 227, 230,231, 234 incompleteness proof 234 incompleteness theorem 230 Gaifman, C. 552 Gallin, D 350 Gamut, 320 Gazdar, G. 354,358,503,536,541,542,547 Geach, P T 552 Geach rule 552 Generalization 154-164, 171-174, 358,407, 408,419,421,426 Generalized quantifier vi, 358-360,373, 374, 376, 377, 387, 390, 391, 395,565 theory 375, 385 Generative grammar 94 sytax 94 transformational theory 433 Generator 454 Ginsburg, S. 555 Glottochronology vi
o f subjects
649
Goldblatt, R. 250, 254 Grttzer, G, 250, 303 Gradual membership 6 Grammar 227, 234,337,353, 363,364,415, 433,435,437-439,446-455,473-477, 481,482,486,492-494,496-498, 500-502,505, 515-517, 525, 527, 529-532, 534, 536-539, 542,546, 547-550, 552, 553,555, 556,559, 565, 573,629,632 generative 94 Montague 7 transformational vi Grammatical constituent 138 Graph of function 31 Greatest element 50, 51,53 Grelling’s Paradox 26 Groenendijk, J. 358 Group 257, 261-263, 266, 271-274,311,312, 314,434, 506, 580, 606, 612,614 operation 258 theory 263, 281 Halfway membership 6 Halting problem 512, 513,522-524 Halvorsen, P. 334 Hao Wang 205 Hasse diagram 279 Hayashi, T. 542 Head grammar 548, 549 Head language 548 Heim, I 389 Henkin, L 227,228,350 Heterological adjective 26 Heuristics 123 Heyting, A., v, 215, 297, 303,305,308,309, 414,415,612 algebia v, 215,297, 303,308 lattice 612 Hierarchical categorization 503 Hierarchy 436,452,453,535, 561 Chomsky 451 Higginbotham, J. 411 Higher order analysis 374 logic 231, 234 quantifier 571, 572 system 232 HUbert, D. 94, 220, 223, 224 program 94 Hoeksema, J. 390 Hofstadter, D, 231 Homomorphism 253,254, 271, 274, 286-289, 295, 299, 302, 313, 314,334-337, 605,
650
In d e x
606
Hopcroft, D. 463,466,492,530,531,533, 536 Hughes, G. 413 Hukaii, T, vii Hypothesis 517, 531,558 continuum 65, 66 Hypothetical syllogism 118,156,163,165, 172,593 Ideal 287-289, 298 Idempotence 309 Idempotent 21, 251,274, 284, 293,577 law 18,22,112, 281,607,609, 610 Identity 262,266,269, 273, 311,403-405, 421,428,429,475,576,577,479, 585, 603, 604,606,607,612 and cardinality, set-theoretic 8 element 215, 252,257, 258,261,26:3-265, 274,434 function 34,35 law 18,19,21,22,112, 114 relation 35, 39 set-theoretic 9 Imperative 95 Imperfective paradox 247 Implication 224 Inclusion relation 561 Inclusive disjunction 12,103 Incompleteness theorem 231 Inconsistency 228 Increasing 380-382, 385,389,399-401 determiner 383 Independence 202, 203, 220, 224 law of quantifier 150 of axiom 224 Index 533,537,547,621 Indexed grammar 536,541,542,547,549, 553 Indexed language vi, 540-542,546,547 Indexical 427,428 Indexicality 427 Indices 414-419,422-429,536, 538,540,549 Indirect proof 122, 123,133, 580, 595, 601, 602 Individual constant 138 variable 138 Induced partition 47, 52 Induction 235, 276, 309,434,596, 608, 609 Induction, principle of mathematical 195-199, 217, 269, 270 step 197,198 Inductive proof 194-199
of subjects
Inference 221,224-226, 235,375,404,405, 409,437 productions of 186,189 rules of 45, 95,118,119,122,151,153, 154,162,172, 185,196,197, 220 Infinite cardinal number 65,73 Dedekind 195 denumerably 71 numbers 72 sequence 69 set 6, 9,55-60, 65,70, 86, 212 countably 59 denumerably 59 60 equivalent 58 nondenumerably 69 vs unbounded function 69, 70 Infinity 218,219 concept of 55 Informal proof 173-175, 180 formal and 170 Information-state 306,307,309 Initial state 455,457,458,461,469,475,476, 479,484,488, 507,510, 517, 520 Initial symbol 437 tree 543, 544, 547 Input 453-457,461,463,484,489, 512,514, 519,529-531,567, 573 alphabet 484,485,491 string 477,483,489,490,508, 521,532 symbol 491 tape 487,488,507 Instantiation 154-174,183, 184,196,197, 617 Integer 62, 64,73, 75,78,79,177, 178, 196-200, 235, 253, 257,258, 263,265, 267, 269-273,311,313,434, 527, 556, 575, 576,581,585, 591, 606,608, 609, 613 distinct pair of 43 extension to the set of all 78 negative 62,71,79, 80 nonnegative 80 odd 60 ordered pair of 81 positive 6,9,49,59-61,79, 80 relation o f 85 relation of positive 43 set of 31, 32 positive 55,56 positive and negative 51 Integral domain 266-272,275,276,312,313 power 71 Intended model 91
In dex
