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(final states)
6 (p, a, Y) (q, ~A),
of the general model.
A preset pda is an 8-tuple
F (stack-symbols) , q0 F
it will be removed
When we return and find that
We give a formal d e s c r i p t i o n
~=
*
location and the p r o c e d u r e - b o d y
be pushed in the next one. recursive
*
s2
al; a 2
is a recursive
(top-most)
*
with Q (states),
(initial state) , Z 0 (initial stack-contents),
as usual,
and for all
a finite set of instructions
(qr AB)
7. (inputs),
(A, B e F)
finite s e t of instructions
p e Q, a E 7 u {i}, Y e F of the form
and, similarly,
of the form
(q, #),
(q, #),
6top( p, a, Y) (q, A).
(q, A), a
180
There can be more general d e f i n i t i o n s that allow for w r i t i n g longer w o r d s per move, but it can be shown no gain in power.
(see [55]) that there is
O b s e r v e that this version is close to the p r o g r a m -
c r u n c h e r we m o t i v a t e d it with, and b e y o n d the p e r m i s s i b l e limit.
6to p
for instance n e v e r w r i t e s
The given version is a normal form.
Here is an e x a m p l e of a preset-pda, w i t h notions of acceptability defined
(as usual)
by empty store and final state.
2 Example. follows.
A p r e s e t pda for
Say the stack is p r e s e t at
marker,
with instructions
6(state,
a, Z) = {(state,
and push a
Z
6(state, ~Z)}
to the top.
{(state, ~)} l-input.-
L = {a n
Z0
w o u l d o p e r a t e as
with
as b o t t o m -
Z0
a, Z0) = {(state,
Z0Z)},
it will make k-i moves on input
It reverses with
and returns to the
Then it lifts
k,
I n ~ I}
(first)
Z0
6top(State,
a
a, Z) =
it finds d o w n w a r d s on
two l o c a t i o n s h i g h e r
(on l-input)
repeats the same cycle as before over and over again.
When,
and
lifting
Z 0 , it w o u l d have passed the preset limit it stops in a n o n - a c c e p t i n g state, o t h e r w i s e it goes on until at last maximal l o c a t i o n and sweeps down on {(final state,
is lifted into the
6top(State,
a, Z 0) =
#) }.
At a successful (an integer)
Z0
termination
moves on i n p u t
a
k +
(k - 2) + ... + 1 =
were made,
.k+l.2 ~-~--)
and the machine will
accept all and e x a c t l y all the squares.
I n s t a n t a n e o u s d e s c r i p t i o n s of a p r e s e t pda are of the form (state, r e m a i n i n g input, p u s h - d o w n contents, n), w h e r e
n
is the
p r e s e t t i n g r e l e v a n t and c o n s t a n t for the p a r t i c u l a r computation.
Definition.
A language
only if a p r e s e t pda
~
L
is called a p r e s e t p d a - l a n g u a g e
as above exists such that
if and
181
L=
{xE
~
I 3qeF(q0,
x, Z0, n)~-
for at least one
(q, i, ~, n)
n > i}.
In the definition
it is e m p h a s i z e d
might actually be appropriate As usual accepting
that more than one p r e s e t t i n g
for acceptance.
for n o n - d e t e r m i n i s t i c
machines
and rejecting computations
there may be both
for an input-string,
but it
will still be counted in the language.
3.
THE LANGUAGES OF PRESET PDA'S Here we will develop a generative
description general
general
of preset pda's in terms of their languages.
theory rightaway
grammars
(as o p p o s e d to analytic)
developed
in
leads into the algebraic
F-iteration
[57] and made more e x p l i c i t in
results proved
for various parallel
from a few Iuheorems in this area of the abstract concepts
([57]).
The
[103].
rewriting
Most
systems
follow
Here we only give a few
that were developed
and found useful.
A family of languages will be defined as usual, but in addition we always assume closure under isomorphism.
Definition.
Let
e x t e n s i o n of ~J T k k>0 ($) n
F * ~ ,
F
be a family of languages.
The h y p e r - a l g e b r a i c
consists of all languages of the form where
T
is an
F-substitution
and
Z
an
alphabet. *)
When R
F
is a q u a s o i d
(i.e. containing
and finite substitution),
an AFL closed under iterated
C
and closed under
the h y p e r - a l g e b r a i c F-substitution.
extension becomes
Several
*) Footnote. In later generalizations h y p e r - a l g e b r a i c were defined differently and more widely!
algebraic
extensions
182
results were obtained. The characterization languages
theorem we will prove for preset pda
is an interesting
for context-free
analogue of a classically
languages
known theorem
(which can be described as the algebraic
extension of the family of regular languages~)).
THEOREM.
The family of preset pda languages
is the hyper-algebraic
extension of the regular languages. Proof. Let
T
be a regular substitution,
a finite automaton for
T(a). ~ a
we will in fact for simplicity empty stack-locations
~a
=
may have
A-transitions,
assume that always
will have a contents
qa ~ Fa"
and Non-
as shown below.
q;a I P where
q E Qa'
representing
we got as far as
q
that in generating
in the finite automaton
The idea is not to generate
a member of
T(a)
for it.
the word from T(a)
immediately
as
a whole, but symbol by symbol and each time a next symbol is generated to use it to expand its
T-image to the preset top first before
proceeding with the next symbols. strictly for the
Thus we generate
T(S)
in
leftmost manner and use the finite automata as a finite code (possibly)
arbitrarily
long strings we can rewrite individual
symbols with.
.a s
/"
,q
in ~ a
J . . . . . . ~ b[ //
%
*) Footnote. See van Leeuwen, J., "A generalization of Parikh's theorem in formal language theory", Tech. Rep. 71, Dept. of Comp. Sciences, SUNY, Buffalo, 1974.
183
Here is how the instructions look like. basic schemes. entirely on
We only give their
First realize that expansion to the present top is
A-input, only
6to p
6(state, A, [q; ~ a ]) = {(state,
checks input. [r; ~ a ] )
Therefore
I r ~ 6a(q, A)}
t2
{(state, ~) I ~rCFa r c 6a( q, A)} {(state,
[r; OZa] [qb; % ] )
{(state, ~ [ q b ; ~ b ] )
I b e Z
I b e 7
and
and
~rCFa
r c 6a( q, b)} r E 6a(q,b)}.
Note that we may or may not stop in a final state when generating a word of
T (a).
When the preset maximum
how far we iterate) {(state,
is reached, then
(which actually stands for
~top(State, b, [q; % ] )
=
[r; 6~a]) I r c ~a(q, b)}
{(state, #) I 3 r e F and the obvious rules on A-transitions.
a
r ~ 6a( q, b)} A-input for traversing possible
Successful termination now is actually on empty
store but as standard we may non-deterministically
send the machine
in a final state at the same time. The formal proof of the reverse is tedious but follows the same lines as the restricted case worked out in detail in [55].
Here
we give the basic type of constructs. Let
~q =
recognizing
L.
The iterated
be a preset pda
(regular) substitution
T
for
L
will have basic symbols of the form [~, A, B, q]
(p, q e Q, A e F, B e F u {~})
with the following semantics. [p, A, B, q] pointer,
(B £ F):
when in state
continue on the ~, q]
with
(after a local move) it will write
upwards, later returning in state
(p, A,
p
q
A
under the B
and move
(which will then
B) :
similarly, but (after a local move)
it will empty the location and move upwards, later
184
sweeping down in state q. Thus
"variables"
will represent recursions.
symbols will actually be coded as well-known terminal
cycle
range)
a ÷ a + ~
a,
(where
and later finish with the ~ ÷ ~
to enforce synchronous
and
~
outside
termination.
problem there when subsequent expansions on
The "terminal"
and
There is a
6top-instructions
A-input, which in the iteration would be an untimely
of that branch.
However,
the hyper-algebraic
termination
(from a more
general result in [57]), and we may as well let the machine l
read
$
all the time and erase it later from the language.
Replacement
rules become
[p, A, B, q] ÷ {wq[Pl, A I, B I, ql]w2 starting in state a (local)
p
with
computation on BA 1
writing
wi
for local computation
to
Pi
with
Ai
~
will do
input and recur in state then all possible
from
on the stack
qi-i
on
Bi_ 1
strings
leading
(note the final return
returning {wl[Pl , A I, q]
..- wk[Pk , A k, B k, qk]W--k+l I
but now
tation from
qk
I wI
assumption
The symbols
[p, A, q]
q}
emptying
the location and
v a local computation
leading to state
that the
[p, A, B, q]
that by previous
also inducing a local compu-
Bk
inducing
eventually
The construction
Wk+ 1
on
in state
It is straightforward strings by which
on the stack
I
u
similarly,
A
... wk[Pk , A k, 4, q]}
on the stack,
{wI[Pl , A I, B 1, ql]w2
on
A Wl
Pl
state)}
language.
are
extension of regular
languages is closed under arbitrary homomorphisms
instead of
the
may be rewritten
now all
wi's
represent when
p
Pl and stacking BAI}°
(possibly infinite)
goes through for
from
set of
is a regular
B = ~
as well.
Note
are ~ X. ~top
has to be used.
185
The rules are [p, A, q] ÷ {w I (unbarred this time) computation with
6top
from
w
p
induces a local
and
the location and returning in state
A q
finally emptying (w
is M ~
by
assumption)} With appropriate start rules
(a presetting 1 is to be treated
separately, but there is closure under union) language is exactly
L.
It follows that preset pda languages AFL).
the generated iteration-
form an AFL
(even a full
Later we will see when or when not you get an AFL with the
restricted preset pda-models. COROLLARY.
Preset pda languages are indexed,
included in the context-sensitive
4.
languages.
and
(hence) strictly
(see [56])
DETERMINISTIC PRESET PDA'S As opposed to classical machine-models
this time it is not
immediately clear when a preset pda should be called deterministic. Obviously
6
and
6to p
have to yield applicable instructions
unambiguously and should be chosen as for deterministic pda's, with regards to the present modifications.
i.e.,
But what about presetting?
It can be shown that as far as computational power is concerned,
the
stack in preset pda's need never grow more than linear in the length of input strings and that would reasonably limit choices when we were to preset the stack functionally in the input it may be different). "determinism"
(in global theory
We will argue that another form of
is more useful here, but illustrate it with an example
first.
Example.
A preset pda for
L = {ww R I w e Z }
would operate
186
as follows. reached,
It copies input in the stack until the presetting
then with
6top
is
it changes mode and removes symbols from
the stack while checking against input.
It is important can be only
to observe
that for any word
ww R c L
there
one presetting of the given machine which would lead to 2 In fact, the machine for {a n I n > i} designed before
acceptance.
had the same feature and was s t r u c t u r a l l y s t r o n g the appropriate
presetting of the stack.
Definition.
is called a weakly deterministic
L
if there is a preset pda
~
enough to "enforce"
preset pda language
with deterministic
6
and
6to P
*
accepting for some
L
such that ~ w e L ~ ! n (q0' w' Z0' n) ~--
q E F,
(q, l, 7, n)
with the preset maximum location reached at least
once.
All deterministic
context-free
languages
are weakly deter-
ministic preset pda, but we get many more. The next result is a generalization deterministic
THEOREM.
of a classical
theorem on
pda's.
The family of weakly deterministic
preset pda languages
is closed under complementation.
The main step in the proof is to eliminate non-terminating computations
on
l-input at r~un-time
(there can be no pre-calculation
as is the classical proof for ordinary deterministic stored in all locations
to record past symbol-state
that location are used to check for periodicities.
pda's).
Tables
configurations
at
187
5.
LOCALLY FINITENESS AND FINITE RETURNING Returning to the original motivation,
preset pda's as formalized
so far may not yet be what computer scientists would call "practical". Further restrictions that are necessary were extensively discussed in [55], and basically relate to the original idea that in the stacklocations procedure-bodies would be stored.
Implementation
leads to
bounds on size and, in slightly informal terminology, we would like preset pda's to have the following properties: locall~ finiteness - a fixed bound on the length of local computations,
i.e., with non-moving pointer.
finite returnin 9 - a fixed bound on how many recursions
there
can be from a base-location. Here is the relation to developmental systems.
THEOREM.
The family of languages defined by locally finite preset
pda's which have the finite return property coincides with the family
CROL
(or EOL).
The proof is in [55] but follows the same lines as the general result in section 3. This representation
for CROL-languages
is very powerful,
and of
interest also from a schematologist's point of view. Finite returning imposed alone would permit machines to do arbitrary long calculations on the same location.
THEOREM.
(Representation theorem)
L
can be accepted with a preset
pda that has the finite return property if and only if there is a l-free regular substitution L = T(L').
T
and a language
L' e CROL
such that
188
The virtue of this theorem, representation
and the previous
that it directly uses the machineresult,
contrasts with more abstract
approaches. In particular, easily.
the following
They were independently
results
can now be derived very
also given in
[7] in an entirely
different approach.
THEOREM.
The least AFL e n c l o s i n g CROL at the same time is the least
such full AFL.
THEOREM.
(Representation
theorem)
enclosing CROL if and only if A-free regular s u b s t i t u t i o n
L
belongs
L = T(L')
to the least full AFL
for some
L' e CROL
T.
The results were found through the machine-approach.
and
RECURRENCE SYSTEMS G. T. HERMAN Department of Computer Science The State University of New York at Buffalo
The purpose of this paper is to demonstrate that recurrence systems provide a useful approach to the theory of developmental systems and languages.
The demonstration is done by providing a sim-
ple proof of the important theorem of Ehrenfeucht and Rozenberg on the equivalence of the family of E0L languages with the family of images under coding of 0L languages.
i.
Introduction.
Recurrence systems have already been introduced elsewhere
[42,
alternatively see 45], a n d we are not going to repeat the detailed biological motivation behind the definition.
Our purpose in this
paper is to demonstrate the usefulness of recurrence systems as a tool for simplifying the proofs of mathematical developmental languages.
theorems concerning
By the way of demonstration we shall provide
an alternative proof of an important theorem of Ehrenfeucht and Rozenberg
[20].
We begin with a discussion of the Ehrenfeucht and
Rozenberg theorem and its significance. E0L languages were introduced in [35], as those languages which are the intersections of 0L languages and This corresponds
4*, for some alphabet
4.
to introducing the notion of a "terminal alphabet"
into L system theory.
E0L languages were found mathematically
tractable and interesting and have been discussed in a number of other papers.
In particular,
it was shown in [39] that E0L languages
are a natural extension of ALGOL-like languages.
190
Although such theorems are interesting from a mathematical point of view, their biological
significance is not inherent in their
statements, but are a consequence of the Ehrenfeucht and Rozenberg theorem.
This is because,
even though 0L languages are well moti-
vated from a biological point of view and the creation of a language , from another language by intersection with ~ is a standard procedure in formal language theory, the intersection of a 0L language ,
with
~
does not seem to make any biological sense.
should we exclude certain developmental
After all, why
stages from the language
of
an organism just because of the appearance of certain cellular states? The answer is provided by the following discussion. Clearly,
the experimental verification of our developmental
systems should inlcude,
among other features,
the individual cellular states
the identification of
(corresponding to the symbols of the
alphabet) which occur in the model.
In some cases, no morphological
or biochemical distinction can be made among the cells in the course of development,
and thus only the temporal and spatial distribution
of the cell divisions can be tested against observations.
In other
cases, there are irreversibly differentiated cells in the organism, and the spatio-temporal occurrences of such cells can be ascertained and compared with those given by the model.
It should be the goal of
developmental studies that eventually all cellular states, not only the irreversibly differentiated ones, postulated by the models should also be experimentally distinguishable. from being realized,
The goal is however very far
and in the meantime we must ask the question:
what class of developmental
systems
(class of 0L systems,
for instance)
will satisfy a certain observed distribution of cell divisions and of differentiated cells? ing way.
This problem can also be phrased in the follow-
Our observations provide us with "images" of the actual
developmental sequences.
We are trying to find the class of develop-
191
mental systems which generates a sequence whose "image" is the observed developmental sequence. For example,
in a description of the development of a compound
branching pattern in [42] we denoted each cell by c, i.e., we made no distinction between cells in different states.
Even though it is
quite possibl~ that experimental observations do not provide us a method by which we can distinguish states, it is clearly reasonable to postulate the existence of different states when trying to explain the development.
We have therefore proposed a 0L system with ten
different possible cell states. the observed development,
This system is sufficient to explain
inasmuch that if we replace each digit in
the sequence generated by the system by c,then we get exactly the observed sequence.
Definition.
We can formalize this as follows.
A coding of an alphabet
Z
is a homomorphism which maps a single letter of letter of
into an alphabet Z
A
into a single
A.
Our discussion above can now be restated as follows.
Due to
our lack of ability to distinguish always cells in different states, when presented with a language
L
describing an organism as observed,
we should look not only for systems which generate
L,
systems which generate a language
is the image of
under a coding.
In particular,
K
such that
L
but also for
the family of all languages which are
images of 0L languages under coding is in some sense biologically more important than the family of 0L languages. A major justification
for studying E0L languages is provided
by the following result of Ehrenfeucht and Rozenberg
Theorem.
[20].
A language is an E0L language if, and only if, it is
the image of a 0L language under some coding.
K
192
This is the theorem we are going to prove using recurrence systems.
2.
Definition of basic recurrence
systems.
For our proof we do not need the full generality of recurrence systems.
We therefore
a basic recurrence
Definition.
define
a more restricted
concept which we call
system.
A basic recurrence
system
(BR system)
is a 5-tuple
S =, where (i)
Z
is a finite non-empty
set of symbols
(2)
~
is a finite non-empty
set
(3)
A
is a function,
set
A(x)
(4)
F
(of axioms)
is a function,
non-empty (5)
x,y
F(x)
x e ~
A(x) C
(of recurrence
x e ~
a finite
x c ~
a finite
Z +,
associating with each
~ e ~ (the d i s t i n g u i s h e d
For any L
set
(the index set),
associating with each such that
(the alphabet),
formulas)
such that F ( x ) c ~+,
index).
and for any positive
integer y, we define
(S) as follows. Lx,I(S) If
= A(x).
y > i, L
(S) =
~_J Lk l,y-I (S) - - -Lkf,y_l (S). kl'''k f c F(x)
x,y Then
L(S) = is said to be the language erated by a BR system,
0 L ~,y(S) y=l generated by S.
A language w h i c h is gen-
is said to be a BR language.
193
(In the t e r m i n o l o g y of
[42], a BR s y s t e m is a
A-free depth 1
r e c u r r e n c e s y s t e m in w h i c h the r e c u r r e n c e formulas do not contain constants. )
Example.
Let
S = <{a, b},
A(1) = A(2) = A(3)
{i, 2, 3, 4, 5}, A, F, i>,
where
= @, A(4) = {a}, A(5) = {b},
F(1) = {35, 45}, F(2) = {2}, F(3) = {22,
33, 44},
F(4) = {2}, F(5)
=
{i}.
The f o l l o w i n g are the top four lines of a s e m i - i n f i n i t e which gives the values of
L
x,y
(S).
1
2
3
4
5
1
~
~
~
a
b
2
ab
~
aa
~
3
~
~
aaaa
~
4
aaaaab
f(n) =
ab
aaaaaaaa
It is easy to prove by i n d u c t i o n that where
table
L(S) = {af(n)b
I n > i},
(22n - 1)/3.
Definition.
A BR s y s t e m
S =
~-free if and o n l y if,
for all
L(S)
~-free BR language.
is said to be a
x e ~, A(x) # ~.
The e x a m p l e given above is not a the l a n g u a g e can be g e n e r a t e d by a in the n e x t section.
In such a case
~-free BR system.
However
~-free BR s y s t e m as w i l l be shown
The more c o m p l i c a t e d BR s y s t e m
b e l o w to d e m o n s t r a t e our results.
is said to be
S
w i l l be used
194
3.
Some results on basic recurrence
Lemma i. alphabet
If
Z0' d
F
is a finite set of non-empty strings over an
is a non negative
S i =L = F U ~L(Si) i=l Proof. generality
~ @,
If
that
L
L
and,
~-free BR system, is a
for
1 ~ i ~ d,
such that
~-free BR language.
F
is not empty. for
S =
We may also assume that the
1 < i < d.
~,
A,
F,
~i's
We are now going to construct such that
w>
L = L(S).
i~=dOZi . =
~J a i U i=l
A(~) = F,
{~},
where
A(x) = Ai(x)
L,I(S) for all
if
~d },
It is easy to see that shown by induction
w ~
¢
a i-
i=l
F(~) = {~, e I . . . . .
and,
integer,
is not empty we can assume w i t h o u t loss of
#-free BR system
~=
is a
then
are pairwise disjoint, a
systems.
x e ~i"
F(x) = Fi(x) S
is a
if
x e ~i" It can be
~-free BR system.
that
= F, y > l, Y0 1
L ,y(S) = F Q
d ~ L (S i) z = l i = l ~i 'z
Therefore, L(S) =
d y~--l= L ,y (S) = F L] ~J L(S i) = L. i=l
Note that the proof of Lemma 1 is constructive.
Lemma 2.
Proof.
Every BR language
If
L
S =
is a
is a BR language, such that
~-free BR language.
then there exists
L = L(S).
a
BR system
1 9 5 ,'
For
y > i,
we define E(y) = {x I LX,y(S)
= @}.
It is easy to see from the definition of E(y l) = E(y 2),
then
there must be a Let p < q
p
E(y I + i) = E(y 2 + i).
Yl
and
and q
Y2
such that
be the smallest positive
integers
Si
=
Li =
if
t L~,p+i+zd(S) z=0
~
L i ~ @, then -
is
Yl M Y2"
such that
either
Since
F
and
F i (x)
a
where,
finite sets of strings over
Fd(x) C
~,
~+
Si
is a
is not empty.
E(p + i).
- E(p
+
Lx,q+i(S)
+
i).
=
i),
Fd(x)
x c ~ - E(p + i),
in
~
by non-
integer
k,
of the same kind. ~+.
We define,
= Fd(x)/3 (~ - E(p + i)) +.
#-free BR system all we need A i (x)
The former is obvious
To see the latter,
x £ ~ - E(p
x
Fi(x)
to show is that for all x ~ ~ - E(p + i), F i (x)
for
for any positive
for any
x e ~ - E(p + i),
In order to show that
@-free
@-free BR system
of elements of
is well defined and is a finite substitution
0 < i < d - i,
or is a
is defined as follows.
is a finite substitution
in particular,
@
L i = L(S i)-
E(p + i), Ai, Fi, ~>,
A i (x) = Lx,p+ i (S) ,
for
and
~,
to show that for
We shall do this by c o n s t r u c t i n g
such that
Thus,
E(y) C
E(p) = E(q). Let d = q - p. ~i d-i L(S) = L ,y(S) tJ ~ (0 L~,p+i+zd(S))y=1 £=0 z=0
BR language.
Fk
Also, since
and
0 _< i < d - i,
empty
that if
x,y
E(y I) = E(y 2) ,
In view of Lemma i, it is s u f f i c i e n t
Si
L
is not empty and
from the d e f i n i t i o n of
all we have to show is that for all
contains
at least one string over
But if this was not the case we w o u l d have kl...kf~ F ~
ting the fact that
d(x) LkI'P+i(S)
"'" Lkf 'p+i(S)
x ~ E(p + i) = E(q + i).
= ~' contradic-
196
EaCh of the
Si
is therefore
easy to show by induction
a
f-free BR system.
that for all z > 0
It is now
we have that,
for all
x ~ ~ - E(p + i), Lx,p+i+zd(S)
= Lx,z+I(Si).
In particular,
°
©
Li = z--~0=L~'p+i+zd(S) This completes is constructive, finite set
F,
S~u, ..., Sd_ 1
=
z=0 Lx'z+I(Si)
the proof of our lemma.
given a BR system a positive
S,
integer
d
= L(Si)" We note that the proof
we can e f f e c t i v e l y and
produce
a
f-free BR systems
such that L(S)
= F t)
U
L(Si).
i=0
From these we can use the c o n s t r u c t i o n BR system
S'
such that
We demonstrate
of Lemma 1 to produce
a
f-free
L(S) = L(S').
this on the example of the last section.
E(1) = {i, 2, 3}, E(2) = (2, 4, 5}, E(3) = {1, 2, 4}, E(4) = {2, 4, ~5}. Thus,
p = 2, q = 4
L0 = 0 L~,2+zd(S) z=0
L1 =
F0(3) = {3333}.
begins with
d = 2.
= L(S 0) ,
~ L ,3+zd = @. z=0
{i, 3}, A 0, F 0, I>, and
and
Hence where
F = L ,I(S) = @.
where
SO
will be defined below.
L(S) = L(S0). A0(1)
where
= {ab}, A0(3)
The semi-infinite
S 0 = <{a, b},
= {aa}, F0(1)
table associated w i t h
= {331} SO
197
1
ab
aa
2
aaaaab
aaaaaaaa
3
a21b
a 32
4
a85b
a 128
In the case struction
of! this
of L e m m a
Those or
45],
example
with
appreciate
corol!ax ~.
to m a k e
use o f the
con-
i.
familiar
will
we did not have
the d e f i n i t i o n
the
Every
following
recurrence
of
recurrence
consequence
language
systems
of L e m m a
is a
~-free
[42,
2.
recurrence
language.
Lemma language
3.
Every
under
Proof.
some
If
BR system
define
a 0L s y s t e m
integer
BR language
is the i m a g e
is
a
S =
~-free
BR l a n g u a g e ,
~, A, F, ~>
G = <4,
P,
a>
such
as
such
is a w o r d
definition
of
a
~-free
of
BR language,
there
exists
L = L(S).
We
L
of
there
where length
k k.
is at l e a s t
is a p o s i t i v e (By o u r one
such
integer.) = S I S 2 ... S kP
consists
S 1 ÷ <~, in
of
the
al><~, Z
such
a2>
following ...
that
<~, ala 2
a
follows.
~ { S I, S 2 . . . . , Sk},
there
then
that
= ~ x Z ~ ~--~ that
of a 0L
coding.
L
~-free
#-free
productions. at>,
for
all
... a t e A(~).
a I, a 2,
...,
at
198
S i ÷ A, for
2 <_ i _< k.
<x, a> + y,
for all
where
f
x e ~,
is a
w i t h every
are in <x, a> ÷ A, Let
T
a I a 2 ... a k
y e f(F(x)),
the set of all strings
...,
and
for all
and
finite s u b s t i t u t i o n w h i c h associates
z e ~
a e 7.
where
al, a2,
..., a t
a I a 2 ... a t e A(z).
x e ~
and
a c 7.
be a w o r d of length
k
in
L.
Let
h
be
the coding d e f i n e d by h(S i) = ai, h(<x,
for
a>) = h(<x,
We c l a i m that
1 <_ i <_ k, a>) = a,
for all x E ~
L(S) = h(L(G)).
and
This claim has a r e a s o n a b l y
s t a n d a r d inductive proof and is therefore omitted. the proofs of Lemmas 2 and 3 in claim on the BR s y s t e m In this case <3, a>, < - I ~ ,
~,
SO ,
[39].)
a e 7..
(See for e x a m p l e
Instead, we d e m o n s t r a t e our
w h i c h is d e s c r i b e d f o l l o w i n g L e m m a 2.
A = {, <3, a>, , <3, b>,, b>, S l, S 2}
and a = SIS 2.
P
has the f o l l o w i n g
productions : S 1 ÷l, S2÷
A
+ <3, a><3, a><3, a><3, a>÷ <3, a><3, a><3, a><3, a> <3, a> ÷ <3, a><3, a><3, a><3, a><3, a><3, a><3, a><3, a> ÷ ~ < 3 , <x, u> ÷ A
for
A d e r i v a t i o n in
a><3, a><3, a><3, a><3, a><-3, a><3, a> x ~ {1, 3} and G = <&, P,
~>
u e {a, b}. looks as follows:
sis 2<3, a><3, a><3, a><3, a><, ~ < I , (<3, a><3, a>)4(<3,
b>
a><3, a>)4(<, ~ < 3 ,
a>) 2
199
If the coding
is now applied to this d e r i v a t i o n
h
(h(S I) = a, h(S 2) = b)
w e get the f o l l o w i n g strings ab ab aaaaab
a21b 4.
Proof of the E h r e n f e u c h t - R o z e n b e r ~ Theorem.
It has been p r o v e d in
[42, C o r o l l a r y 7], that the image of a
0L language u n d e r any h o m o m o r p h i s m an E0L language.
(and hence under any coding)
We are n o w going to prove the converse.
Using the results of
[42]
(especially T h e o r e m 7 and C o r o l l a r y
it is easy to show that for every E0L language BR s y s t e m a
S,
is
such that
L(S) = L - {A}.
L
3)
there exists a
By Lemma 2,
L - {A}
is
~-free BR language and hence, by Lemma 3, it is the image of a
0L language under some coding. or
(L - {A}) ~
{A}
Hence
L, w h i c h is e i t h e r
L - {A},
is also the image of a 0L l a n g u a g e under some
coding. This c o m p l e t e s the proof of the theorem.
We note that it can
easily be made constructive. It should also be p o i n t e d out that even though we have shown that for every E0L l a n g u a g e
L
L = h(L(G))
h,
proof
for some coding
there is a the G
0L s y s t e m
G
such that
we have p r o d u c e d during the
(see proof of Lemma 3) is b i o l o g i c a l l y s o m e w h a t undesirable.
It has the p r o p e r t y that in every step a cell e i t h e r dies or divides into step.
(usually) m a n y cells, m o s t of w h i c h
(usually) die
in the n e x t
The same c r i t i c i s m applies to the o r i g i n a l E h r e n f e u c h t and
R o z e n b e r g proof of their t h e o r e m
(see
[20]).
It w o u l d be i n t e r e s t i n g
to see w h e t h e r or not a similar t h e o r e m w o u l d still be valid r e g a r d i n g a b i o l o g i c a l l y r e a s o n a b l e r e s t r i c t i o n of 0L systems.
200
5.
Conclusion.
R e c u r r e n c e systems
theorems
provide us with a powerful
about developmental
Acknowledgement.
G. Rozenberg,
systems and languages.
The research for this paper has been sup-
ported by NSF grant GJ 998. the cooperation
tool for proving
The work has been highly influenced by
of the author with Professors
this cooperation
A. Lindenmayer
and
has been supported by NATO grant 574.
A D U L T L A N G U A G E S OF L SYSTEMS AND THE CHOMSKY H I E R A R C H Y Adrian Walker D e p a r t m e n t of C o m p u t e r Science State U n i v e r s i t y of New York at Buffalo
Introduction The c o n c e p t of an L system was [59, 60] as "a theoretical
first i n t r o d u c e d by L i n d e n m a y e r
framework w i t h i n w h i c h i n t e r c e l l u l a r
r e l a t i o n s h i p s can be discussed,
computed,
and compared".
The concept
has p r o v e d to be a fruitful one, and has o p e n e d up a new area of i n t e r d i s c i p l i n a r y research.
M u c h of the w o r k to date on L systems
is r e p o r t e d in Herman and R o z e n b e r g
[45].
The m o t i v a t i o n for the
p r e s e n t p a p e r is the thought that, since L systems are p r o v i n g so useful as a framework for s t u d y i n g b i o l o g i c a l growth and development, perhaps
they can also be used to study the ways in w h i c h o r g a n i s m s
achieve and m a i n t a i n r e l a t i v e l y stable adult states. emphasis
Thus w h i l e the
in work on L systems to date has been on all the strings
derivable
from an initial string, we shall focus in this paper on
just ~%ose strings w h i c h renew themselves d y n a m i c a l l y once they have been derived.
Notation We w r i t e string set a
V.
a
l
(e.g. If
u
Ill = 0), and
we w r i t e sym L for a
#V
lul
for the length of a
for the number of e l e m e n t s in a
is a string we w r i t e the set of symbols o c c u r r i n g in
as sym ~, e.g.
the symbol
for the empty string,
sym abbac = {a, b, c}. ~
If
L
is a set of strings
sym u. We w r i t e the n u m b e r of o c c u r r e n c e s of
in a string
~
as #a(U), e.g.
#a(abbac)
= 2.
202
We abbreviate c o n t e x t free grammar, linear b o u n d e d automaton, respectively. then
and T u r i n g m a c h i n e as CFG, CSG, LBA and TM
We require that if
lel <_ 161-
context sensitive grammar,
e + 6
is a p r o d u c t i o n of a CSG
We w r i t e the classes of c o n t e x t free, c o n t e x t sen-
sitive languages not c o n t a i n i n g l, guages as L(CF), L(CS)
and L(RE)
and r e c u r s i v e l y e n u m e r a b l e lan-
respectively.
O t h e r w i s e we use the
n o t a t i o n of H o p c r o f t and U l l m a n % for phrase structure grammars. If
6
is a m a p p i n g from a set of strings into a set of sets
of strings, we say that 6 i+l(~) = 66 i(~).
60(e) = {~},
We say that
and for each
6"(~) =
i > 0
0 6 i(~). i=0
Definitions A 0L system is a 3-tuple bet, V
S e V
and
6
is a m a p p i n g from
defined as follows.
Q ~V
x V
6(I) = {I}
If for each p r o d u c t i o n
V
There is a table
such that for each
e Q.
H =where
b e V
and for
V
is an alpha-
into the finite subsets of Q
of p r o d u c t i o n s
there is a
6 e V
such that
~ = al...a n, 6(e) = Q(a I)
e Q
6 ~ i
we say that
H
...Q(an).
is a
p r o p a g a t i n g 0L system, or P0L s y s t e m for short. A 2L s y s t e m is a 4-tuple are as in a 0L system, from
V
g
is a symbol not in
into the finite subsets of
is a table
Q
of p r o d u c t i o n s
such that for each abc e V VV g g
H =
6> e Q.
Q ( a 4l_ l)a 4JaJ4J+
V, and
there is a
a0
where
6 ~ V
~ = al...an,
where
6
V
and
and
S
is a m a p p i n g
defined as follows.
Q C VgVVg x V ,
6(1) = {l}, and for ... Q ( a n _ l a n a n + I)
V
where
There
Vg = V U {g}
such that 6(~) = Q ( a 0 a l a 2 ) ... an+ 1
stand for
g.
