Systems of reductions

Lecture Notes in Computer Science Edited by G. Goos and J. Hartmanis

277 B. Benninghofen S. Kemmerich M.M. Richter

Systems of Reductions I

Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo

Editorial Board

D. Barstow W. Brauer R Brinch Hansen D. Gries D. Luckham C. Meier A. Pnueli G. Seegm(Jller J. Stoer N. Wirth Authors

Benjamin Benninghofen MBB D-8012 Ottobrunn, FRG Susanne Kemmerich Technische Hochschule Aachen Lehrstuhl f~JrAngewandte Mathematik, insbesondere Informatik Templergraben 64, D-5100 Aachen, FRG Michael M. Richter Universit~t Kaiserslautern, Fachbereich Informatik Postfach 30 49, D-6750 Kaiserslautern, FRG

CR Subject Classification (1987): R2.2, 1.2.3, R4.1, E4.2 ISBN 3-540-18598-4 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-t8598-4 Springer-Verlag New York Berlin Heidelberg

This work is subject to copyright.All rights are reserved,whetherthe whole or part of the material is concerned, specificallythe rights of translation,reprinting,re-useof illustrations,recitation, brcadcasting, reproductionon microfilmsor in other ways, and storage in data banks. Duplication of this publicationor parts thereof is only permitted underthe provisionsof the German Copyright Law of September 9, 1965, in its version of June24, 1985, and a copyrightfee must always be paid. Violationsfall underthe prosecutionact of the German Copyright Law. © Spr[nger-VerlagBerlin Heidelberg 1987 Printed in Germany Printing and binding: DruckhausBeltz, Hemsbach/Bergstr. 2145/8140-543210

Introduction

Systems

of r e d u c t i o n s

called)

enjoy

science. tional

this

have

algebra;

also

become

these

and have

common

areas

much

is

the

equality

is the only predicate. with

a

fragment

class

of

tool

anyway An

are

not

In practice, problems

often

computer computa-

very

well

branch

logic".

of p r e d i c a t e

restricted as well as more general.

in

important

"equational

speaking,

concerned

a useful

in common.

background

as they

in t h e o r e t i c a l

are

strictly

is

rules

a growing popularity

They

separated

(or r e w r i t e

This

logic, equality

which

of is,

where logic

are

more

Typical are implications

of the form

Z~p Here

~ is

however,

a

universally

quantified

set

need not to be an equation

of first order predicate

of

equations;

and not

P,

even a formula

logic.

An example of the latter type of problems "Is each algebra in which ~ h o l d s

is:

finite?"

or equivalently "Is the free algebra defined by ~ finite?"

The

answer

one

expects

is an a l g o r i t h m i c single

questions

in equality

one.

but

That

whole

logic to

means,

classes

one

such questions

does

not

of questiQns.

consider These

may

arise in different ways:

i)

is fixed and P varies. An example is the word problem,

where ~ defines the

algebra and P varies over all equations

2)

s = t.

varies and P is fixed. An

example

algebras

is

the

(which

infinity

are

under consideration).

defined

problem by

the

for

a

axiom

class system

of

IV

3)

and P both vary. An example is the uniform word problem for a class of algebras.

Algorithmic mainly

decision

in the

theorem

context

proving.

in p a r t i c u l a r ago,

at

were

computable. abstract

of

more

In

algorithmic

algebra,

time

arise

of abstract

The

in

"the

problems

problems

less

first

tradition

much

half

older.

the

methods

There was still an enormous

interest

computability

problems,

decision

and automated

A

hundred

solutions

of

science

in mathematics,

automatically

and non-constructive

and

computer

data types

Kronecker,

or

the

is

in

to

algebraic

expected

twentieth

became

years

to

century

more dominant.

in principal aspects of but

constructive

me-

thods did not play such a big role in the every-day-work mathematicians.

This has radically changed under

ence of the development of computer science. tive

methods

are

not

only

certain applied situations; relevant

regarded

Among the defining properties one.

The basic

as

for

direction.

groups

1912-14

which

being

useful

in

of equality reflexivity idea of a reduction

to

of a specific for of

course,

search

on Dehn's

which

aspects

one-sided

than some

full

use.The

idea

and

fifty

years.

of the

power

algorithm was of

Dehn

and

Dehn's algorithm is the application

system of reductions more

M.

word problem

and which was one of the first decision problems In fact,

by

is to give

of equals by equals

is as old as the

considered

is the

in

studied

was

idea

Today construc-

A.Thue

be formulated.

looses,

This

the influ-

insights.

up symmetry and allow only replacements in one

of

their main purpose is to provide

structural and combinatorial

only trivial

be

Giving

are

regaining

up

with the

The

of

re-

question

preserved part

were

symmetry

of equality.

concerned

equality of

its properties

the

by

its power

p r o v i d e d by symmetry by systematically adding new reductions came

up m u c h

later.

This

leads

to

concepts

of

systems

of

reductions like the Church-Rosser and the weak Church-Rosser property

(which

are

known

under

finite t e r m i n a t i o n p r o p e r t y ;

various

names)

and

in c o n n e c t i o n w i t h the

the

latter

t h e r e is g r o w i n g interest in p a r t i a l and total w e l l - f o u n d e d orderings on the terms.

The

most

ensures

useful

that

property

is

certainly

e a c h t e r m t reduces

i r r e d u c i b l e t e r m t#;

to

completeness,

it

a uniquely determined

t # is the canonical form of t. The aim

of the completion a l g o r i t h m is to enlarge a given system of reductions system

in

order

to

(if it exists)

obtain

a

complete

one.

A

complete

can be regarded as a link b e t w e e n the

finite system of equations and the algebra defined by these equations which is a set-theoretic,

There one

are two main studies

criteria

the

which

to

terminate

systems.

lines of research here. completion

ensure

a l g o r i t h m in many

its

algorithm

termination.

leads

to

cases these

the

and As

searches the

investigation

can be

object.

On the one hand

(one is tempted to say "most")

this

In many

often infinite,

for

completion cases fails of

infinite

finitely d e s c r i b e d and

are as useful as finite systems.

The other type

of investigations

is concerned with the use

of complete systems. A complete system certainly provides an answer

to

the

word

problem

but

unravels

much

structure of the algebra under investigation.

more

of

the

This turns out

to be most apparent in the case of groups.

Most

of

the

years after partially

material

in

these

notes

obtained

in

1978 at the Technische H o c h s c h u l e Aachen;

contained

in

the

dissertations

Patrick Horster and Susanne Kemmerich; wrote

was

of

Hans

the

it is

BOcken,

Patrick Horster also

section IV.2.Very useful for computer experiments was

Vl

an i m p l e m e n t a t i o n semigroups

as w e l l

implementation Cyber

175.