Intension 414—429 Intensional construction vi equivalence 417 logic 429 Intensionality 403,413,414 Interpretation 409,410,412,414,415,422, 425,428,429 Interrogative 95 Intersection 12, 19,20, 26, 300,302,308, 382,388,477,481,498,499, 533, 542, 546, 547, 5783,576 definition 17,19, 20 operation o f 255 set-theoretic 14 union and 11 Into function 32,34,443 Intransitive 45 relation 41,42,44 Invalid argument 115,116, 119 Inversal instantiation 592,593,603 Inverse 30, 34, 35, 37, 218, 225, 252, 253, 261-263, 265-267, 274, 275, 286, 311, 313,434, 578,603, 604,606, 607 definition of 174,175 element 255, 257 law 613,614 of complement 36 o f function 33 of relation 29 properties 45 relation and complement 44,52 schema 192 Irrational numbers 64 Irreflexive relation 39-51, 85, 205 Irreflexivity 208, 209, 213,391-393,429, 442,444,578,579 Isomorphic model 206, 207 subsystem to the old system 78 system 204 Isomorphism 80, 81, 203-205,253,254,267, 271, 272, 274, 275, 285, 286, 292, 300, 301, 303, 311, 378, 476, 605, 612 Isotone 285 Jacobson, P. vii James, W. 72 Janssen, Th. 250, 334, 352 Joshi, A. 535, 542,546, 548,553 Kalish, D. 318 Kamp, H. 424, 425,427 Karttunen, L. vii Keenan, E 354,361, 379,388-390
of subjects
651
Kernel 271 Kleene, S. 241-243,464,465,466,469,471, 477,498,518, 533,542,547 closure 542 star 464-466,469,471,477,498, 533,547 Kripke, S.. 254, 306-309,414,415,419,429 semantics 254, 306,308 valuation 307 Kuroda, Y. 556 Labeling 443,446,447, 545, 624 function 444 Ladusaw, W. 334,385,390 Lambda 351,352,354-367,374 abstraction 338,341, 348 Lambda calculus 317,339, 348-351 conversion 353 rule 347 Lambek, J. 206, 553 calculus 553 Landman, F. vii, 303 Langacker, R. 445 Language 67,104, 115,137,138,140,155, 230, 231, 242, 250, 254, 277,300,303, 306, 309,317-324,326, 333,334, 338, 340- 347, 351-353, 355,360,373, 375, 378-380, 383,385, 389, 392,393, 395, 399,401,403, 405,406,409,413^tl5, 421,423, 424,425, 427,429,433, 435-438, 448,451^57,460,462-472, 474- 484,486-489,491, 492,494, 496-501, 503, 505-507, 511-513, 515517,519, 520, 522-525, 527,529-535, 540-542, 545-549, 556-558,561-563, 572, 573 categorial vi formal vi, 95 indexed vi logical 100,102 meta 92,93 natural 3, 7, 92, 95, 96,100,101,103 object 92 of logic 93-97 of ordinary arithmetic 93 programming 7, 8, 93 tree adjoining vi Lasersohn, P. vii Lattice 277, 280, 282-297,302, 304, 306, 308,309, 314, 389,609-612 theory 281,283,390 Law 148,149, 267, 268, 275, 282, 291 associative 18,19,21,22 commutative 18, 20, 22
652
In d e x
complement 18,19, 22 DeMoigan’s 18,20-22 distributive 18,19,21,22, 25 idempotent 18,22 identity 18, 19,21, 22 of double negation 148 of quantifier dependence 150 of quantifier independence 150 of transivity 275 of trichotomy 268 Laws and prenex normal form, quantifier 148 logical equivalence, logical consequence and 110 o f quantifier distribution 149 of quantifier movement 151 of set theory 45 of statement logic 112 Leaf 441,444,447,449,556, 557 Least element 50,51, 53,212,232,441 number 395 Left identity element 251, 252 increasing 398 inverse 252 linear grammar 474 tree structure 541 monotone decreasing 381,382 increasing 380-382,400,620 zero 253 Lemma 299,302,303 Level of representation 334 Levy, L. 542 Lewis, D. 463,466,492 Lexical rule 364—367 Lexicon 347,356,364,550 Lindenbaum algebra 301 Linear bounded automata 529-533 function 529-531 grammar 474 list 67 order 51,66, 69,76,210,434 sequence 62, 64 Linearly ordered set, finite 51 system 86 Linguistic object 5 theory 437, 559 Link, G. 334,390 Lisp 93, 214,351 List, linear 67 notation 4-6 Listing, effectively 62 Lo Cascio, V. 425 Lobachevsky, N. 90,91 Logic vi, 95, 140, 142, 143, 147, 148,153,
of subjects
169,175, 177, 183, 190,199, 212, 214, 219, 220, 223, 225-232, 234, 236,241, 246, 307,308,317-330, 333, 348, 350, 359,360, 362, 368,369, 373, 378,382, 385,403,405,408,409,413-415,417, 419-421,423,429 and formal systems, basic concept of 89 and predicate logic, statement 95 first-order 93 formal v, 96 language o f 93-97 laws of statement 112 of statement 140,144, 239, 241,250,307, 335 predicate vi, 95-97,99, 100, 102,118,119, 130 system of 92 Logical connective 102, 103 Logical consequence 111,116 and laws, logical equivalence 110 equivalence 111, 114, 113,118,120,121, 172,417, 418 language 100,102,104,137, 300,550 negation 101 omniscience 422 paradox 7 predicate 138 Logics 306 Machine translation vi Machover, P. 231,301,303 MacLane, S. 52, 250, 302 Manipulation, symbol 95 Many to one function 32 Map 31 Mapping 31,33,34,36,57,58,63, 200, 234, 249,253, 254, 271, 272, 285-287, 302, 312, 313, 326, 334-336, 346,373,434, 443, 555, 556, 581, 605 function 62 one to one 56 single-valued 32 Markov, 518 Marsh, W. 542 Mass terms v Massey, G. 236 Mate, clause 445 Mathematical and linguistic analysis 3 induction 596, 608, 609 principle o f 195-199 Mathematics, discrete v, 82 of computation vi of wave theory vi