% Hopcroft, J. E., J. D. Ullman, Formal L a n g u a g e s and their Relation to Automata, A d d i s o n - W e s l e y , Reading,Mass., 1969. F r o m now on we refer to this book simply as H & U.
203
If for each p r o d u c t i o n
8> e Q
8 @ I
we say that
H
is a
P2L system. If
H
is an L system w i t h m a p p i n g
6
and initial symbol
S,
.
we define the adu!t l a n g u a g e of H as
A(H) = {e e 6 (S)
1 6(e) = {e}}.
Phrase Structure Grammars We now s u m m a r i z e some results about phrase structure grammars w h i c h we shall need later. We follow Aho and Ullman + in saying that a CFG G =(i)
for each
(ii) e i t h e r S ÷ i
is P r o p e r if A e VN P
it is not the case that
has no p r o d u c t i o n s of the form
is the only such p r o d u c t i o n and
of a production, (iii) •
S
A ~> A, A ~ I, or
never appears on the right
and
for each
B e VN
there e x i s t
~, 8, 7 £ V T
such that
,
S => ~By => e~y. The following result is o b t a i n a b l e by algorithms
2.8-2.11 of Aho and
Ullman % . Lemma 1 CFG
G
T h e r e exists an a l g o r i t h m w h i c h takes as input any
and p r o d u c e s as o u t p u t a p r o p e r CFG
G'
such that
L(G) = L(G').
Lemma 2 grammar
G
T h e r e exists an a l g o r i t h m w h i c h takes as input any
and p r o d u c e s as o u t p u t a grammar
G'
such that
if
~'+ B'
is a p r o d u c t i o n of
G'
then
I 'I
(ii) if
U' + I
is a p r o d u c t i o n of
G',
then
Ie'l = i,
(i)
e {i, 2},
% Aho, A. V., J. D. Ullman, The Theor[ of Parsing, T r a n s l a t i n g and Compilin_gg, volume I, P r e n t i c e Hall, E n g l e w o o d Cliffs, 1972.
204
(iii)
if G is a C S G
(iv)
L(G)
Proof
then so is
Let
tion in
Then P
G =
to c o n s t r u c t
having
m ~ 3, (i)
by a s s u m i n g
i
AI...A m ~ BI...B n
where
Q contains
A I A 2 ÷ BIC2,
(ii) C i A i + 1 ÷ B i C i + 1 C n A m ÷ BnZ,
if
and
if
Z + I,
in
Q.
a n d in a d d i t i o n and
m ~ n;
and
(n < i ~ m - 2),
and
m ~ n + 2.
Ci's
are n e w s y m b o l s ,
g i v e rise to s u b s e t s
Q2
2 directly
(2 ~ i ! n),
if
the
then
m = n + i;
C i A i + 1 ÷ ZCi+ 1
In this c o n s t r u c t i o n
e ~ I
Ai, Bj e (VN ~ V T)
(2 ~ i ~ n - i)
(iii)CiAi+ 1 + BiCi+ 1
C m _ i A m ÷ Z,
or
(2 ~ i ~ m - 2)
Cm_IA m ÷ Bm_l...Bn,
Q1
t h a t if
It is e a s y to
G' =, p l a c e e a c h p r o d u c -
CiAi+ 1 ÷ BiCi+ 1
PI' P2 e P
be a g r a m m a r .
a l e f t s i d e of l e n g t h
For each production and
and
= L(G').
see t h a t w e l o s e no g e n e r a l i t y I~I = i.
G'
QI' Q2
of
and if p r o d u c t i o n s
Q,
then the
C z.'s
in
are d i s t i n c t .
It is s t r a i g h t f o r w a r d
to c h e c k t h a t o u r c o n s t r u c t i o n
has
the
required properties.
Adult Languages In o r d e r we
first derive
order
to c h a r a c t e r i z e a property
for a s t r i n g
necessarily e.g.
if
6(~)
= {B}
a + ab,
b ~ c
w e shall w r i t e
the a d u l t
languages
o f the p r o d u c t i o n s
to m a p o n l y
the case t h a t
of 0L S y s t e m s
into itself.
a ~ a and
Note
for e a c h l e t t e r
c ÷ ~, then
simply
which
6(~)
~(abc)
= B.)
of 0L s y s t e m s , m u s t h o l d in
t h a t it is n o t in s u c h a s t r i n g • = {abc}.
(When
205
Lemma
3
If
H =
6, S>
is a 0L system,
Z = s y m A(H),
and
e
m = #7.,
6*6m(a)
then for e a c h
a e 7.
t h e r e is a u n i q u e
8 e 7.
such
that
= g.
Proof i.
For each
then t h e r e e x i s t
a ~ Z, #6 (a) = 1
el' ~2 e Z
and
such that
6 (a) c Z :
if
a E
6 ( e l a e 2) = elaa2 =
6 (~i) 8 (a) 8 (~2) .
exist Hence
2.
If
el'
~2 e Z
if
a e Z
#a ~(a)
then
#a6(a)
such that = k
e {0, i}:
if
then there
6 ( ~ l a ~ 2) = u l a e 2 = 8 ( ~ I ) 8 ( a ) 6 ( ~ 2 ) .
#a 6 i ( e l a e 2 ) -> k i
then
a ~ Z
for e a c h
i -> 0 "
Since
*
6 (~lau2)
= alau2
must have
k < i.
3. for all 7
If
a e Z
i > 0,
s u c h that
and
#a(elae2)
and
6 i(a)
#aS(a)
# I.
6(y) = Y.
<
= 0
Since
Since
lulau21
it is o b v i o u s
then
a e x
~m(a)
= l:
we have
#a 6 (a) = 0,
that we
suppose
that
a ¢ s y m T for some
w e can w r i t e
either
*
(i) Y
= u6(a)
v a w
for s o m e
u, v, w ¢ Z ,
or
(ii) Y
= uavS(a)w
*
for some
u, v, w £ 7. .
Hence,since
16i(T) I >_ 1 6 0 ( a ) . . . 6 i ( a ) l
for e a c h
160(a) ---61YI (a) l > IYI,
6 (Y) = Y i >_ 0.
a contradiction
m u s t be the c a s e t h a t for s o m e
i > 0,
e a s y to s h o w by p a t h l e n g t h a r g u m e n t s 4. 8 e 7.+ 8(a)
such that
= ~a~
i > 0 that y
For e a c h
a e 7 6*6m(a)
for s o m e
such
that
that
since
6(y) = 7.
is i m p o s s i b l e So there
that
less t h a n or e q u a l
m
= ~.
= ~.
= i,
there
= 1
If
6i(a~)
since
is an
i
is a u n i q u e
we c a n w r i t e ~ ~
for all
there exists a
occurs
s u c h that
~m(u~)
= ~.
Let
such
that
~r(u)
So it
F r o m this i t is
#a6(a)
£
161YI (y) I >_
6 IYI (7) = Y.
6m(a)
#aS(a)
to see t h a t for any
which
and h e n c e b y 3. w e h a v e integers
8i(a)
that
= ~:
B u t then
since
~, ~ e (~ - {a})*
then it is e a s y 16J(a) I > £,
such
we can s h o w t h a t
r, s
a
j
such
in a s t r i n g
~i(u~)
= X,
be the g r e a t e s t
~ ~, 6r+l(~)
= ~,
206
6s(~)
# A,
8 = 6r(~) now
and ...
follows
6 s+l(~)
60(~)a~0(~)
from
We s h a l l
Lemma 0L s y s t e m
4
H
£ A(H) Hence
such
there
~ ~2n(s)
such
~ e 6m(~)
and h e n c e
that
such
b £ s y m L.
W e can u s e in w h i c h
6(a)
Let
if w e w r i t e The
lemma
for any 0L s y s t e m
takes
as i n p u t
any
and let
L = {~ e 6i(s)
I i ~ 2 n + m,
if
e e ~ that
Then
(S) ~
there
{u
I 6(u)
u e ~i(s)
~ ~ ~i(s), Hence
sym ~ = sym u
and
there
and
is an
u E 6i+m(S).
exists
by Lemma 6(~)
= u}.
an
3 there
= ~.
So
exists ~ e L
•
two l e m m a s
for e a c h
letter
to p u t a
which
any
0L s y s t e m
occurs
in a f o r m
in the a d u l t
language.
Lemma 0L s y s t e m A(G)
G
= A(H)
Proof
5
There
exists
and p r o d u c e s and
for e a c h
Let
G =
7 G = s y m A(G) , and l e t
H.
s y m A(H).
b E Z.
s y m e = s y m ~.
the l a s t
= a
= B.
be a 0L s y s t e m ,
So
such
that
that
which
the set
Suppose
i > 0
to c h e c k
that
~ = s y m L.
b e s y m u. an
6*~m(a)
an a l g o r i t h m
n = #V.
s y m L C Z.
exists
to see
to f i n d s y m A(H)
8, S>
and
that
that
then
as o u t p u t
H =
We claim
6s(~)
how
exists
m = #Z,
it is e a s y
4. •
to k n o w
Let
B u t it is easy
an
3, and
There
Obviously
Then
...
and p r o d u c e s
= s y m A(H), = ~}.
2,
need
Proof
6(~)
= ~.
an a l g o r i t h m
as o u t p u t
a 0L s y s t e m
a ~ s y m A(H),
6 G,
m = #ZG"
S>
which
be
~H(a)
takes H
as i n p u t
such
= a.
a 0L system.
Let
that
any
207
If H =
ZG = @
~H' S>
then we are done,
be a 0L s y s t e m
so s u p p o s e
constructed
7 G # ~.
from
G
Let
as follows.
*
Define
a mapping
0: V ÷ V
by
[ a, %(a)
extend
0
to d o m a i n
if
a e V - ZG
[
6G(a),
if
V
by
e(X)
a £ ZG, = I
and
0(as)
= 0(a)8(u),
and
*
further
to d o m a i n
6H: V * 2 v
2v
in the o b v i o u s
manner.
Then define
by t
186G(a),
[a,
6H(a) By lemmas
For e v e r y
an
a £ V
straightforward 2. Let
if
a £ V - 7G
a e Z G.
3 and 4, H is w e l l - d e f i n e d .
i. exists
if
•
t > 0
such
to p r o v e
A(G) C A(H):
and
We c l a i m
8 e V , t
that
u e 6G(S)
by i n d u c t i o n
that
A(G)
8 £ ~ (S)
iff
and
on
0(a)
Since
there
= 8: this
is
t.
Let ~ = a l . . . a n e A(G).
6G(a.3 ) = ~m+l vG (aj) = 8j.
= A(H).
6G(~)
Then
= e,
aj c Z G.
w e have
BI...8 n = ~,
*
and so
0(~) = 81...8 n = a.
Since
e e 6G(S)
•
0(~)
e 6H(S).
follows
we h a v e
from i. that
*
from
But
0(u)
= ~,
so
the c o n s t r u c t i o n
*
u e 6H(S).
of
6H
that
Also,
6H(U)
since
= ~.
~ ~ 7G
it
Hence
e A(H). *
3.
A(H) C
A(G):
Let
8 e A(H).
Then
8 e 6H(S).
So it fol-
*
lows
from I. that t h e r e e x i s t s
8(u)
= 8.
and
n >_ 0. •
8j £ 7 G
an
u
such
Let a = U0Alel...~n_iAnen, Since
such
~j £ Z G m
that
the ~ e f i n i t i o n
of
6G(aj ) = -G 0
that
where
it follows ~m+l
8(a)
that
~ e ~G(S)
uj £ ZG'
from L e m m a
(ej) = 8j.
3
and
Aj e (V - 7~G), that there
It follows
= 8 0 A l S l . . . S n _ i A n 8 n.
from this
is a and
Since
*
89 e ZG' we have
from the c o n s t r u c t i o n
Since
we h a v e
89
8 e A(H),
it is c l e a r
that
6H(8)
= 8-
6H(A j) = Aj.
of Since
Since
6H
that 6H(8)
6H(8 j) = 8j. = 8
and 6H(8 j) =
Aj ~ ZG' if yj e 6G(A 9)
208
then if
8(yj)
E 6H(Aj),
yj = Aj.
Hence
and so
8(yj)
6 G ( A j) = Aj.
8 = 8(a) = 8 o A l S l . . . S n _ i A n 8 n, 6~(~j)
= Sj
m 6G(Aj)
and
6 (~) c 6G(S),
and so
adult languages If
~G(8)
we have
establish
use L e m m a s
6
= 8.
and
Moreover,
(~) = 8. H e n c e
o u r c l a i m that
A(H)
4 and 5 to c h a r a c t e r i z e
of 0L s y s t e m s .
~G: V ÷ 2 v
6G(8 j) = 8j
since
8 = 6G(8)
8 e A(G).
G =
mapping
B u t this is o n l y p o s s i b l e
Now since
we have
= Aj,
2. a n d 3. t o g e t h e r
we shall
= Aj.
= A(G).
the class A(0L)
F i r s t we n e e d the f o l l o w i n g
is a C F G w i t h
[]
VN U VT = V
of
notation.
we define
a
by t
I a,
[
~G (a) and we e x t e n d ~G(a)~G(~).
6
H
to d o m a i n
and p r o d u c e s
Let
and a s s u m e w i t h o u t may also a s s u m e
V
H =
~G(1)
that
L(G)
a e V N, = i
a CFG
6H, S>
t h a t for e a c h
and
takes
that
as i n p u t any
= a.
A(H)
= L(G).
let Z = sym A(H),
S £ V - Z.
a e ~, 6H(a)
By L e m m a
5 we
L e t G =
f r o m H, w h e r e P = {A + ~ Z
T.
G s u c h that
be a 0L system,
4 w e can c o m p u t e
~G(a~)=
= ~G(S) ~ V
an a l g o r i t h m w h i c h
loss of g e n e r a l i t y
By L e m m a
if
by
as o u t p u t
be a CFG c o n s t r u c t e d
E 6H(A)}.
a e VT
1 a ~ 8}
There exists
Proof
P, S>
{8
It is e a s y to c h e c k
Lemma 0L s y s t e m
~G
if
I A e V - Z
and
f r o m H, so o u r c o n s t r u c -
tion is e f f e c t i v e . N o w i t is e a s y to c h e c k i i >_ 0, 6H(S)
= ~ G (S).
~G(S) /~ 7 .
But since
that L(G)
A(H)
= 6H(S) ~
= ~G(S) ~
Hence 6H(a)
from our construction * 6H(S) = a
* = ~G(S). for e a c h
Z , and it is a p r o p e r t y
7 , hence
A(H)
= L(G).
[]
So
t h a t for e a c h
* 6H(S) ~
~* =
a e Z, it is e a s y of o u r n o t a t i o n
to see ~G that
209
We can p r o v e
Lemma CFG
G
7
the c o n v e r s e
There
and produces
Proof that
G
as o u t p u t
is p r o p e r .
G
V = VN U
V T,
and
{
Clearly
the c o n s t r u c t i o n
6: V ~ 2 v
i (S) = 6H(S). G
~} = 6H(S) ~
that
Hence
1
A(0L)
Proof
Immediate
if
the f o l l o w i n g
{L U {~} result
is p r o p e r
~ e V T. So
= L(G).
L(CF)
Hence
A(H)
i > 0, f r o m the fact from our
= {~ e 6H(S) I 6H(~)
L(G)
= ~G(S) ~
VT
of
•
of a d u l t l a n g u a g e s
of c o n t e x t
of
free l a n g u a g e s .
= L(CF).
from Lemmas
L e t us say of two c l a s s e s Ll ~ L2
G
N o w it f o l l o w s
the c l a s s A(0L)
in terms of the c l a s s
Theorem
a ~ VN
t h a t for e a c h
= e iff
A(H)
from
is a 0L system.
f r o m the p r o p e r t y
we h a v e
1 we may
be c o n s t r u c t e d
and s i n c e H
e e V T.
We can n o w c h a r a c t e r i z e 0L s y s t e m s
so
* = 6H(S).
~G(~)
if
By L e m m a
= A(H).
a e VT
defined,
= ~ iff
V T. ~G'
if
is e f f e c t i v e ,
* ~G(S)
Hence
6H(~)
our notation
H =
from our construction
is p r o p e r
construction,
be a CFG.
I a ~ ~ },
a,
is e v e r y w h e r e
It f o l l o w s
that
=
t a k e s as i n p u t any
a 0L s y s t e m H s u c h that L(G)
Let
{~
6H(a)
6.
an a l g o r i t h m w h i c h
L e t G =
assume by
exists
of Lemma
Ll
6 a n d 7.
•
and
of l a n g u a g e s
I L £ i I} = {L U for p r o p a g a t i n g
(%}
i2
I L ~ L2}.
0L s y s t e m s .
that
Then we have
=
210
Theorem 2
Proof
A(POL)
~ L(CF).
By L e m m a 6, we have A(POL) C
Then there is a p r o p e r CFG such that
L(CF).
L = L(G).
Suppose
L C L(CF).
It follows from
Lemma 7 and the c o n s t r u c t i o n in its proof that we can c o n s t r u c t a P0L system
H
such that
(L - {~}) = A(H), i.e. such that
L = L(G) =
A(H) U {4}. •
Thus we have e f f e c t i v e c o n s t r u c t i o n s w h i c h take us from any 0L s y s t e m to a c o r r e s p o n d i n g CFG,
and vice versa.
We have also shown
that the p r o p a g a t i n g r e s t r i c t i o n makes little d i f f e r e n c e for adult languages of 0L systems,
i.e. A(0L) ~ A(POL).
We shall see h o w e v e r
that the p r o p a g a t i n g r e s t r i c t i o n is very i m p o r t a n t in 2L systems.
A d u l t L a n g u a @ e s of 2L Systems We now look at adult languages of 2L systems w i t h and w i t h o u t the p r o p a g a t i n g restriction,
and their r e l a t i o n s h i p to the phrase
structure languages of the Chomsky hierarchy.
Lemma 8 grammar
G
There exists an a l g o r i t h m w h i c h takes as input any
and p r o d u c e s as o u t p u t a 2L s y s t e m
L(G) = A(H).
Proof
H
such that
M o r e o v e r if G is a CSG, then H is a P2L system.
Let
G =
be a grammar.
By Lamina 2
we may assume w i t h o u t loss of g e n e r a l i t y that if
~ ~ 8
lel c {i, 2}
We shall show how
and that if
to c o n s t r u c t from
G
a ~ i
a 2L s y s t e m
then H
lel = i. such that
then
A(H) = L(G).
The
idea b e h i n d the c o n s t r u c t i o n is as follows. Our c o n s t r u c t i o n will be such that if y = CIC 2. . .Cn
and
Y ~ L(G),
then a string
S ~> y, w h e r e ~C l C 2 . . . C n
is derivable
211
in H.
The ÷ will then move to the right along the string,
local rewriting
according to the p r o d u c t i o n s
single symbol on the left. to
When
string,
it changes
÷.
The
string,
allowing local rewriting
÷ ~
reaches then moves
according
which have two symbols on the left.
When
the string,
=>.
it changes
the above process
to
÷
or to
is repeated.
things can happen.
yielding
string contains
VN,
Formally, grammar
to
A)
our construction
to the left along the
to the productions ÷
reaches
=>
=> =>
~, then
then two moves all the If the
moves as far as that
continues
2L system
as above. H
from the
where
where
g
X
is a symbol not in
is a symbol not in
V N t2 V T-
V.
^
B)
~, ~, ~> and
individually
V
are mutually
disjoint
from
disjoint sets, w h i c h are
V t) {g}, defined by
-- {I J A ~ v} V>= {X>i A ~ v} ^
V = { [C7]
I AB ~ C7
where A, B, C e V
and
*
7~v} c)
w:vu~u~u~>u$ Wg =
W U {g}. *
D)
Q1 = {LAB ÷ 7
A, B C V, L £ Vg, 7 e V
, and
A ~ 7 }
*
Q2 = {LAg + ~y
P
the left end of
a string in A(H).
then
of a
of
is as follows.
V = V N V V T t) {X}, V g = V U {g},
the right end of the
VT, + then
÷, and rewriting
G =
which have a
If the change is to
way to the right and vanishes,
symbol F then changes
P
If the change is to
If the string is in
a symbol from
of
allowing
I A, B, C E V, y E V , and A ~ C7}
Q3 = {LAg + X
L e Vg, A e V, A ~ I}
Q4 = {L~R ÷ I
L, R e V g }
212
Q5 = {LAB +
[Cy]
I A, B, C E V, L e Vg, y e V
and
AB ~ Cy} Q6 = {L[Cy]B ÷ ~
i B, C, e V, L e Vg, and
Q7 = { [Cy]BR ÷ 7 I B, C e V, R e Vg, and Q8 = {ABR ~ B I A, B e V
E)
and
B c V N}
QII = {LAB ~ I
L £ V
and
A, B e V}
QI2 = {ABR ~ B
A, B ~ V
QI3 = {gIR ~ ~>
A e VT
QI4 = {gIR + I
A e V
QI5 = { L ~
L, R e V
R e Vg}
and g
R e Vg} and
A E V}
and
R E Vg}
A £ V, B e V N
QI8 = {LAR ÷ A
L, R e Wg, A £ W,
U
such that
Suppose
LAR + I,
R e Vg}
and there is no 17 (LAR ÷ y) e k~__ I= Qk }
where
6
is defined by
since our construction
there exists a
It remains
and
ok
H = <W, 6, g, S>,
detailed proof that
Q~.
and
R e Vg}
QI7 = {X~R +
From the construction
form
and
A, B e V T
H is a 2L system,
gating.
g
016 = {X~R ÷ B>
k=l
LAR e WgWWg
A, B e V}
L £ Vg
Q =
F)
and
QI0 = {LBg +
yeW 18
[Cy] e V}
and R e Vg}
Q9 = {LAB ÷ A I L E Vg
+ A
[Cy] e V}
7 e W
such that
G
G
contains
is a CSG.
But then it follows
LAR ~ Y.
If
to write out a
(A full proof is given in Walker%).
to be shown that if
then by
is such that for each
it is s t r a i g h t f o r w a r d
L(G) = A(H).
Q.
Q
is a CSG then H
a production of the
inspection this production from the construction
is propa-
is in
Q1 U Q4 U
that there is a
T Walker, A. D., Formal Grammars and the Stability of Biological Organisms, Ph.D. thesis, Department of Computer Science, State University of New York at Buffalo, 1974.
213
production
(* ~ 8 for w h i c h
lul >
161, a contradiction.
In the n e x t lemmas we shall M
is an L B A we denote
T
is a TM
we denote
Lemma P2L s y s t e m A(H)
9
the language the language
There
H
use the f o l l o w i n g
exists
a c c e p t e d by accepted
M
by
LBA
notation. as L(M),
T
an a l g o r i t h m w h i c h
as
If
and
if
L(T).
takes
M
•
and p r o d u c e s
as o u t p u t
an
Let
6, g, S>
be a P2L system.
as input any
such that
= L(M).
Proof
H =
an L B A c o n s t r u c t e d The
from
tape of
M
on the top track of
H has
to o p e r a t e three
the tape,
M
Let
M
be
as follows.
tracks.
If a string
decides w h e t h e r
e
is p l a c e d
or not
~ £ L(M)
in
the f o l l o w i n g way. (i) below.
M
tests w h e t h e r
If not, (ii) M
ministically,
M rejects writes
or not
e
S
6(~)
If so,
M does
(ii)
and halts.
in the m i d d l e
to see if
= e.
e E ~
(S),
track
and proceeds,
using
the lower
(S),
then
nondeter-
track as w o r k -
, space.
If
M
halts.
If,
in s i m u l a t i n g
~k e 6k(s)
discovers
M finds
that
that
~ c 6
a derivation lekl
>
I~I,
F r o m the above d e s c r i p t i o n write
down
formally
to show that
Lemma 2L s y s t e m A(H)
H
= L(T).
L(M)
10
There
~
accepts
~
"'''
where
ek
and halts.
it is a s t r a i g h t f o r w a r d constructs
and
M
from
task to H,
and
•
exists
and p r o d u c e s
el'
M rejects
an a l g o r i t h m w h i c h = A(H).
S = ~0~
M
an a l g o r i t h m w h i c h
as o u t p u t
takes
a Turing machine
as i n p u t T
such
any
that
214
Proof
is similar to that of L e m m a 9, except that in step
there is no limit on the length of an i n t e r m e d i a t e string not every c o m p u t a t i o n by way in w h i c h case that
L(T)
T
terminates.
P2L
Hence
However, because of the
is d e f i n e d for a Turing machine
T,
it is the
A(H) = L(T).
We can now c h a r a c t e r i z e the classes of
ok.
(ii)
A(P2L)
of adult languages
systems and
A(2L)
of adult languages of
2L
systems in
terms of the classes
L(CS)
of context sensitive languages and
L(RE)
of r e c u r s i v e l y e n u m e r a b l e languages.
Theorem 3
Proof
A(P2L) = L(CS).
That
that for each
A(P2L) C L(CS)
LBA
M
there is a CSG
e.g. H & U, T h e o r e m 8.2. L(CS) C
follows from Lemma 9 and the fact G
such that
L(M) = L(G);
see
It is immediate from L e m m a 8 that
A(P2L).
Theorem 4
Proof that for each
A(2L)
That TM
= L(RE).
A(2L) C L(RE) T
there is a g r a m m a r
see e.g. H & U, T h e o r e m 7.4. L(RE) C A(2L).
follows from L e m m a i0 and the fact G
such that
L(T) = L(G);
It is i m m e d i a t e from Lemma 8 that
g
This completes our c h a r a c t e r i z a t i o n of
2L
systems.
We note
that while the p r o p a g a t i n g r e s t r i c t i o n made little d i f f e r e n c e for systems,
in the sense that
difference
for 2L systems,
A(0L) ~ A(POL), since
it makes a fundamental
A(P2L) C A(2L).
#
0L
2t5
Conclusions Theom~ms 1 - 4 give us a satisfactory analysis of L systems from the point of view of the adult languages they generate,
for
they establish direct correspondences with three of the four main classes of languages in the Chomsky hierarchy. is that of the regular languages,
The remaining class
and it is an easy exercise to
restrict the form of the productions of a 0L system to ensure that its adult language is regular.
In Walker % it is shown that the
result for 2L systems can be extended to Herman and ~3zenberg k + £ ~ l, PL
L
systems
(see
[45] for the definition of such systems) with
and that the result for systems with
P2L
systems can be extended to
k, £ > i.
From the point of view of formal language theory, we have given a new characterization,
by totally parallel grammars, of each
of the classes of languages in the Chomsky hierarchy.
From the point
of view of biological model building, we have gained access to many of the established results of formal language theory.
Acknowledgements The author wishes to thank Professors G. T. Herman, A. Lindenmayer,
and G. Rozenberg for their help and encouragement.
This
work is supported by NSF Grant GJ 998 and NATO Research Grant 574.
% Walker, A. D., Formal Grammars and the Stability of Biological Organisms, Ph.D. thesis, Department of Computer Science, state University of New York at Buffalo, 1974.
STRUCTURED OL-SYSTH~MS
Department of Applied Analysis and Computer Science University of Waterloo Waterloo, Ontario, Canada and Geselschaft fur Mathematik und Datenverarbeitung Bonn
ABSTRACT
A new type of Lindenmayer systems, called Structured 0L-systems (SOL systems)~is studied which gives a formal tool for investigation of structured organisms and structurally dependent developments. It is sho~na that a restricted version of SOL-systems is equivalent to codings (length-preserving homomorphisms) of OL-Languages. The properties of unrestricted SOLsystems are then studied. It is for example shown that the languages generated by them properly include the languages generated by extended table OL-Systems.
I. Introduction. Lindenmayer systems have been the object of extensive study during recent years. The systems, also called developmental systems, were introduced in connection with a theory proposed to model the development of filamentous organisms. The stages of development are represented by strings of symbols correspondings to states of individual cells of an organism. The developmental instructions are modelled by grammar-like productions. These productions are applied simultaneously to all symbols to reflect the simultanity of the growth in the organism. This parallelism is the main difference between Lindenmayer systems and ordinary generative grammars. Another difference is that in most of the versions of Linde~nnayer systems only one type of symbols (terminals) is considered which means that all the intermediate strings in a derivation are
217
strings in the generated language. OL-Systems
[61] in which every symbols is rewritten independently
When we compare 0L-systems
and the corresponding
grammars, we see that OL-systems grammars,
The simplest type of Lindenmayer
systems are the
of its neighbours.
class of grammars,
are missing one important
context-free
feature of context-free
namely they are not structuring the generated strings. A derivation
context-free
grammar can be represented
as a derivation-tree
structure of a string with respect to this derivation. derivation tree for a derivation are labelled by terminal
in a
which describes the
We can consider the analogous
in an OL-system but in this case the branching nodes
symbols rather than nonterminals
the tree does not reflect the possibly ihteresting
(grammatical
categories)
and
structure of an organism.
In this ps@er we introduce Structured OL systems
(SOL-systems)
which not only
allow to describe the structure of generated strings but also give a formal tool to study the cases when the development
is structurally dependent.
A simple example is
the case when all the stages of a developing organism consist of certain fixed number of partes and there are different
development
rules for every part of the organism.
From a mathematical
point of view the SOL-systems
context-sensitivity
in parallel rewriting
is represented
give another interesting type of
(compare with
IC01T).
A structured organism
in an SOL-system as a labelled tree. The labels of the leaves of the
tree represent the individual cells of an organism, the labels of its branching nodes represent the structural An SOL-system developmental
"units" of the organism.
is given by a single starting structure and a finite number of
rules. At every stage of the development the rules are applied
taneously to all cells and structural units
(nonterminals)
of an organism.
the rules each structural unit can change its state or disappear every cell can be replaced by a substructure,
simul-
According
(but not divide)
and
i.e. it can divide into several parts
each of which can be either a single cell or another structured part. At every step a cell can divide only into a limited number of subparts but there is generally no limit on the number of subparts of a structural unit which can be created during the
~ICOI : K. Culik II and J. Opatrn#~ Context in parallel rewriting,
in this volume.
218
development,
i.e. there is generally no limit on the number of sons of a branching
node. Types of structural units will be represented by labels from an alphabet which corresponds
to the nonterminals
in such a way that
rewriting
of a context-free
of a nonterminal may depend on its father but not on
its sons in a tree. We consider parallel all labels in a tree must be
grammar. We will define SOL-systems
rewriting~
simultaneously
at every step of a derivation
rewritten.
The language generated by an
SOL-system is the set of 8~1 frontiers of the generated trees. We will consider a special subfamily of SOL-systems, in which essentially only labels on leaves
called simple SOL-systems,
(individual cells) are rewritten,
i.e. in a
simple SOL-system it is possible to create new structural units but once a unit is created it never changes its state. We will show that the languages generated by simple SOL-systems
are exactly length preserving
homomorphisms (coding) of OL-languages
[10,20]. Then we will investigate the properties ted SOL-systems
of the family generated by unrestric-
(SOL). We will show that SOL is closed under Kleene
operation (u,.,~)
and under e-free homomorphism but not under intersection with a regular set. The closure result will help us to establish the relations of SOL to other known families of languages. closure results,
It is easy to show that SOL ~ TOL [81] and then, using the
that SOL ~ ETOL [89]. The fact that the later inclusion and therefore,
of course, also the former is proper follows from the result that SOL is incomparable with the family of context-sensitive recursively
enumerable
languages.
set over T with an end
Actually it will be shown that any
marker can be expressed as an inter-
section of an SOL language over an extended alphabet with the set of all the terminal strings with the endmarker.
.
219
2. Preliminaries. We assume the knowledge of the basic notions and notation of formal language theory, see e.g.
[HU+,Io2]. We start with a slightly modified, but equivalent, defi-
nition of extended table OL-systems
Definition:
(i) (ii) (iii)
~I]
[3~] ~ d ~OL-systems [81].
~oL-system~
4-tuple
[89], involving as special cases OL-systems
An extended table L-system without interaction (ETOL szstem) is a
G = (V,T, @,~)
where
V is a finite nonempty set, the alphgbet of G, T [ V, @
the terminal alphabet of G,
is a finite set of tables. @ = {PI,...,Pn} V )( V ~ .
Pi
Element
(u,v)
of
is usually written in the form
for some
Pi' I < i < n, u + v.
following (com~letness) condition:
Every
For each
n ~ I,
where each
is called a production and
Pi' I < i < n, a { V
there is
satisfies the w [ V~
so that
(a,w) 6 Pi' (iv)
~ e V +,
Definition: (i) (ii) (iii)
An ETOL-system
a TOLusystem if
an OL,system if
P'I 6 @
is called
{PI};
V = T
and @ = {P~}.
Given an ETOL-system
a I ~. ..,ak E V
for some
G = (V,T, @,a)
V = T;
an EOL-system if @ =
Definition: exist
the axiom of G.
and
G = (V,T,P,~)
Yl ~. "''Yk £ V ~
so that,
we write
x ~
y
if there
x = al...ak, y = yl...yk
and
, aj ÷ yj ~ Pi' j = 1,...,k.
The transitive and reflexive closure of binary relation
~ G
Definition : by G is denoted by
is denoted by ~--->~. G
Let
G = (V,T, ~ ,q)
be an ETOL system. The language generated
L(G)
and defined as
L(G)
=
{w ~ T ~
: c
~
w}.
G Notation:
A language generated by an XYZ system, for any type XYZ will be called an
XYZ-lamguage. The family of all the XYZ languages is denoted by XYZ.
i"IHUI : J.E. Hopcroft and J.D. Ullman: Automata, Addison-Wesley, 1969.
Formal Languages and their Relation to
220
Before we can define SOL-systems we need to introduce a notation for labelled trees
(forests).
We will recursively
define labelled rooted ordered forests and ex-
pressions denoting them. We are not interested in names of particular nodes in a forest, i.e. we actually consider the equivalence
classes of isomorphic
consider forests with labels of leaves from one alphabet (nonleaves)
and labels of branch nodes
from another distinct alphabet.