Most

that

by

part

the

interest

Klaus of

Dittrich

these

material

is

of the a u t h o r s

applications

however.

these

forward-backward

notes part

algorithm.

This

in

on

Pascal

was

of

his

are

is in g e n e r a l

are

in

T h e r e are s e v e r a l familiar

feeling

for d i f f i c u l t i e s

general

universal

group

possibilities

reasons

structures and

algebras.

for

and

importance

Another

interesting

and is also u s e f u l

order

general

by

Habilita-

to

reader with

the

for this. one

has

r e a s o n is that

is

is

understanding

somewhat

included.

meant

of

and

One

is

a better

of r e s u l t s

connections

volume

an at least

completeness

semigroup

than

in

in t h e s e

This gives comparisons

for a fruitful competition.

make

material

principles.

and

areas c o m p u t a t i o n a l m e t h o d s are well e s t a b l i s h e d .

In

written

a

at the U n i v e r s i t y of K a i s e r s l a u t e r n .

concrete

theory,

done

last

Otto;

tionsschrift

as the

was

The

Friedrich

The m a i n

of the c o m p l e t i o n a l g o r i t h m for g r o u p s and

the

The

almost in

i d e a was

complete

the

rest

self-contained

sense

of t h e

much

to p r o v i d e

introduction.

that

suffices

material.

It

the Here

for

is

an

natural

that m a n y aspects h a d to be left out.

The

authors

remarks.

have

O n the

independently situation

was

also

been

one hand,

many

by d i f f e r e n t not

so

very

reluctant

results

authors.

clear

to

us

with

historical

s e e m to b e

On the that

obtained

o t h e r hand,

we

dared

to

the make

s t a t e m e n t s on p r i o r i t y questions.

There

are

several people

w h o m the

u s e f u l h e l p and d i s c u s s i o n s students Dittrich,

of

Aachen

influenced

are i n d e b t e d

over the years.

would

name

Hans

P e t r a Z i m m e r m a n n and Tom Beske.

w a n t s to m e n t i o n early

we

authors

A m o n g the former BQcken,

Woody

Bledsoe.

Klaus

One of the a u t h o r s

D a l l a s L a n k f o r d in p a r t i c u l a r ; by

for

Later

he was also on

useful

VJl

discussions Buchberger,

took Richard

place

with

Mike

Ballantyne,

G~bel,

Deepak

Kapur,

Wolfgang

Bruno

K~chlin

and J. N e u b ~ s e r .

All

authors

are

indebted

to

Mr.

v.

N~kel

for the e x c e l l e n t

preparation

Last

not

our

thanks

are

Forschungsgemeinschaft

without

their

this

least

research

Hehl

and

Mrs.

Scarlet

of the m a n u s c r i p t .

due

to

the

support

over

w o u l d not have b e e n possible.

Deutsche the

years

I.

General

Concepts

Algebra

1

Terms and Substitutions

1

I.l

Algebras,

1.2.

Some Concepts Languages

II.

from Universal

of the Theory of Formal

and Automata

14

1.3.

Decidability

24

Finite

Sets of Reductions

26

II.l.

First Concepts

26

II.2.

The Completion Algorithm

38

II.3.

The Ground Case

48

II.4.

First Analysis

II.5.

The Special Word Problem

of the Completion Algorithm

Small Cancellation II.5.1.

for Groups

and

Theory

82

Superpos-Deduction-Chains Criteria

and

for the Solvability

of

the Word Problem II.5.2.

84

The Small Cancellation

Conditions

and the Condition K 11.6.

Relations

III.

Infinite III.l

96

between the Completion

and the Todd-Coxeter

Procedure

Algorithm

106

Sets of Reductions

Regular III.l.l.

Infinite III.l.2.

116

Systems Regular

116 Systems as Special Systems

Applications

116

of Regular Reduction

Systems to Sets of Words III.l.3.

The Undecidability

121

of the Church-

Rosser Property III.l.4.

A Possible

III.2. Forward-Backward III.3. The Church-Rosser Backward

Systems

69

137

Church-Rosser

Test

Systems Property

140 146

of Forward153

IV.

Automata and Reductions

170

IV.l.

General Aspects

170

IV.2.

The Complexity of Reductions Algorithms

181

IV.3.

The Cycle Structure and the Growth Function

190

IV.4.

Effective Aspects of Gromov's Theorem

202

IV.5.

A Relation between the Growth Function and the Completion Algorithm

V.

Deciding Algebraic Properties of Finitely

210

218

Presented Monoids by Friedrich Otto V.I.

Monoid Presentations and Tietze Transformations

V.2.

218

Markov Properties of Finitely Presented Monoids

221

V.3.

Automata for Reduction Systems

228

V.4.

Deciding Algebraic Properties of Monoids Presented by Finite Complete Reduction Systems

V.5.

234

Deciding Algebraic Properties of Monoids Presented by Finite Monadic Complete Reduction Systems

243

References

256

Subject Index

264

List of Symbols and Abbreviations

265

I.

General

1.1.

Concepts

Algebras,

Although algebra

we we

from_Universal

Terms

and Substitutions

assume recall

concerned

with

Algebra

familiarity some

of

the

finitar Z

with

the

basic

definitions.

alsebras,

We

i.e.

concepts will

only

algebras

of be

of

the

ni ~

0 a

form A

where

A fi:

the

Ani

natural

number.

it w i t h

an e l e m e n t

signature

fix

it t h r o u g h o u t

of A

from

i.e.

I i ~ i ~ m>

are

ni-ary

we

from

A, t h e c a r r i e r

of

A ~ B

call

f~

operations,

ni=0

most

h:

B,

A

is the

Homomorphisms to

~

If

The

A

= < A , f ~±

sequence the

a constant

and

identify

of A°

~ = ~(A)

= ;

we

discussion.

are

structure

h ( f ~ ( a I ..... a n i ) )

=

preserving

mappings

f ~ ( h ( a I) ..... h ( a n i ) )

holds. Isomorphisms

are

homomorphisms A s B means

that

relation

for

R ( a k , b k) The

homomorphisms

that A and B are

A congruence such

i-i

and

endomorphisms

are

A ~ A.

congruence

isomorphic.

for A is an e q u i v a l e n c e

relation

RcA 2

l~k~n i implies class

R { f ~ ( a I ..... a n i ) , f ~ ( b I ..... b n i ) ) .

of a ~ A m o d u l o

R is d e n o t e d

by

lair or

2 simply

by

[a];

the q u o t i e n t

A/R

the c a n o n i c a l Defining

of A m o d u l o

= ( [a] I a ~ A};

mapping

7:

the o p e r a t i o n s

A ~ A/R

the q u o t i e n t

The c o n g r u e n c e

(i)

a l g e b r a A/R.