Index
Matrix 141 Maximal element 50, 51,53 May, R. 334, 338 McCawley, J. 361 McKinsey, J. 305 Meaning 403,429 postulate 364—366 Member 465 first 27 o f set 3, 5, 7, 8,10-12,15, 21, 24,25, 27, 30, 32, 36,43,52,53,56,58-60,62, 63, 66,69,70,75,76,142,143,155,171,173, 182,184, 237, 262 second 27, 28 Members, first and second 44 nonidentical 42 Membership 500,501, 510,533, 573,578, 579, 581, 582,596 gradual 6 halfway 6 multiple 6 question 479 relation 4 status 6 Memory 453 Meta-language 92,93, 230,319,320, 322-324,424 Miller, G. 463,481 Minimal element 50,51, 53 Modal logics 306 Modality 413,418 Model 90, 91,143-146,168,170, 206,207, 209-211,214, 221, 223, 224, 232, 237, 246,266, 267, 269, 300, 301, 303,307, 325- 327, 329-332, 337, 338, 346-348, 351, 368, 369, 376, 378,380, 383, 389, 391, 398, 399,401, 414-417,421^t25, 428,429, 433, 535, 594, 617 intended 91 standard 91 theoretic semantics v theory 92,94, 95, 181, 194,200, 201, 203, 205, 208, 215, 228, 233, 353,374, 375, 387,414,415, 417 Modified head grammar 548,549,553 language 549 Modular law 290, 291 Modularity 293 Modus Ponens 116-118,121,122, 155,161, 174, 182-184, 196, 197, 220, 221, 226-228,586, 587,599, 600, 602, 603 Modus Toiiens 118,123,164,586,587,599, 601
of sub jects
653
Monadic predicate logic 229 Monoid v, 215, 263, 264, 266, 271, 274, 275, 434, 606 Monotone 285, 383 decreasing 383,384,389,620 determiner 381 set 385-387 increasing 383, 384, 389, 399,401, 620 Monotonicity 381-383,385,397,400 Montague Grammar 7, 234, 250, 363,415 Montague, R 7,233, 234, 250, 317, 318,333, 334,338, 339, 352, 353, 359,363, 366, 374,415,425, 550 Montague semantics 233 Morpheme 203, 351,433 Morphism 253, 254, 271 Morphology 365,424 Morse, S. 93 Moss, L. 390 Move 491,492 produces in one 459-462 Movement, laws of quantifier 151 Multiple 575 membership 6 Multiplication 20,21,31, 73,76-78, 80, 81, 204, 215 law 267 N-place relations 30 n-tuple, ordered 27 set of 62 Named object 5 Natural deduction 115,154,170 Natural language 3, 7, 92,95, 96,100,101, 103, 104, 115, 137, 138, 230, 231, 242, 250, 254, 277, 303, 306,309, 317, 322, 333, 338, 351, 360, 373,375, 379,380, 383, 385, 392-395,399,401,403-406, 409,413-415,421,423,427,429,433, 436,453,482, 503,535,542, 550,551, 557, 558 formal languages and 93 Natural number 3, 9,11, 36,51, 56-60, 62, 66,75,76,78-80, 85, 86,194-196, 200, 216, 217, 230, 237, 263 Negated determiner 566 Negated quantifier 565 Negation 99, 101, 122, 123,125, 126,135, 159, 167,220, 223, 228, 231, 236, 239, 240, 243, 307,308, 351, 352, 368,383, 384,385,400,567, 602 law of double 148 symbol 100
034
Index
Negative function 59 integers 62,71,79, 80 rational numbers 62,86 set of positive and 51 strong 387,388 New York Times 65 Newton, I. 89 91 Newtonian system 90 91 ‘No’ automaton 567 Node 440-447, 544, 545,547,567,568, 581, 619 Node admissibility condition 450 Non-Abelian group 612 Non-determinism 490-492,497,499,512, 515, 516, 530-533 Non-deterministic automata 457,485,486 finite automata 460,462,477,484 language 562 linear bounded automaton language 562 pushdown automaton language 562 Non-distributivity 291-293 Non-empty set 46,206,219,232, 281, 288 Non-fmite automaton 462,463 of primitives 92 Non-empty subset 283, 287 Non-modularity 291,292 Non-monotone 382 Non-reflexive relation 39,40 Non-terminal alphabet 437 symbol 439,446,451 Non-triviality 379 Non-Turing acceptable language 562 Nonconnected relation 42,43 Nonconnectedness 578,579 Nondenumerable set 62,64,69 infinite 69, 86 Nonidentical first and second members 42 Nonnegative integers 80 Noraeflexive relation 51 Nonsymmetric 45 relation 40,43,44 Nonsymmetry 578,579 Nontangling condition 442-444 Nontransitive relation 41-^4 Nontransitivity 578,579 Normal form, prenex 151 quantifier laws and 148 Not every automaton 566 Notation 34,135,136,234,236, 239 binary 66,69 empty list 9 list 4-6 predicate 6,11, 12,14,16, 28, 31,33
of subjects