Definition:
Let
T,N
be two alphabets,
reserved symbols not in
(i)
forests. We
T ~ N = ~,
and let
[
, ~ and k be
T u N.
is a forest expression
and denotes the empty tree (forest), i.e. the tree
with no nodes. (ii)
For
a ~ T, a is a forest expression and denotes the tree with a single node
(root)labelled by (iii)
If
e],e 2
from trees
a.
are forest expressions ~1'''''am
and
denoting nonempty forests
B 1,...,Bn,
respectively,
then
expression and denotes the forest consisting from trees (iv)
If
A ~ N
of trees
~ , ~ consisting ele 2 is a forest
~1,...,~m,S1,...,Sn.
and e is a forest expression denoting nonempty forest ~ consisting ~1,...,an
then
A[e]
is a forest expression
and denotes the tree
with the root labelled by A and the sons of the root,from left to right,the the roots of subtrees Notation.
al,...,mn.
The set of all the forest expressions
(forests) over alphabets N (labels
of branch nodes) and T (labels of leaves) is denoted by
(N,T)~. Let (N,T 4
(N,T)~ - {k}.
will be called structures
(over N,T)
In the following the elements of
(N,T)~
and we will not distinguish between an expression
=
and the forests denoted
by it. Notation.
Let a be a particular occurence of symbol a in forest expression ~. The
label at the father of the node labelled by the considered occurence of a will be denoted by Father
(a).
This notation will be used in such a way that there will be
no confusion of which oceurence of a is being considered. Notation. (N,T~
Mapping Y (yield) maps k to empty string (denoted by s) and a forest in
to the string of the labels of its leaves
(from left to right).
221
3. Structered
OL Systems.
Now we are ready to define formally Definition:
A structured OL-schema
(SOL-system)
N is an alphabet
of nonterminals
T is an alphabet
of terminals,
xiE
(ii)
We write
~ ~
x i £ {I,I}
if
x. E T u N
A . x i ÷ wi e P
then
where
is obtained
the set
XXY,X[Z]
and
A.a ÷ ~
where
B 6 (N,T)~
where
8' = w ] w 2 . . . w n
~ = x I x2...x n
so that for
x. + w. E P l
l
or
A[X] by XY,Z or X,
such that
and reflexive
respectively,
until either
the condition
closure of relation ~
generated by G is denoted
. T(G)
and defined to be
L(G) = (Y(~)
; ~ET(G)}.
G is called a structured OL-system
of completness:
of all the
For every
~ ~ T(G)
(SOL-system)
there exists
of completness
requires that for every structure which can be
from the starting structure there is a "next step" in development.
some definitions restrictive
B = X
8.
Our definition developed
of the
~}.
A structured OL-schema
~ =~ G
I < i < n:
of X in ~.
generated by G. Formally
if it satisfies
for
or
The language generated by G~ denoted by L(G), is the set of the yields
D_efinition:
A , B E N,
Ai = f a t h e r ~ ( x i ) ;
The set of structures
T(G) = {~ : a ~
structures
x (N,T)~,
in the following form (all
from B' by repeated replacing of each subexpression
be the transitive
Definition:
k~(Tu(N×T))
w i = xi;
or there is no occurenee Let ~
where
are special reserved delimiters.
8 for ~ ~ ( N , T ~
then either
I
form
"."
] < i < n, if there exists
if
(iii)
A ÷ C, A . B + C, a + ~
a ~ T, ~ e (N,T)~ and "÷"
NuTv{[,]]}, (i)
( N u (N×N)) × ( N v ( k } )
from
of an SOL-schema will be written
possible types are given).
Definition:
G = (N,T,P,a)
, the initial structure.
The productions
C ~ N u{~},
is a tuple
(states of structural parts),
P is a finite set of productions ~ (N,T)+
structred 0L systems.
and auxiliary results
definition
of completness
After
it will be shown that the choice of a more (strong completeness)
requiring
existence
of a
222
"next step" f o r e v e r y structures
does not change the families of sets of
or languages generated by SOL-systems.
Definition:
An SOL-system
P ~ {s ÷ s } u where
structure in (N,T~
G = (N,T,P,~)
is called full if
((N×~) × (~u{~})) u((N×T) × (~,T)~,
N = N - {S), S being the root of o, i.e. the father-context
appears on the left
side of every production with the exemption of the production which keeps unchanged the (reserved) label of the root. Definition: Lemma ]:
SOL-systems G I and G 2 are called equivalent if
For every SOL-system there exists an equivalent full SOL-system.
Proof. Given an SOL-system G' = (N',T,P',~') arid
L(G]) = L(G2).
where
G = (N,T,P,~) N' = N u {S)
P' = {A-X+w : A . X ÷ w E P } Clearly,
Definition:
we construct a full SOL-system
for a new symbol S not in
u {A-X÷w : X÷w 6 P
and
N u T, ~' = semi,
A ~ N'} u (S÷S}.
L(G') = L(G). An SOL-system
G = (N,T,P,~)
e (N,T)+ there exists B such that
~ ~
is called strongly complete if for every ~.
G Theorem I:
For every SOL-system G there exists an equivalent
strongly complete SOL
system. Proof.
By L e n a
I we may assume that
system
G' = (N~T,P'~a) where P' = P u {A.X÷X : A 6 N,X 6 N u T
production of the form
G = (N,T,P,o)
L(G') = L(G).
Now we will study a special case of SOL-system, in which essentially
a derivation.
and there is no
A.X÷W in P}.
Clearly, G' is full and
systems),
is full. We construct the SOL-
called simple SOL-system
(SSOL-
onl~~ terminal symbols are rewritten at every step of
We will show this subclass of SOL-systems
generates
exactly the same
family of languages
as several other types of systems known already to be equivalent~
namely FMOL systems
[9]~ E0L systems
languages
[35], and length preserving homomorphisms
EIO,20]. This does not mean SSOL-systems
of OL
are without interest, on the
contrary they give an alternative mechanism for the description
of languages from a
very natural class with the advantage to exhibit explicitly the structure of generated objects. They also contribute another evidence to our opinion that the family of the
223
length-preserving Definition: P = {A+A
homomorphisms
AN SOL-system
G = (N,T,P,q)
: A ~ N} u P' where P' c
In the following SSOL-system
(N×TuT)
simple
Lemma 2 :
COL c SSOL.
Let
G = (Z,P,~)
productions"
be an OL-system
SSOL system
loss of generality
G' = ( Z u { S } , T , P ' , o ' )
that
(ii)
Let
for
i = 1,2,...n.
....
an [ h ( a n ) ] ] .
o = a I ...an, a i c Z
o' = Sial [ h ( % ) ] a 2 [ h ( % ) ] (iii~
P' = { a ' h ( a ) + b 1 [ h ( b ~ ) ] % [ h ( b 2 ) : a,b 1,b2,...,b k E
Clearly,
h(L(G'))
SSOL
Let
G --; (N,T,P,o)
(a i)
homomorphism}.
from
Z~ to T *.
Z ~ T = ~.
Then
] ... bk[h(b~)]
:
= L(G).
be
an SSOL-system.
x
P ~_ (NxT)
E (N,T)~ to the string for
G' = (Z,P',o') A.a÷
an
~ and a+blb 2...b k ~ P}.
(N,T)~
and let g be the mapping
father
whenever
c COL.
assume that
Z = N×T
A÷A
where
Z u T.
clearly
of form
and h be a homomorphism
S is a new symbol not in
Proof.
if
× (N,T)~.
(i)
3:
(SSOL)
: L E OL, h is a length-preserving
we can assume withnout
Construct
Lemma
class of languages.
is exhibited.
Let COL = {h(L)
Clearly,
is a very natural
is called
we omit the "identity
Notation.
Proof.
of OL-languages
where
with "identity"
from
(N,T).
(A 1,a I) ... (An,a n )
I < i < n.
In pa-~icular
o' = g(o)
and
By modification
of Lemma
productions
into Z * which maps such that
g(k) = s.
1 we
can
omitted.
Let
a structure
a 1...a n = Y(~) and A.m = We construct
OL system
P' = {(A,a)-~g!AIB I) : A E N, a E T
and
S ~ P}. Note that if forest
then
g(AIBI)
h((A,a))
= g(S).
= a for every
B does not include
Let h be the homomorphism
Proof.
~:
from Z
to T
from a single node only, defined by
(A,a) E Z.
It is easy to verify that Theorem
a tree consisting
SSOL = COL = E0L
h(L(G'))
= L(G).
= FMOL.
By Lemma 2 and 3 we have the first eq[uation; the definitions
and other results
are in
[9,20]
of FMOL languages
224
h. Closure Properties
of SOL.
To show the relation of SOL to other known families of languages we will need some closure results which will be shown first. The most interesting of SOL under s-free homomorphisms preserving homomorphisms) Theorem 3:
which in particular means that codings
(length-
do not increase the descriptive power of SOL-systems.
Family SOL is closed under s-free homomorphisms.
Proof. Given an SOL language L over T and a homomorphism h from assume by Lemma I that L is generated by full SOL system construct SOL-system Let
is the closure
G' = (N',Z,P'o')
N' = N u T v ( N × T )
Let f he the homomorphism
G = (N,T,P,o)
we may and we
as follows.
u {A : A e N} u {Q} from
T * to Z ~
where Q is a new symbol not in N u T .
(N,T)~ into (N',Z)~
defined as follows.
The forest
expression f(a) is obtained from expression e by replacing every terminal symbol from
T, say a, by subexpression Let (i) (ii) (iii) (iv)
(v) (vi
e' = f(~)
a[alQ[a2]Q[a3]
and productions
P'
...
Q[an~ ] where h(a) = a l a 2 . . . a n .
be constructed
as follows:
A+A is in P' for every A in N. A.a+(A,a)
is in P' for all A in N and a in T.
a.t÷t is in P' for all a in T and t in Z. S÷S is in P'. If A.B÷X is in P then A . ~ X
is in P' for all A,B in N and X in N u {l}.
If A.a÷~ is in P and h(a) = ala2...a n then (A,a)-a1÷f(a)
is in P' for all A
in N and a in T. (vii
(A,a)÷l is in P' for all A in N and a in T.
(viii
Q÷~ is in P'
(ix)
Q't÷~ for every t in Z.
Clearly, (iv)-(ix)
in any derivation in
G'
can be used only in alternative
later in even steps of any derivation. that
(i)-(iii) and the productions
steps, namely, the former in odd steps and
Realizing this it is straigthforward
to verify
L(G') = h(L(G)).
Theorem 4: Proof.
the productions
Let
N I m N 2 = ¢.
The family SOL is closed under union,
concatenation
and star.
G I = (N],T~,PI,~ 1) and G2= (N2,T2,P2,o 2) be SOL-systems.
Assume that
225 To show the closure under union we construct an SOL-system follows. Assume
Let
+o2,Q.ai÷X
for
Clearly,
with S and Q new symbols not in
N 3 = N t u n 2u{S,Q}
Y(o I) = al...a n.
Let
(N3,T I ~ T2,P3,o 3)
o3
:
S[aiQ[a2...an] ] and P3 : P I U P 2 U
as
N I u N 2 u T I u T 2. {S'a1÷ol, S.a1+
i = 2,...,n}. L(G3) = L(GI) u L(G2).
To show the closure under concatenation we construct SOL-system (N4,T ] u T2,P4,~4) as follows. Assume
N 4 = NIu N 2U{S,A,B,C,D,E,F,H,Q}
Y(o I) = a1'''an and Y(o2) = bl...bm.
P4 = (I)
PI u P 2 U P¼
A÷A
where Let
S,A,...,Q
are new symbols.
~4 = s[AEc[al]E[a2"''an]]A[D[bl]E
where P~ consists of the following productions:
(II)
A÷B
A" C÷C
B. C+F
A-D+D
B'D+H
A •E÷E
B •E+Q
C'a1+a I
F'a1+o I
D'b1+b I
Q-a.÷l 1
E.a ÷a.mm for i = 2 , . . . , n
H.bl*o 2
E.b.÷b. for i=2,...,n 1 1
Q.b.+X 1
for i=2,...,n
for i=2,...,m.
Productions of group (I) allow to delay the start of the generation of strings in
L(G I ) or L(G 2)
L(GI).L(G 2)
to assure that even by parallel generation all strings in
are obtained. Once the production
A÷B
is used the productions of group
(II) assure the start of generation from o I by productions PI or from a 2 by productions P2" It is straightforward to verify that
L(G 4) = L(GI).L(G2).
Finaly, to show the closure of SOL under star we construct SOL-system (N5,TI,P5,a 5) bols not in
as follows. Let N I v T I.
Assume
= A[CEalbEa2%...%]].
Let
tions:
N 5 = N I~ {S,A,B,C,D,E} where S,A,B,C,D,E are new symO1 = ala2"''a "n Let
o 5 = S[aiQ[a2a3...an] ] and
P5 = PI U P' where P' consists of the following produc-
226
S+S
A.C+C
S.a1+e~ 5
A.D+D
S.a1÷l
B.C÷E
Q+~
B. D+Q
Q.ai+l for 2 < i < n
C-a1+a I
A÷A
D.a.÷a. 1
A÷B
for 2 < i < n
1
E.a1÷~ l
It is straigthforward
to verify that G 5 generates only strings in
see that all such strings are generated we observe that
L(G I).~ To
S ------~ S[~S[~S[~...S[e]]...]] G5
and that ~_____~ka~------~B[C[al]D [a2a3...an]]~--->B [E [al] Q[a2a3 . , ,an]]~>B[E [~I]]~----->B[E[8]] for all
k >0
and 8 in
T(GI).
w~ have ~__~ki a--------~> B[E[~ i]] I < i < m,
Therefore for any for
I < i < m
we can " s y n c h r o n i z e d " d e r i v a t i o n
m>
1
and
B I ..... 8m in T(G I)
and by suitable choice of
S ~*
S[B[E[BI] ] S[B[E[62] ]
ki,
...
. . .
Since ~5 ------->~ 5.
also
e g L(G5).
Relation of SOL to other Families First we show that SOL-systems
of Languages.
can simulate TOL-systems
sure of SOL under e-free homomorphisms
and then using the clo-
this result will be generalized to ETOL-sys-
tems. It is easy to see that these results can be further generalized to (E)TOL-systems with some restrictions
on the sequences of productions which may be used in a
derivation. Lemma ~:
TOL ~ SOL.
Proof. Given TOL system
G = (T,{PI,...,Pn},~)
G' = (T,N,P,~') where N = {O,l ..... n}, ~' = 0[~]
we construct SOL-system and
P = {i.a÷w : a÷w ~ P . } U l
U{i+j
: ig
N, j ~
Clearly,
N - {0}} U {O-a÷a : a a T}.
L(G') = L(G)
and therefore T0L ~ SOL.
The following lemma shows that in certain restricted manner every recursively enumerable Lemma 5:
exists
set can be represented by an SOL-system. Let L be a recursively
era. SOL language
enlmlerable
L' c (Z u { $ } ) ~ }
set over Z and let @,@ be not in Z. There
such t h a t
L { # } = L' f] 2m{@}.
227
Proof.
It
follows
from results
in
~fLG,H~P]; --
can be generated by a phrase-structure
see for
grammar
example
Lemma 1
G = (N,T,P,S)
in "~LpjCthat L
with productions
only
of the form A÷B, A÷BC, AB÷AC, A+a or A÷s for A,B,C ~ N and a e T, i,e. there are only context-free system
G' = (N',T',P',o)
Q not in (i)
or "left-context-sensitive"
NuTu{$,#},
where
productions
in P. We can then construct SOL
T' = T U {$,~}, N' = N u N 2 ~ { A
0 = S[$ #]
: A e N} u {Q}
for
and P' is defined as follows.
For all A,B,C in N and a in
T u{$}
the following productions are in P'.
A.B÷B
(A,B) .a÷$
(A,s)-C+(B,C)
~
(A,~). (C,D)÷(B,C)
Q.a÷X
A.(mC)÷B
A. # ÷ #
A.~+B
~. # ÷ $@
(A,B).~+(B,C)
(A,B). #+~ [#]
A" a-~$ (ii)
If A+a, B+b are in P and d is in T u {$}
then the following productions
are
in P'. A-d+a
(A,~)-~÷a QIhl L#+~# . (iii)
If A÷a is in P then D.A÷Q is in P' for every D in N. If A÷B is in P then D.A÷B is in P' for every D in N. If A÷BC is in P then D.A÷(B,C)
is in P' for every D in N.
If AB+AC is in P then A.B+C is in P'. Let h be the homomorphism
from
(NuN 2U{A
: A6N}) ~
into N ~ defined by
h(A) = h(A) = A for every A in N and h(A,B) = AB for all A,B in N. It can be verified (by induction on the length of a derivation)
that if
o ~
~
then ~ must be of the
G' IGI: A.V. Gladkij, Formal grammars
and languages
(in Russian) Mir, Moscow,
IHI: L.H. Haines, A representation for context sensitive languages, the Amer. Math. Society, to appear. IP]: M. Penttonen,
LCS = CS,
to appeaP
in InfoPmation
and Control.
1973
Transaction
of
228
form
X1[tiX2Ft2X3[...Xk_1[tk_iXkFtk~]7...7]]_ ~ . . ~ ~ ~-~
i = 1,...,k, X.l is in N u N 2 for i = I,...,k-I
h(X1X2...X k)
is in L(G). Moreover, i f
in L(G). Thus
t.l is in
where
Tu{$}
and X k is in N u N 2 u { A
for : A~N}
t l t 2 . . . t k is in T* then also
and
t l t 2 . . . t k is
L(G')g_L(G)nT*{#}.
To show the reverse inclusion we observe that every string x in L(G) can he S ------->~w -------~> x
generated by a derivation productions
of the form
A÷a
use only such productions (i)
If
in
such that
in the derivation w -------> x.
and on the other hand we
Further we can see that
tt,...~tn~
S l , . . . , s m in Tu{~)}
%[tl%[t2 A~b"%-~[tn-~ % [ t n ~ ] ' ' ' ] ] ] • "- ~-1 [Sm-l~[Sm~]'"]]l" If
S -----> w
and we do not use the
AIA2...A n ------->BIB2...B m for A. in N for I < i < n and B. in N for I < j < m G m U
then there exist
(ii)
w ~ N~
A.÷a. £ P for I < i < k 1
such t h a t
G, Bl[sl%[s2 h L . .
then
1
%[tl%[t2"' "~-1 [tk-l~[tk~l-]""'~] 7, %[a~%[a2"""%-~Ea~_~%[%~]]...]1. Therefore
every derivation
in G can be simulated by a derivation
in G' and
L(G)~ T*{#} ~ L(G'). Theorem 5. Family SOL is incomparable
with the family of context sensitive languages
(CSL). Proof. Every SOL languages is clearly exponencially
dense in the terminilogy
of [CO]~
i.e. for every SOL language L there exist constants p,q such that for every string u in L of length n, n > p there is string v in L of length m so that are context-sensitive
languages not satisfying this property,
~ < m < n.
There
e.g. the language
{a22n: n ~ O} therefore CSL ~ SOL. CSL is closed under intersection with a regular set therefore sensitive language L' the language
L'N Z~{~)
for every context-
is again context sensitive.
Thus the
assumption SOL ~ CSL is in contradiction with Lemma 5. Corollary:
Family SOL is not closed under intersection with a regular set.
Proof. By Lemma 5 and Theorem 5Theorem 6.
ETOL ~ SOL u [{~}}.
% ICOl K. Culik II and J. Opatrny,
Context in parallel rewriting,
in this volume.
229
Proof. In ~ 7 ]
it is shown that ETOL -{{E}}is equal to the closure of TOL under
length-preserving
homomorphisms
(codings). Therefore it follows by Lemma 4 an8 Theo-
rem 3 that ETOL q SOL ~ {{s}~ In [7] it is shown that ETOL is included in the family of indexed languages
[A] T. Thus our inclusion is proper by Theorem 5.
From Lemma 4 and results in [CO] ~ it follows that SOL is included neither in the family of languages generated by L-systems with interaction of predictive context languages
~0]*.
[88] nor in the family
We conjecture that both these families are
incomparable with SOL.
f IAI: A.V. Aho, Indexed grammars - an extention of context-free grammars, JACM 153 (1968), 647 - 671. * IC01
K. Cullk ii and J. 0 P a t r n ~
Context in parallel rewriting,
in this volume.
IN PARALLEL REWRITING* C O N T E X T K. ~ulik I I and J, O~atrn~ Department of AppITe-d Analysis'and Computer Science University of Waterloo Waterloo, Ontario, Canada
Abstract Three new types of context sensitive parallel rewriting systems, called global context L-systems, rule context L-systems and predictive context L-systems are introduced in this paper.
We investigate the generative power of these new
types of context sensitive parallel rewriting systems and we compare i t to the generative power of TOL-systems ~I], L-systems with interaction [92], regular grammars and context sensitive grammars. I.
Introduction Parallel rewriting systems were introduced in [59, 60] as a mathematical
model for biological developmental systems. Most of the papers related to parallel rewriting have dealt with rewriting systems of coDtext free type, e.g. OL-systems [6~, TOL-systems [8~, and their generalisations [9], [B~]. A generalisation of context sensitive grammars with parallel rewriting known as L-systems with interactions has been studied in [ 9~.
L-systems with
interactions have the same basic rules (productions) for rewriting as OL-systems, but with restriction on their use given by right and l e f t "context".
A rule may be
applied only in the given context. However, in the case of parallel rewriting i t is quite natural to consider different forms of "context".
Since we are replacing all symbols at once, we
may r e s t r i c t the use of a rule, a ÷ m say, by the context adjacent to m after simultaneously replacing all the symbols in a string rather than by the context adjacent to a before the rule was applied.
We w i l l call this kind of context,
predictive context. Even more generally, the restriction on the use of a rule may concern rules used on adjacent symbols. We w i l l call this type of restriction rule context. Clearly, a l l these generalisations make sense only in the case of parallel rewriting. We can also consider restrictions on the use of rules, which in distinction to the above are of a global rather than a local character.
In a global
* The research was supported by the National Research Council of Canada, Grant No. A7403.
231
context L-system, in addition to the set of labeled rules, a control set over their labels is given. We can only rewrite a string with a sequence of rules with labels from the control set. The new types of context sensitive L-systems introduced in this paper also have a natural biological motivation.
The development of a cell might be
completely independent of the other cells, i . e . in OL-systems, or i t might depend on the configuration around the cell before the development takes place i . e . in L-systems with interactions, or i t might be restricted in such a way that only compatible cells can occur adjacently, i . e . in predictive context L-systems, or only compatible developments can occur adjacently, i . e . in rule-context L-systems, or even the development of an organism as a whole is r e s t r i c t e d by c e r t a i n p a t t e r n s , e.g. the development can be d i f f e r e n t i.e.
in c e r t a i n parts of the organisms,
in global context L-systems. In t h i s paper we i n v e s t i g a t e the generative power of these new types of
L-systems.
Among other r e s u l t s i t
is shown t h a t global context L-systems w i t h
r e g u l a r c o n t r o l sets ( r e g u l a r global context L-systems) are e q u i v a l e n t to r u l e context L-systems.
We also show t h a t the f a m i l y o f r e g u l a r global context
L-languages p r o p e r l y contains the f a m i l y of languages generated by L-systems w i t h i n t e r a c t i o n s and the f a m i l y of TOL-languages. 2.
Preliminaries
We s h a l l assume ~hat the reader is f a m i l i a r w i t h the basic formal languages t h e o r y , e.g. [ I 0 2 ] . Now, we w i l l
review the d e f i n i t i o n s
L-systems w i t h i n t e r a c t i o n s
[ 9 2 ] , and we w i l l
of OL and TOL-systems [9~, [8~, and introduce some n o t a t i o n used
throughout the paper. Definition (i) (ii)
I.
A t a b l e OL-system (TOL-system) is a 3 - t u p l e G = ( Z , P , { ) , where:
Z is a f i n i t e , P is a f i n i t e
nonempty set, c a l l e d the alphabet. set of t a b l e s , P = {PI,P2 . . . . . Pn} f o r some n m I , where
each P i ' i : 1,2 . . . . . n is a f i n i t e
subset of Z × Z*.
Element
(a,~) of P i ' 1 ~ i ~ n, is c a l l e d a r u l e and is u s u a l l y w r i t t e n in the form a ÷ m.
P must s a t i s f y the f o l l o w i n g c o n d i t i o n of completeness.
For each a E Z and i , 1 ~ i ~ n, there e x i s t s m ~ Z* so t h a t (a,m) ~ Pi" (iii)
~ c Z+, the i n i t i a l , s t r i n g of G. Given a TOL-system G = (S,P,~), we w r i t e m~> 6, where m ~ Z+, 6 E z * ,
i f there e x i s t k m I , a l , a 2 . . . . . a k c Z, and 61 ,62 . . . . . 6 k c Z* so t h a t = a l a 2 . . . a k, 6 = 6162...6 k and f o r some t a b l e Pi ~ P' aj ÷ #j ~ Pi f o r 1 ~ j ~ k. The t r a n s i t i v e denoted by - > * , G
and r e f l e x i v e closure of the binary r e l a t i o n ~> is
232 The language generated by a TOL-system G is denoted by L(G) and is defined to be the set {~ ~ % * : ~ > * ~}. Definition 2.
A TOL-system G = (%,P,o) is called an OL-system i f P consists of
exactly one table of rules, i . e . P = {Pl }. Notation. Throughout the paper i f r is any binary relation, then r* denotes the reflexive and t r a n s i t i v e closure of r, without repeating i t specifically in every case. Notation. by l~I.
The empty string is denoted by E.
The length of a string c is denoted
For any string ~ and k ~ l , we define Firstk(~) and Lastk(~) as follows.
Firstk(m)
= i_f_f Iml m k then f i r s t
k symbols of
el se m. Lastk(m)
= i_[_f Iml m k then l a s t k symbols of else m.
For any string ~, we define Firsto(~) = ~, First (~) =
Lasto(m) Definition 3. (i) (ii)
: E, Last (m)
{Firstk(m)}, k=l I~I = k=l {Lastk(m) }.
A context L-system is a 3-tuple G = (Z,P,o), where
z is a f i n i t e , nonempty set of symbols, called the alphabet. P is a f i n i t e subset of {#,c}.Z* x S × %*.{#,E} × Z*, called the set of rules, where # is a symbol not in Z called the endmarker. A rule (~,a,8,y) c P is usually written as <~,a,8> ÷ ¥.
(iii)
~ ~ Z+, the i n i t i a l string. Given a context L-system G = (S,P,~) we write ~ >
8 for ~ E S+,
6 E £*, i f there exist k z O, al,a 2 . . . . . ak E Z and 81,B2 . . . . . 8k ~ z* so that = ala2...a k, B = BIB2...~ k and for every i , l ~ i ~ k, there exist m,n z 0 such that (Lastm(#ala2...ai_l),ai, Firstn(ai+lai+2...ak#),Si ) ~ p. Context L-system G must be strongly complete, i . e . for any ~ c Z+ there exists ~ c Z* such that ~ >
8.
The language generated by a context L-system G is denoted by L(G) and is defined to be the set {~ E Z * : ~ > * ~}. Note.
The d e f i n i t i o n of a context L-system given above is a simplification
and an unessential generalisation of the d e f i n i t i o n of an L-system with i n t e r action from [92]. generative power.
I t is obvious that both types of systems have the same
233 Notation.
We say t h a t a language L is a ~-language (where ~ may be OL, TOL,
context L, e t c . ) i f there e x i s t s a h-system G such that L = L(G). The family of context L-languages w i l l be denoted by ~. I f f is a mapping from Z to subsets of A*, then f can be extended to strings and languages over Z as follows. (i) (ii)
f ( s ) = {s}. for a ~ ~, ~ ~ S*, f(~a) = f ( ~ ) - f ( a ) ,
where " . " is the operation of set
concatenation. (iii)
f o r L E ~*, f ( L ) = {~:~ c f(B) f o r ~ ~ L}. We w i l l use these extended mappings l a t e r on without repeating the
process of extension in every single case. 3.
Context s e n s i t i v e p a r a l l e l r e w r i t i n g systems Now, we w i l l define three d i f f e r e n t types of context sensitive p a r a l l e l
r e w r i t i n g systems.
A l l of them are using only one type of symbols, i . e . we are
not considering any nonterminals. F i r s t we w i l l give the d e f i n i t i o n of global context L-systems. context L-system has, s i m i l a r l y as an OL-system, a f i n i t e however, each r u l e has a f i n i t e
number of l a b e l s .
A global
set of context free r u l e s ,
The use of rules in a global
context L-system is r e s t r i c t e d by a language over labels, called the control set. D e f i n i t i o n 4.
A global context L-system is a 5-tuple G = (S,F,P,C,a), where:
(i)
~ is a f i n i t e ,
nonempty set of symbols, called the alphabet.
(ii)
F is a f i n i t e ,
nonempty set of symbols, called the labels.
P is a f i n i t e ,
nonempty subset of p(F) x Z x ~*, where p(F) denotes the
(iii)
family of nonempty subsets of ?.
Element (B,a,~) c P is called a rule
and is usually w r i t t e n in the form B:a ÷ ~. (iv) (v)
C E ?*, called the control set. ~+ a c , the i n i t i a l s t r i n g . Given a global context L-system G = (Z,F,P,C,a), we w r i t e ~ >
~ for
~ Z+, ~ c Z*, i f there e x i s t k ~ I , a l , a 2 . . . . . ak c Z, BI,~ 2 . . . . . ~k E Z* and BI,B 2 . . . . . Bk E p(F) so that ~ = a l a 2 . . . a k , ~ : B182...~k, ( B j , a j , ~ j ) j = 1,2 . . . . . k and BIB2...B k n C ~ @ I
E P, f o r
The language generated by a global context L-system G is denoted by L(G) and is defined to be the set {~ ~ ~*:~ =>* ~}.
i BIB2...B k is the concatenation of sets BI,B 2 . . . . . Bk.
234 A global context L-system G is said to be a X global context L-system i f i t s control set is of the type X.
In t h i s paper only regular global context
L-systems w i l l be studied and t h e i r control sets w i l l be denoted by regular expressions. The family of regular global L-languages w i l l be denoted by ~. Example 1 Let G1 be a regular global context L-system, G1 : { { a } , { S I , S 2 } , P , C , a } , where P = { { S l } : a ÷ aal{s2}:a ÷ aaa} and C is denoted by regular expression s I s2 • C l e a r l y , at any step in a d e r i v a t i o n , we can apply e i t h e r the production a ÷ aa to a l l symbols in a s t r i n g , or the production a ÷ aaa is used throughout the s t r i n g .
Therefore L(G I ) = { a 2 i 3 J : i ~ O, j ~ 0}.
Since we may consider an L-system as a model of the development of a filamentous organism, i t is natural to require
that f o r any stage of the
development there exists a next stage of the development.
Therefore, a condition
of "completeness" is usually included in d e f i n i t i o n s of a l l versions of L-systems. Now, we w i l l give the formal d e f i n i t i o n s of the completeness and strong completeness f o r regular global context L-systems. D e f i n i t i o n 5.
Let G be a regular global L-system with an alphabet ~.
complete i f for any ~ E L(G), ~ # c, there exists B E D e f i n i t i o n 6.
so that ~ >
G is B.
Let G be a regular global L-system with an alphabet Z.
strongly complete i f f o r any ~ ~
there e x i s t s ~ c
so that ~ >
G is B.
Note that in [92] only strongly complete systems were considered (and called complete).
However, t h i s is u n n e c e s s a r i l y r e s t r i c t i v e , there is no
biological motivation to require that a next stage of the development is defined also f o r configurations of c e l l s which can never occur in the development.
More-
over, i t follows from the next lemma that every complete regular global context L-system can be modified to an equivalent strongly complete regular global context L-system. Lemma I .
For any regular global context L-system G, there e f f e c t i v e l y e x i s t s an
equivalent regular global context L-system G' which is strongly complete. Proof. finite
Let G = (~,?,P,C,~) be a regular global context L-system.
Let f be a
s u b s t i t u t i o n on r * defined by a ~ f ( k ) i f ! a n d only i f there exists a rule
(B,a,~) ~ P so that k ~ B.
Let R = f ( C ) , l e t R1 = S*-R.
Since regular languages
are closed under f i n i t e s u b s t i t u t i o n and complement, R and R1 are regular languages. I f ~ E R, then there exists ~ c complete.
such that ~ >
~.
I f R1 = @ then G is strongly
235 Suppose t h a t R1 ~ @.
Let s be a new symbol not in F.
homomorphism defined by h(a) = s f o r any a i n Z.
Let h be a
Let G' = ( Z , F ' , P ' , C ' , ~ ) ,
where F' = F u { s } , C' = C u h ( R l ) , and P' = P u { ( { s } , a , a ) : a
c ~}.
From the
c o n s t r u c t i o n of G' f o l l o w s t h a t G' is s t r o n g l y complete and i f m E R, and m~> f o r some B E %*, then m G=~ B, and i f m c R1 then m G=~ m. Lemma 2.
Therefore L(G') = L(G). D
I t is undecidable whether a r e g u l a r global context L-system is
complete. Proof.
We w i l l
show t h a t f o r any instance of Post's Correspondence Problem Do~
there e x i s t s a r e g u l a r global context L-system which is complete i f and only i f the instance of Post's Correspondence Problem (PCP) does not have a s o l u t i o n . Let Z = { a l , a 2 . . . . . a n } be a f i n i t e lists
alphabet, and l e t A and B be two
of s t r i n g s in Z+ with the same number of s t r i n g s in each l i s t .
A = ml,m2 . . . . . mk and B = #I,S2 . . . . . Bk.
Let G = (Z',F,P,C,$)
L-system, where Z' = Z u { $ , ~ } , F = {Sl,S2,S3,S 4} u { r i : i P = { ( { S l } , $ , ~ i $ S~):i = 1,2 . . . . . n} u { ( { r i } , a i , s ) : i u {({s3},ai,ai):i
= 1,2 . . . . . n} u { ( { s 2 } , $ , ~ ) }
Say
be a regular global
= 1,2 . . . . . n},
= 1,2 . . . . . n} u
u { ({s4},~,~)},
where Sri denotes
the reverse of ~ i ' and C is denoted by S3SlS 3 . * + s3s2s + +3 + s~s 4 + s4s ~ + s~s4s ~ + + s 3 r l s 4 r l s 3 + s3r2s4r2s3 + . . . + S3rnS4rnS 3. C l e a r l y , $ ~>* mi
B~~j sTl j _ l . .. B~i I
..... [ BT "' ~ ..... i~12 ~ l j $ B l j l j _ 1 • l I G> ~ l l ~ l 2
for j ~ I, il,i 2 ..... ij
mi ~ j
being integers smaller or equal to k.