= h(b));

The

interrelations canonical

kernel (ii)

it is a l s o

h: A ~ B

called

If

h:

A ~

The

R h,

intersection

ence

relation.

a smallest

This

is c a l l e d

R ~

the kernel

implies each

R

is

congruence

is a h o m o m o r p h i s m

a

surjective

of c o n g r u e n c e

Therefore

the

A 2 there relation

operator sense

each

relation

congruence

the

with

is a g a i n

relation

generated

a congru-

R c A2

s.t. R [ by

R;

there



the

c A 2.

operator

operator.

relation

In s o m e

A/R

homomorphism

relations

for

congruence

closure

A ~

then A / R h s B.

R(h(a),h(b)) c

are:

is R.

B

is a c l o s u r e

congruence

between these concepts

mapping

of w h i c h

kernel

A

[f~(a I) .....f ~ ( a n i ) ]

h.

The b a s i c

is

=

r e l a t i o n i n d u c e d by a h o m o m o r p h i s m

is R h = ((a,b) I h(a) of

is g i v e n by ~(a) = [a].

on A / R by

f ~ / R ( [ a l ] ..... [ani])

one o b t a i n s

R is

R

for

each

is R

is

the c

fu_!l!z i n v a r i a n t endomorphism smallest

(R~;

the

h.

fully

operator

if

R(a,b)

Again,

for

invariant R ~ (Rm

is

a

too.

the m a i n

g a t i o n of this o p e r a t o r

topic

of t h e s e

notes

f r o m the a l g o r i t h m i c

is the

investi-

p o i n t of view.

For a set

X an a l g e b r a

free a l g e b r a

for

generated ~

every

nature

free

extension

equivalent

If

(ii>

all

l~i~m>

mapping

(of the

h:

same

sig-

X ~ B there

is a

of h to a h o m o m o r p h i s m

possibility

k,

of

h:

describing

A

an

~

B.

absolutely

{iii)

A

the

{0)°

In t h i s

algebra

isomorphism, A(X)

various

the

which

An absolutely

free

an alphabet

For

absolutely

tk=s k

all

operations

f~

and

all

<~,s>

of

conditions

algebra

the

generated

motivates

by

Xo

X u {fi~

free

of A

t h e n B = A.

(i) -

the

natural

by the (iii)

numbers

singleton

set

are

the

alternative

just

name

Peano

b y X is u n i q u e

up to

free algebra.

generated

It

of c o n s t r u c t i n g

in X.

If

i=j,

h e n c e w e can r e f e r to the a b s o l u t e l y

not

induction

then

i.e. if B is a s u b a l g e b r a

of a b s o l u t e l y

generated ways

b y X,

operation

case

instead

for

algebra

successor

axioms

X

X ¢ B,

is t h e

with

f ~ ( s I ..... Snj),

t k-

such t h a t

Peano

~

is g e n e r a t e d

example

=

l~k~n i,

f ~ ( t I .....tni) arguments

over

every

f ~ ( t I .....tni)

for

bra

=
a l g e b r a A is:

(i)

An

B

A) a n d

X c A is c a l l e d an absolutel_~

X iff

algebra

as

unique

Another

A with

always it,

A

and

there

are

e.g. as s t r i n g s or as t r e e s

l~i~m)

algebras

exists

free a l g e -

where

the

the

fi a r e

principle

of

symbols

structural

holds:

a

set

S

c

A

contains

t I .....t n i

e

S

implies

operations

f~ and all t e r m s

all

generators

f ~ ( t I .....tni) t k,

then

E

S

S = A.

and for

if all

In p a r t i c u l a r a)

two

special

cases

are

of

interest:

X = ~: A(0)

is

words

called

and

there

the

initial

is d e n o t e d

are

algebra

by W.

constants

or

the

(It is o n l y

because

it

of

algebra

of

interest

if

would

be

empty

otherwise). b)

X

= Vat

In

=

this

terms

case

(with

x e Vat

When

tions

allow

define

us

of

can

the

carrier

T) or

the

algebras

we

s in

regard

the

algebra;

T ~

for

A.

that

set

is

the

we

of

in

a!Hebra

of

term

algebra;

the

make

the

explicitly

strings gives

"s is

or

the

refer

general construc-

as

trees.

The

possibility

a subterm

also

each

of

the

the

as

called

is e x a c t l y

a quotient is

words

of

to

t"

to

or

"a

particular

subalgebra terms.

of

the

While

the

with

the

h:

~

correspondence

one

homomorphism

algebra

word

only

a

ground

one-one

particular,

identity

as

can

also

are

Vat ~ A there

empty

the

set:

t.

mappings

In

like

t";

are A

The

induction

algebra

words

homomorphisms

each

terms

concepts

of

called

constants.

regard

in

infinite

variables.

have

x occurs

occurrences

is

structural

familiar

variable

term

to

= A(Var)

initial

we

a countably

T

the

about

that

principle

We

are

talking

assumption

{x o, x I .... } is

generated

algebra.

endomorphism

We of

W

by

A

the

also

note

the

word

algebra.

The

concept

class If over

K

of is

of

an

absolutely

all

algebras.

any

class

of

free

algebra

then

A

algebras

refers

is c a l l e d

free

to

the

for

K

X iff

A

•

for sion

K,

c__ A a n d

X

every of

mapping

for h:

any

B

X ~

h to a h o m o m o r p h i s m

• K,

B

B there

=
h: A ~ B.

l~i~m>

a unique

and

exten-

For

arbitrary

classes

In connection

with

K free

equational

sufficiently

many

that

algebras

An

two

free

e~uation

is

denoted

by

between

ground

~

t;

h:

T ~ A

a

X are

<s,t>

~round

always

there

are

it

easy

Again

K over

pair

not

is

exist.

however to

see

isomorphic.

of

terms

e~uation

is

and

will

an

be

equation

terms.

h:

= h(t);

for

will

theories

algebras.

ordered

A homomorphism h(s)

free

an

s

algebras

T ~ A

s ~ t

is

satisfies

satisfies

true

an

in A

s ~ t. T h i s

equation

iff

last

every

s ~ t

iff

homomorphism

situation

is d e n o t e d

by

A I~ s ~ t.

The

models

for A

which

set

A I~ s ~

semantic

equations tion

for

t

E

of e q u a t i o n s

holds

theory

are

K

variet[)

for

all

are those

s ~ t e E

algebras

(denoted

by

where tant

Hence

of

A i~ E i m p l i e s

equations

are

those

A I~ s ~ t; t h e

nota-

of

equations

a class

closed

K of

under

algebras

consequences.

iff

K

definable

is d e f i n e d

by

some

E;

in t h i s

by

quotient

theory

A

= { A I A ~ E).