set-theoretic 4 Notion, primitive 27 Noun 231,439,440 Noun phrase 102,395, 399,404-408,410, 411,418,419,426,439,440-443,480, 504 Null set 4,10, 19, 21,123,189 string 192 Number 230,263, 267,274, 340,395 cardinal 56,59, 62,63, 73 extension to the set o f all rational 80 extension to the set of all real 82 finite 72 infinite 72 infinite cardinal 65,73 irrational 64 natural 3, 9,11,36,51, 56-60, 62,66,75, 76,78-80, 85, 86,194-196,200 negative rational 86 positive rational 86 rational 60, 64, 81-83 real 51,64, 65,69, 70 set of real 66 systems, set-theoretic reconstruction 75 theoretic tree 379,380 theory 16,400 unique real 69 Object 3,4, 11, 16, 28,29, 31 language 92,319 linguistic 5 named 5 Odd integers 60 Oehrle, R. 553 Omniscience 422 One to one conespondence 32-36, 55-63,67, 69,76,78, 86, 198, 200, 204, 205, 253, 271,311,579, 580,612 function 32, 34, 37,56,62 mapping 56 Onto function 32,34, 37 Opacity 405,406,408^15,418^21,423, 425,426 Open formula 142,208 statement 138,139,144 Operation 250,252,253, 257-261,263, 264, 266, 267, 271, 273- 275, 281-284, 297, 311,312, 314, 318-321, 324, 325,334, 335, 348, 350, 362-364,420,612 complementation 477 intersection 255,477 order and v set-theoretic 12
In d e x
Operator, unary 100 Order 50, 208-213, 232, 264, 267, 278, 280-283, 306, 363, 391, 393,434,435, 442-444, 580,588, 607 and operation v chronological 86 linear 51, 66, 69,76 partial 51 strict 47-49,51 strong 47 theory 277, 285 total 51, 53 weak 47-51 partial 53 Ordered n-tuple 27 set of 36f pair 28, 29, 34,39,40,41,44,47,48,60, 62, 79, 81, 143,145, 146, 174,186,313, 438,444 and Cartesian products 27 set of 30,31 quadruples 30, 62 set, finite linearly 51 partially 51 system, linearly 86 partially 86 triple 27, 30, 62 Ordering 47,79,81 Ordinary arithmetic, language of 93 Ordinary set 27 Output 453,454,565 Pair 313 ordered 28, 29,34, 39,40,41,44,47,48, 60,62,79, 81,143, 145, 146,174,186 reversed members of 40 Pairing 218 Papadimitriou, Ch. 463,466,492 Paradox 427 Grelling’s 26 logical 7 Russell’s 7, 10, 26,6.3 Paradox, set-theoretic 8 Parallel Postulate 89, 90 Parameter 395-397,422,425,427-429 Parse tree 494,495, 501, 502 Parsing vi Parsons, 334 Partee, B.H. v, vii, 334, 352,254, 257, .358, 366, 379,427, 555 Partial function 31,32 order 51, 277, 278, 280,282,306, 393 weak 53
of su b jects
655
recursive function 520 Partially ordered set 51 system 86 Partition 135 equivalence relation and 45 induced 47,52 trivial 46 Partitioning 46 Pascal, B 93 Passive 363,367 Peano, G 194, 195, 197, 200, 212, 215-217, 230,232, 234, 237, 269 axiom 194, 197, 200, 212, 215-217, 230, 2.34, 237 axiomatization 2.32 fifth postulate 269 Performative 95 Permutation invariance 572 Persistent 381 Peters, S. 320, 334, 352,415, 555-558 Phoneme 3,52,433, 578 Phonemic overlap 578 Phones 52 Phonetics, acoustic vi Phrasal conjunction 103,352, 354 passive rule 363 rule 364 Phrase 102,480 structure 322 tree 474,515 PNF 152, 153, 159,178,229 Polarity 386 reversal 385 Polish notation 135,136,234, 236 Pollard, C 548 Polygons, regular 70 Poset 277-285, 288, 295,306,308, 609 Positive and negative integers, set of 51 closure 533 element 268, 269 even integers 56 function 59 integer 6,9,49,59, 60,61,79,80, 2.35, 265,434 relation of 43 set of 55, 56 rational numbers 86, 257 strong 387, 388, 394 Possessive 390 Post, E 518 Post correspondence problem 525 Postulate 364—366 parallel 89, 90
656
In d e x
Power, finite 73 integral 71 set 11,62, 63-65, 185, 218, 219, 232, 235, 275,280, 295, 300-302, 376,435, 596 lattice 304 Precedence 441,442 relation 444,446 Predecessor 49-53,76, 217,442 Predicate 8,23,42,138,139,140,143,144, 166,170, 171,173,176,178,182, 183, 185,195,196, 199,205, 216,242 calculus 95,142,144,147,160, 341, 343 logic vi, 96,97,137,140,142,143,147, 148,153,169,175,177, 212, 214, 225-231, 234,236,241, 246, 307,308, 318, 323, 324-328, 330, 333, 350, 359, 360,362,363, 368,369, 373,382,385, 405,408,409,413-415,419-421 statement logic and 95 notation 6,11,12,14, 16,28, 31,33 Prefix 435 Premise 115,117,119, 120, 122-126,129, 155, 156,160-168,171,173,174,197, 221,236,270,412,418-421 Prenex normal form, quantifier laws and 148, 151 Preorder 209 Presupposition 242,243,398 Prime 99,128,188 Primitive 94, 217,220, 237,319,414,415, 434 non-empty set o f 92 notion 27 predicate 216 Principal filter 620 ideal 288 Principle, consistency 18,20, 22 of compositionality 254, 317, 318 of finite induction 269,270 of mathematical induction 195-199, 235, 596,608, 609 Probability and statistics vi Problem 512,513, 523,524 Produces in one move 459,460,462,491, 492,510,511 Product 498 Cartesian 28,29, 30, 60, 62 Production 187,191,192,536,537,549,596 schemata 187,189-191,193,197 Productions of inference 186, 189 Program 455 Chomsky’s 94 Hilbert’s 94
of subjects