I f ma ~ aS ~ L(G), where m,~ ~ %*, a ~ %, then ma ~ abe-> m ~ S.
I f ma ~ bS ~ L(G),
where m,B E %*, a,b ~ % and a ~ b then ma ~ bB~> ma ~ bS is the only possible d e r i v a t i o n in G "From ma # b~. PCP has a s o l u t i o n .
PCP does not haw~ a s o l u t i o n . Now we w i l l
Thus i t is not decidable whether G is complete.
give the d e f i n i t i o n
context L-system has a f i n i t e number of l a b e l s .
Therefore $ ~>* ~ i f and o n l y i f the instance of
Since s 4 ~ C, G is complete i f and only i f the instance of
of a r u l e context L-system.
set of context free r u l e s , each r u l e having a f i n i t e
For each r u l e p there are r e s t r i c t i o n s
on what r u l e s might be
used on the symbols adjacent to the symbol on which p is used. are s p e c i f i e d by a f i n i t e D e f i n i t i o n 7.
(i) (ii) (iii)
A rule
These r e s t r i c t i o n s
number of t r i p l e s .
A r u l e context L-system is a 5-tuple G = (S,F,P,C,a), where:
Z is a f i n i t e ,
nonempty set of symbols, c a l l e d the alphabet.
F is a f i n i t e ,
nonempty set of symbols, c a l l e d the l a b e l s .
P is a f i n i t e
subset of p(F) × E × Z*, c a l l e d the set of r u l e s .
(B,a,~) i n P is u s u a l l y w r i t t e n in the form B:a ÷ ~.
Rule
236 (iv)
C is a f i n i t e
subset of { # , c } F * × F × r * { # , s } , c a l l e d the context set,
where # is a special symbol not in F, c a l l e d the endmarker. (v)
a ~ ~+, the i n i t i a l
s t r i n 9.
Given a r u l e context L-system G = (Z,F,P,C,a), we w r i t e ~ > ~ z +, B ~ S* i f there e x i s t k ~ I , a l , a 2 . . . . . a k ~ Z Sl,S 2 . . . . . s k ~ F so t h a t ~ = a l a 2 . . . a k ,
B for
BI , B 2 . . . . . Bk ~ Z* and
B = BIB2...B k and f o r every i , 1 ~ i ~ k,
there e x i s t ( B i , a i , B i ) ~ P and m,n ~ 0 so t h a t s i ~ Bi and ( L a s t m ( # S l S 2 . . . S i _ l ) S i , Firstn(Si+isi+2...Sk#))
c C.
The language generated by a r u l e context L-system G is denoted by L(G) and is defined to be the set {~ ~ Z ~ =>* ~}. G The f a m i l y of r u l e context L-languages w i l l Example 2.
be denoted by ~.
Let G2 be a r u l e context L-system, G2 = ( { a } , { S l , S 2 , S 3 , S 4 } , P , C , a } ,
where P = { { S l } : a ~
a3,{s2}:a ÷ a , { s 3 } : a ÷ a 4 , { s 4 } : a ÷ a 2} and
C = { ( # , s I , # ) , ( # , s 4 , # ) , ( # , s I ,s 2 ) , ( s I , s 2 , # ) , ( s 2 , s I ,s 2 ) , ( s I ,s2,s l ) , ( # , s 4 , s 3 ) , (s3,sl,#),(Sl,S4,S3),(s4,s3,sl),(s3,sl,s4)}. Let ~ be a s t r i n g in a*. I f the length of s t r i n g ~ is d i v i s i b l e by 3, then according to control set C we can apply on ~ only rules w i t h labels Sl,S3,S 4 and the only s t r i n g we can derive in G from ~ is the s t r i n g ~ .
I f the length o f ~ is even then we can derive in G
from ~ only the s t r i n g ~ . From the i n i t i a l s t r i n g of G2 we can derive s t r i n g s aa and aaa. Therefore L(G 2) = {a2n:n ~ O} u {a3n:n ~ 0}. Now, we w i l l
show t h a t the f a m i l y of rule context L-languages is
equal to the f a m i l y of r e g u l a r global context L-systems. Theorem I . Proof.
~ = ~. Let G1 = ( s , r , P , C , a )
integers such t h a t i f
be a r u l e context L-system.
Let k,m be p o s i t i v e
(~,a,B) c C, then I~i < k and IBI < m.
L = F i r s t ( # F k - l ) u Fk, R = Last(Fm-l#) u Fm.
Let
Let A be a f i n i t e
A : ( K , F , 6 , q o , F ) , where K = (L x F x R) u {qo }, F K
automaton,
n ((F u { # } ) * × F × { # } ) ,
and ~ is defined as f o l l o w s . (i) (ii)
I f (#,p,B#) c C, where B c r * then (#,p,B#) c ~(qO,p)I f (#,P,B) c C where B ~ F*, then ( # , p , B y l # ) ~ ~(qO,p) and (#,p,~y2) ~ ~(qO,p), f o r every y i , Y 2 ~ ~* such t h a t By1#, By2 ~ R.
(iii)
I f (ms,p,~) ~ C, where m ~ F* u {#}F*, B ~ F* ,s,p ~ F then (Lastk(YlmS),p,By2q)
~ ~((ylm,S,PBy2),p), and
(Lastk(YlmS),p,BY3#) ~ ~((ylm,s,pBY3#),p) f o r any q ~ F u {#},y2,Y3 ~ ~*, Yl ~ #F* u F+ such t h a t B¥2q,~¥3# ~ R and ylm~ L.
237
(iv)
I f (c,p,B) E C, where p E r , B E ?*, then (Lastk(YlS),p,BY2 q) ~ 6((Yl,s,PBy2), p) and (Lastk(YlS),p,By3#) ~ 6((Yl,s,PB¥3#), p) for any s c ?, q c r u {#}, Y1 c L, Y2' Y3 c ?*such that BY2q,By3# E R.
(v)
I f (ms,p,B#) ~ C, where m c ?* u {#}?*, B c ?*, s,p E ? then (Lastk(YlmS),p,B#) c 6((ylm,s,pB#), p) for any Yl ~ #?* u ?+ such that yl m E L.
(vi)
I f (E,p,B#) c C, where B c r * , p ~ ?, then (Lastk(YlS),p,B#) ~ 6((Yl,S,pB#), p) for any Yl ~ L. L(A) is a regular language and, c l e a r l y , m is in L(A) i f and only i f
is a string of labels of rules which can be simultaneously applied to a string in S* according to context set C.
Therefore, the regular global context
L-system G2 = (Z,r,P,L(A),~) w i l l also generate language L(Gl ) and thus ~ £ ~. Now, we w i l l show the other inclusion. regular global context L-system.
Let G = (~,?,P,Q,~) be a
Let A = (K,?,6,qo,F) be a f i n i t e automaton such
that 6(q,e) = @ for any q c K and L(A) = Q.
Let G3 be a rule context L-system,
G3 = (Z,?3,P3,C3,~), where ?3 = ? × K, P3 = {(A × K,a,~):(A,a,~) c P}, and C3 is defined as follows. (i) (ii)
I f 6(qo,a) ~ @, where a E ?, then (#,(a,qo),E) E C3. I f r ~ 6(q,a) and 6(r,b) ~ @where a,b ~ ? and q,r E K, then ( ( a , q ) , ( b , r ) , ~ ) E C3.
(iii)
I f r ~ 6(q,a) and r c F, where q E K, a E ?, then (E,(q,a),#) ~ C3.
I t can be easily v e r i f i e d that ~ - > B i f and only i f ~ > G3
B.
Therefore
L(G3) = L(G).
Let the completeness and strong completeness is defined for rule context L-systems in the same way as for regular global context L-systems.
Since
rule context L-systems are e f f e c t i v e l y equivalent to regular global context Lsystems, Lemma 1 and Lemma 2 also hold when replacing in them a regular global context L-system by a rule context L-system. Since any t r i p l e in the context set in a rule context L-system i m p l i c i t l y includes also a r e s t r i c t i o n on the adjacent symbols, i t is quite obvious that the family of rule context L-languages includes context L-languages. We w i l l show in the next theorem that t h i s i n c l u s i o n is proper.
238
Theorem 2. Proof.
~ ~ @. Let G = (Z,P,~) be a context L-system.
We construct a r u l e context
L-system G' = ( S , Z , P ' , C , ~ ) , where P' = { ( { a } , a , B ) : ( ~ l , a , ~ 2 , B ) ~I c {#,E}%*,~ 2 E ~ * { # , ~ } }
, and C = { ( ~ , a , B ) : ( ~ , a , B , y )
~ P, a c Z, B c ~*,
E P f o r some y c Z*}.
We have constructed the r u l e context L-system so t h a t a l l r u l e s f o r a symbol a in % have the same label a, and the context set of G' allows to o b t a i n in G' e x a c t l y the same d e r i v a t i o n s as in G.
Therefore L(G) : L ( G ' ) .
Thus we have shown t h a t
~ @ and i t remains to show t h a t the i n c l u s i o n is proper.
In Example 2 the
language L = {a2n:n ~ O} u {a3n:n ~ O} is generated by a r u l e contex L-system. I t has been shown in [ 9 2 ] , t h a t L is not in ~. Now, we w i l l
Q
give the d e f i n i t i o n o f a p r e d i c t i v e context L-system.
In a p r e d i c t i v e context L-system the use of a r u l e is r e s t r i c t e d by the context of the r i g h t hand side of the r u l e a f t e r the simultaneous replacement of a l l the symbols in a s t r i n g . D e f i n i t i o n 8. (i) (ii)
A p r e d i c t i v e context L-system G is a 3 - t u p l e (Z,P,~), where
~ is a f i n i t e , P is a f i n i t e
nonempty set of symbols, c a l l e d the alphabet. subset of ~ × {#,~}Z* x z* x S* { # , ~ } , c a l l e d the set of
r u l e s , where # is a special symbol not in Z, c a l l e d the endmarker. A r u l e (a,Bl,m,B 2) in P is u s u a l l y w r i t t e n in the form a +. (We assume t h a t "<" and ">" are symbols not in Z.) (iii)
~ E ~+, the i n i t i a l
string.
Given a p r e d i c t i v e context L-system G = (%,P,~), we w r i t e m -> B f o r ~ %+, B c ~* i f there e x i s t k ~ I , a l , a 2 . . . .. ak c ~ and BI,B2 . . . .. Bk ~ ~* so that m = a l a 2 . . . a k, B = BIB2...B k and f o r every i , l ~ i ~ k there exist m,n m 0 such that (ai,Lastm(#BiB2...Bi_l),Bi,Firstn(Bi+iBi+2...Bk#)) ~ P. The language generated by a predictive context L-system G is denoted by L(G) and is defined to be the set {m E Z * : ~ > * m}. The family of predictive context L-languages is denoted by ~. Example 3. Let G be the predictive context L-system ({a,b,c},P,abc), where P = {a ÷ <~,a,bc>, b ÷ , c ÷ , c ÷, a ÷ <~,aa,bb>, b ÷ , c ÷ , a ÷ <~,a,a>, b ÷ , c ÷ . Using the f i r s t s t r i n g (abc) m, m m I .
f o u r rules in P we can generate from the s t r i n g abc the
I f we decide to use a r u l e which would double a symbol,
then, c l e a r l y , we have to double each symbol throughout the whole s t r i n g (abc) m.
239
Therefore, (abc) m ~> (a2b2c2) m and from any s t r i n g of the form ( a i b i c i ) m, where i > I , m m 1 only the s t r i n g ( a i + I b i + I c i + l ) m can be generated. L(G) = {Caibici)m: i m I , m m I } .
Thus
We can define the completeness and strong completeness f o r p r e d i c t i v e context L-systems in the same way as f o r regular global context L-systems. We can prove that i t is undecidable whether a p r e d i c t i v e context L-system is complete.
However, in t h i s case we cannot show that f o r every p r e d i c t i v e context
L-system i t is possible to construct an equivalent strongly complete p r e d i c t i v e context L-system. We can only show t h a t every complete p r e d i c t i v e context Lsystem can be made strongly complete. {a312j Lemma 3. The language L = :i ~ O, j ~ O} is not a predictive context L-language. Proof: Since the proof of this lemma is very tedious, we w i l l present i t only informally. Suppose that there exists a predictive context L-system G = (Z,P,~) such that L(G) = {a312J:i ~ O, j ~ 0}.
Then there exists exactly one integer j ,
j ~ 0 such that a ÷~ P for some integers i , h . j ~ I.
To be able to generate a l l string a31 for i ~ l , j has to be a power of
three, i . . e . j = 3p, p ~ I.
a 21 f o r i ~ I . Theorem 3. Proof:
Since L is i n f i n i t e ,
But then we cannot generate in G a l l strings
D
~ ~ ~. Let G : (Z,P,~) be a p r e d i c t i v e context L-system.
Let k, m be natural
numbers such that I~I < k, IYl < m f o r any (a,~,B,y) ~ P. Let A be a f i n i t e automaton, A = (K,P,6,qo,F), where K : (First(#% k - l ) u Zk) x (Last(zm-l#) u Zm) u u {qo }, F = K n (% u {#}7* × (#), and 6 is defined as f o l l o w s . (i)
I f p = (a,#,B,y#) E P where a E ~, and
B,y ~ S*, then
(Lastk(#B),y#) E 6(qO,p). (ii)
I f p = ( a , # , ~ , y ) c P, where a c z, and B,y c ~*, then (Lastk(#B),y61#) E 6(qO,p) f o r any 61 ~ Z* such that 1811 ~ m~Iyl-I
and (Last(#~),y~2) c 6(qO,p) f o r any 62
such t h a t 1621 = m-IYl. (iii)
I f p = (a:,~,B,y) c P, where ~ ~ S* u #~+ and
B,y ~ E*, then
(Lastk(Yl~B),yy26) ~ 6 ( ( y l ~ , B y y 2 ) , p) f o r any Y1 ~ S* u #S*'Y2 ~ ~*' c Z* u Z*# such that ( y l ~, By2 ) ~ K and 161 = I ~ I , and also ( L a s t k ( Y l ~ ) , Y Y 2 # ) E 6((yl~,~yy2#),p) f o r any Yl E z* u #S*, and Y2 E %* such that (yl~,~yy2 #) E K.
240
(iv)
If p
=
(a,m,~,y#) E P, where m c Z* u #Z+ and B,y E ~* , then
(Lastk(YlmB),y#) E ~((ylm,By#),p) for any Yl E ~ u #Z* such that (y]~,By#) E K. I t follows from the construction of automaton A that i f plP2...pn ~ L(A), where Pi ~ P' Pi = ( a i ' ~ i ' S i ' Y i ) for l ~ i ~ n, then ala2...a n ~ vice versa.
8iB2...Sn and
Therefore, the regular global context L-system G' = (Z,P,P',L(A),a),
where P' = {({a,B,~,y},a,~):(a,B,~,y) E P}, generates also the language L(G). Thus ~ 5 ~. In Example l we have shown that the language L = {a3i2J:i z O, j z O} is a regular global context L-language. context L-language.
However, by Lemma 3, L is not a predictive
Thus, the inclusion is proper.
0
I t has been shown in [92] that the family of regular languages is included in the family of context L-systems.
I t is easy to modify this proof to
show that all regular languages containing a nonempty string are also included in the family of predictive context L-languages. Let the family of regular languages be denoted by REGULAR. Theorem 4. Proof.
REGULAR-{{~},$} ~ ~. Similar to the proof that REGULAR-{{~},@} is included in ~ in [92]. Now, we w i l l compare the generative power of TOL-systems with that of
context sensitive L-systems. The family of TOL-languages w i l l be denoted by TOL. Theorem 5. Proof.
TOL ~ R. TOL does not include all f i n i t e sets as shown in ~I].
Therefore, i t
follows from Theorem 4 that ~ ¢ TOL. We have shown in Lemma 3 that the language L = {a312J:i z O,j z O} is not a predictive context L-language. generated by TOL-system G = ({a},{{ a ÷ aa},{a ÷ aaa},a). TOL ¢ ~.
Therefore,
0
Theorem 6. Proof.
However, L is
TOL ~ ~. Let G = (Z,P,~) be a TOL-system, where P = {PI,P2 . . . . . Pn}.
Let G' = (z,?,P',Q,~) be a regular global context L-system, where r = {Sl,S 2, .
,Sn}, . Q is denoted . by. sT + s~ +.. +sn,+ and P' is defined as follows.
P' = {(A,a,~):a c Z, ~ c S*, (a,~) c Pi for some i , l ~ i ~ n and A = {sj c r:(a,~) c Pj}}, i . e . a rule p has label sj i f and only i f p is in the table Pj.
Since the control set Q allows to use at one step in a derivation
only rules which all are from the same table of P we have L(G) = L(G'). TOL 5 ~.
Thus
241
I t f o l l o w s from Theorem 3 and Theorem 5 t h a t the i n c l u s i o n is proper.
D
Lemma 4.
The language L = {(anbncn)m:n,m ~ I } i s not a context L-language.
Proof.
We w i l l
give here only an informal proof to keep the paper short.
Suppose t h a t there e x i s t s a context L-system G = (~,P,~) generating the language L = {(anbncn)m: n,m ~ I } . Since G can generate a l l s t r i n g s anbnc n f o r n ~ O, there e x i s t s e x a c t l y one integer i , i ~ 1 such t h a t÷ a i is in P f o r some i n t e g e r k , j .
Similarly,
f o r rules i n v o l v i n g only symbol b and
only symbol c. Therefore, there e x i s t s a constant c such t h a t f o r any n ~ c, i f (anbncn)m~> * ~ then ~ = (akbkck)m f o r some integer k ~ n. Thus there e x i s t s an i n t e g e r j ~ 1 such t h a t the f o l l o w i n g holds. There e x i s t i n f i n i t e l y many n s t r i n g s in L of type (aJbJcJ) m f o r some i n t e g e r m and (aJbJcJ)m~> (aJlbJ2c ~2) I , (aJbJcJ)m~> (aJ2bJ2cJ2) m2 and mI ~ m2, Jl # J2"
Then we can generate in G also
s t r i n g s not in L, which is a c o n t r a d i c t i o n to L = L(G).
D
Since we have shown i n Example 3 t h a t the language L = {(anbncn)m:n,m ~ I } is a p r e d i c t i v e context L-language, i t
is c l e a r t h a t
context L-languages do not include a l l p r e d i c t i v e context L-languages. Theorem 7. Proof.
~ ¢ ~ . It follows directly Now~ we w i l l
from Lemma 4 and Example 3.
D
compare the generative power of context s e n s i t i v e grammars
w i t h t h a t of p r e d i c t i v e context L-systems and r e g u l a r global context L-systems. Theorem 8.
For each type 0 language L over alphabet T, there e x i s t s a p r e d i c t i v e
context L-system G such t h a t L = L(G) n T*. Proof.
Let L be generated by a type 0 grammar G1 = (N,T,P,S).
Let
G = (Z,P',S) be a p r e d i c t i v e context L-system, where = T u N u { ( p , p ) : P c P} u { ( p , A ) : p ~ P and A ~ N u T } , and P' is constructed as f o l l o w s . (i) (ii)
I f A ÷ ~ c P, where A ~ N, ~ c (N u T ) * , then A ÷ ~ ~ P'. I f p = A I A 2 . . . A n ÷ BIB2...B m ~ P, where AI,A 2 . . . . . An,BI,B 2 . . . . . Bm~NUT, m ~ n, then Ai + < ( P , A I ) B I ( P , A 2 ) B 2 . . . ( P , A i _ I ) B i _ l , ( p , A i ) B i , ( P , A i + I ) B i + I ( P , A i + 2 ) B i + 2 . . . ( P , A n _ I ) B n _ I ( P , A n ) B n B n + I . . . B m ( P , P ) > ~ P' f o r 1 ~ i ~ n - l , and An + <(P,AI)BI(P,A2)B2...(P,An_I)Bn_ I , (P,An)BnBn+l...Bm(p,p),~>
c p'
242
(iii)
I f p = AIA2...A n ÷ BIB2...B m E P, where AI,A 2 . . . . . An , BI,B2, . . . . Bm E N u T, 1 ~ m < n.
Then
Ai + < ( P , A l ) B l ( P , A 2 ) B 2 . . . ( P , A i _ l ) B i _ l , ( P , A i ) B i , ( P , A i + l ) B i + 1 (P,Ai+2)Bi+2...(P,Am)Bm(P,Am+l)(P,Am+2)...(P,An)>
E P' f o r
1 ~ i ~ m, and Ai + <(P,Al)Bl(P,A2)B2...(P,Am)Bm(P,Am+ I ) (P 'am+2)" " "(P 'Ai -I ) ' (p 'Ai ) ' (p 'Ai+l )(p 'Ai+2)" "" (p 'an)> E P' f o r m+l ~ i ~ n. (iv)
I f p = AIA2...A n ÷ ~, where AI,A 2 . . . . . An E N u T, n > I , then Ai ÷ <(P'AI )(P 'A2)" "" (P 'Ai -I ) ' (p "Ai ) ' (p ' a i + l )(p 'Ai+2)""" (p 'an)> E P' forl ~i~n.
(v) (vi)
(p,A) + ~ c P', and (p,p) ÷ c E P' f o r any p ~ P, A E N u T. A ÷ A E P' f o r any A c N u T. I t follows from the construction that i f ~AIA2...AnB ~>El~BIB2...BmB,
where AI,A 2 . . . . . An,BI,B 2 . . . . Bn ~ N u T,~,B E (N u T)* using the rule AIA2...A n ÷ BIB2...Bm,m ~ n, then ~AIA2...AnB~> ~(P,AI)BI(P,A2)B2... (P,An)BnBn+l...Bm(p,p)B~> ~BIB2...Bm~, and i f ~ > (P,An)BnBn+l...Bn(p,p), of rules of G1 are used. S ~>* GI ~.
then y = AIA2...A n.
~(P,AI)BI(P,A2)B2...
The same can be shown i f other types
Therefore S ~>* ~, where ~ E (N u T)* i f and only i f
Thus L(G I ) = L(G) n T*,
D
Let the f a m i l y of c o n t e x t - s e n s i t i v e languages be denoted by CS. Theorem 9. Proof.
~ ~ CS. Suppose t h a t ~ ~ CS.
Since context s e n s i t i v e languages are included in
recursive languages and recursive languages are closed under i n t e r s e c t i o n , L n T* is a recursive language f o r any L in ~ and any alphabet T. contradiction to Theorem 8.
This is a
Therefore, ~ ~ CS.
We have shown in Lemma 3 t h a t the language L = { a 3 i 2 J : i ~ O,j ~ O} is not in ~.
However, L is c l e a r l y a context s e n s i t i v e language.
CS ¢ ~.
D
Therefore,
Now, we would l i k e to compare the f a m i l y of context s e n s i t i v e languages to the f a m i l y of regular global context L-languages.
I t ~s
clear from the
previous theorem and from Theorem 3 t h a t the f a m i l y of regular global context L-languages is not included in the f a m i l y of context s e n s i t i v e languages.
To
243 prove that the family of regular global context L-languages does not contain a l l context-sensitive languages we introduce the concept of exponentially dense languages. D e f i n i t i o n 9.
Language L is called exponentially dense i f there e x i s t constants
c I and c 2 having the following property: For any n ~ 0 there exists a string in L such that c l e ( n - l ) c 2 ~ I~I < Cl eric2. Lemma 5.
Any regular global context L-language which is i n f i n i t e is exponentially
dense. Proof.
Let L be an i n f i n i t e ,
regular context L-language.
be a regular global context L-system generating L. d 2 = max { I y I : ( A , a , y ) Since L i s i n f i n i t e ,
c P f o r some A~F, a c Z and y c ~*}. d2 > I .
nc 2
Since L i s i n f i n i t e ,
an integer, 1 ~ j < k such that IB
Lemma 6. Proof.
c I ec2.
Let n
there e x i s t s ~ E L
As ~ ~ L and i~l > I~I there e x i s t k > l and
BI,B 2 . . . . . Bk ~ L so t h a t ~i -> ~i+l f o r l ~ i ~ k - l , nc 2
such integer j e x i s t s .
(Z,F,P,C,~)
Let c 2 = log d2.
I f n = 0 t h e D , c l e a r l y , c I ~ l~I <
be an a r b i t r a r y f i x e d i n t e g e r , n > 0. such t h a t I~I ~ c I e
Let G
Let c I = 1oi,
<
cle
~l = ~ and Bk = ~. nc 2
and IBj+II m c I e
.
Let j
Clearly,
Now we have i. Bjl ~ I ~ j + l l / d 2 ~ c I enC2/d2 = c I e (n-l)c2
The language {a22n:n m O} is not a regular global context L-language. 2n The language {a 2 :n m O} is not exponentially dense and therefore by
Lemma 6 is not a regular global context L-language. Theorem I0.
CS ~ ~. @
Proof.
By Theorems 3 and 9, ~ is not included in CS.
The language
L = {a22n:n m O} is a context sensitive language, however, L is not in ~ by Lemma 6.
be
D
NONTERMINALS
AND CODINGS
IN D E F I N I N G
VARIATIONS
OF OL-SYSTEMS
Sven Skyum Department
of C o m p u t e r
University
Science
of A a r h u s
Aarhus,
Denmark
Summary T h e u s e of n o n t e r m i n a l s terns i s s t u d i e d . low generative
capacity
high generative Finally
versus
the u s e of c o d i n g s in v a r i a t i o n s
It is s h o w n t h a t the u s e o f n o n t e r m i n a l s in d e t e r m i n i s t i c
capacity
produces
s y s t e m s w h i l e it p r o d u c e s
in n o n d e t e r m i n i s t i c
it is p r o v e d t h a t the f a m i l y of c o n t e x t - f r e e
w h i c h w a s o n e of the o p e n p r o b l e m s
c a n be f o u n d in [ 7 1 ]
a comparatively
systems. l a n g u a g e s i s c o n t a i n e d in
the f a m i l y g e n e r a t e d b y c o d [ n g s on p r o p a g a t i n g O L - s y s t e m s axioms,
of OL-sys-
a comparatively
in [ 1 0 ] .
with a finite
A l l the r e s u l t s
s e t of
in t h i s p a p e r
and ~ 7 2 ] .
1. D e f i n i t i o n s By definition~ and
an E O L - s y s t e m
/k a r e a l p h a b e t s w i t h
relation
~
o n the s e t
letters
ai
Z;*
for every
and
a i 4 czi
is a p r o d u c t i o n
transitive
closure
x~
conyield
y h o l d s i f f t h e r e i s an i n t e g e r
el ~ 1 = < i =< n, s u c h t h a t
y =C~ l . . . ~ : n 2 in P , f o r e a c h i = I , . . .
of ~ .
~
is the re-
I co=~" w } .
is an O L - s y s t e m
t h e r e is e x a c t l y o n e p r o d u c t i o n
, n. T h e r e l a t i o n
T h e l a n g u a g e L ( G ) g e n e r a t e d b y G is d e f i n e d b y
L(G) = {w E A*
The EOL-system
productions
l e t t e r of ~,~ a n d 0o E Z;+. T h e d i r e c t
is defined as follows:
and words
x = a1...an,
flexive
P , W, /k>, w h e r e
A_c Z;, P is a f i n i t e s e t o f c o n t e x t - f r e e
t a i n i n g at l e a s t o n e p r o d u c t i o n
k => 1,
is a q u a d r u p l e G - - < ~
iff
A=~.
for every
It is d e t e r m i n i s t i c
l e t t e r of
ted: P ) i f f t h e r i g h t s i d e of e v e r y p r o d u c t i o n s
(abbreviated:
]C. It is p r o p a g a t i n g
is d i s t i n c t
f r o m the e m p t y w o r d
We m a y a l s o c o m b i n e t h e s e n o t i o n s a n d s p e a k , f o r i n s t a n c e ~ o f P D O L -
D) i f f
(abbrevia~..
or EPOL-
systems. We a l s o c o n s i d e r replacing
the axiom
generalizations
co b y e f i n i t e s e t
of the systems defined above obtained by ~
of a x i o m s .
The language generated by such
a s y s t e m c o n s i s t s of t h e ( f i n i t e ) u n i o n of the l a n g u a g e s g e n e r a t e d b y the s y s t e m s o b tained by choosing each element n o t e d b y the l e t t e r F .
Thus,
co E ~
to be the a x i o m .
This generalization
we may s p e a k of E P D F O L - s y s t e m s .
is d e -
245 For" any c l a s s of s y s t e m s ~ w e u s e the s a m e n o t a t i o n f o r the f a m i l y of l a n g u a g e s g e n e r a t e d by t h e s e s y s t e m s ,
E,g,,
EPDOL
d e n o t e s the f a m i l y of l a n g u a g e s
g e n e r a t e d by i - P O O L - s y s t e m s . [By a c o d i n ~ w e mean a l e n g t h - p r e s e r v i n g literal
homomorphism).
The prefixC
cates that we are considering 2. D e t e r m i n i s t i c
(often also called a
c o d i n g s of the l a n g u a g e s in the f a m i l y .
systems
We s t a r t by e x a m i n i n g the r e l a t i o n d i n g s in d e t e r m i n i s t i c Theorem
homomorphism
a t t a c h e d to the name of a l a n g u a g e f a m i l y i n d i -
b e t w e e n the u s e o f n o n t e r m i n a l s
and co-
systems.
~.. 1
EDOL ~ CDOL
and EPDOL
~ CPDOL.
Proof We w i l l o n l y p r o v e the f i r s t
inclusion.
T h e s e c o n d o n e is p r o v e d
in the s a m e
way. N o w let G = < , ~ ,
P~ e3~ A > be an E D O L - s y s t e m .
constructionofaDOL-system
G =
A such t h a t L ( G ) = h ( L ( G t ) ) .
For
pI
The following
c~1 ~ t > a n d a c o d i n g
describes
h from
~1
the into
a w o r d x~ m i n ( x ) d e n o t e s the s e t of l e t t e r s o c c u r -
r i n g in x . Consider There
the s e q u e n c e of w o r d s f r o m
exist natural n u m b e r s
n
and
m
]C~ g e n e r a t e d by G,c~=c~ I , co2, co3, . . . . .
such that m i n ( ~ m) = m i n ( ~ m + n ) ~ w h i c h
im-
plies that for a n y i > 0 and any j~ 0___ j < n:
(1) Let
min(C~m+j ) = m i n ( ~ m + n i + j ) . dk
d e n o t e the c a r d i n a l i t y
of min(O~k) ~ 1 --< k <
m+n. D e f i n e
N A = { k e N t 1 --< k < re+n, m i n ( ~ k) c A J . For" a n y
k E N A i n t r o d u c e n e w s y m b o l s n o t in
]Ck = { a k j I 1 -----j S and define isomorphism
fk mapping
fk(a) = akj iff
a
dk}, min(0~k) o n t o ~k~ w h e r e
i s the j~th s y m b o l o f min(c0k} , k ( N A
N o t e t h a t the f k * s a r e d e f i n e d f o r s o m e f i x e d e n u m e r a t i o n s g o i n g to be the u n i o n of the a b o v e d e f i n e d
~;klS:
o f the s e t s m i n ( ~ k ) . ~
is
246
~l
=
U ~k kEN A
D e f i n e k 1 and k 2 as the m i n i m a l and maximal e l e m e n t s of N&, F o r any of the l e t t e r s
akj
where
k ~ k 2 d e f i n e p r o d u c t i o n in P~:
akj ~ fki(e) where
k ~ is the s m a l l e s t element in N A g r e a t e r than k and ~. is the s t r i n g d e -
rived from
f~l(akj)~
in ( k l - k ) steps in G.
1~ f o l l o w s f r o m (1) that L ( G ) is f i n i t e if I~ < m. If this is the case then d e f i n e f o r any j~ 1 < j <--dk2 p r o d u c t i o n in p1:
ak,aj -~ ak2 j. O t h e r w i s e ~ let k 3 denote the minimal element in N A g r e a t e r than1 o r equal to m~ and d e f i n e p r o d u c t i o n s in p i some s t r i n g e E A*
f o r any j~ 1 -< j-< dl~ ~ f o r w h i c h f~..~(al~ j ) ~
derives
in ( n - k 2 + k 3) steps in G:
ak.a J -~ fka (~) N o t e that the use of fk,~
.
is w e l l d e f i n e d s i n c e min(~k2 +(n-k,~ +k3 )) = min(~ks ) (from
the f a c t that k~ ~ m and ( I } above}, F i n a l l y d e f i n e the c o d i n g h f r o m ~1 into A in the way that f o r e v e r y
akj E C1: h(akj) = f:-l(akj)'K T h e n
L(G) = h ( L ( G ' ) ) where
GI =<~,
pi
fkl,
( ~ k l ) , FA>, and this p r o v e s the i n c l u s i o n of the t h e o r e m .
T h e i n c l u s i o n is p r o p e r because
{ a n I n-> | } E C D O L \ E D O L .
We have the f o l l o w i n g t h e o r e m as an immediate c o n s e q u e n c e of theorem 2 . 1 . Theorem 2.2 E D F O L ~ CDFOL. and E P D F O L ~ C P D F O L . 3. N o n d e t e r m i n i s t i c systems We w i l l now e x a m i n e the r e l a t i o n s between the n o n d e t e r m i n i s t i c femilies~ c o r r e s p o n d i n g to those o c c u r r i n g in T h e o r e m s 2.1 and 2 . 2 . T h e f o l l o w i n g two t h e o r e m s c o r r e s p o n d to T h e o r e m 2 . 1 . T h e p r o o f of the
247
first
o n e c a n be f o u n d in E2,0~.
Theorem
3, 1
COL = EOLo
T h e o r e m 3.2, CPOL
~ EPOL.