Z

the

(or case

a

the

K(E).

congruence TE

relation

= T/<E >

T / < E > is

the

is

generated

generated

canonical

by

mapping.

It

by XE is

E in T = ~E(X) impor-

notice:

varieties

a class

E of

e~uationall

E is t h e

T ~

set

is

denoted

~E:

which

a

algebras

~

to

of

E ~ s ~ t.

sets

iff K

is

<E>

is

defines

class

then

s ~ t for

E

class

consequences

this

Theories

If

a

t~ E).

The

A

of

of

T E is

free

for

have

free

algebras;

algebras

has

free

K(E)

over

X E.

moreover

algebras

iff

it

we

mention

that

is a v a r i e t y .

In

the

if

E contains

K(E)

same

are

way only

the

in g e n e r a l In f a c t

WE

ground

same

more

{s

as

Because

the

word

we

~

= <E>

can

refer

algebras

to are

also

assume

added The

the

initial of

has

gruence "below"

algebra and

the

this

these

up

to

algebra

of

data

data

type,

which

contains

<E>

defined

relation

in

is as

follows.

only

D c A x A we put

D

are

invariant

endomorphism

isomorphisms

K(E).

Such

types,

E

Here

we

we

initial

is

called

recall

principle

{s

-- t

as

{
information

is a g a i n by

consequences

=

the

smallest

E.

A

has

e

for

be

=

E(D)

= D u (l<s,t>

of e q u a l i t y

In

fact,

in t h e

<E>

con-

invariant

con-

E:

t}.

fully

of

arbitrary

cases

R(D)

A

~E~

s>l

<s,

from

algebras

= T and

A

A)

E

D and

h is a n

t>

of A}

(iii)

S(D)

= D u {
(iv)

C(D)

= D u { < f A ( t I ..... t n i ) , f A ( s l ..... S n i ) ~ < t k , S k >

~ D}

from = D u

A

= W.

endomorphism

T(D)

a

that to

contained

construction

in t h e

t> I t

E.

of

I E I~ s

It w o r k s

interested

u

all

generated

above

For

(v)

in

case.

information

=

we

(ii)

fully

non-trivial

abstract

WE

but

(i)

no

initial

semantic

been

are

in T E a n d

true

E.

unique

signature

<E>

(Em

are

abstract

relation

just

in K(E)

specification.

K(E)

congruence tains

the

a fixed

to

logic

of

algebra

equations

is t h e

by

ground

called

The

true

=: ~

has

algebras the

initial

<E>.

generated

in t h e

W E as

specification we

just

algebra

initial

an

equations

~ t I T E ~ s ~ t) relation

Because

is

equations.

the

than

congruence

have

= W/<E >

( ( s , t ) I 3 t I ..... t n,

D for tl=s,

l~k~n i,

l
tn=t,

e D for

l
R(D),

S(D)

xivity,

and

T(D)

symmetry

gruence

are

and

relation

and

In t h i s t e r m i n o l o g y

The

E(D) the

for

C(D)

and

their

generation.

the

T(D)

reflexive

these

tant

cases: a)

b)

Zn o r d e r

illustrate

only

S

=

a binary

resp.)

(resp.

(resp.

{s i

E

til

F G ( a I ..... a m ) / < S > defined

by

that

1

from

for

E and

applications

recursively

of

enumerable

will

be

considered

below

no n o n - t r i v i a l

endomorphisms;

we

endomorphisms

".",

l~k~m

monoids

an

occurs

the f a m i -

and

a constant

and

impor-

which

we consider

presented)

groups.

e,

a con-

additional

unary

case. x.(y.z)

the ~ro~)

word on

= (x.y)-z, algebra

e-x

= x

(and

is

the

free

W E

m generators,

denoted

by

F G ( a I ..... a m ) ).

i ~

are

rules

= <E> b e c a u s e

contains ~

as

S(D),

have ~

under

are

by

presentation

viewed

E(D),

where

has

group

free

sets

iterated

is

operation

then

the

is

in t w o

generator, the

the

be

the terminology

equations

F M ( a I ..... a m ) If t h e

case

(finitely

in

relation

It c a n be o m i t t e d

applications.

each

con-

enumerable.

in

.-i.

= e

(Em

substitions.

ground

of

by

often

for

operation

t

closed

have

ak

~

is

we

a

invariant°

c a n be o b t a i n e d

endomorphisms

of

fully

way

~m

E

Here

monoid

t

If

examples

refle-

show.

means,

algebra

to

under

guarantees

congruence

obvious

the w o r d

liar

x-l.x

That

In the

it

C(D)

formation

in an

under

we discuss

If t h e

the

E is r e c u r s i v e l y

when

stant

to

In p a r t i c u l a r

closure

operations

= T(C(S(E(R(E)))))

equations

rules.

whenever The

can

makes

generated

is s t r a i g h t f o r w a r d

conditions

closure

transitivity;

(Em which

the

n),

called

presentation.

additional then the

ground

FM(a I ..... monoid

equations

am)/<S>

resp.

the

and group

In t h e form the

case

ui

groups

= si.(ti )-I

words

The

of

u i,

the

~

e,

of

F G ( a I ..... a m ) / N

generated

by

the

elements

of

the

contains

form

tot; These

a

denoted

by

such x.

0(x)

= s

be

For

word

N

u

is

<S> the

is

given

then

normal

we

by

the

subgroup

group

identify

in

words

the

or

e

or

of

a genera-i b i • bi+ 1 •

they

of

identity

such

b i is

simply

carrier

each

either

addition

b l . b 2 ... bk;

the

words;

is

and

b i • e and

have

as

classes

which

where

endomorphisms

0(x)

= x for

substitution

for

which

in

a term

way.

0:

all

is

s • x,

this

is

are

usually

words.