Programming language 7,8,93,348,351,491 semantics 8 Projection onto first and second coordinates 28 onto first coordinate 29 onto second coordinate 29 problem 243 Prolog 93, 214 Proof, conditional 120-122 formal and informal 170 Proper ideal 289 subset 10 Properties 251 inverses and complements 44 of complements 45 of connectedness 44 of inverses 45 of operation 250 of reflexivity 44 of relations 39,45, 51, 52 of set 7 of symmetries 44 of transitivity 44 truth-functional 101 Proposition 100,162 Propositional calculus 95,135,138,142,144, 145, 163 logic 142 Pseudo-complement 304,611 Pseudo-complementation 305-309,314 Pseudocleft 358 Pullum, G. 503, 542 Pumping lemma 571 theorem 471-473,480,481,494,496,497, 499, 502, 505, 506,545 Push and copy rule 536 Pushdown automata 487-493,497,499,506, 508,512,529, 530,541,571-573 Quadruple 438,491, 508, 510,520, 526,548, 551 ordered 30, 62 Quantification 395,418,424 vacuous 140,141 Quantified variable 140 Quantifier 138-156, 159-161,163,164,167, 169,170, 177, 208, 225, 229, 231-233, 308,318, 323, 324, 358-360, 373,374, 375-380, 383-385, 387,388, 390-393, 395,397-400,409,416,424,426,565, 567,571,572, 591 automata 565 dependence, law of 150
Index
distribution 172 laws of 149,161,171 generalized vi independence, law of 150 law, first 148 laws and prenex normal form 148 movement, laws of 151,154 negation 159, 164,595 laws 154,173 word 97 Quantity 378-380, 383, 392,394-397, 618,621 Quasi-reflexivity 620 Quasi-model 223 Quasi-order 209 Quaternary function 36 relation 30, 36 Question 479 Quine, W.V.O, 239,406 Quine’s dagger 239 Quintuple 458,462 Rabin, M 463 Range 29,30, 36,56 of f unction 32,34 Rasiowa, H, 306 Rational numbers 60,64,75, 81-83, 257, 267, 274 extension to the set of all 80 negative 62, 86 positive 86 subsystem of 80 Real numbers 51,64, 65,69,70,267,340 extension to the set of all 82 set of 66 unique 69 Reconstruction of number systems, settheoretic 75 Reconstruction, set-theoretic 79 Recoverability 556-558 Recursion 555 Recursive 354 clause 320, 321 definition 181-186,188,194, 196, 234, 236, 270 function 513,514 language 532,533,563 rule 8, 23, 335 set 519, 520,531,558 Recursively enumerable language 507,558, 563 set 520,524, 531,556,557, 562 Recursively enumerated set 514,516
o f sub jects
657
Reduction, set-theoretic 29 Reference 404 time 427 Reflexive 45, 272,277 clitic 351 relation 41,43,44, 46,47,48, 85 transitive closure 461,462 Reflexivity 39,42,43,52, 208-210, 212,213, 279,282, 305, 391, 392,394,440,459, 460,492,511,578, 579, 581,588, 614, 615, 621 properties of 44 symmetry, transivity and connectedness 39 Regular expr ession 466,485 gr ammar 451 language 455,464-466,477,478,480,492, 499,503, 533, 547, 562, 563 polygons 70 Regularity 218 Reichenbach, H. 427 Rejection with 0 state 569 with 1 state 569 Relation 28-31,36,37,44,78,80,134,135, 174,175, 180,186, 204-209, 211-213, 218, 219,232, 267, 268, 272, 277-280, 286, 301, 311,313, 317, 329. 337, 365, 374, 389, 391-394, 398,401,409, 414, 415,418,424,426,436,440-446,452, 459,460,462,492, 511, 551,561, 565, 578, 579, 582,583,588, 613-615, 620 abstract 28 and complement, inverse 52 and function 27 and partition, equivalence 45 anti-symmetric 41,44,47,48 asymmetric 40,42-^4,47,49, 85 binary 30, 39,40,47 complement 44, 29,33, 36 connected 42,43,44 dense 51 diagram of 43 equal 76 equivalence 46,47,52,56,71,79 identity 35,39 intransitive 41,42,44 inverse of 29 irreflexive 39,40,43,47,49, 51, 85 membership 4 n-place 30 non-reflexive 39,51 nonconnected 42,43 nonsymmetric 40,43,44 nontransitive 41,42,44
658
In d e x
of integer 85 of positive integers 43 properties of 39,45,51,52 quaternary 30,36 reflexive 41,43,44,46,47-48,85 set-theoretic 29 subset 28,48, 85 symmetric 43,44,46,47, 85 ternary 30, 36 transitive 41,43,46-49,85 Relative clause 355,357,373 complement 15 Replacement 218, 219 Representation 334 ideal 288 Restriction 398 Reuland, E. 389 Reversal 434,435 Reversed members of pair 40 Review exercises 85, 245,311,573 Rewrite rules 94 Rewriting, string 95 Riemann, B. 90 Right identity element 251,252 increasing 398 inverse 252 linear grammar 451,474,492 language 562 tree structure 541 monotone decreasing 381, 382, 399 increasing 381,382,384,385,400,620 zero 253 Rigid designator 419,421,426,429 Ritchie, R 555-558 Roach, K, 548 Roberts, C„ vii Rogers, H. 518 Rohrer, Ch. 425,427 Root 441,445,447,494 Rooth, M, 354 Ross, JR. 71,361 Rosser, J. 350 Rozwadowski, K. vii Rule 318-320,322, 324,325, 328, 329, 331-333,3.35,344-346,352,353,356, 357, 359, 362-366,367,474,476,493, 536,575, 576, 629 for Beth tableaux, construction 128 of function application 551,553 of implication 224 of inference 45,94,118,119,122,151, 153,154,162, 172,185, 196,197,220, 221, 224-226, 235,404,405,437