The
inclusion
Proof is
easily
checked.
l a n g u a g e L = { a n b n c n I n>_ 1 } U l d 3n b e l o n g to C P O L .
The inclusion
because the
but is does not
T h e p r o o f f o r t h e l a t e r s t a t e m e n t c a n be f o u n d in [1 0~ o
Notice here that while the generating capacity was weaker
is p r o p e r
i n ~ 1 } b e l o n g s to E P O L ,
than t h e g e n e r a t i n g
ting OL-systems~
the c o n v e r s e
capacity
d u e to t h e u s e o f n o n t e r m i n a l s
d u e to c o d i n g s
in d e t e r m i n i s t i c
propaga-
is t r u e if y o u a r e d e a l i n g w i t h n o n d e t e r m i n i s t i c
pro-
pagating OL-systems.
The following
Theorem
theorem corresponds
to T h e o r e m
T h e p r o o f c a n be f o u n d in ~ 7 2 ] .
3.3
CFOL
= EFOL
and CPFOL
It is an o p e n p r o b l e m w h e t h e r
A somewhat related problem
or
c_ E P F O L .
not the inclusion CPFOL
is whether or
i s i n c l u d e d in t h e f a m i l y C P F O L .
Theorem
2,2.
~ EPPOL
is proper.
n o t t h e f a m i l y of c o n t e x t - f r e e
languages
I n d e e d w e h a v e t h e f011owin9 t h e o r e m .
3.4
T h e f a m i l y of c o n t e x t - f r e e
l a n g u a g e s is p r o p e r l y
i n c l u d e d in t h e f a m i l y
CPFOL.
Proof Let G =
~, P~ S > be a c f - g r a m m a r form (i.e.~
the p r o d u c t i o n s
of a language not containing
S u p p o s e t h e r e a p e no u s e l e s s s y m b o l s in Vo For
each
A E V
we choose
w A e {w e =* I A G w, l wl minimal t. w A will~
Define
in t h e r e s t o f the p r o o f ~ be f i x e d f o r e v e r y
k : V-e N
X in
a r e of the f o r m A-+ a o r A - ~ a A l . . . A n ) .
b y k ( A ) = [WAL , and f u r t h e r m o r e
letter
A E V.
248
s(A) = I xw x E z;k(A)v *
I A=~" xwx, left
re(A} = I x E Z;k ( A ) I :J w e v * S i n c e the g r a m m a r w a s in G r e i b a c h n o r m a l f o r m ,
I xl = k(A) } and
: ×w e s(A)} s(,Zk)
and re(A) a r e f i n i t e s e t s of
strings. Let n : V-~ a
be d e f i n e d a s n ( A ) = { n u m b e r o f s t r i n g s
We w i l l u s e m ( A ) a s an o r d e r e d Now we can construct
H:<su
LJ AEV ! -< L
in m ( A ) } .
set.
a P F O L - s y , s t e m H a n d a c o d i n g h such t h a t h ( L ( G ) ) = L ( G ) :
A I , p',t
'
I
t
22
S 1S2...Sk(S}
s 2k(S) , "'''
,$1S2...
SI(S)S~(S)
....
=nlS)l k(S)~ >
P~ i s d e f i n e d as f o l l o w s :
1) 2)
For
all
a E Z;~ a 4 a
is in
pi.
For
a l l A E V~ 1 < j < n ( A ) , and 1 < i < k ( A ) - I , A ! ~ , a!
where
3)
For
a! is the i l t h t e r m i n a l in the jTth s t r i n g i a l l A E V and 1 --< j < - n(A)
• . a2k(82)- • •
k k • ..BqlqBq2q0.. is in
pi
for all
The coding
h
B 1 ~B 2 , . . .
e S ( A ) and
is d e f i n e d by
We p r o v e that L ( G ) c h ( L ( H ) ) .
Let
, Bq
×
for all
exists a derivation
of
T h e other" i n c l u s i o n
w
in G
in
re(A).
a E Z;~ a n d
w E L(G).
There
k B q k (qB q )
a n d I <-%k i <-- n(B i) w h e r e
i s the j l t h s t r i n g
h(a) = a
pi
k2
• k1 kI k! AJk(A)~ aJ(A)Bll B12 "''81k(81]
xB 1B 2...Bq
is in
in m ( A ) .
such t h a t
i s s h o w n in the s a m e w a y .
249
S = Ai ~ x~ A 2 ; A 3 . . . A n left
4~ x~×p21 ... a2q2xp31... B3q3. • . X n B n ! I
I1
I
II
=~ x~x2, xgl . . . X2;qgX 3 x 3 1 . . . w h e r e x t. E m(A i)_ l
xlf
3q 3.
• .X
X II
n nl""
.
X II
:
W
nq n
fop a l l I/ <__ i <-- n(A) and B i_j ~ x!! for" 2; ~< i < n and I --< j <-- qi • Ij
It s u f f i c e s then to show that t h e r e e x i s t s a n
s vI s 2 I" ' " S/,S)~
I
...Bnq n
I I I a x i o m S 1 S 2 ; . . . S k , S,/J
in H such that:
H
. t A k2;A kg k2; k3 k3 k3 - I " ' 2 1 "'22 " ° ' A g k ( A 2 ; ) A31 A 3 2 " • ' A 3 k ( A 3 )
•''An!
k nA k n A kn n~. •• • nk(An)
and k. k. k. A j t JAJ2;J" •" AJkJ(Aj ) H :--'->x!w.jJ f o r a l l 2--< j--< n but that is e x a c t l y how H is c o n s t r u c t e d •
~00KING
£0R A GENERAL FRAMEWORK I
ITERATION
GRAMMARS AND LINDENMAYER
AFLIS
Arto Salomaa D e p a r t m e n t of C o m p u t e r S c i e n c e University Aarhus,
O n e of t h e f i r s t
of A a r h u s Denmark
observations concerning L-systems
w a s t h a t the c o r r e s p o n d -
ing l a n g u a g e f a m i l i e s h a v e v e r y w e a k c l o s u r e p r o p e r t i e s ,
in f a c t , m a n y o f the f a m -
ilies are anti-AFL~s,
operations.
i.e.,
c l o s e d u n d e r n o n e o f the A F L
However,
t h i s p h e n o m e n o n is due to the l a c k o f a t e r m i n a l a l p h a b e t r a t h e r t h a n to p a r a l l e l i s m w h i c h is the e s s e n t i a l f e a t u r e c o n c e r n i n g L - s y s t e m s . mOL-languages
is an a n t i - A F L ,
w e w i l l see h o w L - s y s t e m s sure properties
can be u s e d to c o n v e r t
into full AFLIs
E.g.,
w h e r e a s the f a m i l y E T O L
the f a m i l y T O L of a l l is a f u l l A F L .
L a t e r on
language families with weak clo-
in a r a t h e r n a t u r a l w a y .
T h e b a s i c n o t i o n in t h i s p a p e r , K - i t e r a t i o n
g r a m m a r ~ is a s l i g h t g e n e r a l i z a -
t i o n of the n o t i o n i n t r o d u c e d b y v a n L e e u w e n [5'77 . T h e m o t i v a t i o n fop such a n o t i o n is t h r e e - f o l d : i)
It p r o v i d e s a u n i f o r m f r a m e w o r k their context-free
for discussing OL-systems
and a l l of
general i zations.
ii)
It s h o w s t h e r e l a t i o n b e t w e e n O L - s y s t e m s
iii)
It a s s o c i a t e s w i t h e a c h f a m i l y K o f l a n g u a g e s ( h a v i n g c e r t a i n m i l d c l o s u r e properties)
some full AFLIs,
o b t a i n e d f r o m K in t h e I I L i n d e n m a y e r w a y II.
We m a k e the f o l l o w i n g c o n v e n t i o n s , families discussed are non-trivial, taining a non-empty word.
and ( i t e r a t e d ) s u b s t i t u t i o n s .
i.e.,
valid throughout
this paper. All
language
t h e y c o n t a i n at l e a s t o n e l a n g u a g e c o n -
( A l a n g u a g e f a m i l y is u n d e r s t o o d as in Et 021. ) T w o g e n e -
r a t i v e d e v i c e s a r e t e r m e d e q u i v a l e n t if t h e y g e n e r a t e the same l a n g u a g e o r e l s e t h e l a n g u a g e g e n e r a t e d by one d e v i c e d i f f e r s f r o m the l a n g u a g e g e n e r a t e d by t h e o t h e r t h r o u g h the e m p t y w o r d
X. ( T h u s in t h i s s e n s e , f o r a n y c o n t e x t - f r e e
t h e r e is an e q u i v a l e n t c o n t e x t - f r e e We i n t r o d u c e f i r s t The mapping 0
(1)
K1
is a m a p p i n g 0" f r o m s o m e a l p h a b e t V i n t o K .
is e x t e n d e d to l a n g u a g e s o v e r V in the u s u a l f a s h i o n . F o r
o r K~ = T O L ,
respectively~
f a m i l i e s (1) a r e c a l l e d m a c r o - O L
.
and m a c r o - T O L
fam-
and d e n o t e d b y K z M O L and K 1 M m O L . M a c r o s w e r e i n t r o d u c e d
in [?7 and [ 9 7 , w h e r e e s p e c i a l l y the c a s e s K : = F and K z = R
language
and K ~ , w e d e f i n e
S u b ( K 1 , K ~ ) = {Cr(L) I L e K2 and0" is a K 1 - s u b s t i t u t i o n }
If K a = O L ilies,
g r a m m a r w i t h no X - r u l e s . }
s o m e s t a n d a r d t e r m i n o l o g y and n o t a t i o n s . L e t K be a f a m -
i l y of l a n g u a g e s . A K - s u b s t i t u t i o n
families
grammar~
( t h e f a m i l y of r e g u l a r
( t h e f a m i l y of f i n i t e l a n g u a g e s )
languages) were investigated.
U s i n g the f a c t
251
(cf.
[55])that
t h e f a m i l y of E O L - l a n g u a g e s
phism,
it i s e a s y to s h o w t h a t
(There
s e e m s to be no s h o r t d i r e c t
FMOL
Similarly,
= RMTOL
2 n
h-l{a
is in t h e d i f f e r e n c e
[ n->_ 0}
i n c l u d e d in R M O L b e c a u s e H e r m a n l s
w i t h h(a) = a ,
RMOL-FMOL,
cf.
[35~.
grammar
is a q u a d r u p l e
cf.
is t h e s m a l l e s t f u l l
[9~ o r [551 . It is a l s o
G = (VN,
...
0"t
1
(S)
VT,
i
and e a c h
A
S , U ) , w h e r e V N and V T a r e S E V + with V = V N U V T (ini-
is a f i n i t e s e t of K - s u b s t i t u t i o n s
that, for each
= U cr i
L e t K be a f a m i l y of l a n g u a g e s .
and t e r m i n a l s ) ,
The language generated by such a grammar
where
The family RMOL
the basic definition.
t i a l w o r d ) a n d U = {0"1 , . , . , 0 " n }
L(G)
h(b) = X ,
including the family OL,
alphabets (of nonterminals
with the property
(g)
language
of F--MOL u n d e r i n v e r s e h o m o m o r p h i s m .
We w i l l n o w p r e s e n t
disjoint
_c E O L . )
= ETOL.
is p r o p e r l y
( a n d the s m a l l e s t A F L )
K-iteration
proof fop the inclusion FMOL
one can prove that FMTOL
the c l o s u r e
homomor-
= EOL.
O n the o t h e r h a n d , F M O L
AFL
is c l o s e d u n d e r a r b i t r a r y
d e f i n e d on V a n d
a E V , 0"t (a) is a l a n g u a g e o v e r V . is d e f i n e d by
N VT*
k
the u n i o n is t a k e n o v e r a l l i n t e g e r s k>_- 1 and o v e r a l l o r d e r e d
k-tuples
(i 1 , . . . , i k ) w i t h 1 -< i t - n. T h e f a m i l y of l a n g u a g e s g e n e r a t e d b y K - i t e r a t i o n gram(t) m a r s is d e n o t e d b y K i t e r . B y K~.ter w e d e n o t e the s u b f a m i l y of K l t e t , g e n e r a t e d b y such grammars,
where U consists
The different work.
Consider
OL-families
of at m o s t
t
elements,
f o r s o m e t_-> 1.
c a n n o w be e a s i l y c h a r a c t e r i z e d
the s p e c i a l c a s e K = F .
within
this frame-
Then
K (iter I ) = =(1 ) --iter = E O L = F M O L .
(Note that it suffices
to c h o o s e ,
f o r e a c h a E V , 0"(a) to be the l a n g u a g e c o n s i s t i n g
o f t h e r i g h t s i d e s o f the p r o d u c t i o n s
F~ter = E T O L
The families
s u b f a m i l y of F ~ t ~ r ,
(= F M T O L
with D and/or
o"s are homomorphisms,
with
a
on the l e f t s i d e . ) S i m i l a r l y ,
= RMTOL).
P are characterized
as f o l l o w s .
P means that the o's are~.-free.
obtained by such grammars
Thus,
D m e a n s that the EPDTOL
where all substitutions
is the
cr a r e ~ -
free homomorphisms. If o n e w a n t s to c o n s i d e r
families
without E (OL,
TOL,
etc.),
then o n e
252
s i m p l y a s s u m e s that V N is empty ( w h i c h means that the i n t e r s e c t i o n w i t h V T *
in
(2) is s u p e r f l u o u s ) . N o t e that in the g e n e r a l c a s e the g e n e r a t i v e c a p a c i t y is not a f f e c t e d by a s s u m i n g that S E V N. F i n a l l y , o b t a i n e d by K - i t e r a t i o n
the m a c r o - f a m i l i e s K M O L and K M T O L a r e
g r a m m a r s s a t i s f y i n g the f o l l o w i n g c o n d i t i o n . T h e r e is a
s u b - a l p h a b e t V I of V N such that, f o r each 9ua9 e o v e r V N . F u r t h e r m o r e , each
f o r each
i
and each a E V I, o t (a) is a f i n i t e l a n -
i and each a E V T , 0"i ( a ) is empty and, f o r
i and each a E V N - V l , o's(a) is a l a n g u a g e (in K) o v e r the a l p h a b e t V T -
( H e r e i t i s a s s u m e d that K c o n t a i n s a l l f i n i t e l a n g u a g e s . ) T h u s , a l l c o n t e x t - f r e e L - s y s t e m s f i n d t h e i r c o u n t e r p a r t in t h i s f o r m a l i s m . N o t e , h o w e v e r , that so f a r ( a p a r t f r o m r e g u l a r m a c r o s ) one has not c o n s i d e r e d in the t h e o r y of L - s y s t e m s c a s e s mope g e n e r a l than K = F . T h e b a s i c tool needed in p r o o f s f o r c l o s u r e r e s u l t s is the f o l l o w i n g T h e o r e m 1. We s a y that a K - i t e r a t i o n
g r a m m a r is X - f r e e i f f each of the s u b s t i t u t i o n s o't is X - f r e e .
T h e o r e m 1. ( [ 5 5 ~ ,
E|03~) A s s u m e that K is a l a n g u a g e f a m i l y c l o s e d u n d e r
[57~,
f i n i t e s u b s t i t u t i o n and i n t e r s e c t i o n w i t h r e g u l a r l a n g u a g e s . T h e n f o r each K - i t e r a tion g r a m m a r ,
t h e r e is an e q u i v a l e n t X - f r e e K - i t e r a t i o n
Applying standard AFt-theory
grammar.
and the t e c h n i q u e used to p r o v e M e d v e d e v l s
T h e o r e m f o r f i n i t e a u t o m a t a , one can e s t a b l i s h the f o l l o w i n g r e s u l t s :
T h e o r e m 2. A s s u m e that K s a t i s f i e s the h y p o t h e s i s of T h e o r e m 1
and, f u r t h e r m o r e ,
z ( t ) , fop any c o n t a i n s a l l r e g u l a r l a n g u a g e s . T h e n a l l of the f a m i l i e s K~.ter , ['"f.ter
t > = I,
K M O L and K M T O L a r e f u l l A F L ' s .
T h u s , T h e o r e m 2 can be a p p l i e d w h e n e v e r K is a cone ( a l s o c a l l e d a f u l l t r i o ) . S i n c e the f u l l A F L l s
a s s o c i a t e d w i t h K ape o b t a i n e d by p a r a l l e l r e w r i t i n g ~
naturally called Lindenmayer A~LIs. _
_
t h e y ape
A p a r t f r o m the o b v i o u s i n c l u s i o n s _,--(I)
K M O L c K M T O L c Kiter , K M O L c ~it~r
v e r y l i t t l e is known a b o u t t h e s e A F L I s ,
e.g.,
~_
(2)
c
K~ter - •
..
~_
Kiter ,
a b o u t the s t r i c t n e s s of the i n c l u s i o n s .
It is shown by van L e e u w e n ( " N o t e s on p r e - s e t p u s h d o w n a u t o m a t a 't , t h i s v o l ume) that ,.~ter e q u a l s the f a m i l y of l a n g u a g e s a c c e p t e d by p r e - s e t p u s h d o w n a u t o m a ta. (In v a n L e e u w e n ~ s t e r m i n o l o g y , Riter c o u l d be c a l l e d t l h y p e r - a l g e b r a i c m u l t i e x t e n s i o n of regular l a n g u a g e s u . ) A n a t u r a l n o t i o n f r o m the p o i n t of v i e w of L - s y s t e m s
in A F t - t h e o r y
is t h a t of
a hyper-AFL
. B y d e f i n i t i o n , a f a m i l y K s a t i s f y i n g the h y p o t h e s i s of T h e o r e m 2 is
a hyper-AFL
iff K~t~r z K . H y p e r - A F L I s
a r e d i s c u s s e d in the p a p e r by P . A .
s t e n s e n ( t h i s v o l u m e ) . T h i s a p p r o a c h s h o w s t h a t , among the L - f a m i l i e s ,
Chri-
the f a m i l y
E T O L has a v e r y i n t e r e s t i n g m a t h e m a t i c a l p r o p e r t y . I t e r a t i o n g r a m m a r s h a v e been g e n e r a l i z e d by D e r i c k Wood ( " A n o t e on L i n d e n -
253
m a y e r systems~ s p e c t r a and e q u i v a l e n c e l ' ~ M c M a s t e r U n i v e r s i t y T e c h n i c a l R e p o r t N o . 7 4 / 1 ) to c o v e r L - l a n g u a g e s an e x a m p l e of h o w the u n i f o r m f r a m e w o r k eralize specific results. in E O L -
Computer Science
with interactions.
He also gives
of i t e r a t i o n g r a m m a r s c a n be u s e d to g e n -
T h e e x a m p l e c o n c e r n s the u l t i m a t e p e r i o d i c i t y
and E T O L - s y s t e m s ,
[2,0],
of s p e c t r a
[ 2 , 7 ] . W o o d ' s r e s u l t s h o w s t h a t the s p e c i f i c
r e s u l t s m e n t i o n e d d e p e n d o n l y on the m e t h o d of i t e r a t e d s u b s t i t u t i o n and n o t at a l l on the f i n i t e n e s s of the s u b s t i t u t i o n s .
HYPER-AFL~S
AND ETOL SYSTEMS
P, A. C h r i s t e n s e n D e p a r t m e n t of C o m p u t e r S c i e n c e U n i v e r s i t y of A a r h u s Aarhus~ D e n m a r k
T h e n o t i o n s of K - i t e r a t i o n
g r a m m a r s and of h y p e r - A F L %
a r e i n t r o d u c e d in
[57~ and [103~. T h e n o t a t i o n f o l l o w s that of [103~. Theorem
ETO--
S k e t c h of a
O,,o
h yper-AFL.
TO--
' °-
proof
It is o b v i o u s f r o m the d e f i n i t i o n of. an i t e r a t i o n grammar that ETOL
g ETOh(~e)r -c E T O L i t e r . To prove ETOLiter
9rammar w i t h
U=
g E T O L ~ let G = (VN~ VTJ S~ U) be an E T O L - i t e r a t i o n
t ~1,...
, ~ n } , w h e r e each
Assume that ~ j ( a i) = L ( G i , j ) , w h e r e s y n c h r o n i z e d v e r s i o n s of E T O L - s y s t e m s
~
isan ETOL-substitution.
Gi, j = ( V ~ J , V T U V N , T i , j, S i , j) a r e
and the alphabets
VI~ j a r e p a i r w i s e d i s -
joint.
We define a new E T O L - s y s t e m :
V~ = {$}
U i~j new symbols. ~X
,/i,j ( --N
U {~i,jt
= {'~ I a E V x }
G' = ( V 6 , V ¥ ,
X ' , ~ ), where
= T U ~N ) U ~T U ~NU V
and ~ X
= { a I a E VX}
where
for
$ and a l l
X = N and X = T
sets of new symbols. If X
is a s t r i n g of symbols
=~l'''~m
X = bt,..bk~
" T h e axiom of G 1 is defined as ~
Finally
T ~ c o n s i s t s of the tables:
a i + Si~ j f o r each
i and each j
to: A ~ For
$ f o r any o t h e r symbol A.
1 --< j < n t h e r e is the table:
then ~ =
S
B1 , . . B k and
in e x a c t l y the same way.
. are
l~J
are
255
Si, j ;
Si, j e
Si,.j
for each
i
t.:
J
4 A
For
-*
a
;
$
1 --- j < - n
e 4
where
for each
a E V NU V T
for any other symbol A.
and
there
| ~ i_< I V N U V T I
which consists I,j
a
of all tables from
the table with the terminal
changed to produce
barred
is t h e s e t o f t a b l e s :
T. productions
terminals
instead.
( G i , j is s y n c h r o n i z e d }
is
In a l l t h e s e t a b l e s w e add t h e
productions: ~e
a
for each
a E V NU V T
for each
k
m
-~ S k , j
A-'S
Finally
for any other new symbol.
t h e r e is t h e t a b l e w i t h t h e t e r m i n a l
productions:
¢=
a -* a f o r e a c h A
4
a E VT
$ for any other symbol A.
T h e c l a i m is n o w t h a t L ( G ) = L ( G ~ ) .
T h e r e a s o n f o r t h i s is ai
-~ S i , j
that r e w r i t i n g
is the same as choosing
a double-
the s u b s t i t u t i o n
is t h e n p e r f o r m e d
via the tables
formed,
is again double-barred.
the word
forth until we finally
bol is introduced
tj
and
u s e the t e r m i n a l
be u s e d in the Iline o f d e r i v a t i o n in t h e s t r i n g ,
barred
~'k,j
word via t0's productions
~ __
t o be u s e d .
The substitution
~ a n d w h e n the s u b s t i t u t i o n
is p e r -
We c a n c h o o s e a n e w s u b s t i t u t i o n
t a b l e to r e a c h a t e r m i n a l
are not chosen according
word.
If the t a b l e s to
to t h i s s c h e m e ,
a n d f r o m t h i s it i s i m p o s s i b l e
and so
a
S-sym-
to r e a c h a t e r m i n a l
word. Therefore ETOL
ETOL
is a f u l l A F t ,
= ETOL(~t )r = ETOLiter
we conclude
that ETOL
, a n d s i n c e it i s w e l l - k n o w n
is a hyper-AFL.
that
256
Corollary If
1 K
i s a f a m i l y o f l a n g u a g e s s u c h tha~: F g K _c E T O L
then
Kiter
= ETOL.
Proof ETOL
: Fiter
g Kiter
_c E T O L i t e r
Thus we have for instance proved ETOL
that:
= Flier : Riter : C F i t e r = E O L i t e r
Since each hyper-AFL Corollary
is a f u l l A F L
= ETOLiter
and s i n c e E T O L
.
= R iter ~ we conclude:
9.
ETOL
is the s m a l l e s t
hyper-AFL.
In [.56] it is s t a t e d t h a t t h e f a m i l y L by pre-set-pushdown Corollary
= ETOL.
R (i t| )e r
is e x a c t l y
the family of languages accepted
a u t o m a t a ~ a n d w e a r e n o w a b l e to p r o v e :
3
R (') iter
~ Riter
-- E T O L
Proof B y definition hyper-AFLls coroltary~ Let
~(I) r~iter c_ Rite r and w e k n o w that Riter = E T O L .
Furthermore~
are easily seen to be closed under substitution; in order to p r o v e the - - i(t1e)r is not closed under substitution: it s u f f i c e s to p r o v e t h a t N 2n 2m L I = {a I n ~ 0} and L 2 - - l a b I m-> 0}, then obviously
L I~ L 2 E O L c E O L = F ( I ) c R ( I ) iter iter " -
Define the substitution a b 2nz a b 2he . . .
a b 2nk ~ w h e r e
Define the finite
substitution
[35]
of the non-closure
t(#(L
1)) ~
finite
substitution~
J-(a) -- L 2.
each t by
t(a) = { a I
and
inverse
. It is f u r t h e r m o r e
therefore
Then
J - ( L 1) is t h e s e t of a l l w o r d s
n i >- 0 a n d t h e r e e x i s t s
of EOL under
E O L = F (1) iter
a pumping ,emma,
Tby
I >- 0
t(b) = {~., b} t h e n the p r o o f
homomorphism well-known
{
Rli
in
shows that
that EOL
~-(L 1) ~ F (iIt)e r " B u t s i n c e i n f i n i t e
obvious that
s u c h t h a t k = 21 .
is c l o s e d u n d e r
regular
r and the coro,,ary
sets fulfil
proved.
257 Finally K
w e m e n t i o n t h a t w e h a v e p r o v e d the e x i s t e n c e o f f u l l A F L I s
such that: K(1) iter
and
K
~, K i t e r
but
~
Kiter
= ~'(1 ) iter
= ~'iter
(Kiter)iter
= Kiter
namely
K =R
and ~" = E T O L ,
K
and
SYSTEMS*
~-OL
A n d r e w L. S z i l a r d D e p a r t m e n t of C o m p u t e r S c i e n c e University
Definition
An the is
elements
n-ary
domain
of ~ are
the arity
operator.
Definition
is
a set
called
o f ~.
~ with
operators,
If a(~)
We write
a mapping
~(n)
=
a
a n d if ~ ,
= n, w e
say
{~e~la(~)
: ~ + N; then
that
a(~)
~ is an
= n}.
1.i.2
L e t A be
a set
~-algebra
structure
Thus
each
with
Ontario
l.l.1
operator
called
of W e s t e r n
and
g an o p e r a t o r
on A is a f a m i l y
~e~(n)
we
associate
domain,
then
of m a p p i n g s
an n - a r y
an
~(n)
operation
÷ A An,
lIEN.
on A
: An÷A
Definition
The
1.1.3
set A w i t h
~-algebra
and
A is c a l l e d
Definition
For
is d e n o t e d
the
If n=0, constant *An
extract
carrier
by
(A,~)
structure or A
on A is
called
an
.
of A~.
1.1.4
any
~-algebra
to an n t u p p l e We write
an ~ - a l g e b r a
this then
A~
and
( a l , a 2 , . . . , a n) clement
this
any
that
the
from A gives
in postfix
means
~(n),
~eA.
Polish These
application
an e l e m e n t
notation ~'s
are
from
"ON THE A L G E B R A I C
the
author's
FOUNDATIONS
called
thesis: OF D E V E L O P M E N T A L
of A.
a I a 2 ...
operators.
SYSTEMS".
of
an~.
25g
In w h a t follows, we assume a fixed ~ s t r u c t u r e throughout,
and w i l l o m i t s o m e t i m e s
the s u b s c r i p t ~.
Definition 1.i.5
G i v e n ~ - a l g e b r a s A n and B~, a m a p p i n g f : A + B
and
~eQ(n), we say f is c o m p a t i b l e w i t h w, if for all a l , . . . , a n c A
f(al)f(a2)...f(an)~ = f(ala2...an~).
If f is c o m p a t i b l e w i t h each ~c~,
then f is said to be a
h o m o m o r p h i s m from A to B.
Definition
i.i.6
Given any two ~ - a l g e b r a s A~ and B~, w e say that BQ is a s u b - a l g e b r a of A~ if
B
i.e.,
if the c a r r i e r of B~ is a
s u b s e t of the c a r r i e r of A~.
Definition
1.1.7
G i v e n a family
(Ais)i~I of ~-algebras,
~ - Ai~ , the iai
a s s o c i a t e d d i r e c t p r o d u c t is d e f i n e d as follows.
Let P be the C a r t e s i a n p r o d u c t of the A ' s z projections
K
l
: P +A., 1
with
then any e l e m e n t p s P is u n i q u e l y
d e t e r m i n e d by its c o m p o n e n t s
K. (p) and any choice of e l e m e n t s l (a a A ) d e f i n e s u n i q u e l y an e l e m e n t p s P by K (p) = a i l iel l l
for all ia!.
C o n s e q u e n t l y if pl,P2 .... 'Pn a P and ~s~(n), we
can d e f i n e p l P 2 . . . p n ~ by the formuli
260
Ki(plP2...pn~)
= Ki(Pl)Ki(p 2) ...Ki(Pn)~
This p r o c e d u r e
defines
an ~ structure
all o p e r a t i o n s
componentwise.
for all i ~ I
on P simply by doing
Consequently
we have
T h e o r e m 1.1.7 D i r e c t p r o d u c t of ~-algebras
is an £-algebra.
We note that the A. 's need not be d i s t i n c t
and that the
1
projections
Definition
are homomorphisms.
1.1.8
Let ~ be an o p e r a t o r alphabet auxiliary
representing alphabet
domain and let ~ be a shadow
the o p e r a t i o n s
disjoint
from ~.
a l p h a b e t of free variables. ~-words,
W~(X),
If w ~ ( 0 ) , allen ~ W
2.
If xsX,
3.
If a l , a 2 , . . . , a n cW~(X)
4.
Nothing
X is called the
We define the language
as the following
i.
of ~, let X be an
subset of
of the
(XU ~)~
(X) .
then x £ W~(X) .
else belongs
and ~s~(n),
to W£(X)
then a l a 2 . . . a n ~ £ W~(X).
unless
from a finite nu~Lber of a p p l i c a t i o n s 1,2 and 3.
its being
so follows
of the set of rules
261
Definition
1.1.9
Let X and ~ be giyen as in D e f i n i t i o n define
an n - a l g e b r a
(n, (XU ~)*)
for any al,a 2 ..... a n ~ ( X U ~ ) *
1.1.8,
then we
as follows
and any men(n)
m(al,a 2 .... ,a n ) = ala2...an~. Theorem
i.i.i0
The a-words W~(X) the s u b a l g e b r a
of
for a given set X, form an a - a l g e b r a
(n,(XU ~)*)
g e n e r a t e d by X.
Proof: By D e f i n i t i o n under
1.1.8,
the ~ o p e r a t i o n s
the language of ~-words
and is generated by X.
C o n s i s t e n t w i t h this theorem, Definition
is closed
we have the following.
l.l.ll
The a - a l g e b r a
of the set of S-words W~(X)
is c a l l e d an
a-word algebra. Definition
1.l.12
Given the set of a-words relation i)
<~< over W~(X),
WQ(X), we define
called the ~-subword
For all nsN and all aie W
the following
relation,
(X)p 1 ! i ~ n, if ~s~(n),
a.! <~< b iff ala 2...an~ = b.
then
262
Furthermore,
a
2) for all a,b,c ¢ ~ ( X )
<~< a and
3)
if a <~< b and
i.e.,
b <~< c, then a <~< c.
<£< is the r e f l e x i v e
defined
by
Definition
a mapping
h
= 0
for each
h(x)
= 0
for each x s X
h(ala2...an~)
wEn(n)
These
branches
÷ N as follows
~£(0)
{h(ai) I l
the h e i g h t of the R - w o r d
definitions
rooted
trees,
are the
leaves
of the tree,
labels
of the i n t e r n a l
are e m i n a t i n g
distinction
operator between
o n l y nominal.
the o t h e r
nodes.
If ~ e ~(n), labelled
by w.
and n u l l a r y
is e n c o u r a g e d
to draw
as
and n u l l a r y
a-operators
the root of the tree,
free v a r i a b l e s
The r e a d e r
~-words
the free v a r i a b l e s
from the node
labels
in the examples.
ao
allow us to r e p r e s e n t
labelled
right-most
trees
= l+max
: W~(X)
and a £ W n(x) .
W e call h(a)
are the
of b.
1.1.12.1
h(~)
operators
of the r e l a t i o n
then w e say that a is an ~ - s u b w o r d
We d e f i n e
ordered,
closure
i).
If a <£< b,
where
transitive
then n The
and the
operators ~-words
is as
263
Definition 1.1.12.2
Let v be an ~-word of W~(X), correspondence non-negative
then w e d e f i n e a
£v f r o m the ~ - s u b w o r d s of v into the
i n t e g e r s as follows
£v(V) = 0, and for any ~ - s u b w o r d s b and ai; b , a i <~< v, £v(ai)
= l+£v(b)
if a l a 2 . . a n e = b, for all i° l~i<_n, and
all mE~(n).
We note that Zv is not n e c e s s a r i l y different occurrences d i f f e r e n t values.
functional,
of the same S - s u b w o r d may have
We say that a c e r t a i n o c c u r r e n c e of
an ~ - s u b w o r d u of v is on level n if ~v(U)
The f o l l o w i n g
since
= n.
s t a n d a r d theorems are stated w i t h o u t
proof.
The r e a d e r is e n c o u r a g e d to c o n s u l t any of the f o l l o w i n g e x c e l l e n t i n t r o d u c t o r y r e f e r e n c e texts:
A.G.
KUROSH, L e c t u r e s on G e n e r a l A l g e b r a ,
P.M. COHN,
U n i v e r s a l Algebra,
Our n o t a t i o n a l parallel
Cohn's
Harper-Row
Chelsea
(1963,
1965).
(1965).
c o n v e n t i o n s are chosen in a t t e m p t to
symbolism, w h o s e w o r k is r e m a r k a b l y
for clarity and consistency.
impressive
284
Theorem
1.1.13
For
any
isomorphic, are
the
W~(X)
= W£(Y),
iff
the
£-algebras
cardinalities
are of X a n d Y
same.