FM

resp.

element

T ~ T

except

thus

e

FG

with

is

also

The

of t h e t e r m

finitely

determined

x a variable, set

a substitution

of

many

by

and

alge-

all

will

variavalues

in g e n e r a l

substitutions

is

closed

always

V 2 are

two

02(V 2)

the

variables

The

notion

the

a

two

in of

of

variable

of

renaming

= 6;

replacement

0 the

term

0(t)

is c a l l e d

an

one-one

then

is

t.

a renaming

If V 1 a n d always

t and

of

0(x)

need

is

composition.

called

n

resp.

are

that A

instance If

in t h e

i.

defined

under

presentation

congruence

written

these

Substitutions

bles

are

case

will

of

written

l~i~n.

unique

are

we

set

bra

FM

semigrouR

usual

the

u i,

group

terms

called As

the

be

modulo

where

(bl*(b2.(...)...)

in

the

F G ( a I ..... a m )

quotient

class

hence

can

ISd_~n.

quotient

The

equations

0

is

0

variables. finite

sets

of v a r i a b l e s

substitutions

this

case

V 1 and

V 2.

substitution

of c e r t a i n

following

and

we

say

does

subterms

terminology:

81 a n d that

not

82

such

01

and

cover

of a t e r m

then

there

that 02

the

by o t h e r

are

81(V I)

separate

concept terms.

of We

i.

Definition:

(i)

(ii)

A

place

is

numbers;

the

The

a

empty

subterm

defined

tupie

of

l

=

place

a

term

(i I ..... i n )

( ) is a l s o t at

place

))

inductively

= sbt(ti,X) and

replacement

in a t e r m

of

t by

the

subterm

a term

for

undefined occuring

r is i n d u c t i v e l y

r e p ( t , ( ),r)

l
otherwise. at a place

defined

X

by

= r

r e p ( f ( t I ..... t n) , ( i , ~ ) , r )

= f(tl, . . . , r e p ( t i , k , r ), for

2.

admitted.

X is

= t

s b t ( f ( t I ..... t n ) , ( i , ~ ) )

The

natural

by sbt(t,(

(iii)

of

l_
t i + l , .... tn )

undefined

otherwise.

Definition:

(i)

(ii)

A

unifier

of

such

that

S is

called

A

most

a

unifier

If

S

e(s)

set =

0

is

= ~

notion unifiers

of

some

{s,t) of is

S of

0(t)

of

such

by

we ~

justified are

all it S

that

unique

a

substitution

s,t

E

S.

a

unifier.

(m.g.u.(S)

for A

indicate =

is

has

substitution

then S

iff

unifier S

terms

for

unifiable

general

there

fier

a

all

by

the to

short)

unifiers

satisfying

the

most

m.g.u.(s,t).

up

in

fact

This that

a renaming

of

S

~ = A-e:

general

uni-

functional most

of

d

is

general

variables.

10

We quote

3. T h e

the fundamental

Unification

Let

S be

(i) (ii)

set

is d e c i d a b l e

Most

most

general

unifiers.

of t e r m s . whether

S is u n i f i a b l e

(iii)

about

Theorem:

a finite

It

theorem

iff

general

S is u n i f i a b l e .

it has

unifiers

a most

can

be

general

unifier.

constructed

effec-

tively.

In

fact,

of

unifiers

tive

there

case,

is

and cf.

Subsitutions

an

algorithm

constructs [Ro

are

a most

cases

called

reductions.

The

volume

and

introduced

be

Reductions

provide

bras.

computations

Such

relations

on

the

decides

general

the

one

existence

in

the

posi-

65].

special

will

which

a

of

latter

basis

are in

for

are

more

general

the

chapter

main

topic

of

this

II.l..

computations

closely

replacements

in

connected

term

with

alge-

ordering

terms.

4. D e f i n i t i o n :

Suppose the

~

term

(i)

is

an

algebra

~

is

0(s) i

~

(iii)

s

a

~

8(t)

~

is

or

t ~

tI

and

and

transitive

orderinq all

relation

~ t2

over

for

~

r

if

iff t h e r e

all 8

we

sbt(r,A) are

terms have = t.

no i n f i n i t e

~ ....

total

holds.

iff

substitutions

rep(r,l,s)

well-founded

called s

term

t

is c a l l e d

chains

and

T.

called

s,t,r,

(ii)

irreflexive

iff

for

all

s • t either

s ~ t

11 In

a

term

because

ordering

x ~

y

transitivity

In

gives

other

ordering An

implies

particular,

On the

no

to the

sequel;

two by

x ~

it w a s

can

substitution

ordering

can

cases

terms

of

x,y

a contradiction

in m a n y

ground

class

variables the

x~

term

hand,

important

the

no

such

be

be

comparable

property to

y ~

x;

irreflexivity.

total.

the r e s t r i c t i o n

of t h e t e r m

is total. orderings

introduced

in

will

be w i d e l y

used

in

[Kn-Be].

5. D e f i n i t i o n :

Let A

T

=
weight

l~i~m>

function

~

for

~:

(i)

~(fi ) > 0

(ii)

~(x)

The weight

be a t e r m T

is a f u n c t i o n

{fil

for

l~i~m)

ni=l,

N

u Var

l~i<m

= m i n ( ~ ( f i ) I ni=0)

function

algebra.

and

for

f o r all

is e x t e n d e d

such that

ni=0,

l~i~m

x e Vat.

to T by m

~(s)

where

n(x,s)

fences

The

so

depend

of

=

Z n(x,s).~(x) xeVar (resp.

x in s

called on t h e s e

n(fi,s))

(resp.

KB-orderings weight

of

+

Z n ( f i , s ) . ~ ( f i) i=l

denotes fi

number

of

occu-

in s).

(which

functions.

the

will

be

defined

next)

12

6.

Definition:

The

KB-ordering

with

s ~

a weight

i

= ~(~)

function

on ~

a

term

algebra

is d e f i n e d

T

recursively

(i)

~(s)

<

~(t)

and

n(x,s)~n(x,t)

for

all

xEVar,

(ii)

~(s)

= ~(t)

and

n(x,s)=n(x,t)

for

all

xeVar

and

either (~)

s = x

and

t = fm(fm(...fm(X).°.) for

or

(~)

s = f k ( S l ..... Snk), such

(a)

k < j

or

(b)

k = j

l
recursive

direct

ordering

7.

call

is

in t h e

inspection is a t e r m

we

last

get

clause

some

x E Var

t = f j ( t I ..... tnj)

that

and

l~i~n k,

By

by:

iff

t

or

The

associated

some

sl=t I for

and

of

for

i,

all

s i ~ t i.

this

immediately

definition.

that

each

KB-

ordering.

Proposition:

(i)

The

KB-orderings

(ii)

The

restriction

is

total.

For

the

proofs

not

contain

any

we

refer

are of

to

surprises.

well-founded.

a KB-ordering

[Kn-Be];

they

to t h e

are

ground

lengthy

terms

but

do

13

Additional

There

remarks:

have

axioms

E

been

into

various

the

substitutions.

t i o n of the u n i f i c a t i o n

A r e two t e r m s

An

important

case

associativity [Sti

81].

given

in

Besides

and

the

leads

where

of

E

consists

a

modifica-

commutativity

which

overview

E ?

the

was

equations

carried

of u n i f i c a t i o n

in

for

numbers, terms

general

orderings

for a g e n e r a l i z a t i o n

the

chapter

terms

more

of

out

in

theory

is

are The

is a v e r y

ground IV).

terms

(such a p a r t i a l

Well-founded

isomorphic order simple

total

(as o r d e r e d

type one,

of

type,

all

Dershowitz

e.g. t h e

recursive

(cf.