o f subjects
of substitution 114,120, 220, 221, 224, 225 recursive 8,23 rewrite 94 well-formedness 94 Russell, B. 7,10, 26,6.3, 216, 220,224, 339 paradox 7,10,26,63 Salomaa, A. 536, 542 Schema 114, 187-193, 225-227, 2.35, 354, 488,617 Schieber, S.. 504 Scott, D, .351,463 Second coordinate, projection onto 28, 29 elements, first and 29 member 27,28 first and 44 nonidentical first and 42 Self-dual determiner 384 Semantic automata 565 component 93,95 rule 319, 346 tableaux 124,130 value 142,143,155 Semantics 89,101,104,115,125,138,140, 142,144,148, 200, 231,241,250, 254, 281, 297,306, .308, 309,317-321, 323-327, 329,332-335, 338-348, .351, 35.3-359, 361, .362,364-368, .370, 373-375,377, .379,380, 383-385, .387, 390,395,399,401,403,404,406-410, 412-415,419,422-429, 504,550, 551 formal 95 programming 8 syntax and 94 Semi-Thue system 191-193, 2.36,596 Semigroup 263-265, 274,275,606 Semilattice 280,283-286, 295,609 Sense 404,405,407,415 Sensitive, context 449-451 Sentence 104,142,295 atomic 100 constituent 102 declarative 95, 100,102 set of 67,68,70 Sentential conjunction 103 connective 95,97,100,102,103,145 negation 101 tree 546 Separation 219 Sequence, infinite 69 linear 62,64 Set 3,5,19, 23,52, 57, 85,134,142,143, 170-174, 176, 181- 185,189,198, 205,
In dex
206, 209, 211, 213, 219, 227, 228, 230, 232, 235, 237,249,251, 258, 262,265, 266, 273, 275, 281, 288, 295, 301,302, 306,307,308, 311, 324, 325, 333,335, 338, 339, 340,342, 343, 376,377,379, 385-388,390,391, 396-399,415,417, 422-425,433,435,440,442-444, 455-458,463,466,474,477,487,491, 492,510, 514,516,519, 520, 524,526, 529,531, 535, 536, 553, 556-558,578, 580, 582,596, 597, 603-605, 607, 611, 620, 622 and cardinality, equivalent 55 arbitrary 12,13,14 Cantor’s theory 7 class of finite 59 complement of 16 countably infinite 59 denumerability of 58 denumerable 59 denumerably infinite 59 60 disjoint 73 distinct 9 empty (null) 4,9,75 equal 55 equivalent 56,57, 62 finite 6,9, 56, 58,70, 86 linearly ordered 51 fuzzy 6 infinite 6, 8, 9, 55-60, 65,70, 86 member o f 5,7, 8,10—12,15, 21, 24, 25, 27,30,32,36,43,52, 53, 56,58-60, 62, 63, 66, 69,70,75,76,155 non-empty 46 nondenumerable 62, 64,69 nondenumerably infinite 69, 86 null (empty) 4,10,19,21,123 of all integers, extension to the 78 of all rational numbers, extension to the 80 of all real numbers, extension to the 82 of axioms 92 of cardinality 82 of elements 203 of entities 375 of integers 31,32, 267 of languages 68 of n-tuples 62 of ordered n-tuples 36 of ordered pairs 30, 31 of positive and negative integers 51 of positive integers 55, 56 of primitives, non-empty 92 of real numbers 66
o f subjects
659
of rules 437 of sentences 67,68,70,295 of sets 12 of statements 245,307 of strings 186,214,438,464,481,484 of transition rules 483 ordered pair s of 30 ordinary 27 partially ordered 51 power 11, 62-65 product 498 properties of 7 specification of 4 subset of 10 theory 3, 6-8, 30, 80, 82,85,93,148,171, 208, 212, 217- 219, 251, 254, 255,274, 277,279, 280, 295, 299,300, 302,348, 381,383, 384, 388,392,400,434,477, 606 and logic v, vi axiomatization of 8 basic concepts of 3 concepts of 45 laws of 45 uncountable 69 universal 19, 21 unordered 28 well-ordered 51 Set-theoretic complement 17 concept 55 construction 79 difference 15,16 equality 17, 18, 25, 26 expression 21 identity 9 and cardinality 8 intersection 14 notation 4 operation 12 paradox 8 reconstruction 79 of number systems 75 reduction 29 relation 29 union 12,13 universe 17 Sextuple 491 Shamir, E. 552 Sheffer stroke 240 Sikorski, R. 306 Simple decreasing 389 increasing 389 Simplification 118,121,127,156,159,161,
660
In d e x
162, 172-174, 183, 184,362,586,587, 592,595, 599,600-603 Single root condition 441,443 Single-value 31 mapping 32 Singleton 4, 12,27 Sister 441,442,444 Situation 458-460,462 Skolem, function 23,232 normal form 229,232,233,237 Solutions 575 Soundness theorem 308 Specification of set 4 Square o f opposition 566 Stack 487,489,490,536, 542, 571, 572 alphabet 487,491 automata 541 symbol 491 Standard model 91 notation 135,136 theory vi, 555, 557 Star 464-466,471,477,547 Kleene 469,498,533 State 306,307,453,454-459,461,463,466, 469,471,472,474- 477,479,485, 487-489,492,507-510,512, 517, 520, 521,529, 567-569,572, 573, 625 diagram 457,461,476,479,482,484 transition 478 Statement 106,107,110, 111, 113,115, 117, 119,120,122, 123- 132,137-140, 142-145,147-150,154, 155,157,159, 161,162,164- 166,168,170, 171, 176, 180, 182,183,185,191, 206, 207, 216, 219-221,223, 227, 239, 241,242, 245, 250, 252, 254,270,283, 300, 301,306, 307, 319, 322,323,327,335,350,359, 364, 367,368,392,403-405,407,408, 413,414,418,420-422,425,428,429, 444,489,519,520, 588, 596,617 atomic 99,100 calculus 95, 137, 234, 235 logic 99,100, 102, 118,119, 130,190,199, 219,220, 223, 226, 229, 319-322 and predicate logic 95 laws of 112 syntactic component 101 Statistics, probability and vi Status, membership 6 Steedman, M. 553 Step 510,511 Stokhof, M 358 Stone, M F. 301,303