Theorem
i.i.14
The f
X and Y the corresponding
composition
: A ~ ÷ B e and
Theory
g
of h o m o m o r p h i s m s
: B~ ÷ CR
~-algebras
is a h o m o m o r p h i s m
fog
: Ae
÷ C
.
1.1.15
The
image
of a h o m o m o r p h i s m
f
: A~ ÷ B~
is a D - s u b a l g e b r a
of B~.
Definition
1.1.16
An equivalence a subalgebra
Theorem
there
Definition
The A n by
p.
of A £ × A ~
on an R - a l g e b r a
is c a l l e d
a congruence
A~ w h i c h
is a l s o
relation
on A S •
1.1.17
Let A then
relation
b e an is
~-algebra
a homomorphism
and
p a congruence
natp
: A~ ÷
relation
on A
(A/p,~).
1.1.18
a-algebra
(A/p,~)
is c a l l e d
the q u o t i e n t
algebra
of
,
265
T h e o r e m 1.1.19
For any ~ - a l g e b r a s A , B and any g e n e r a t i n g
set X of A,
a h o m o m o r p h i s m h of A into B is u n i q u e l y d e t e r m i n e d b y its r e s t r i c t i o n hlX.
Theorem
1.1.20
Let A~ be any n - a l g e b r a and X any set, e
then any m a p p i n g
: X ÷ A may be u n i q u e l y e x t e n d e d to a h o m o m o r p h i s m : W~(X)
÷
A m.
T h e o r e m 1.1.21
Any ~ - a l g e b r a A is a h o m o m o r p h i c a l g e b r a W~(X)
Definition
image of an ~ - w o r d
for some set X.
1.1.22
F, a s y m m e t r i c set of d e s i g n a t e d p a i r s of ~ - w o r d s in the n-word a l g e b r a W~(Y),
is called a set of i d e n t i c a l r e l a t i o n s
that h o l d in W~(X).
£ ~W~(Y)
× W~(Y),
(Wl,W 2)
E £ is w r i t t e n as W l = W 2.
Two L-words v and u in WD(X) r e s p e c t to F, v = F =
are called e q u i v a l e n t w i t h
u, iff there is a finite s e q u e n c e of
t r a n s f o r m a t i o n s v i ÷ vi+ 1 i = l , . . . , n - i there are L-words Pi and qi' respectively,
: v=v I and u=v such that
~ - s u b w o r d s of v i and vi+ 1
Pi
and vi+ 1 is o b t a i n e d
from v i by r e p l a c i n g Pi by qi' furthermore,
Pi and qi are the
266
homomorphic
images of w I and W 2 r e s p e c t i v e l y u n d e r an
a r b i t r a r y m a p h i : Y + W~(X) where
(Wl,W2)
e x t e n d e d to a h o m o r p h i s m ,
~ £.
T h e o r e m 1.1.23
For any ~-word a l g e b r a W~(X) relations
F ~ W~(X) × W~(X),
and any set of i d e n t i c a l
the r e l a t i o n =£= is a c o n g r u e n c e
r e l a t i o n on W~(X).
Proof :
O b v i o u s l y =~F= is an e q u i v a l e n c e relation, but since the rel~tion
<S< is t r a n s i t i v e and since subwords may be r e p l a c e d
by e q u i v a l e n t subwords,
it is also a c o n g r u e n c e relation.
In fact, let a i =F= b i for all i, l<_i<_n, then to show that for any e ~ ( n ) ,
al...an~
=F= bl...bn ~, we are r e q u i r e d to show
that there is a finite s e q u e n c e o f a l . . . a n e to b l...bn~. transformation
transformations
from
S u p p o s e the length of the s e q u e n c e of
from a i to b i is
mi,
r e q u i r e d s e q u e n c e of t r a n s f o r m a t i o n s
then the length of the n is at m o s t [ m.. i=l
D e f i n f t i o n i.I. 24
W(~,X,F)
w h i c h denotes the f a c t o r a l g e b r a
(W~(X)/=F:,Z)
(or any a l g e b r a i s o m o r p h i c to it), is c a l l e d the free ~ - a l g e b r a o f the v a r i e t Y F g e n e r a t e d by X.
Note the g e n e r a h o r s of this algebra are the sets of =F= e q u i v a l e n t words,
e q u i v a l e n t to the elements of X.
267
In p r a c t i c e ;
when
£ is u n d e r s t o o d ,
=£=
is w r i t t e n
simply
as =.
Example
1.1.25
Consider
the f o l l o w i n g
free
~-algebra
GdI, G d I = W ( ~ , X , £ ) ,
where e(i) (2)
= % =
{o}
X = {a}
Gd I is c a l l e d
The
the
and
free g r o u p o i d
following
lexicographically,
for all i+2
are the
first
within
length
following
free
F=~
generated
by a singleton.
few e l e m e n t s and height.
a aao aaoao aaaoo aaoaaoo aaoaoao aaaoaoo aaaaooo
Example
1.I.26
Consider
the
a-algebra
Sg 1.
of Gd ! o r d e r e d
268
S g I = W(e,x,r), where ~(i)
= %
(2)
=
for
and
S g I is c a l l e d
the
free
The
are
the
form
ordered
by
i~2
{o}
X = {a}
following
all
semigroup first
length.
highest
form
i n the
e-words
representing
£ = {xyozo
= xyzoo}
generated
few elements
By normal
a singleton.
of Sg I in normal
form we mean
lexicographical an e l e m e n t
by
order
among
here
the
equivalent
o f S g I.
a aao aaaoo aaaaooo
We may and
note
= aaaoaoo
To show
the
chain,
Example
= aaoaoao
equivalence we
are equivalent
forms
= aaoaaoo.
of the
use
t h e m a p h.
the
following
: w(e,x,r) , where P2
= aaoao
first
h(x)
and
last elements
= a,h(y)
= ~h(z)
1.1.27
Consider G
aaaoo
so a r e
aaaaooo
the
that
free
~q-algebra G
P
2
in
= aao.
269
n (0)
=
{l}
(i)
=
{-1}
~(2)
=
{o}
(i) = #
i>2
X = {a,b} F = {xyozo = xyzoo, x-lxo
=
I,
x!o
xx-lo = 1, =
x,
ixo
=
xt
x-ly-lo = yxo ~I, x -I-I = x}
Gp
is the free group g e n e r a t e d by two elements.
F is not
2 minimal,
but a c o n v e n i e n t
set of identical
relations
for
a free group.
The following
are the first few elements
of G
in P2
normal
form o r d e r e d
lexicographically
within
length.
l,a,b,a-l,b -I, aao,abo,bao,bbo, ab-lo, a-lbo ,ba- Io, b- lao, a- la-!o, a-lb- io ,b-la-lo,b- lb- lo, aaaoo,
aaboo, ....
By n o r m a l form we m e a n here the highest form in the lexicographical length.
order among e q u i v a l e n t
~-words
of m i n i m a l
270
Definition
1.2.1
Given a set A, we consider operations
the set of all finitary
on A, d e n o t e d by a(A), An
(A) :
where
U A n~N
= U a n(A) n~N
the set of n-ary o p e r a t i o n s
~n(A)
' ~n(A)
We define
= A An.
the c o m p o s i t i o n
n-ary operation, g am(A)
on A is d e n o t e d by
of ~ l , m 2 , . . . , m m
8, for each e l , m 2 , . . . , m m
with
~ as an
~ an(A)
and
as follows
6 : An + A
~(x) =
(Xml) (xm2)''" (x~m)m
for each x ~ A n.
Any o r d e r e d
set of operations,
requirement
for c o m p o s i t i o n
define
is said to be conforming.
for all n > ~ n n-ary o p e r a t i o n s
Knl,Kn2,...,Knn,
[ni
w h o s e arities obey the
An
where
namely
for any given i, l~i
Kni
:
Kni
(a l,a 2 .... ,a i,---a n ) = a i
selects
in an(A),
We
÷ A for all aj E A,
the i-th element in an n-tuple of elements
l<j
of A.
271
Kni is c a l l e d the i-th c o m p o n e n t p r o j e c t i o n of an n-tuple.
{Kni I l
is c a l l e d the set of n-ary p r o j e c t i o n o p e r a t o r s .
A set 6 of o p e r a t i o n s on A is c a l l e d a c l o s e d set of operations
on A, or a clone on A in short,
contains
if and only if
the p r o j e c t i o n o p e r a t o r s and the c o n f o r m i n g
o r d e r e d s u b s e t s of ~ are c l o s e d u n d e r c o m p o s i t i o n .
E x a m p l e 1.2.2
~(A)
is a clone,
an(A)
is a clone.
D e f i n i t i o n 1.2.3
Let A be an Z-algebra, ~-operators
then the clone g e n e r a t e d by the
on A is c a l l e d the clone of a c t i o n of 9 on A.
T h e o r e m 1.2.4
Let A be an m - a l g e b r a and 8 the clone of a c t i o n of Z on A, let x s A n and 6 ({x~l~ ~ Bn},
n
the set of n-ary o p e r a t o r s
in ~, t h e n
~n ) is the Z - s u b a l g e b r a g e n e r a t e d by the
e n t r i e s of x.
Proof:
The n-ary p r o j e c t i o n o p e r a t o r s will p r o v i d e and the o p e r a t i o n s
the g e n e r a t o r s
in 8 w i l l p r o v i d e the ~ - a l g e b r a structure,
the c o m p o s i t i o n of the n-ary p r o j e c t i o n o p e r a t o r s with e l e m e n t s of ~ are the set of all n-ary o p e r a t o r s
in ~.
272
We may see this in a d i f f e r e n t Theorem
form in the following.
1.2.5
Let ~ be an o p e r a t o r
domain,
let n ~ N +, let W~(X)
an ~ - w o r d a l g e b r a g e n e r a t e d by X = {Xl,X2,...,Xn},
be
let 8
be the clone of action of ~ on X, let E n be the vector of n-ary p r o j e c t i o n
~n = denotes
operators
on X, namely
(~nl,Kn2,-'-,Hnn) "
8(Kn)
the set of n-ary o p e r a t i o n s
Kn by r e p e a t e d operations
compositions
o f W~(X)
that can be obtained
of the projections
in ~, furthermore,
the elements
then from
and the
there is an i s o m o r p h i s m between
and 8(~n).
Proof: Let us define the operations
a mapping
in 8(~n),
¢ from the elements of W~(X)
and show that this m a p p i n g
to
is an
isomorphism.
i)
~(xi)
= ~ni = Eni(~n )
Since the arity r e q u i r e m e n t
for c o m p o s i t i o n
are the same as the arity r e q u i r e m e n t may p r o c e e d i n d u c t i v e l y
of operators
for forming
on the h e i g h t of S-words
in
~-words,
we
as follows:
273
Suppose
¢ is d e f i n e d for Yk" the [~-words of h e i g h t
less than k, so that ~(y)
s 6(~ n) for e a c h y s Yk"
Consider
an a - w o r d x of h e i g h t equal to k such as x = y l Y 2 . . . y m ~ , w h e r e ~ £ 9(m). therefore,
The Yi'S are
by h y p o t h e s i s ,
of h e i g h t less than k,
each %(yi ) e 6(~ n) and
¢(yl ) , %(y2) ..... %(ym ) w i t h ~ form a c o n f o r m i n g o r d e r e d s u b s e t of o p e r a t i o n s ,
2)
t h e r e f o r e we may d e f i n e in K
n
~(x) = e(%(yl ) , %(y2 ) ..... ¢(ym ))
Since i) defines ~ for a-words of h e i g h t O, the i n d u c t i o n is complete.
F r o m i) and 2) it is clear that ¢ is a h o m o m o r p h i s m . C l e a r l y every 9 o p e r a t i o n can u n i q u e l y be so s i m u l a t e d and the c o m p o s i t i o n of f u n c t i o n s w i t h a p r o j e c t i o n o p e r a t o r w i l l not p r o d u c e an a d d i t i o n a l f u n c t i o n t h e r e f o r e ¢ is an i s o m o r p h i s m . The f o l l o w i n g e x a m p l e is i n c l u d e d simply to show an i n s t a n c e of the i s o m o r p h i s m and to m a k e the n o t a t i o n m o r e familiar..
274
Example
1.2.5
Let
X
=
(0) ~(i)
(2)
~(i)
Let
an
e-word
algebra
WR(X)
be
given
as
follows:
{a,b}
= ¢ = {.,-i}
=
{o}
= #
for
H21(a,b)
Consider
the
i>2
= a and
H22(a,b)
= b.
~-word
x = abo-la.o.
¢(x)
=
¢(abo-la.o)
= o(¢(abo-l),
= o(-l(¢(abo)),
.(¢(a)))
= o(-l(o(#(a),
= o(-l(o(H21(a,b)
¢(b)),
~(a.))
=
=
.(n21(a,b)))
, H22(a,b))
=
, *(H21(a,b)))
275
A few remarks all,
in T h e o r e m
at this point
are in order.
First of
1.2.5 the concept of an i s o m o r p h i s m
u s e d in the r e l a x e d ~-word x is graph
sense that the parsing
theoretically
tree of the operators
in ~(x),
isomorphic
is
tree of the to the c o m p o s i t i o n
as labeled o r d e r e d rooted
trees. Secondly, ( X U R),n ÷
the o p e r a t i o n
(XUn)*,
Thirdly, the o p e r a t i o n
#(x)
is an n-ary o p e r a t i o n
from
w h e r e X = {Xl,X 2 ..... Xn},
Theorem
1.2.5 is simply the formal notion of
of s u b s t i t u t i o n
for the free variables
in a
given ~-word. On the basis of these remarks,
Definition
we adopt the following:
1.2.6
Let x be an ~-word in the ~-word algebra W~(X), X = {Xl,X2,...,Xn}.
We denote the value of the n-ary
operation
: (X~)*n
~(x),
proof of T h e o r e m
X(Zl,Y2,...,Zn)
~(x)
( x U n)*, given in the
1.2.5 as
for any n-tuple
(XO ~)* and any x £ W~(X). word x w i t h the o p e r a t i o n formula.
÷
where
(Zl,...,y n) of strings
This convention it represents
identifies
in the
as a s u b s t i t u t i o n
276
Example
1.2.6
Following
Example
1.2.5, we obtain
x = abo-la*o
x(a,b)
= abo-la*o
x(a,a -I)
= aa-lo-la*o
x(a-l,abo)
= a-laboo-la-l*o
x(b,oa)
= boao-lb~o
abo-la*o(b,a)
= bao-lb*o
We now introduce of c o m p o s i t i o n s
operations
of operations
To define o p e r a t i o n s a potentially variables. explicit
infinite
transformations,
variables,
put constants
of arbitrary
projection ment
operators
is given
we use the scheme of
i.e., we permute for variables
or identify
and add new variables,
by c o m p o s i t i o n
and substitution.
as follows:
arity, we w i l l need
for the free s u b s t i t u t i o n a l
an operation,
all this may be a c c o m p l i s h e d
images
in an a-word algebra.
alphabet
To s p e c i a l i z e
that are h o m o m o r p h i c
with the
Formally
the develop-
277
Definition
1.2.7
Given the X1 = let
an i n f i n i t e
sequence (x l)
of o r d e r e d
, X2 =
~ be
be
the
{Xl,X2,X3,...}
subsets
domain,
, consider
of S.
(Xl,X2),...,X n =
a operator
algebras
set X =
(Xl,X2, .... Xn),
l e t the
corresponding
... ~-word
sequence
w~(x l) , w~(x 2) ..... wn(x n) .... let x be denoted
a given operation
x
let
: W~(Xn)
~' b e
d
in W ~ ( X n) w i t h
X ( X l , X 2 .... ,x n) n
mapping,
the
correspondingly
= x, w h i c h
is
a homomorphism
÷ W~(X)
an o p e r a t o r
preserving
such
9-word
domain
and
let d be
a given
arity
i.e.
: ~ ÷ £'
that d[m]
l e t A~,
be
= m'
implies
an s ' - a l g e b r a ,
and
a l , a 2 , . . . , a n e Ag, l e t h be
the h o m o m o r p h i c
a(~)
= a(e')
for any
, extension
of
the m a p
'b
then we-define
the
,
operation ~:An÷A
x as
follows
h ( x i)
= ai,
278
x(al,a 2 ..... a n ) = d[x] (h(Xl),h(x 2) .... ,h(Xn))
(In practice
d is bijective
The definition commuting
and ~ and ~' may be identified).
o f ~ may be exhibited
diagram
by the following
of homomorphisms.
W~ (X n) n
h
~_
An
xI
I
w~ (x n)
~
A
h
x is called is denoted
a principal
derived
by the expression
For any integer
arguments
Example
=
a derived
we chose
for notational
X(Xl,X2,...,Xm,bl,b2,...,bk), n-ary operation
to specialize
convenience
on A~,.
the last k
only.
1.2.7
Consider A
,, and
k
is called
In this definition
on A
~(Xl,X 2 ..... Xn).
b l , b 2 , . . . , b k ~ A, the operation where m=n-k,
n-ary operation
({l,a,b}
the ~l-algebra ,
{o,l,-l}),
given by where
a(o)
= 2, a(1)
= 0, a(-I)
= 1
279
1
a
b
1
1
a
b
a
a
b
1
b
b
1
a
,,1
It is easy to see that AR1 is the cyclic group on {l,a,b}.
<(x,y)
Consider
= x-lyo(x,y)
the subgroup
closure o p e r a t o r
= o(-l(Kx(X,y)) ,Kx(X,y)) ,
where K (x,y) = x
and
~y(x,y)
= y.
X
The o p e r a t o r
~ is a p r i n c i p a l
derived binary o p e r a t o r
on
A~I" Examp!e
1.2.8
On the other hand, A~2 =
({l,a,b},
table
for K is given as d e r i v e d
consider
l(x,y)
_
o(x,y)
where
the o p e r a t i o n s
= <(H
--
-I(x,Y)
{K}),
consider
(x,y) X
= K(H
= <
x
(-i
the ~2-algebra,
a(K)
= 2 and the m u l t i p l i c a t i o n
in the previous
o, I, and -i, w h e r e
Kx(X,y)) ;
(x,y)
l(x,y))
(x,y)
, Hy(X,y))
given by
and
example,
then
280
The o p e r a t i o n s
~(x,y)
, __-I(x,Y) and o(x,y)
derived binary operations
llll
a
-I
b
li a
1
are p r i n c i p a l
on A~2
1
a
b
1
1
1
1
1
b
b
b
1
1
a
&
a
oillb i
i
!
a
b
a
b
1
:b
i
..... illl
a
and clearly they are r e s p e c t i v e l y e q u i v a l e n t to the o p e r a t i o n s i, -i and o in A n in the p r e v i o u s example,
in fact,
they
are g i v e n as the f o l l o w i n g d e r i v e d operations:
i = <(i,i)
,
-l(x)
Definition
1.2.9
= K(x,l)
For an ~ - a ! g e b r a A
o p e r a t i o n ~ is p r i n c i p a l , d e r i v e d t r a n s l a t i o n on A n. , f(f(a))
o(x,y)
= o(x,y) .
, the d e r i v e d unary o p e r a t i o n s
called derived translations
S = {a,f(a)
and
on A~.
are
If a d e r i v e d u n a r y
then f is c a l l e d a p r i n c i p a l A subset S of A such that
, f(f(f(a))) .... } is called a s p l i n t e r
in A~ or an S - a l g e b r a i c s p l i n t e r in A for any d e r i v e d translation
f and any e l e m e n t a in A.
S p l i n t e r s play an
i m p o r t a n t role in the fundations of m a t h e m a t i c s , s t u d i e d by Ul!ian,
Myhill, Y o u n g and Rogers;
f to be any r e c u r s i v e function. disguise
they c o n s i d e r
S p l i n t e r s also turn up in
in the study of a u t o n o m o u s
of c y c l i c s u b m o n o i d s
and w e r e
automata,
in the study
of the m o n o i d of e n d o m o r p h i s m s
of
281
e-algebras
and in the study of fixed p o i n t theorems
in
general.
Definition
1.2.10
L e t A~ be an ~-algebra, w e d e f i n e t h e Y ' - a l g e b r a ~ ( A ) e, as follows.
For any m ¢ ~(n), we d e f i n e m' as follows
for all i, l
~' (XI,X 2 .... ,X n) = {~(Xl,X 2 ..... x n) I xi%X i
We i d e n t i f y ~ and ~'.
(A) s is called the ~-set a l ~ e b r a on h. p r e s e n c e of the b i n a r y o p e r a t o r
U
In ~
(A)~ the
is t a c i t l y assumed.
an i m m e d i a t e c o n s e q u e n c e of the definition,
we have the
following :
T h e o r e m 1.2.11
For any e-set a l g e b r a
a ( X I , X 2 ..... X i ..... Xn) U
m(Xl,X2,...,Xi ~
~(A) n and any ~ a ~(n)
e(XI,X 2 ..... X~ ..... X n) =
X' ...,X n) i'
w h e r e the X i and X' are i
subset~ of A.
Proof : The l e f t - h a n d side and the r i g h t - h a n d side are same as sets.
As
.
282
Example
1.2.11
Consider
~2
the f r e e m o n o i d
= W(~,Y,F), where
a(l) = 0 and a(o)
~ = {~,o}
, Y = {a,b}
= 2, F = {xyozo = xyzoo,
The first few n o r m a l i z e d order w i t h i n
g e n e r a t e d by a d o u b l e t o n
elements
lxo = xlo}.
of W 2 in l e x i c o g r a p h i c a l
length are
X,a,b,aao,abo,bao,bbo,aaaoo,aaboo,abaoo,abboo,...
Here the binary
operation
and the following
is an instance
o(aao,b) The o p e r a t i o n
o is called the string c a t e n a t i o n of that
= aaobo = aaboo.
symbol may be o m i t t e d
the a-set a l g e b r a ~ ( ~ 2 ) a l g e b r a of languages set-catenation
in the forms.
~ corresponding
to~,
Consider
namely the
over the binary a l p h a b e t w i t h o p e r a t i o n
or set-product.
The following
is an instance
of this operation:
o({a,aa},
Because identity o(A~B,
{b,ab,aab})
of the previous
= {ab,aab,aaab,aaaab}.
theorem,
for any set A , B , C , D CUD)
in
we have the following ~(~2
)
= o(A,C) ~ o ( B , C ) ~ o ( A , D ) ~ o ( B , D ) .
283
A set v a l u e d set m a p p i n g f is called m o n o t o n i c p r e s e r v e s the i n c l u s i o n relation, f(X) ~ f ( Y ) .
i.e. X ~ Y
if it
implies
U s i n g this t e r m i n o l o g y , we can state T h e o r e m
1.2.11 e q u i v a l e n t l y as follows:
Corollary
1.2.11
For any ~-set a l g e b r a
~(A)~
any d e r i v e d t r a n s l a t i o n
is monotonic.
Proof:
Let X ~ Y ~ A
and w i t h o u t
t h a t the d e r i v e d t r a n s l a t i o n
loss of g e n e r a l i t y assume
f is given as follows
f(V) = m ( V , A I , A 2 , . . . , A n _ I) for all V ~ A ,
where A~
and ~ is in the clone of a c t i o n of ~ such that a(~) then X C Y
implies
that Y = X ~ Z, w h e r e
Z~
~oo,
f u r t h e r m o r e by
T h e o r e m 1.2.11
f(Y) = f ( X ~ Z) = ~ ( X ~ Z , A I , A 2 ..... An_ 1 ) =
~ ( X , A I , A 2 ..... An_l) ~ ~ ( Z , A I , A 2 , . . . , A n _ I )
= f ( X ) ~ f(Z),
therefore f ( X ) ~ f(Y) .
Corollary
1.2.12
For any ~-set a l g e b r a f o l l o w i n g i n c l u s i o n holds:
~(A)
and any w s ~(n),
the
284
~ (XI,X 2 ..... X i .... ,Xn) ~ ~ (XI,X 2 ..... X i ..... X n ) ~
~ ( X I , X 2 ..... X i ~ where
Xl ..... X n)
,
t h e X i's a n d X~ are subset5 of A.
Proof :
The right-hand
s i d e is i n c l u d e d
intersection
on the left,
consequently
it is i n c l u d e d
by the p r e v i o u s
is p r o p e r
~ ( x I . . . . . ¢ . . . . . x n)
=
in g e n e r a l
if X c Y ,
(X,f(X) ,f(f(X)) ,...
) as a set s e q u e n c e .
then the m a p p i n g
translation S[f,a,A~]
on A~ g e n e r a t e d
S[f,a,A~]
the s p l i n t e r
1.2.13
Let f be a derived
sequence
and that
t h e n the s p l i n t e r
) dominates
of A,
We note
¢.
(Y,f(Y) ,f(f(Y)) ,...
Definition
theorem,
in t h e i r i n t e r s e c t i o n .
that the inclusion
Furthermore,
in e a c h t e r m of the
on A
, let a be an e l e m e n t
is c a l l e d
b y a.
: N ÷ A~
S [ f , a , A ~ ] (o) = a S [ f , a , A ~ ] (n) = f ( S [ f , a , A ~ ] (n-l))
an f - s p l i n t e r
285
In w h a t splinter
follows,
to d i r e c t
we mean
a direct
we generalize
powers
the
concept
of ~ - a l g e b r a s .
product
in w h i c h
all
of
the
By d i r e c t
factors
are
power
identical.
Thus 1 A~n = A
Definition
2 ×A~
1.2.14
~, an n - t u p l e ~-algebra all
the
called
n A
×...×
A~
Of d e r i v e d
n-ary
a principal
operations
T =
Y
An
:
on A nn-
operations
are p r i n c i p a l ,
then
derived
( t l , t 2 , . . . , t n)
->
transformation
we
define
of
Am
{a,T(a),
S[Am,-~,a]
and
ti
A,
: A
If
it is
n o n A~.
l
then
An .
Definition
Let
an
transfo~lation
n Let
on
a derived
is c a l l e d
derived
n-ary
1.2.15
• be
a derived
an m - a r y
~(TCa)),
will
transformation
s~!inter
as
the
m on Ae and
following
T ( T ( Y ( a ) ) ) .... }
denote
the
following
mapping:
set
m a ~ A~,
then
of e l e m e n t s
286
m S [A n , T ,a]
m
:N÷A~
m S [ A R , T , a ] (o) =
a
S [ A -m~,T,a] (n) =
Let
T be a derived
following
for n>0.
(S [A~, T, a] (n-l))
transformation
s e t of t r a n s f o r m a t i o n s
o n A~,
is c a l l e d
t h e n the
a splinter
of
o n Ama .
transformations
{~m,T,T2,...},
where
T
n
. is d e f i n e d
as f o l l o w s :
Am ÷ Am
n
n =.S[A nt~,a]
Tn ( a )
Because
(n)
of t h e a s s o c i a t i v i t y
a c t i o n of the p r o j e c t i o n
of c o m p o s i t i o n
operations,
and the identity
we obtain
the
following:
Theorem11.2.16
Given splinter
an ~ - a l g e b r a
A n a n d let S = { K m , ~ , ~ 2 , . . . }
of t r a n s f o r m a t i o n
composition
forms
on A~,
a cyclic monoid
then S with generated
by
the o p e r a t i o n T.
Proof:
Clearly,
a ~ m °T
ao~ = ~
from what we said
T a+b = Tao~ b and
= Ta m
For consistency
we have
the f o l l o w i n g :
be a
287
Definition
1.2.17
For any splinter
transformation
~ on A ~
Hm = ~0 Definition
1.2.18
Let Ws(X) and let y =
W(X),
=
be an ~-word algebra,
(yl,Y2,...,yn)
where
X = {Xl,...,x n}
be an n - t u p l e of S-words
let
(~(yl) , ~ ( y 2 ) , . . . , # ( y n )) be the c o r r e s p o n d i n g
n-ary
in
operations,
where
~(yi ) : ( X U ~ ) *n ÷
for each i, l~i<_n,
n-tuple of
~(yi ) is given
(X~)*,
as defined by
Definition
1.2.6,
then
T is called an n-ary {T0,Tl,T2,...}
S-word transformation,
and the set
is called an n-ary ~-word t r a n s f o r m a t i o n
splinter.
Definition
1.2.19
Let WD(X)
be an D-word algebra w i t h the finite o r d e r e d
set of generators transformation~ H :(WD(X] ,T,X)
X =
(Xl,...,Xn) , let T be an n-ary
let x be an ~-word, is called an OL ~-word
then the triple system.
D-word
288
L[H~], defined L[Hn]
the language
of the ~-OL word
as the following
system,
is
set:
= {x(~m(x)),[ m>_O}
WS [H~], the R-word
sequence
defined
by H
is the following
mapping :
WS[H~]
:
N ÷ W~(X')
WS[H~] (m) = x(Tm(x))
Theorem
1.2.20
Given any
an OL ~-word
(non-empty)
non-negative equality
finite
integers
system H~ = (W~(X),T,x), ordered
alphabet,
where X is
then for any
such that i+j = m, the following
holds :
WS[H~] (m) = x(Ti(x) (TJ (X)))
Proof : Let n be the cardinality a e (xU ~)*n and for any n-ary
of X, then for any n-tuple transformation
x(X) (a) = x(a) , therefore yi(x) (~J(x))
: ~i($J(x))
= ~i+j(x)
= sin(X)
hence x(~i(x) (TJ(x)))
= x(Tm(x))
= S[H~] (m)
X,
289
Corollary
1.2.20
WS[H~] (ra) = x(ym-l(x) (~(X))
WS[H~] (m) = X(T(X) (T
Corollary
m-i
(X))
if m>O.
1.2.21
L e t X = (Xl,X2,...,Xn) , and let Hke = (W~(X),~,x k) for all k, l
i
(Vj))
for each i,j and m
such that i+j=n. In p a r t i c u l a r , WS[He] (m) = x(T(Vm_I)) if m>O. Proof : Vj = TJ(x).
= x(T m - l ( V I))
290
Example
1.2.22
Let X =
~(~,~)
(a,b)
(~b(~,~)
=
, ~ =
{0}
, a(o)
, O(Hb(a,8)
=
2,
, Ka(~,6)))
where e
~a(~,~)
= e
Let H = From
and
(W~(X) ,T,x),
this w e o b t a i n
.r2 ( c ~ , 6 )
=
-c3 ( ~ , 6)
=
~b(~,6)
= ~
where
for all
~,6 in
(XU ~)
x -- abo.
the following:
(O(K bCa,S) ,Ha(a,S)) ,o(o(~ b(a,6) ,Ha(a,~)) ,K b (~,6)))
(0(0 (Eb+(0~,~),IIa(C~,6)) ,~b (a,(~)) , O ( O ( O ( ~ b(~,~) ,Ha(~,~)),~ b(~,6) ,o(~ b(~,6) ,Ra(~,6))))
(x)
=
(b,o(b,a)) =
(b,bao)
T 2 (x)
=
(o(b,a) ,o(o(b,a) ,b))
T 3 (x)
=
(o(o(b,a) ,b) ,o(o(o(b,a)
=
(bao,o(bao,b)) ,b) ,o(b,a)))
= =
(o(bao,b) ,o(o(bao,b) ,bao) ) = (baobo,o (baobo,bao)) ~2 (T (x))
= ~2(b,bao)
-c (-c 2 Cx)) =
=
(bao,baobo)
=
(baobo,baobobaoo)
(baobo,baobobaoo) =
(baobo,baobobaoo)
(bao,baobo)
291
WS [H] (3) == x ( b a o b o , b a o b o b a o o )
x(~(X) (~2(X)))
= abo((b,bao)
= baobobaobobaooo
((bao,baobo)))
=
= abo (b (bao,baobo) ,bao (bao,baobo))
= abo ( b a o b o , b a o b o b a o o )
=
= baobobaObobaooo
x(~ 2(X) (~(X)))
= abo((bao,baobo)
((b,bao)))
= abo((bao(b,bao)
,baobo(b,bao) ) ) =
= abo(baobo,baobobaoo)
=
=
= baobobaobobaooo.
The called
image
R-0L
system.
Theorem
1.2,20
commute
h and
is
as
of an O L ~ - w o r d
may T.
It is
clear
be phrased
that
about
One way we may
under
a homomorphism
a statement
the
image,
interpret
similar
h, to
since
we may
Corollary
1.2.20
follows:
In a d e v e l o p m e n t a l pattern the
system
on the h i g h e s t
first
stage
The m a c r o - c o s m o s
of
system level
without
is i d e n t i c a l
the d e v e l o p m e n t ,
reflects
interaction, with
the p a t t e r n
o r in c o s m i c
the m i c r o c o s m o s .
the g l o b a l
terms:
of
is
BOUNDED PARALLELISM AND REGULAR LANGUAGES D. WOOD Department of Applied Mathematics,
McMaster University
CONTENTS Section I:
Introduction and Overview
Section 2:
n-parallel
finite state generators
Section 3:
Variations
on the basic model of n-PFSG's
Section 4:
An alternative formulation, properties
Section 5:
Characterisation Theorems for
Section 6:
Concluding Remarks
Section i:
Introduction and Overview.
n-PRLG's,
and closure
~ [ I and
~irl
If we examine the rewriting systems of Ibarra (I), the so called Simple Matrix Grammars
(SMG), we see that rewriting has the following
three facets, namely (i)
rewriting occurs in PARALLEL,
(2)
the parallelism is, a priori, BOUNDED,
(3)
the rewriting is, a priori,
Secondly,
CONTROLLED.
if we examine the rewriting systems of Lindenmayer
[61], the so called E0L systems, (!)
and
and compare and contrast with the
Ibarra, O. H., Simple matrix languages, 17 (1970), 359-394.