[Der

or

e.g.

82]

[Der

been

ordering

orderings

on

structures)

to

KB-orderings

namely

however,

have

is not to r e q u i r e

on

t h a t of the n a t u -

ral n u m b e r s . T h e r e h a v e b e e n u s e d o r d e r i n g s order

to

modulo

is

One way

even

ground

ground

This

in a d d i t i o n a l

s and t u n i f i a b l e

KB-orderings

occur

ordinal

built

[Si 84].

investigated.

will

to

problem:

A comprehensive

the

totality

attempts

of a m u c h b i g g e r

path

85]).

orderings

of

14

Some Concepts of the Theory of

1.2

Formal Languages

and

Automata

In t h i s of

paragraph

the

theory

be u s e d

in

pose

this

of

tools

and

order

to

tory

text

(cf.

[Ho-Ul],

l~i~m.

The

is

symbols

of

words

over

finite

acceptor) of

initial

set

found

all

of a w o r d

by

word

e or

by

l0

in

words

listing

the

about

in

developintroduc-

and

automata

derivatives

of

[Br 64].

usually w

be

denoted

= bl...b m,

denoted

by

of

length

0 is t h e

Notice

pur-

in any

w~

X + := Z * \ { e }

(or

A is a 5 - t u p l e Z is

lw[,

by

Z,

b i e Z for

is the n u m b e r ~[Z

word,

that

we

always

have

the

set

of a l l

non-empty

F!Q

is

function

set

mapping

with

discrete

inputs

inputs.

(Q,Z,~,qo,F),

the

automaton

determines

nondeterministic

a finite

a finite

quent

be

main

furtheron

languages

results

will

detailed

occurs

formal

The

of

automaton

state,

which

of

can

Thus

Q)

section

needed A

which

The

and

concepts

eEZ

.

y~.

states,

transition

contained.

of t h i s

w. T h e

denote

self

78]).

chapters.

theory

{a I .....a n ) w i l l

the

denoted we

set

[Ha

automata,

terminology

automata

theory

of the b a s i c

and

following the

of

book

the

length

Hence

A

the

some

languages

the

material

to

review

setting

alphabet

Z

mostly

of

expressions

while

of

formal

is

make

finite

will

principles

of the

A

of

several

ment

regular

we

finite

where

Q is a f i n i t e

i~P_q~ ~ l _ ~ h ~ h ~ , of Q

final x X to

states the

is a m a t h e m a t i c a l

and

a finite

the

behaviour

control of

the

state

qoEQ and

power model

is

the

is

the

6 set

of

P(Q).

a

system

(the set of

states

system

on

subse-

15

A

finite

automaton

diaqram,

a

labelled

correspond arrow

to

from

the

the

transition

If

the

range

the

The

of

and

w E Z

A

6(qo,W)

language,

A

finite

n

of

of

control

a more

of

stack

is

tion

a

•

labelled qj

the

vertices

automaton with

one-element

Q, A is c a l l e d 6

transition

and

an

a is c o n t a i n e d

E 6(qi,a).

only

@.

by

can

be

sets,

i.e.

if

6

deterministic.

extended

alphabet, F[Q

to

Z

in

that q'~6(q,a)

Z

is

languages

can

is

all

the

]

T

be

memory,

the

is the set

of

mapping

Q

accepted

by

initial final

x

besides

stack.

the

denoted

by

if t h e r e

is

the

Q

state,

ZoEF

and

so

finite

A pushdown

where

x F to

by

to the

alphabet,

states

(Xu{e})

is

recognized

include

input

A

i.e.

= L(A).

which

finite

automaton

words,

regular

(Q,Z,F,6,qo,Zo,F) , a

qoEQ

of

called

that

powerful

by the f i n i t e

automaton

A such

is the

function

such

set

a finite

automata

Z

The

T c

7-tuple

states,

symbol,

F

automaton

pushdown

set

iff

finite

to be a c c e p t e d

words

class

B

where

I BqE6(qi,w)

is said

called

maton

graph, the

its

a e X.

accepted

set

wider

diagram

:= { q'

A word

some

by

way:

for w E Z

A

of

function

6(qi,wa)

L(A).

states

set Q x ~ into

following

described

directed

6 contains

transition

iff

often

s t a t e qi to s t a t e qj

in

maps

is

auto-

is

a

finite

F is

a

finite

is the

6 is the finite

start

transi-

subsets

of

QXFo The

language

of

all

to

enter

called those

input

words

a final

for

which

which

state;

context-free.

a pushdown

it

Again

c a n be a c c e p t e d

some will

automaton choice

of

be denoted

context-free by s o m e

B

moves

the

are

set

causes

b y L(B)

sets

pushdown

is

and

B is

exactly

automata.

16 The

Turinq

machines,

enumerable a finite the

sets.

set of

states

well

as

contain ning

of

infinitely not

started

at

possible input

with

the

exactly

the

symbol

in

! ~

are

~ ,

blank

input

cells the

with

of

scanned

by

the the

of

it

special

machine

state

state

as

remai-

a

B. A T u r i n g in

is

is input-

while

symbol

that

and a tape

connected

inscribed

(i.e.

state and FcQ

symbol

alphabet,

dependency

aEZu{B},

start

one

many

input

cells

are

symbol

the

leftmost

moves

the

recursive

control

at one time. The tape

many

in Z,

the

has a finite

qo6Q

finitely

Z,

just

an i n f i n i t e

containing

medium;

symbols

symbol

Q,

scans one cell

output

accept

machine

states),

into cells

head w h i c h

least,

A Turing

s e t of f i n a l

divided

at

qo;

qeQ

tape

is its

and

the

head,

the

following: replace

-

moving -

move

a

where move

changing

a Turing

function

symbol head If

machine are

as

that

state

without

to

above

leftmost

6 (q,ai) =(p,aj, left)

and

6 is t h e

next

TM

is d e n o t e d by

~lq~2,

state of T M and ~i¢2 is the s t r i n g

tape

al...ai_lqai...a n

{Q,Z,6,qo,B,F),

to Qx(Zu{B})x{left, right}.

of the tape

up to the r i g h t m o s t the

the state

is a 6 - t u p l e

machine

is the c o n t e n t s

scans the

resp.