of subjects
representation 301,303 Strict linear order 211 order 47^9, 51 partial order 209,210, 443,444 total order 444 String 181-183,186-192,200, 214, 215,234, 311,312,433-438,447-449,451, 453-455,457-460,462-469,471^73, 476-481,483- 485, 488,489,491^94, 496-498, 500-502, 506,507, 509-514, 517, 519, 521, 523-525, 527, 530-536, 538,545,546, 548, 549, 551- 553, 556, 558,5 73, 574, 583, 596,612,613,622, 626,629 concatenation 213 empty 58 finite 66,71,86 rewriting 95 unique 58 Stroke 240 Strong 387,388, 394 connectedness 391,392 conservativity 377 order 47 Structure 322 algebraic v continuous v syntactic 101 Subalgebra 301 Subformula 113,114, 329 Subgroup 263-265, 274,311, 313,605,606 Sublattice 280,283, 284, 287,292, 302 Submonoid 265 Subposet 284 Subscript 99,138 Subset 9-11, 24, 28-31,45,46, 51,53, 57-58,63,65,66,68,69,72, 80,123,143, 157,171, 173, 182,185,209, 212, 219, 231, 264, 265, 268, 273,283, 287,288, 299, 301,306,338, 374, 378, 381,433, 435,462,481,491, 503,505, 536,562, 565, 581, 582, 583,606 definition of 10,17,20 of domain 32 of set 10 proper 10 relation 28,48, 85,452 symbol 10 Substitutability 418 Substitution 115,116, 124,196, 220, 221, 224,225,404,405,421,555 rule of 114,120 Substring 435,437,449, 546, 574
In d e x
Subsystem, isomorphic to the old system 78 of rationals 80 Subtraction 77, 78, 80, 81 Subtree 447,544,558 Successor 49,76-78, 195, 196, 217, 232, 234, 237, 279, 348,350, 598 Suffix 435 Sum 36 Suppes, P. 213 Surface structure 555,556 Surjective 253 Syllogism 118,120-122 disjunctive 173 hypothetical 156,163,165,172 Symbol 437-439,446,448,451,455-457, 459,461,471,474,476,487,488,490, 492,493,497, 507, 515, 517,531, 532, 536, 541, 543, 544,547, 548, 551,553, 556, 557,571, 573,580, 629 auxiliary 6 connective 126 manipulation 95 negation 100 subset 10 Symmetric 45 difference 25 relation 43,44,46,47, 85 Symmetries, properties of 44 Symmetry 39,40,42,43, 52, 255, 263,266, 272, 274, 275,313,391, 394, 396,398, 400,424,444,459,578, 579, 588,603, 606,608,612,614,615, 621 Syntactic component 95 statement 101 constituent 114 rule 318, 319 statement 115 structure 101 Syntax 92,95, 99,137,140,181,200, 232, 254, 281,300,307,317-320,322-325, 328, 330, 331, 333-335, 337, 338, 340-345, 349- 360, 362, 364, 367, 369-373,375, 383,389,390,410,414, 415,423,425,439,443, 535,550, 555 and semantics 94 generative 94 System 188-194, 202-207, 210-212,227, 232, 235, 304,312, 318,333, 350, 357, 360,404,414 axiomatic vi, 187 basic concept of logic and formal 89 coordinate 31 formal vii, 91,92, 94,181
of sub jects
661
linearly ordered 86 Newtonian 90, 91 of logic 92 partially ordered 86 theorems of the 93 well-ordered 86 Table 368 truth 101-106,108, 110, 111, 113-116, 118, 119, 125-127,131, 229, 240, 241, 250 Tableau 166-168,170,588, 589 Beth 12.3—129,135,165 semantic 124,130 Takahashi, M. 542 Tarski, A. .305 Tautologies, contradictions and contingencies 106 Tautologous conditional 111 Tautology 107,107,110, 111, 113,115,116, 123,132,143, 144,162, 220,223, 229, 2.36,242, 301, .388,405,420-422, 584 Temporal connotation 102 order 306 Tense 423,426,427 Ter Meulen, A. v, vii, 389, .390 Term 137,138,142,144,160,324 Terminal alphabet 437 symbol 438,447,451 Ternary f unction 36 relation 30, .36 Theorem 94,185,186,189-191,19.3,195, 198,200, 218-220,226, 229,2.30, 234, 236, 260-264, 267,269, 270,276, 289, 295, 299, 302, 308,311,313,350, .351, 385, 387, 388,392-394,437,444, 466, 471-473,480, 496,497,499,502, 505, 523, 524, 545, 572, 608-610, 620 Cantor’ s 62,64, 73 provers 123 Theory 212,215, 217,218, 228,233, 250, 251, 254, 26.3, 274,277, 279-281, 283, 285, 295, 299, .300, .306, 333,334, 338, 3.39, 348, 351, .353, 374, .375,381, .384, 385, 387, .388,390,392, .395,400,404, 414,415,417,433,434,437,477, 502, 518, 555, 557,558, 606 model 92,94, 95,181,194, 200, 201,203, 205, 208 number 16 of automata 565 of functions 349 of generalized quantifiers 380