Information and Control
293
SMG's w e find: (1)
rewriting
(2)
the p a r a l l e l i s m
(3)
the r e w r i t i n g
Later (a)
again occurs
extensions
is,
a priori,
is, a priori,
to E0L systems
'RULE-CONTEXT-FREE' [81];
rules
rules
from a given
restriction the other (b)
of rules
have
- Rozenberg's
set of tables. rule
is d e p e n d e n t applied
-
[ii].
upon
can be c o n s i d e r e d
CONTEXT-FREE' results
CONTROL.
on the e f f e c t
In the
the r e w r i t i n g
This
is the i n v e s t i g a t i o n
lelism
is bounded,
the d e v e l o p m e n t deprives erating
systems,
w h i c h we g e n e r a l i s e n-parallel
Section
2:
right
of "food".
n-parallel
of a partic-
time. simple m a t r i x or
'RULE-
a survey of
organisms
where
the paral-
to be the i n v e s t i g a t i o n under
We consider
state g e n e r a t o r s
systems
an e n v i r o n m e n t
two d i f f e r e n t
and right
finite
linear
kind of gengrammars,
state g e n e r a t o r s
and
grammars.
finite
A f~nite state generator
state generators.
(FSG)
G,
is a q u i n t u p l e
of
which
Definition
where
of
the s e q u e n c e
'RULE-CONTEXT'
of E O L - l i k e
to give n - p a r a l l e l
linear
independently
(3) for SMG to:
and can be c o n s i d e r e d
finite
this
is U N C O N T R O L L E D .
of f i l a m e n t o u s
the cells
facet
Systems
within
in E0L systems,
either
of C O N T R O L
at the same time.
f o l l o w i n g we p r e s e n t
of c h a n g i n g
(3)
However,
The a p p l i c a t i o n
at a p a r t i c u l a r
to have
Tabled
its context w i t h i n
In the light of these d e v e l o p m e n t s grammars
and
from one table of
is a p p l i e d
that are applied CONTROL
and EXHAUSTIVE,
lead to two kinds
EXHAUSTIVELY
a particular
rules
UNBOUNDED
UNCONTROLLED.
CONTROL
are a p p l i e d
'RULE-CONTEXT' ular rule
in PARALLEL,
(N,T,E,S,F)
294
N
is a f i n i t e
set
points,
of
alphabet,
T is
a terminal
E is
a set
S
N
is a s e t
of
entry
F
N
is
of
exit
Example
edges,
of
a set
E
VxT*xV, points,
and
points.
1 N
>aa
"b
=
{1,2},
T
=
E =
{(l,aa,2),
S =
{i}
and
{a,b},
(2,b,l)}
F =
{2}.
Definition Given (u,x,v)
G,
then
we
write
u÷vx
if u , v
in N,
we
write
u÷ivx,
Xl,...,x i such
that
uj÷uj+iXj+l,
0~j
x = Xl...xi, We write
i>0
u = u 0 and
+
u÷ vx
*
if
there
if t h e r e
exist
sequences
and
if
an F S G
either
u÷ vx
u 0, .... ,u i
v = u i. exists
i>0
such
that
+
u÷ Vx
x in T*
in E.
Similarly, and
a n FSG,
or u = v a n d
x =
e.
i u ÷ vx,
language
The
and we
write
generated
by
G is w
L(G) We
say
L~T
G such the
=
family 1
L(G)
We
u+ vx,
is a
that
Example
{x:
in F}.
finite 8tare language
L = L(G). of F S L ' s
We
usually
by ~
say
L
(FSL) is a
iff
there
exists
regula~ set,
an F S G
so w e
denote
.
(cont.) =
{aa,aabaa,aabaabaa,...}
=
{ a a ( b a a ) i:
can
now
operating
peating
a generalised
informally.
i~0}.
consider
n FSG's
model
u in S, v
n-parallel
synchronously form
of
FSGs.
and the
The
basic
in p a r a l l e l .
notation
above,
idea
Rather we
is than
develop
to h a v e rethe
295
Consider
a 2-PFSG
Initially both
(2-parallel
these
alive and idle.
nously
FSG)
two p r o c e s s o r s
G:
(or generators)
(G 1 and G 2) are
We start them up s i m u l t a n e o u s l y
(by w h i c h we m e a n we m o v e
along edges
and synchro-
at the same time).
Thus we have: point
reached
and w o r d g e n e r a t e d
G1
G2
1
2
la
2b
= 2
laa
2bb
= 3
laaa
2bbb
time = 0 =
1
If the two p r o c e s s o r s then we c a t e n a t e both points
the two words
time = 0
so far.
at the same time,
Here we have,
i
ab
2
a2b 2
3
a3b 3
that:
by G so far
o ~ o
we say L(G)
= {albl:
i~0}.
In a similar way we can g e n e r a t e
languages
called
the family of n-PFSL's.
n-PFSL's.
Let ~
n
denote
We have: Result
i:
Result
2:
since
E
o o o
n~l,
an exit point
generated
1 and 2 are exit points word generated
Thus,
e a c h reach
so far:
~'~= For
~i' n~l,
trivially. ~ n ~ ~n+l"
given by an n-PFSG,
296
Proof:
Simply add an extra p r o c e s s o r Gn+ 1 to G, w h i c h generates n o t h i n g but the empty word, giving an
Result 3: Proof:
(n+I)-PFSG,
i.e.
~-edges do add g e n e r a t i n g power.
C o n s i d e r L = {alb3:
0~i~j}, L cannot be generated w i t h o u t an
c-edge o c c u r r i n g in at least one processor. Result 4:
For n~2, L n = {a!i ...a n i :
i~0} is in ~ n but not in
n-l" Corollary:
For n>l,
~
n ~
~n+l:
an infinite h i e r a r c h y of lan-
guages. Result 5:
~2
~
family of o n e - c o u n t e r languages.
Result 6:
For n~l,
~
Result 7:
For n~3,
~
n n
~
family of c o n t e x t - s e n s i t i v e
languages.
and the family of c o n t e x t - f r e e languages are
incomparable. These results will be found in Wood (2).
Section 3:
V a r i a t i o n s on the basic model of n-PFSG's.
In the previous section,
the t e r m i n o l o g y alive and idle w i t h
respect to n-PFSG's was introduced.
I f we say that an FSG dies w h e n
it reaches an exit point, then a word is o u t p u t by an n-PFSG w h e n e v e r all n processors die at the same time.
active when it is alive and n o t V a r i a t i o n i: cessors
idle.
Given m FSGs, m>0 and n, 0
(at most)
then allow n live pro-
to be active at one time, and no longer require
that n p r o c e s s o r s die together. initially,
We also say a p r o c e s s o r is
A w o r d is g e n e r a t e d by G I , . . . , G m if
they become alive s i m u l t a n e o u s l y and n of t h e m become
active for one time step, then another n become active for the next (2)
Wood, D., Properties of n - p a r a l l e l finite state languages, Utilitas M a t h e m a t i c a 4 (1973), 103-113.
297
time step, and so on until all m processors nate the m words together Let ~ ( m , n )
GI,...,G m. model.
left to right,
are dead.
Then we cate-
to give a word generated by
be the family of languages
defined by this
Then we have:
Result 8:
~
= ~(m,n)
0
That is the model degen-
proceeding
from left to right be-
for all m,n,
erates. Variation
2:
Let the m processors
come active n at a time
(if there are less than n, then the remaining
number of processors),
and all n die simultaneously.
note the corresponding
family of languages.
Result 9:
For m>0, 0
Let ~
0Kq
~
de-
m,n
m,n
=
( ~n)P~q° Result i0:
~
Result ii:
m,n=~ ~
m+l,n'
m,n and
~
3:
0
m,n+l are incomparable,
Note that ~ m , m Variation
for all m,n,
= ~
m and
for all m,n,
0
~ - m , 1 = ~ ' f o r a l l m>O.
We can impose more structure on the n-PFSG by intro-
ducing a traffic cop, who by making use of a book of regulations termines which edges are allowable
de-
edges for the n FSG's at this
time instant. Example2
~
a
b a
~
~
V
e
n
the 2-PFSG G:
with the rule book {<(l,a,l),v(2,a,2)>,
<(l,b,1), (2,b,2)>}
cessor G 1 can only traverse the a-edge whenever verses an a-edge,
thus L(G) = {ww:
usually called a control is denoted ~ Result 12Result 13: (3)
n
set,
processor
w in {a,b}*}.
the corresponding
then proG 2 tra-
The rule book is
family of languages
C
" ~ n ~ ~ n c' for all n>l. ~ n C = ~ [ n ] , the family of n-~ight
see footnote
i.
linear SML (3)
298
These results can be found in W o o d (4)
Section 4:
An a l t e r n a t i v e formulation, n-PRLG's and c l o s u r e p r o p e r ties.
Definition For n~l,
let G =
(NI,...,Nn,T,S,P)
w h e r e Ni, l~i~n, are d i s j o i n t
n o n t e r m i n a l alphabets, T is a terminal alphabet, S is a s e n t e n c e symbol, S not in
n ~N
i = N,
P is a finite set of rules of the form: (i)
S÷Xl...Xn,
X i in Ni,
(ii)
X÷aY, X,Y in Ni, some i, a in T*, and
(iii)
X÷a, X in N i, some i, a in T*.
G is an n-parallel right linear grammar, n-PRLG. We w r i t e x => y iff either x = S and S÷y in P, or x = Y l X l . . . Y n X n , y = Y l X l . . . Y n X n , Yi in T*, X i in N i, x i in T*uT*N i and X l÷ x .i in P, l~i~n. In the usual w a y we obtain x ~+ y and x => y, n o t i c e that either a v a l i d s e n t e n t i a l form, other than S itself, has either no nonterminals or exactly n nonterminals°
L(G) = {x:
S =2 x, x in T*}, and
L ~ T* is an n - P R L L iff there exists an n - P R L G G such that L = L(G). Let the family of n-PRLL's be Result 14: Result 15:
~i
n
= ~. ~n~n,
for all n>l. /1
Result 16:
(4)
C l o s u r e and n o n - c l o s u r e results for ~ n-
and ~ n ,
n>l.
Wood, D., Two v a r i a t i o n s on n - p a r a l l e l finite state generators, M c M a s t e r U n i v e r s i t y CS TR 73/3 (1973).
299
operation
~n
union
No
Yes
homomorphism
No
Yes
finite substitution
No
Yes
substitution
No
No
catenation
No
No
intersection
No
No
complementation
No
No
intersection with regular set
No
Yes
a-NGSM maps (accepting states)
No
Yes
These results
Section 5:
are detailed
~n
in Rosebrugh
Characterisation
Theorems
and Wood (5) and Wood (6) .
for ~
iI and ~
i i.
Definition Let
~
O ~ i and ~ i i = O ~ and <M> denote an i=l i= 1 l infinite sequence MI,M2,... where M ~ i T*. Define L(M) = {x: x in Mi, some i~l}.
ii =
A sequence
<M> is a regular s~quence
FSG G such that L(M) = L(G) Given two sequences
and M
1
= {x:
u+ivx,
iff there exists an u in S, v in F}.
<MI> and <M2> define <MI> ~ <M2>, the
synchronised product of <Ml> with <M2>, as the sequence MIIM21,MI2M22, ....
Define <MI> G < M 2 > ,
<M2>, as the sequence, Result 17:
(5) (6)
union of <Ml> with
MllUM21,MI2UM22,...
For n>0, L i n ~ <M>, 13
the synchronised
l~i~m,
n iff there exist mxn regular
sequences
l(j~n for some m)l such that
Rosebrugh, R. D., and Wood, D., Restricted parallelism and right linear grammars, McMaster University CS TR 72/6 (1972). See footnote 2.
300
L = L(M) where <M> = <MII > Q <MI2> G
°.. <Mln >
~ < M 2 1 > ....
®<Mml>®. O<Mmn> Result 18:
For n>0, L in ~
iff there exist n regular sequences
n
<Mo>I such that L = L(M) where <M> = <MI> ~ .•. Q <Mn >" Definition Let ~ b e
the smallest family of sequences containing the
regular sequences and closed under Q a n d ~ a n d <M>
in~
let L ( ~ )
= {L(M):
}.
Result 19:
Sequence Characterisation of ~
i i.
Definition Let ~ be the smallest family of sequences containing the regular sequences and closed under ~ Result 20:
.
Sequence Characterisation of ~ I I "
Definition An a-NGSM is a nondeterministic
with accepting states. Result 21:
Let L
n
generalised sequential machine
= (all...a i:
Image Theorem for ~ n
n
i~0}.
an__dd~ n
For all L in /~n, n>0 there exists an a-NGSM M such that L = M(Ln). Result 22:
Since ~ n ~ ~ n
the result holds for ~ n .
Language Characterisation of ~ n ° For n>0, ~ taining L
is the smallest family of languages conn and closed under union and a-NGSM maps.
n For further details see Wood (7) and Rosebrugh and Wood (8) .
(7) (8)
See footnote 2. Rosebrugh, R. D., and Wood, D., A characterization theorem for n-parallel right linear languages, Journal of Computer and System Sciences 7 (1973), 579-582.
301
S e c t i o n 6:
C o n c l u d i n g Remarks.
We close by first p o s i n g two open problems and s e c o n d l y introducing some p o s s i b i l i t i e s Open p r o b l e m I:
for future research.
Is the e q u i v a l e n c e of two n-PRLG's d e c i d a b l e or
undecidable? Open p r o b l e m 2:
Prove%J9 i i is not closed under intersection. p
It
4
is easy to c o n s t r u c t examples of languages w h i c h are formed by the i n t e r s e c t i o n of two n-PRLL, w h i c h intuitively are not in ~ some new
(or adapted)
i I.
But
proof techniques are needed.
Future research possibilities
(i)
Can g e n e r a l i s e n - P R L G
(or n-RLSMG)
in the same way that Salomaa
[103] has g e n e r a l i s e d E0L systems. (2)
R e w r i t i n g in E0L systems has been c o n s i d e r e d as a o n e - s t a t e N G S M map,
Salomaa
[103] has c o n s i d e r e d one e x t e n s i o n of this notion,
h o w e v e r it can be e x t e n d e d in another way by a l l o w i n g more than one state in the NGSM. n-PRLG
This e x t e n s i o n can then be applied to
(n-PFSG and n-RLSMG).
MULTIDIMENSIONAL
LINDENMAYER ORGAN!SMS
B r i a n H . Mayoh D e p a r t m e n t of C o m p u t e r S c i e n c e University
of A a r h u s
Aarhus~ Denmark
Summary Most c e l l u l a r o r g a n i s m s in the r e a l w o r l d a r e not o n e - d i m e n s i o n a l . H o w can we model the g l o b a l d e v e l o p m e n t of such o r g a n i s m s by r u l e s that a r e l o c a l to each c e l l ? It i s not u n r e a s o n a b l e to s u p p o s e t h a t the d e v e l o p m e n t of an i n d i v i d u a l c e i l d e pends on the c e l l s t a t e and of the c e l l .
the c e l l c o n t e x t , the s t a t e and p o s i t i o n of the n e i g h b o u r s
In t h i s p a p e r we p r e s e n t m o d e l s in w h i c h t h i s c e l l i n f o r m a t i o n and the
a t t e n d a n t l o c a l d e v e l o p m e n t a l r u l e s can be r e p r e s e n t e d d i s c r e t e l y .
The precise
ma-
t h e m a t i c a l d e s c r i p t i o n of the m o d e l s is left to an a p p e n d i x .
yon Neumann m o d e l s T h e f i r s t c l a s s of m o d e l s we s h a l l d i s c u s s h a v e b e e n much s t u d i e d (e. g. Cellular
automata~ ed. E . F .
tomata~ ed. A . W .
Burks,
Codd~ A c a d e m i c P r e s s ~ I 968~ E s s a y s on c e l l u l a r a u -
1970). T h e d i s t i n g u i s h i n g f e a t u r e of m o d e l s in t h i s c l a s s
is that the p o s i t i o n s of the c e l l s in an o r g a n i s m a r e f i x e d o n c e and f o r a l l . A s an o r g a n i s m d e v e l o p s the s t a t e s of the n e i g h b o u r s of a c e l l may change~ but the n u m b e r and p o s i t i o n s of i t s n e i g h b o u r s may not. T h e local d e v e l o p m e n t a l r u l e s need o n l y s p e c i f y a new c e l l s t a t e f o r each of the f i n i t e l y many p o s s i b l e c e l l c o n t e x t s . In f i g u r e 1 we c o m p a r e a yon Neumann model f o r the d e v e l o p m e n t of the r e d a l g a e C a l l i t h ~ n i l l t l : R o s e u m w i t h the c o r r e s p o n d i n g O L - s y s t e m .
We s u p p o s e that the
c e l l s of the o r g a n i s m tie on the p o i n t s w i t h i n t e g e r c o o r d i n a t e s in 2 - d i m e n s i o n a l E u c l i d e a n s p a c e so each c e l l has 8 n e i g h b o u r s . T h e s e e d , t h e i n i t i a l c o n f i g u r a t i o n of the o r g a n i s m , has the c e l l at the o r i g i n in s t a t e a and a l l o t h e r c e l l s in the " v a c u o u s " s t a t e . T h e c o n t e x t i n d e p e n d e n t local d e v e l o p m e n t a l r u l e s a r e :
a ~ b~ b-~ b~ c 4
b~ d 4 e~ e e f~ f ~ 9, h-~ h~ h ~ ~ h ~ .
T h e c o n t e x t d e p e n d e n t r u l e s a r e g i v e n in f i g u r e 2. If the r e a d e r c o n t i n u e s the d e v e l opment of the a l g a e in f i g u r e t ~ he w i l l a p p r e c i a t e the f o l l o w i n g d i s a d v a n t a g e s of the y o n Neumann model; - g r o w t h and c e l l d i v i s i o n c a n o n l y occur- when t h e r e a r e c e l l s in the v a c u o u s state~
303
-
u n w a n t e d i n t e r a c t i o n b e c a u s e of the s e v e r , e r e s t r , i c t i o n s on gr,owth direction;
-
the i n f l e x i b i l i t y
c a u s e d by h a v i n g a f i x e d g l o b a l l i m i t on the n u m b e r
of n e i g h b o u r s a c e l l c a n h a v e .
Time
Natur,e
L-model a
VN model
o
[]
a
1
~
bc
2
~
bbd
3
Ib\b/e\d I
bbed
/4-
| b \ b/ ~ e / d
S 6
lb~ b / g \ f / e \ d I i b\ b/ h/ gl f/ el d I
I
bbgfed
bc bbd bbed bbfed bbgfed
bbhgfed
bbh(a)gfed
bbfed
a
'7
L\bJ
~f/e~d I
bbh(bc)h(a)gfed
bbhHgfed c
E
8
b
bbh(bbd)h(bc)h(a)gfed
Ib\bl h\ h'~\g/f\e/d I bbhahgfed ba b d
F i ~ l u r e 1. T h e d e v e l o p m e n t of Callitha~iCII!, R o s e u m
P a p e r , s b y S z i l a r , d ( t h i s v o l u m e ) and L i n d e n m a y e r , [ i n 4 5 ] h a v e i n d i c a t e d h o w the v o n N e u m a n n m o d e l c a n be m o d i f i e d so that t h e s e d i s a d v a n t a g e s a r e s o m e w h a t m i t i gated.
Web m o d e l s Our` s e c o n d c l a s s of m o d e l s w a s i n v e n t e d b y J . L . (,Web 9 r a m m a r s ~ P r o c .
J o i n t Int. C o n f .
1969) f o r o t h e r p u r p o s e s .
on A r t i f i c i a l
The distinguishing
P f a l t z and A . R o s e n f e l d
Intelligence~ Washington,
f e a t u r e of m o d e l s in t h i s c l a s s is t h a t
the p o s i t i o n s of the c e l l s in an o r g a n i s m c a n be r e p r e s e n t e d w h e r e t h e r e is an e d g e b e t w e e n t w o v e r t i c e s i f a n d
b y an u n o r d e r e d graph~
o n l y if the c o r r e s p o n d i n g
ar,e n e i g h b o u r , s . ] - h e p r i c e w e p a y f o r r e m o v i n g the r e s t r i c t i o n
cells
on the p o s s i b l e n u m -
b e r of n e i g h b o u r s is t h a t c e l l c o n t e x t s g i v e no i n d i c a t i o n of the r e l a t i v e p o s i t i o n s of cell neighbours. S u p p o s e w e c a n a p p l y a l o c a l d e v e l o p m e n t a l r u l e to a p a r t i c u l a r particular
context.
cell with a
What s h o u l d it g i v e u s ? T h e n a t u r a l a n s w e r is a l a t e n t o r g a n -
304
f
/
~ _,/,' F~7
A p r o b l e m at t i m e 12.
d e
f d g e b b c b b b bbhhhhhhhgfed b b b a b b b ah f d g.Ze f d e d F i ~ l u r e 2. D e t a i l s of the VN model in fi~lure 1.
--'/4
305
ism, a l a b e l l e d g r a p h j u s t l i k e that w h i c h r e p r e s e n t s
the w h o l e o r g a n i s m .
In t h i s
w a y e v e r y v e r t e x o f the l a b e l l e d g r a p h f o r the w h o l e o r g a n i s m p r o d u c e s i t s o w n l a b e l l e d g r a p h and t h e s e must be j o i n e d by some e m b e d d i n g a l g o r i t h m . L e t us l o o k a t f i g u r e 3 w h i c h s h o w s the w e b m o d e l f o r o u r r e d a l g a e . T h e s e e d , is r e p r e s e n t e d by the g r a p h c o n s i s t i n g of a s i n g l e v e r t e x l a b e l l e d by a. A l l the l o c a l d e v e l o p m e n t a l r u l e s a r e c o n t e x t i n d e p e n d e n t , and to be f o u n d in the t a b leau:
i .
.
.
.
.
.
.
.
.
.
State
Latent organism
!o
b-c
tb:l o lel,l
b
e-d,
Time
Graph
0
a
1
b-c
2
b-b-d
3
b-b-e-d
4
b-b-f-e-d
5
b-b-g-f-e-d
6
b-b-h-g-f-e-d
g_
o 'h'
h-alhl
a a
b-b-h-h-g-f-e-d b
"c gc b-b-h-t~-h-g-f-e-d
b
a b, d
d d b" a b-b-h-h-h-~-g-f-e-d b, b, b, c e
"d d
10
e /
d c b' b" t i b-b-h-h-h-h-h-g- f-e-d b b a
"b
b
#
e d F!gure 3. T h e w e b model fop Callith~i01~ R o s e u m .
306
B u t w h a t is the e m b e d d i n g a l g o r i t h m ?
The literature
on w e b g r a m m a r s s h o w s the
c h o i c e of e m b e d d i n g a l g o r i t h m m a y i n f l u e n c e g r e a t l y the e x p r e s s i v e p o w e r of o u r c l a s s of w e b m o d e l s , E x p e r i m e n t i n g following embedding algorithm a particular
with various possibilities
h a s s h o w n t h a t the
is a r e a s o n a b l y s i m p l e and p o w e r f u l c h o i c e . F o r
web model we have a forbidden
l i s t of s t a t e p a i r s .
U s i n g t h i s and a
c o l l e c t i o n of l a t e n t o r g a n i s m s ~ one f o r e a c h v e r t e x in a l a b e l l e d g r a p h , w e b u i l d a new graph by I ) replacing
the v e r t i c e s
by the corresponding
latent organisms$
2) f o r e a c h e d g e ( v , v f) in the o r i g i n a l g r a p h ~ j o i n i n g e a c h v e r t e x in the l a t e n t o r g a n i s m f o r v w i t h e a c h v e r t e x in the l a t e n t o r g a n i s m f o r vl~ 3) d r o p p i n g the e d g e s a d d e d at s t e p 2, w h i c h w o u l d g i v e a s t a t e p a i r in the f o r b i d d e n l i s t . F o r o u r r e d a l g a e the f o r b i d d e n l i s t is:
(b, d), (f, d), (a, b), (e, g), (a, h), (c, h)
in t h i s p a r t i c u l a r
e x a m p l e w e can a l s o use the c o n c e p t of the s k i n of a l a t e n t o r g a n -
ism to g i v e a s i m p l e r e m b e d d i n g a l g o r i t h m Figure
4 illustrates
[691 .
a w e b model f o r the d e v e l o p m e n t of a l e a f . A g a i n the
seled is r e p r e s e n t e d b y the g r a p h c o n s i s t i n g of a s i n g l e v e r t e x l a b e l l e d b y a. A g a i n a l l the l o c a l d e v e l o p m e n t a l P u l e s a r e c o n t e x t i n d e p e n d e n t . T h i s t i m e the r u l e t a b leau is:
I State
1
I
I Latent or~lanism
a b-a
11I111 b
c
d
e
f
c
d-e
d
f
c-c
and the f o r b i d d e n l i s t is: (a, d), (a, e), (b, e), (c, e), (d, e). We n o t i c e ~ t h a t s t a t e s a and d r e p r e s e n t
the primary
and t e r t i a r y
c e l l s of N a ' g e l i
[61 ]. Web m o d e l s h a v e t h e i r w e a k n e s s e s . O n l e a r n i n g a b o u t them A r i s t i d
Linden-
m a y e r c h a l l e n g e d the a u I h o r to r e p r o d u c e t h e w a y n a t u r e c r e a t e s a c l o s e d s u r f a c e . It so h a p p e n s t h a t w e b m o d e l s h a v e no zip m e c h a n i s m - if t w o c e l l s in an o r g a n i s m are disconnected,
t h e n t h e r e is no w a y f o r a d e s c e n d a n t of o n e of them to be l i n k e d
to a d e s c e n d a n t of t h e o t h e r at a l a t e r s t a g e in the d e v e l o p m e n t , i t is p o s s i b l e to i n t r o d u c e the n e e d e d c o n t e x t s e n s i t i v i t y , a p p l y to s u b g r a p h s not j u s t v e r t i c e s -
in m a n y w a y s - e . g .
by a l l o w i n g r u l e s to
but~ as yet~ n o n e o f t h e s e w a y s can be j u s t i -
f i e d b y a c o n v i n c i n g b i o l o g i c a l a r g u m e n t . N o t i c e t h a t the k i n d of c o n t e x t s e n s i t i v i t y n e e d e d to m a k e a b o x is d i f f e r e n t f r o m t h a t w h i c h is a l l o w e d in w e b m o d e l s . T o see the a l l o w a b l e k i n d o f c o n t e x t s e n s i t i v i t y
consider
t h e f i c t i v e o r g a n i s m ~ the IMIAD~
307
Nature
Time
Graph a
0 1
2
\~a
4
f
e c c \/
e
I
d--d
....... b
\d/\o//a I
f e-e \/ d - - d ,
I c
\ /
,,
d c
/
\
e
c
f--f
I
I
d-d %J d
\/\
c.c \1 d
.....
e I d
,I--I"' d--d
I
e--e
Fi~lure
4.
The
web
model
for" Phascum
b~.
/ \//a
c-" f
l
Ouspidatum.
308 Time 6.
~
~
e~ e i
I
d-d f d-- d-c,~ ~d/ \d/ \b a /I
CC OL-representation;
i
e
(d(de) (de))(d(c)(c))(d(f))(de) (c)ba
Time 6 1/2.
Time 7.
--
C~C
'-'G.
. . .
0,dg;.
/\
I
d-d I
e
f I
e-@ F i g u r e 5.,,,, An epoch,, i,n the life of Phascum CuspidautUm.
309
of f i g u r e
6,
I t s d e v e l o p m e n t is n o t m o n o t o n o u s ~
e v e n a l t h o u g h it is o n l y a t w o c e l l
state fish,
Local
d e v e l o p m e n t Pules.
•
gives latent organism
o--o
if 1 or 3 nei9hbours
state state
o
9ires
latent organism
o--o
othePwise
state
o
gives latent organism
o--o
Forbidden
list,
Time
(o, o)
(e, o)
Nature
Graph
C)
1
.-o
Figure
6. T h e d e v e l o p m e n t of an I M I A D .
Map models For
all their virtues
the n e i g h b o u r s Software
of a c e l l .
Engineering
web models cannot represent
Following
Rosenfeld
and Strong
Vo 2,2 e d , J. Tou~ A c a d e m i c
Press
the r e l a t i v e
positions
of
(A grammar for maps, 1 971 } w e c a n i m p r o v e t h e
310
models by allowing
the e d g e s of the u n d e r l y i n g
g r a p h to be o r d e r e d .
This
influen-
c e s t h e m o d e l in t w o w a y s ; the o r d e r i n g
-
-
in t h e c o n t e x t c a n be u s e d to r e s t r i c t
of a p p l i c a b l e
local developmental rules~
the o r d e r i n g
in t h e g r a p h i n f l u e n c e s t h e i n t e r ' c o n n e c t i o n of t h e
latent organisms Figure
? illustrates
France
produced by the local developmental
rules.
t h e s e c o n d of t h e s e . T h e p o i n t to n o t e is t h a t a n y s u b r e g i o n o f
t h a t t o u c h e s Sp~ It, B e must a l s o t o u c h H . T h i s r e s t r i c t i o n
pressed
i f the u n d e r l y i n g
to g i v e a p r e c i s e
graphs into one large ordered
graph.
graphs cannot express
rule for connecting several ordered
Furthermore,
c e l l s w h i c h t o u c h o n e a n o t h e r in s e v e r a l Ordered
c a n n o t be e x -
g r a p h is u n o r d e r e d .
It is v e r y d i f f i c u l t
cells.
the choice
w e s h a l l l a t e r w a n t to c o n s i d e r
places and even cells which enclose other
this~
and we have a problem.
w a s s u g g e s t e d to t h e author` b y J. T h a t c h e r ` : to c o n s i d e r ` an o r d e r e d function from vertices
to c i r c u l a r
a l p h a b e t A is an o r d i n a r y words
w o r d s on the v e r t i c e s .
a precise
description
an a p p l i c a t i o n splitting
letter.
The circular
=
In o r d e r
56 co ando~ 5, t h e n s u b s t i t u t i n g
and s t r a i g h t
network
bisectors
these
is c o r r e c t l y
that draws a picture
formulated,
the o r g a n i s m
as a l i s t of l i s t s f r o m w h i c h it c a n g e n e -
b e f o r e p l o t t i n g (the i n t e r n a l
of t r i a n g l e
edges).
cell boundaries
In t h e c o m p u t e r
are the per-
the l o c a l d e v e l o p m e n t a l the
life stage. Before
l e a v i n g map m o d e l s l e t us c o n s i d e r
s h o w s t h e map m o d e l f o r P h a s c u m C u s p i d a t u m figur,e 5 and also figure UACER
The
b o u n d a r y e d g e s in the c o m p u t e r p l o t s a r e d u e to the f a c t
r u l e s of t h e w e b m o d e l b e c o m e r u l e s to c h a n g e the l i s t of l i s t s that r e p r e s e n t current
a
of the l i f e s t a g e s in an o r -
b y a map m o d e l . F i g u r , e 9 s h o w s t w o s u c h p i c t u r e s .
t h a t the c o m p u t e r r e p r e s e n t s rate a triangular
8 shows
on the r u l e is met b y
in t h e s k i n of the l a t e n t o r g a n i s m .
has been written
ganism that is described
pendicular
f o r map m o d e l s . F i g u r e
The restriction
to c h e c k t h a t t h e e m b e d d i n g a l g o r i t h m
computer program
smooth curves
( l a s t ) letter` is t h e r i g h t ( l e f t ) n e i g h -
of the e m b e d d i n g a l g o r i t h m
of t h i s e m b e d d i n g a l g o r i t h m .
in t h e t w o c e l l s
w o r d on an
of o n e o r m o r e s u b -
w o r d s a r e u s e d in t h e a p p e n d i x to g i v e
the c o n t e x t o f 4 i n t o the t w o w o r d s ,
words for
graph as a
A circular
w o r d A on A e x c e p t t h a t i t c o n s i s t s
t h a t a r e " c y c l e s It in t h e s e n s e : the f i r s t
b o u r o f the l a s t ( f i r s t )
The way out
two more examples.
in a c t i o n .
organs.
10
It s h o u l d be c o m p a r e d w i t h
11~ w h f c h s h o w s t h e map m o d e l f o r a f i c t i v e
that develops internal
Figure
organism
311
Query:
C a n F r a n c e d i v i d e so t h a t o n e p a r t has o n l y rill n e i g h b o u r s and the
o t h e r has o n l y + n e i g h b o u r s ?
+In4 Ge ,-6
~rc~r~c~
/ S~
rh
It +
In a n s w e r i n g t h i s p r o b l e m it h e l p s to c o n s i d e r the c o n t e x t o f F r a n c e as the Hcircularil word,
5e
\
D
Ge
!
s.
~ ...z{ /
F!gure
?. T h e r e l i g i o u s
war problem.
312 Rule Name
Condition
Latent O r g a n i s m
BUD state P G MoPe p r e c i s e d e c r i p t i o n of the latent organism. cell 1
state: p
context: (2)
cell 2
state: s
context: (~)
skin
C2
Restriction S p l i t touchs o~
An a p p l i c a t i o n of the r u l e . Before: L
~
cel I 5
t4~, old context: 6~,~
s,1-2 new context: 6 _ ~
cell 6
old context:
cell 1
I. o. context: ,-~2~ new context: 5 2.~¢.=
cell 2
I.o. context: 1~.~ new context: 1 " : 5
#1% ¢4~5 new context :~,~76~,
After:
F i g u r e 8. R u l e a p p l i c a t i o n in map models.
313
6
S p l i t t i n g o f c e l l 4- pPoduCes c e l l 8 Splitting of cell 7 produces cell 9 S p l i t t i n g o f c e l l 6 p P o d u c e s c e l l 10
0
Figure
9. A n e p o c h in t h e l i f e o f a l e a f .
314
Rules:
Restriction
Name
Condition
BUD
state p
ANT I
state s
s p l i t t o u c h s ~,,once
PER1
state s
s but not s p l i t t o u c h s ~
STABLE
state t
Lat. org.
cs't 0
split touchs~once
NONE
Life history.
F i £ 1 u r e ! 0. M a p mode,! fo r P h a s c u m C,us,Ridatum,
315
Rules
Name
Condition
BABY
slate p
none
SPLIT
slate s
none
HOLE
stale s
none
STABLE
state
t
Let.
org.
@
Restriction
none
Life, history
(D
~
O~6Y, ~p~l r
BAI~,~, ,~p~.~r, ~ Z l r , , ~ o ~
Figure
II.
Internal
organs:
growth
of a UACER
316
Globe models A t l a s t w e c a n i n d i c a t e h o w to m o d e l t h r e e d i m e n s i o n a l a d e m o n i n s i d e o n e o f the c e l l s o f a t w o d i m e n s i o n a l outside world
as a circle~
could model this world
organisms.
organism.