the

to the right or to the left

desribed

of a T u r i n g

symbol

left

TM

Qx(Zu{B})

q is the c u r r e n t

non-blank

change

the tape and c h a n g e

mapping

A configuration

(Zu{B})

and

the tape head one cell

Q,Z,qo,B,F

where

a'EZu{B}

the tape head

without

Formally

by

head;

symbol

is

resp.

a

we

from

non-blank assume

the

leftmost

symbol

that

the

or the tape

of ~2"

configuration

and

6 (q,ai) = (p,aj, right)

we

write:

al'''ai-lqai'''an

~-TM a l ' ' ' a i - 2 P a i - l a j a i + l ' ' ' a n

al'''ai-lqai'''an

~-TM a l ' ' ° a i - l a j p a i + l ....... an

and say that the second c o n f i g u r a t i o n by one move.

in

results

f r o m the first

17

The

language

L(TM)

L(TM)

:= { w

accepted

I wEX

by

and

a TM

qo w

=

(Q,X,6,qo,B,F)

~--TM ~ I P ~ 2 and

here

~-TM is t h e

Besides by

the

aid

of

machines

generation

A ~rammar

is

a

and

alphabets,

seN

of

pairs

ordered c

T

(u,v)

denoted

by

u ~ v,

closure

of

~ .

According hierarchy called

to

of are

~

L(G)

:= { w

grammars

type-0:

of

and

(unrestricted

in t h e

(context-sensitive

u ~ v e P

a finite

the

elements and

and

w

will

implies

be

by

grammars

Chomsky (the

way:

u e V

,

v e V

grammars) ~

IvI

grammars)

implies

of

transitive

the

following

lul

set

)

allowed, of

of

disjoint

grammars)

implies

(context-free

set

G is d e f i n e d

s ~

classes

i=0 .... 3)

is

productions

productions four

(the

are

reflexive

with

grammars. N

P

NuT,

by a g r a m m a r

u ~ v E P

type-2:

over

languages

languages

where

and

~TM"

of

so c a l l e d

symbol

I weT

of

terminals)

is t h e

u ~ v E P

type-l:

of

called

generated

kind

set

peF

(Zu{B])

kinds

these

(N,T,s,P),

words

L(G)

the

the

some

e

closure

define

start

while

distinguishes

type-i

=

(the

called

, V:=NuT,

language

can

G

~i~2

different

processes,

x V

The

V

is

of

one

4-tuple

non-terminals)

P

transitive

characterization

accepting

the

reflexive~

for

is:

u E N,

v

E

V +

so

18

type-3:

(regular

grammars)

u ~ v ~ P

A

language

is

called

i=0 ....3 if

it

is

A

well-known

type-i

language

generated

theorem

of

implies

u ~ N, v E T

and

by a t y p e - i

the

theory

is

denoted

by

L i,

languages

is:

grammar.

of

formal

L3 c L2 c LI c L0

Furthermore

the

following

equivalences

hold:

type-0

grammars

are e q u i v a l e n t

to T u r i n g

machines

type-2

grammars

are e q u i v a l e n t

to p u s h d o w n

type-3

grammars

are e q u i v a l e n t

to f i n i t e

automata

automata

i.e. L is of t y p e - 0

iff L is r e c u r s i v e

L is of t y p e - 2

iff L is c o n t e x t - f r e e

L is of t y p e - 3

iff L is r e g u l a r

For

regular

tion

which

sets

is

Thus

is b a s e d

which

operations

pressions, defined

that

set

the

the

'u',

'*' a n d only

we the

be

so

as f o l l o w s :

fact

and

way

of

characterisa-

the

class

concatenation of

regular

of

regular

and

Kleene

sets

is

also

complement).

be

defined

achieved '-' to

by

as

many

set

of

X;

of

those

applying each

the

operation

times.

regular

Kesular

the

repeatedly

elements

describe

called

that

class

finitely

can

a third

union,

can

can

m a y be a p p l i e d

Furthermore,

upon

intersection

regular

elements

is e v e n

under

(Notice

under

a

there

closed

closure. closed

sets

enumerable

sets

by

simple

ex-

expressions,

which

are

19 (i)

~,

e

and

noting

a for

~,

{e)

and

T,

then

denoting

formal

notions

left-derivative

according

to a w o r d

t h a t wD(T)

algorithm Dw(T)

which

for

Furthermore, tives Let

w E Z

there

be

a

word

w'E{VlvD(T)=wD(T)} have w

is

w<w',

with

called

pression course,

T

teristic

are

resp.

e T

a

regular

expression

set:

Dw(T)

by:

)

and t h a t

there

is an

expression

wD(T)

resp.

T

finitely

T c Z

and

many

every

word

different

w.

deriva-

T.

with

the

property

for

all

for

all

w'~(VlDv(T)=Dw(T)))

we

to s o m e

finitely

expression

C__HHr(T) w e

the

}

that

ordering

'<' o v e r

word

the

of

a right-characteristic only

:=

are

expression

I tw E T

left-characteristic

regular resp.

I wt

are r e g u l a r

expression

respect

S

i~l.

expressions

a regular

{ t

(resp.

(resp.

there

for e v e r y By CHI(T)

a

:=

Z

S

expressions

for

the r i g h t - d e r i v a t i v e

are o n l y

over

sets

t ~ T ) and

si:=sS i-I

regular

:= { t

regular

regular

regular

is the f o l l o w i n g

generates

for a r e g u l a r

w

of

and Dw(T)

every

de-

[Br 64]) useful.

wD(T)

Dw(T)

Notice

are

I s ~ S,

and

with

(cf.

one o b t a i n s

expressions

denoting

S

:= { st

wD(T)

Analogously

and

S°:={e}

derivatives

regular

respectively.

ST

manipulations

of

are

expressions

ST

where

aEZ

{a),

(S+T),

SuT,

u(si/iE~),

The

and

If S,T are r e g u l a r

(ii)

For

every

many

Z*,

then

regular

word

of

T).

characteristic

exOf

words

T.

denote

the

right-characteristic

set of all words

left-charac-

belonging

to T.

20

Further

we

need

in

chapter

III°l

the

following

terminology

and lemmas:

i. D e f i n i t i o n :

Let

T C Z

be a regular

characteristic

and

expression

fEZ

some

and

IEZ*

some

right-characteristic

left-

word

of

T. W e call

the

the sets:

IS(T)

:= { s

I s D(T)

= 1D(T)

) n IZ

St(T)

:=

[ Ds(T)

= Dr(T)

} N I~ r

{

left-supplement

supplement

s

of T a c c o r d i n g

of T a c c o r d i n g

to 1 resp.

the r i g h t -

to r.