662
In dex
of meaning 403,429 set 3, 6-8,30, 80,82, 85, 93,148,171, 208 type 7, 8 Thomason, R„ 202 Thue, A. 191-193, 236, 596 Time 423 Times, New York 65 Top and bottom 298,301 Total order 51, 53 Transformation 31 Transformational cycle 555 grammar vi, 555-558 language 556,558 rule 555,557, 558 Transition 467,468,475,476, 478,479,483, 487,491,530, 624, 625 function 458, 459 relation 462 Transitive 45, 272, 277 relation 41,43,46-49, 85 verb 28,138 Transitivity 39,41^3, 52, 208-210, 212, 269, 275, 279, 282, 305, 362,391, 393, 394,401,424,442,444,459,460,492, 511, 578-581, 588, 606, 608, 609, 612, 615, 620,621 properties 44 relation 207 Translation 96 machine vi Tree 345,359,362, 367-370,379,380,400, 439,442-450,474,494-497, 501, 502, 515, 537-540, 542-544,546,547, 550, 552, 555,556, 567, 570 adjoining grammar 542-544,546-548,553 language vi, 545-548 diagr am 439-442, 581 of numbers 567 structure 556 Triple 458,459,462 ordered 27, 30,62 Trivial element 262 partition 46 Truth 94 table 101-108,110-119,125-127,131, 229, 240, 241, 250, 254,368 value 101, 102,104-111,113-115,120, 125, 131, 132,142, 144, 146, 151, 165, 243, 250, 320,322, 330, 335,340, 356, 368, 374, 377,405,414,419, 421,428, 429, 584,588 Truth-functional property 101 Turing, A.M. .350,502, 507-5.32,556, 574,
of subjects
632 acceptability 514, 520, 523 acceptable language 513, 519, 522, 562 computability 350 decidability 520, 52.3,525 decidable language 513,519, 522 machine 507-5.32, 556, 574,6.32 theory 502 Turnstile 459,460 Two-sided identity 255 inverse 252, 253 Two-sided zer o 253 Type 340, 343-347,351, 356, .359, 362, 364, .370,374, .381 Type 0 grammar 507, 515-517, 527, 531,532, 556 free character 351 grammar 451 language 5.32 1 grammar 529, 556 2 grammar 492 3 grammar 455,473,475-477,486 raising 552 theory 7, 8, 2.34, .338, .339 Ullman, J. 463,466,492, 5.30,531, 53.3, 536 Ultrafilter 288, 299,301 theorem 299 Unary operator 100 Unbounded function, infinite vs 69 70 Uncountable set 69 Undecidability 502 Union 19,26, 218, 219, 255,300, .302, .308, .314,466,471,485,498, 506, 526, 533, 547, 576,60.3 and intersection 11 definition o f 17, 19, 20 set-theoretic 12,13 Unique real number 69 string 58 value 32,36,56,458 Uniqueness 250 Universal 383, .385 base hypothesis 558 condition 398 determiner 397 generalization 154-156, 158,160-164, 171-174,593, 595, 603 instantiation 154-156, 158, 159,161-166, 171, 172, 174, 183, 184, 196, 197 negation 384 quantifier 1.38, 149, 150, 15.3, 208,397 set 19, 21
In d e x
Turing machine 521 Universe of discourse 16, 25,145,147,149, 155, 157, 166, 168, 169, 176, 594 set-theoretic 17 Unordered set 28 Unquantified variable 139 Unrestricted grammar 515 Upper bound 71, 558 Vacuous quantification 140,141 Valid argument 115-117 Valuation 307-309 Value 31, 70,139, 14.3-146,151, 155,165, .319-323, 327, .330, 333, .335, 340-342, .356, 364, .368,370, .374, .376,377,404, 405,414,415,419,421,427,429,458, 518, 526, 529, 551, 584, 588, 596, 613, 629 of function .34 truth 101,102, 104,106-115, 120,125, 131, 132, 24.3, 250 unique .32, 36,56 Van Benthem, I 374, .378, .379, 383,390, .391, 393, 394,413,424, 553, 565,572 Van Eijck, J. .382 Variable 6,138-140,144,148,150,154,157, 163, 165,171, 175, 177, 187, 188, 192, 212, 220, 223, 224, 226, 228,229, 231, 234, 308, 323,324,326,415,416,425, 529, 591 alphabetic 150-153 convention 345 Variation 379, 380, 388, 392,394, 400, 620 Variety 393, 618 Venn diagram 5,12-15,17, 19,20, 25,400, 606 Verb 231 phrase 102, 424,439-441 transitive 28 Verkuyl, H. 398 Vet, C. 425 Vijay-Shankar, K 547, 548 Vocabulary 433,435,473,486, 525 VP deletion 360-362 Waismann, F. 216 Wall, R, 320, 3.34,352,415 Wave theory, mathematics o f vi Weak linear order 211 non-cardinal 389 order 47-49, 51, 210 partial order 53, 208, 209,211,212,443 quantity 378
o f subjects
663
Weir, D. 548, 553 Weisler, S. vii Well-formed formula 99-102,114,121,122, 183-185, 188, 190, 191, 225, 2.35, 236, 318,320-322 Well-formedness rules 94 Well-ordered set 51, 211 system 86 Well-ordering axiom 269 WesterstShl, D 377, 378,390, 395,396,398 Wheeler, D. 553 Whitehead, A..N. 220, 224 Williams, E„ .352 Witness set .399 Word, quantifier 97 Wrapping 548 Yield 447,448,494,495,545, 546 Zadeh, L. 6 Zaenen, A vii Zermelo, E. 217-219, 348 axiomatization 219 Zermelo-Frankel axiom 217, 218 Zero 79, 80, 86, 253 ZFC 219, 231 Zorn’s lemma 299 Zwarts, F 391,393,394 Zwicky, A, vii
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