It w o u l d s e e t h e
divided into segments by the neighbouring
as a c i r c u l a r
c e l l s of a t h r e e d i m e n s i o n a l
word.
organism,
model this world
using spherical
Revive
It w o u l d s e e the o u t s i d e w o r l d
words
demon for a moment.
circle
into two segments by picking
by adjoining a new segment (consult figure
a circle~
in t h r e e d i m e n s i o n s
then convert
two points~ then convert
There
Just as the circular
Nothing butamap
we described
graphs and functions
t a i l in [ 6 9 1 .
each segment into a
words
We f a v o u r a m o r e s c h e m a t i c a l
words of our map
that represent
words.
All
segments~
representation biological
representations:
t h i s is d i s c u s s e d
mathematical description
is c o m p l i c a t e d ;
- the author cannot draw three dimensional
clouds.
in the p r e l a b e l led in m o r e d e -
of the m o d e l h e r e b e c a u s e :
organisms
k n o w n in d e t a i l ; the p r e c i s e
circle
that represent
of the kind we have discussed
two possible discrete
into circular
- for very few three dimensional
-
The anal-
into two clouds by picking
w o r d s c a n be d e f i n e d in t e r m s of f l a t w o r d s
But what is a flat word. vious section.
of s p h e -
e a c h c l o u d to a s p h e r e b y a d j o i n i n g a n e w c l o u d .
m o d e l s c a n b e d e f i n e d in t e r m s of o r d i n a r y
ordered
It could
It c o u l d s p l i t a
8 a n d the a p p e n d i x ) .
is to s p l i t a s p h e r e
How do we make all this discrete?
s o the s p h e r i c a l
as a s p h e r e ~
cells.
if o n l y w e h a d a s u i t a b l e d e f i n i t i o n
the two-dimensional
circle
ogous process
cells~ and it
N o w i m a g i n e a d e m o n i n s i d e o n e o f the
divided into clouds (regions~ contact areas) by the neighbouring
rical words.
Imagine
organisms.
is the development
317
Appendix
The mathematical
description
of the models
We d e f i n e a m o d e l c l a s s to be a t r i p l e
1) 2) 3)
consisting
of:
a s e t of c o n f i 2 1 u r a t i o n s p a c e s ; a set of rule repertoires; an a l g o r i t h m
~which~
a configuration
given a rule repertoire
space
r ~ provides
A m o d e l in such a c l a s s is (r~ R , rule repertoire~
Y0) w h e r e
a n d Y0 is an e l e m e n t o f
yl a r e c o n f i g u r a t i o n s
in
r
a binary
r
R and relation
on F.
is a c o n f i g u r a t i o n
space~ R i s a
called the seed of the model.
If y a n d
r~ w e w r i t e :
y _~ yt
for
y => y l
f o r the t r a n s i t i v e
~ (R~ r ) h o l d s on ( y , y ' ) ; closure
of
-4, o
We d e f i n e t h e l a n g u a g e of the m o d e l to b e the s e t o f c o n f i g u r a t i o n s
y s u c h that
Y0 =>Y. In a l l o u r m o d e l s w e w i l l h a v e s e t s : Structure~
State~ Context~ Latent Organism
and t h e s e s e t s w i l l b e such t h a t :
4)
State
5)
the c o n f i g u r a t i o n
is f i n i t e ; space
I ~ of t h e m o d e l is the s e t of l i n e a r
w o r d s on S t a t e x S t r u c t u r e ; 6)
the rule repertoire
R o f the m o d e l is a f i n i t e s u b s e t o f
S t a t e x 2C ° n t e x t x L a t e n t O r g a n i s m .
The configurations
a r e d e n o t e d b y (EB~ L X D)~ a n d t h e r u l e s a r e d e -
n o t e d b y (S~ P~ 1). F o r integers~
t h i s to m a k e s e n s e B m u s t be a s u b s e t of
L m u s t be a f u n c t i o n f r o m B to S t a t e ~ D must be a f u n c t i o n
f r o m B to S t r u c t u r e ~
S m u s t be in S t a t e 3 P m u s t be a p r o p e r t y
of
c o n t e x t s ~ and I m u s t be a l a t e n t o r g a n i s m .
Before we fill
in t h i s g e n e r a l f r a m e w o r k ~
we shall need. A circular
word on a set
n i t e s e t o f i n t e g e r s ~ 11' i s a p e r m u t a t i o n We d e n o t e t h e e m p t y c i r c u l a r w o r d on X .
l e t us i n t r o d u c e
X is a t r i p l e
a few concepts
( B , ~'~ F)~ w h e r e B is a f i -
o f B~ a n d F i s a f u n c t i o n f r o m B i n t o X .
w o r d on X b y ~. S u p p o s e C
i f ft is t h e i d e n t i t y p e r m u t a t i o n ~
= (B~ I/~ F ) is a c i r c u l a r
w e c a l l C a l i n e a r w o r d on X .
If G i s
318
a function from X toY,
then we write
G • C f o r ( B , ft', G • F ) ,
a circular
word on
Y . N o w w e c a n c o m p a r e the f o u r m o d e l c l a s s e s d i s c u s s e d in t h e p a p e r :
model class
yon Neumann
Structure
Integer
Context
State m
subset of Integer
m
m u l t i s e t on State
Configuration Space
Latent State
Organism
Map
Web
Globe
circular words on I n t e g e r
circular w o r d s on c i r c u l a r w o r d s on Integer
circular words on S t a t e
circular circular
Configuration Space x Restriction
words on w o r d s on State
Configuration Space X Restrict ion
The yon Neumann model class In a m o d e l of t h i s c l a s s w e h a v e an i n t e g e r rn, and a f i n i t e s e t S t a t e w i t h a d e s i g n a t e d e l e m e n t SO such t h a t : S t r u c t u r e
= I n t e g e r m, C o n t e x t = S t a t e m, L a t e n t :
o r g a n i s m = S t a t e ° In a d d i t i o n w e h a v e d e s i g n a t e d f u n c t i o n s 6 z , 62 . . . t e g e r to I n t e g e r . C o n t r a r y
to w h a t o n e m i g h t e x p e c t , m
6 m from In-
is t h e n u m b e r o f e l e m e n t s
in the c o n t e x t and t h i s is t o t a l l y u n r e l a t e d to the s p a t i a l d i m e n s i o n a l i t y of the o r g a n i s m t h a t w e a r e t r y i n g to m o d e l . A n o t h e r u n u s u a l f e a t u r e o f o u r f o r m u l a t i o n o f a v o n N e u m a n n m o d e l is the d e f i n i t i o n of ( B , L x D)-~ (B I, L I x D I ) . F o r
this we re-
quire 1) B 1 = B U
[ y I x = 6 ~ ( y ) f o r s o m e x E B and s o m e i];j
2) D ' ( y ) = ( 6 1 ( y ) ,
6~(y),...,
6re(Y) f o r y E B I ;
3) f o r e a c h y E B 1 t h e r e is a r u l e ( L l l ( y ) ,
P ( y ) , L l ( y ) ) such t h a t
P ( y ) h o l d s of the c o n t e x t ( L I, 0 6:l.(y), L I, o 6 2 ( Y ) , . . , L t i c ~ ( y ) ) w h e r e L l l ( y ) = i f y E B t h e n L ( y ) e l s e so .
Example
I
m = I, 61(n) = n - 1 . There
are three states: a, b, s o.
T h e seed is [ [17], L o × D o ] w h e r e
I_o(17)= a, D o ( l ? ) =
is: (a, t r u e , b), (b, t r u e , b), ( s o , t r u e , a), (So, t r u e , rations
y
16. T h e R u l e repertoire
So). T h e r e a r e t w o configu~-
such t h a t s e e d • y :
[ [17, 18],
L 1 x Dz~ w h e r e L 1 ( 1 7 ) = b, E l ( 1 8 ) ---a, D 1 ( I ? ) = ( 1 6 ) , D$ (]t8)=(17);
[ [17, 18],
L e X D~] where L e(l?)=b,
E l ( 1 8 ) = S O , D z ( l ' 7 ) = ( 1 6 ) , Dz ( 1 8 ) = ( 1 7 ) .
319
Example 2
(ConwayJs life)
Suppose the points with integer coefficients
in t w o - d i m e n s i o n a l
Euclidean
s p a c e a r e n u m b e r e d in s o m e w a y . T h e n w e h a v e a | :1 f u n c t i o n J f r o m s u c h p o i n t s onto Integer,
a n d we c a n d e f i n e 61, 6 2 , . . . ,
O8 a s t h e f u n c t i o n s f r o m I n t e g e r to I n -
teger given by:
6~(u(x,y)) = J(x-I ,y) 63 ( d ( x , y ) ) = u ( x - i
,
62 (U(x, y)) = J ( x + | , y)
, y-1 ) ,
64 ( J ( x , y ) ) = J ( x + ! , y - i )
65 (u(x, y)) = J(x-1 , y+l ) ,
66 ( d ( x , y ) ) = d ( x + l , y + l )
67 (d(x, y)) = J(x, y+1 )
68 ( J ( x , y)) = U(x, y-1 )
,
Suppose there are two states: sl, (So, B O R N ,
s l ) , (So, m B O R N ,
So. S u p p o s e t h a t t h e r u l e r e p e r t o i r e
Sl) , (Sl,
LIVE,
B O R N C L I V E ) i s t r u e of c o n t e x t s c o n t a i n i n g
s~.), ( S l , m L I V E ,
3 (2 o r 3) o c c u r r e n c e s
is:
s o) w h e r e o f t h e s t a t e s 1.
L e t us d e f i n e t h e s e e d a s ( B o , L o x D o ) w h e r e
B Q
~=
[J(0,0), j(0,1), J(O,2), J0,0), J0,1), j(1,2), j(2,1)].
Lo(x) = s1
for
x E Bo ;
Do ( x ) = ( 6 1 ( x ) , 6 2 ( x ) , . . .
This time there is only one
y
, 6a(X))
such that
for" x e
80-
S e e d - - ) y.
It is (IB1, L 1 x 13:) w h e r e :
B~
= [d(i,j)
L l(x)
= if x = J(0,0) V x = d(0,2) V x = J(-|,l) V x=
] -1 < i , j < 37 - [ J ( 3 , - 1 ) ,
J(2,0) V x = J(2,1) V x=
D ~ i x ) = ( 0 1 ( x ) , O~(x) . . . .
, Oe ( x )
J(3,3)~;
J(2,2)
for xe
then s 1 e l s e so;
B I.
The Web model class in a m o d e l of t h i s c l a s s w e h a v e a f i n i t e Structure
= a subset of Integer,
to I n t e g e r ,
Latent Organism
finite
Context
=Configuration
s u b s e t of S t a t e 2 c a l l e d t h e f o r b i d d e n
( B , L X D)
set State
such t h a t :
= a s e t of f u n c t i o n s Space.
set
By definition
we have:
-~ ( B I, L I X D I)
if a n d o n l y if t h e r e is a f u n c t i o n
p
f r o m B to L a t e n t O r g a n i s m
and a functionT
B to s u b s e t s o f I n t e g e r ;~" s u c h t h a t :
1) if x , y E B ,
from State
tn a d d i t i o n w e h a v e a
then y ~ 1 ( x ) a n d ' r ( y )
is d i s j o i n t
fromT(x)~
from
320
2) f o r each x E B t h e r e is a r u l e ( L ( x ) ,
P ( x ) , p ( x ) ) such t h a t P ( x ) h o l d s
o f the c o n t e x t C ( x ) g i v e n by: C ( x ) ( s ) = n u m b e r of y E D ( x ) such t h a t L ( y ) = s; 3) l"(x) h a s the s a m e c a r d i n a l i t y 4) ( B ' ,
as the f i r s t
L.~ X D ~) = QJ(B, D , p , I", f o r b i d d e n
Clearly
component ofp(x); set).
w e m u s t d e f i n e the f u n c t i o n CLF. F o r
organismp(x) fined order B II
= (Bx~ Lx x D x ) . preserving
function
= u n i o n of T ( x )
each x E B we have a latent
B y c o n d i t i o n (3) t h e r e i s a u n i q u e l y d e h f r o m T ( x ) o n t o Bw. We d e f i n e :
for x E g
L t l ( y ) = L z o hx(y) f o r y E T(x) D"(y)
= hx - z
DIII(Y ) :
Dx
o
hx(y) for y E t(x)
o
[ Y ' I (3 x ' ) ( x I e D ( x ) A yl E T ( x ' ) A (Ltl(y),
Ll~(yl)) ~Eforbidden set)]
(J./(B, D, p , I", forbidden set) = (8 l ' , L ~t X (O 11U Dl1~)). E x a m p l e .3 T h e r e a r e t w o s t a t e s : a~ b. T h e f o r b i d d e n T h e S e e d is ( [ 1 ] ~
s e t c o n s i s t s of: (a~b)~ ( b , a ) .
Lo x D o ) w h e r e L o ( 1 ) = a~ Do (1) = ~ .
We s h a l l n e e d t w o o t h e r
configurations:
a = ([2,3],
L e x D(z) w h e r e Lrv(2) = b, Dcz(2) = [ 3 ] L(:Z(3) = a~ D0((3) = [ 2 ] ;
/3 = ( [ 4 ] ,
LI3 X D/3)
w h e r e LI~(4) = b, D~(4) = ¢ .
N o w w e c a n g i v e the R u l e r e p e r t o i r e : b e t of the language~ w e c h o o s e function
h1
B"
p
(a~ true~ 0~)~ (b~ true~ l~). T o f i n d a n e w m e m -
and 1" such t h a t p ( 1 ) = ~ ,
T(|)=
~'7,9~. O u r
b e c o m e s : h 1 (7) -- 2, h z (9) = 3, a n d w e g e t
= [% 9]
L"(?) =
L(~(2) = b, L " ( 9 ) = L(~(3) = a
D"(?) =
hl - I o DOz(2)= [ 9 ] , D " ( 9 ) = h l - I o Dc((3) = [ ? ]
Seed -~ (B 11, L ~ x D11).
Example 4 T h e r e a r e t h r e e s t a t e s : a, b, e. T h e f o r b i d d e n T h e S e e d is ( [ 1 , 2 , 3 , 4 ] , Lo(1) = b Do(i) = [2,3],
set consists of (a,b),
Lo X D O ) w h e r e : , Lo(2) =b Do(2) = [1,3],
, Lo(3) = e Do(3) = [t,2,4],
, Lo(4) = a Do(4) = F3].
(b,a).
321 We shall need t h r e e o t h e r c o n f i g u r a t i o n s : 0¢ = ( [ t ] ,
LO~ X DCZ) w h e r e L0~(I ) = a, DC~(1) = ¢
= ( [ 1 7 , L~ X D E) w h e r e L~(1) = b, D/3(1) = ¢ ( = ([I,2],
L ( X D ( ) w h e r e L ( ( 1 ) = L ( ( 2 ) = b, D¢(1) = [ 2 ] , D¢(2) = [ 1 ] .
Now we can g i v e the R u l e r e p e r t o i r e : (a, t r u e , i~), (b, trUe~ ~), (e, true~ ~). T o find a new member of the language, we choose /3 and T such that: p(1) = ~
r(T)=Es] Our-functions hi,
/3(2) = ~
p(3) = ~
p(4) = a
~(2)=[6]
r(3)=[7,8]
r(4)=[9].
t ~ , ha, h~ become: h i ( 5 ) = 1
= h2(6) , h a ( 7 ) = 1, ha(8) = 2,
h~(9) = I , and we get: B" = [5,6,?,8,9 7
L " ( 5 ) = L~(1) = b
D"(5) =~
D'"(5) = [6,7,8]
L"(6) = L~(1)=b
D " ( 6 ) =(~
D'"(6) = [5,7,8]
L " ( 7 ) = L¢(1 } = b
D"(7) = [8]
D'"(7) = [5,6~
L " ( 8 ) = L (2) = b
D " ( 8 ) = [?qj
D'"(8) = [5,6]
D " ( 9 ) =(~
D ' " ( 9 ) =~)
U'(9) = Le((1)=a Seed-~ ( [ 5 , 6 , ? , 8 , 9 ] ,
L"
X(D" U D'")).
T h e Map model c l a s s In a model of this c l a s s we h a v e a f i n i t e set S t a t e such that: S t r u c t u r e =
c i r c u l a r w o r d s on I n t e g e r , ganism = C o n f i g u r a t i o n Restriction
Context
= c i r c u l a r w o r d s on S t a t e ~
space x R e s t r i c t i o n = subsetsof Integer
Latent Or-
where 2
x State.
By d e f i n i t i o n we h a v e (B U {0} , L x D) 4 (B' U {0} , L ' x D ' ) if and o n l y if t h e r e is afunction
p
f r o m B to L a t e n t O r g a n i s m , a function(~ f r o m I n t e g e r
g e r ~ )~
and a f u n c t i o n 1. f r o m B to subsets of I n t e g e r
2 to ( I n t e -
such that:
| ) if x , y E B, then 0, y (~ 'r'(x) and T(x) is d i s j o i n t f r o m T(x); 2) f o r each x E B t h e r e is a r u l e ( L ( x ) , P ( x ) , p ( x ) ) such that P(x) h o l d s of the c o n t e x t L *
D(x);
3) 1.(x) U {0} has the same c a r d i n a l i t y as the f i r s t component of the f i r s t component of p ( x ) ; 4) Or is c o n s i s t e n t w i t h r e s p e c t to (B, L, D, p , I') (see note b e l o w ) ;
5) (B' U {0}, L' X O') = ~ (B, L, D, p, 0", "r).
322
C l e a r l y we must d e f i n e the f u n c t i o n ~
. FoP each x E B w e h a v e a l a t e n t o r g a n i s m
p ( x ) = ((gx~ L~ x D~), r x ) . B y c o n d i t i o n (3) t h e r e i s a u n i q u e l y d e f i n e d 1:I o r d e r preserving function
hx
f r o m T(x) U {0} onto B x . We d e f i n e :
B 11
= union of T'(x)
for x E B
L"(y)
= L x o hx ( y )
f o r y E l"(X)
D I I ( Y ) = hx - z * (Dx o h x ( y ) ) f o r
y E 1"(x).
N o w we h a v e to d e f i n e the p a t c h i n g t o g e t h e r of a n u m b e r of c o n f i g u r a t i o n s .
For
each x E B~ y E "r(x) w e d e f i n e (see n o t e b e l o w ) : w(y)
= Flatten
(or~, * D ( x ) ) w h e r e 0 " y ( x w) = 0 " ( y , x ' )
C)t11(y) = S u b s t i t u t e
(w(y)~ DI1(y))
If we set D~'i(0) = ~ a n d L~'(O) = L(0), then we can define /~ (B, L , D, p , 0", I') to
be (B '1W t 0 } ,
L",
Dr").
Note A f t e r a c a r e f u l s t u d y of the t w o e x a m p l e s we a r e a b o u t to give~ the e a g e r r e a d e r s h o u l d be a b l e to g i v e a p r e c i s e d e f i n i t i o n o f : F l a t t e n ~ S u b s t i t u t e ~ 0r is c o n s i s t e n t w i t h r e s p e c t to (B~ L~ D~ p , 1.). H e r e w e w i l l o n l y m e n t i o n : -
the most i n t e r e s t i n g c o n s i s t e n c y r e q u i r e m e n t on or is that w e may not h a v e ( h x - l ( i ) ,
h x - z ( . j ) , L ( y ) ) in r x when w e have
i , j E 1.(x) A O r ( I , j ) = y f o r x , y E B; - F l a t t e n reads around a c i r c u l a r w o r d , and it produces l i n e a r
w o r d s a s it p r o c e e d s ; -Substitute
( w , c ) r e p l a c e s o c c u r r e n c e s of 0 in the c i r c u l a r
word C
by l i n e a r W o r d s f r o m w . E x a m p l e 5 is the d i s c r e t e v e r s i o n of f i g u r e 8. L e t us b e g i n by i n v i t i n g the r e a d e r to d r a w the f o l l o w i n g c i r c u l a r Do(0)
= D"'(0)
Do(4)
=([0, I,2],
w o r d s on I n t e g e r :
=~
~ o , v~.)
w h e r e 1i"o(0) = 1, 1/o (1) = 2, 11'o(2) = 0 v~ ( o ) = o , v~ ( I ) = s ,
v~ ( 2 ) = 6
Do(5)
=([0,1,2],
ITo, v5) w h e r e vs(0) = 0 , vs(1) = 5 , vs(2.) = 4
Do(6 )
=([0,1,2],
~ o , v6) w h e r e v6(0) = 0, v6(1) = /-4, '46 ( 2 ) = 5
D~(0)
= ([0,1], ~T(~, Vo)
D (I)
=([0, I],
1Trv, V l ) w h e r e v l ( 0 )
=0,
vz(1) =2
Dr/(2)
= ([0,1],
II"rv, v 2 ) w h e r - e v2(0 ) = 0 ,
v~(1) =-I
wherelTo~(0) = l , 'r/(~(|) = 0, vo(0) = 1, Vo(1) = 2
323
O~(0)
= (ill,
identity,
identity)
D#(1 )
= ([0],
identity,
identity)
DwII(1 )
= ([0,1,2],
11'0, v { ) w h e r e v { ( 0 ) = 0, vj[(1) = 2, v~[(2) = 8
D,1'(7)
= ([0,1,2,],
11"o, v~) w h e r e vJ/(0) = 0, v.~(1) = 8, v.~(2) = 2
DirT(2)
= ( [ 0 , I , 2 , 3 3 , ~', v~) w h e r e ~Tl(0) -- 1, frl(1 ) = 2, 1T1(2) = 3, ~'1(3) = 0 v~, (o) = o, v~, ( i ) = ?, v~, (2) : 8, v~, (3) = i
= ([0, I , 2 , 3 ] ,
DI"(8)
#', v~)
w h e r e v&(0) = 0, v ~ ( l ) = 1, v ~ ( 2 ) = 2, vsm(3) : ?. N o w w e c a n g i v e the m o d e l . T h e r e a p e f o u r s t a t e s : p, s, t, s o . T h e S e e d is ([0,4,5,6],
L o x D o ) w h e r e L o ( 0 ) = So, 1_0(4) = p, 1_o(5) = t, 1_o(6) = t. We s h a l l
need two other configurations:
e : ( [ 0 , 1 , 2 ] , L e x De ) where Le(0) : So, Ltv(1) = p, Le(2) : s = ([0, I ]
, L~ x DE) where L~(0) = So, L~(1) = t.
The rule repertoire
is: (p, t r u e , (t,
true,
(e, [ ( 2 , 1 ,So)])) , (t3, (~Z~)).
T o f i n d a n e w m e m b e r o f the l a n g u a g e , w e c a n c h o o s e the f u n c t i o n s
p , 1" g i v e n by:
p(4) = (e, [(2,1 ,So)]), p(5) : p ( 6 ) = (#,(~), I'(4) = [I , 2 ] , I"(5) = [ ? ] , I'(6) = [ 8 ] . Our- f u n c t i o n s h e
t%
he
b e c o m e h~.(1) = 1, h4(2) = 2, h5(?) = 1 ,
he(8) = I,
and we
get: B II
= [1,2,7,8]
L'(0)
L"(1)
= Le(1) =p
D I ' ( I ) = D(~(I )
L"(2)
= So
: L_(2) = s
D"(2)
= De(=)
L " ( 7 ) = L/3(1 ) = t
D"(7)
= h5 - z • D ~ ( 1 ) = ( [ 0 ] ,
L"(8)
D"(8)=h6-Z
= L~(1) : t
N o w w e p a t c h the s t r u c t u r e s
* D~(1)=([O],
t o g e t h e r u s i n g a f u n c t i o n 0. s a t i s f y i n g :
id, id) i d , id)
0"(1,2) = 0,
0"(2, 1 ) = 6, 0"(t, 5) = A, or(1,6) = 8, cr(2, 5) = 7, 0 ( 2 , 6) = 8, ~ ( 7 , 6) = 8, c,(7, 4) = 2, 0"(8,4) = 12,, 0"(8, 5) = 7. We n o t e t h a t the o n l y striction
is satisfied
b e c a u s e L ( 6 ) $ s o , a n d c o n t i n u e by c o m p u t i n g :
w(1) = 8, w ( 2 ) = 7 8 ,
Substituting
a p p l i e d Pule w i t h a n o n - e m p t y r e -
w ( 3 ) = B2, w ( 4 ) = 1 27.
t h e s e in D ~l g i v e s : Seed
-* ( [ 0 , I , 2 , 7 , 8 ] ,
Example 6 shows other features P e a d e r to d r a w the f o l l o w i n g
L it, Dltl).
o f map m o d e l s . A g a i n w e b e g i n by i n v i t i n g
circular
words
in i n t e g e r :
the
324
Do (0) Do(11)
Do (1 ~.) Do(1 3)
Do (1 4)
= D'"(0)
=
= ([0,12,,13,14], /r11, identity) where1/~.:l(0) = 14, 1/~.1(14) = 12, 1/19.(12) = 13, 17'11(13) = 0 = ( ~ 1 1 , 1 3 , 1 4 ] , / r ~ , identity) w h e r e 1/:12(1 1 ) = 1 4, /r~ (1 4) = 1 3, 1/z2 (1 3) = 1 1 = ([0,11 ,1 ~-,14], 1/,~, identity) w h e r e /rz3 (O) = 11, 1/1a(11) = 12, 1/~.a(12) = 114., 1/3.3(14) = 0 = ([0,11 , 1 2 , 1 3 ] , 1/za, identity)
D~(~)
= ([4,5~, = ([0,5],
D/./(5) D/3(0)
= ( [ 0 , 4 ] , 1/~5' identity) where 1/cZ5(0) = 4, 1/C~s(4) = 0 = (~1,2,,3], rf~o , identity)
D~(0)
1/(Zo' identity) where1/~o (4) = 5, 1/~o(5) = 4 1/(~/., identity) where 1/rv (0) = 5, 1/n//=(5) = 0
D/~(~)
where/r~o(1) = = ([0,2,3], w h e r e /r~j. (O) = = ([0,1 ,3], 1//~2'
D/~(3)
where1//~2 (O) = 3, 1//5~(3) = 1, 1/#s(1) = 0 = ( [ 0 , 1 , 2 , ] , 1//5a, identity)
D~3(1 )
D~(0) DE(?) DE(8)
D~(0)
D6(6) D0(9) D~1~(1 )
D'II(2,) Dii'(3) D~(4) Dill(5) D'"(6) D"I(?) D'"(8)
2, 17~o(2) = 3, I/#o(3) = 1
9"f~., identity)
2, /r#l(2) = 3, 1//31(3) = 0 identity)
where ~/33(0) = 1 , 1//~3 (1) = 2, IT/~s(2) = 0 = ([7,8] 1/~o, identity) where /rEo (7) = 8, 1/(o (8) = ? =([0,8~ ~ , identity) where fr(.t(0) = 8, 1/¢7 (8) = o
s(O)
1/~ , identity) where/rE, = 7, I/(e (?) = o =([6,9] 1/0o' identity) where1/6o (6) = 9, /r6o (9) = 6 = ( [ o , 9] % e ' identity) w h e r e / r 0 6 ( 0 ) = 9, 11'06(9) = o = ([o,6] %, identity) where/toe(O) = 6, /rOe (6) = o 4 9 5 , 7 , 8 ] , 1/:~, identity) = ([2,3, w h e r e ~r~(8)=7, 1/~(7)=4, 1/~(4)=5, 1/~(5):2, 1/11(2)=3, I/~ (3)=8 = ([1,3, 5], ~/~, identity) where /r2' (5)=3, 1/~,(3)=1, 1/, (1)=5 ([1,2, 5 , 6 , 7 , 8 ] , 1/~, identity) where 7G(5)=6, /r3'(6)=7, /r~(7)=8, 1/,(8)=1, 1/~(1)=2, 1/~(2)=5 = ( [ 0 , 1 , 5 , 7 ] , 1/~, identity) w h e r e ~,(o)=5, 1/~.(5)=1, 1/,~(1 )=7, ~.(7)=o = ( [ o , 1 , 2 , 3 , 4 , 6 ] , /r~, identity) where 1/~ (0)=6, I1"~(6)=3, 1/~ (3)=2, ~ (2)=1 , 1/~ (1)=4, 1/~ (4)=0 = ( [ 0 , 3 , 5 , 7 , 9 , 1 0 ] , 1/~, if i=10 then 0 else i) where I/6I(0)=7, 1/~(7)=3, 11'e(3)=5, 1/4(5)=I0, 1/~(10)=9, 1/~(9)=0 = ([0,1 , 3 , 4 , 6 , 8 ] , 1/~/, identity) where/r-~ (0)=4-, 1/~ (4)=I , l/J/(1 )=8, 11"~(8)=3, 1/.~(3)=6, 1/.~(6)=0 = ([1,3,?], /1'81, identity) where l/'~ (3)=7, 1/&(7)=I, 1/&(1 )=3 =([0,7]
=
325
D'"(9)
= D6(9).
N o w w e c a n g i v e the m o d e l . T h e r e a r e f i v e s t a t e s : a, b, c, d, So. T h e S e e d is ( [ 0 , 1 1 , 1 2 . , 1 3,14],
I-o X D0) w h e r e L 0 ( 0 )
=So,
L o ( 1 1 ) = a, L o ( 1 2 ) = b ,
Lo(13) =c ,
L o ( 1 4 ) = d. We s h a l l need f o u r o t h e p c o n f i g u r a t i o n s - "
= ([0,4,5],
L~ × D~)
where
L
where
L ~ ( 0 ) = so, L ~ ( 1 ) = L/~(2,)=L/3(3) = b
L( X D()
where
L
L 6 X D 6)
where
L 6 ( 0 ) = So, L 6 ( 6 ) = L 6 ( 9 )
# = ([0,1,%3], ~"
= ([0,?,8],
6 = ([0,6,9], The rule repertoire (d, t r u e , tionsp
(6,¢)).
L # X D E)
is: (A,
true,
(0) = So, L.rv(4) = L ~ ( 5 ) = a
(0) = s o , L ( ( ? )
(a¢,.¢)), (b, t r u e ,
(~,¢)),
= L((8) = c
(c, t r u e ,
= d.
((,¢)),
T o f i n d a n e w m e m b e r o f the l a n g u a g e , w e c a n c h o o s e the f u n c -
and I" g i v e n by: p(11)=(~,¢)
p(12)=(/3,¢)
r(11 ) = [ 4 , 5]
1-(1 ~.) = [ 1 , 2 , 3] ~-(1 3) = [?, 8] 1-(1 4) = [ 6 , 9 ] .
With this choice 8"
hll,
h~,
p(13)=(~,¢)
p(14)=(6,¢)
h l a , hi, ~ a r e i d e n t i t y m a p p i n g s and w e g e t :
= [1 , 2 , 3 , 4 , 5 , 6 , ? , 8 , 9 ]
L"(0)
= so
L"(1)
= L/3(1 ) = b
D"(1)
= D/3(1)
L"(2)
= L/~(2) = b
D"(2)
= D~(2)
L"(3)
= L~(3) = b
D"(3)
= D/3(3)
L"(4)
= Lrv(4) = a
D"(4)
= Dry(4)
L"(5)
= Le(5) = a
D"(5)
= Dry(5)
L 1'(6) = L 6 ( 6 ) = d
D"(6)
= D6(6)
L"(?)
= L((?)
D'I(?) = D((?)
L"(8)
= L¢(8) = c
= c
D"(8)
L'T(9) = L6(9) = d N o w w e p a t c h the s t r u c t u r e s
= D (8)
Dll(9) = D6(9) together
0"(2, 3 ) = 11, 0 " ( 1 , 3 ) = 13, 0 . ( 4 , 5 ) =
u s i n g a f u n c t i o n 0. s a t i s f y i n g :
cr(1 ,9.) = 11,
0, 0 ( 5 , 4 ) = 12., 0.(6, 9 ) = 0, 0 . ( 9 , 6 ) =
0"(8,7) = 12, 0"(1,11) = 45, o'(1,14-) = A ,
0, 0 " ( 7 , 8 ) = 12,
0"(1,13) = 87, 0"(2,11) = 5, 0"(2,14) = A,
0"(2, 1 3) = A, 0"(3, 11 ) = 5, 0.(3, 1 4) = 6, 0"(3, 1 3) = 78, 0.(4, 1 4) = A, 0"(4, 1 2) = 1, 0"(4, 1 3) = 7, 0"(5, 1 4) = 6, 0 ( 5 , 1 2) = 321, 0 ( 5 , 1 3) = A, 0"(6, 1 3) = 7, 0 ( 6 , 1 2) = 3, 0"(6,11) = 5, 0"(7,11) = 4, 0"(7,12) = 1, 3, 0"(7,14) = 6, 0 . ( 8 , 1 1 ) = A , 0.(8,14) = A ,
0"(9,13) = A, 0 . ( 9 , 1 2 ) = A, 0.(9,11) = A .
restricitons
and' c o n t i n u e by c o m p u t i n g :
0"(8,12) = 13,
We n o t e t h a t a l l r u l e s h a v e e m p t y
326
wO ) - 8745 w(4) = 1 ? w(6) = ?35
w ( ? ) = 41~ 36
Substituting
, w(2,) = 5, w ( 3 ) =- 5678 w ( 5 ) = 632,1 , w(9)= w(8)
A = 31 ,
t h e s e in D ~ g i v e s ;
Seed
4, ( [ 0 , 1 , 2 , , 3 , 4 , 5 , 6 , ? , 8 , 9 ] ,
L",
D"').
]BIBLIOGRAPHYt BIBLIOGRAPHY ON L SYSTEMS prepared by K. P. LEE Department of Computer Science State University of New York at Buffalo 4226 Ridge Lea Road Amherst, New York 14226 U.S.A. G. ROZENBERG Mathematical
Institute
Utrecht University
Department of Mathematics and
Antwerp University,
Utrecht - De Uithof
Universiteitsplein
The Netherlands
2610 Wilrijk,
UIA 1
Belgium
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