2. Lemma:

The

left-supplement

are

regular

for

every

left-characteristic tic w o r d

Proof:

IS(T)

and

the

regular

word

i resp.

right-supplement

expression every

T

and

Sr(T) every

right-characteris-

r of T.

The proposition

IS(T)

results

from

=

Dr(T)

u

n tZ

rEID(T)

and the c l o s u r e p r o p e r t y .

[]

21

3. Lemma:

Let

T,L a n d

that

for

I be r e g u l a r

all

expressions

leL

I~Z*L\(I,e}Z*.

yields

T = Z IiLi~ i=l

with

the property

n

(i) T c IL

where

IiEI,LiEL

for l ~ i ~ n

T = Z LiI i, w h e r e i=l

LieL,Ii6I

for l
n

(ii) T c LI

yields

n

Proof:

(i) For

(l I ....In):=CHr(T)nL

we

have

T

= Z D]~(T)I~ i=l

n

Hence

have

T

= ~ DIi(T)(Sli(T)nL) i=l

rsET for all

sEL w i t h Ds(T)

Thus we have to show: Assume Since

(a)

there

rEDIi(T)

we

c_ I

such t h a t

r~I.

rl i E IL we have either:

li=lll 2

of the

with

ll~e; which

rllEI

and

12EL

is i m p o s s i b l e

and

by the

therefore assumption

lemma.

r=rlr2,

r2~e,

rleI

r21 i ~ Z L\{r21i,e}

(ii) A n a l o g o u s l y

to

T = Z 1i i=l

and

which

r21ieL

is a g a i n

n

liD(T)

Li=l iS (T)DLc_L ,

and

therefore

impossible.

(i) one obtains:

n

with

for every

= DIi(T).

I i := DIi(T)

is some reDli(T)

IiEZ L\{li,e]

(b)

because

n

= 2 (liS(T)nL)liD(T) i=l

Ii=liD(T)c_I

for

= ~ L~I i i=l

{i I .....in } :=CHI (T)nL.

[]

22

Obviously

4.

the

following

lemma

holds:

Lemma:

Let

P,Q,Pi,Qi

iKi~n.

be

regular

expressions

with

PicP,

QicQ

for

Then n

PQ

n

\ Z PiQ i i=l

=

n

(P \ Z P i ) Q i=l

+ Z Pi(Q\Qi ) . i=l

5. D e f i n i t i o n :

For

a regular (i)

expression

~(T)

:=

{ DIi(T) the

(ii)

Obviously

Finally,

~(T)

let

GSM

for

lar

reduction

A GSM

c(T)

us

short,

0

=

I the

initial

state,

6(q,a) may the

and

from =

(p,w)

w.

I I i e CHr(T) of

liD(T)

¢(T)

are

regular

the

in

FcQ Q x I means

is

mappings chapter with

to t h e

choice

set

for

0 in

of

T.

) of

T.

regular

T.

sequential

which

machines,

describe

our

regu-

III.l.. output;

of

is

a

a GSM

finite

alphabet,

final subsets

state

move)

formally

Q

output

finite

of

parts

parts

where

O the

the

that

end

generalized

(Q,I,O,6,qo,F), and

}

I i i e CHI(T) of

the G S M

!nput-

initial

set

automaton

(as a p o s s i b l e word

set

call:

the

systems

states,

mapping

{

recall

is a f i n i t e

6-tuple

:=

and

T we

states of

q with enter

and

input

state

set

qo

Q x O

is a

is

of the

6 is

a

, where symbol

p and

a

emit

23 The

domain

of

6 can

be

extended

to Q x I

in t h e

following

way:

6(q,e)

= { (q,e)]

6(q,wa)

= {(p,v)

I V=VlV 2 and

for

s o m e p' and

Furthermore, is d e f i n e d

SC_O

mapping

0

determined

(P,V2)£6(p',a)}

by

some

GSM

0

as f o l l o w s :

:= { v

[ (p,v)

0(T)

:= { v

I v ~ 0(t)

GSM i s a

GSM

0(w)

The GSM

For

the

(p',vl)e6{q,w)

mapping

~ 6(qo,W)

is c a l l e d

for

for t E T

some peF

) ; thus

}.

deterministic

if t h e c o r r e s p o n d i n g

deterministic. GSM

mapping

the

sets

0:

~(T)

P(I

resp0

) ~

P(O

~-I(s)

) and

are

regular

always

TcI

regular.

resp.

24

1.3

Decidability

With

each

term

of d e c i s i o n problems case these

out

results

which

form

terms

E. P o s t

and A.

of

the

word

of

groups

by

W.W.

A

an

Boone [Rot

73]) of

two

word

[HNN

decision

so

will

of t h e s e

even

shortly

for t h e s e

definded

in

the

review

notes.

by E.

(although

of

the

suffices

the

much

is

is

in

this

be

~

Does

E I~ s - t

hold

?

(cf.

there

sense

of

disputed).

were

such

father

Britton

is

embedding groups

of

the

even

alge-

only concerned

One

whole

given

58]).

applications

of

Examples

first

L.J.

[Nov

are

the

cases.

behaviour

58].

Neumann

however, For

unsolvability

[Dav

to

Neumann,

It is,

in m a n y

cf.

due

Novikov

some

the

problem

proof

that

properties may

hold

word

algebra.

individual

A IF s = t

independently

Higman,

in

Does

semigroups,

P.S.

problems.

elements

in

for

49])

problem

this

proved

the

generators.

not

algebra

and

with

rested

We

a number

most

unsolvable

background

(a s t r u c t u r e d

(in

science

the

unsolvable

theorem

the

In g e n e r a l

groups.

to:

problem

with

in

braic

E of e q u a t i o n s

recursively

s a n d t:

Markov

consequence

The

be

free

is e q u i v a l e n t

given

set

problem

For a r b i t r a r y This

each

presented

A is the

word

and

is a s s o c i a t e d .

to

finitely

Suppose

The

problems

turn

of

algebra

is e.g.

in

with

computer

often

certain

programs

classes

of

intebut

programs

25 For a l g e b r a i s t s more

in g e n e r a l

algebraic

properties

of A

are of

importance.

Among

the

undecidable

properties

for

groups

we

find

the

following: - Is A f i n i t e - Is A c y c l i c Is A c o m m u t a t i v e

-

Is A s i m p l e

-

Is A s o l v a b l e

-

It

is

natural

questions

have

this

are:

kind

?

to

ask

(cf.

for

restricted

a positive

answer.

- The w o r d p r o b l e m

One

purpose

obtain

more

problems.

of

the

cases

That

There

is no

all g r o u p s

In c a s e are

with

answers

is s h o w n

algorithm

classical

be

which

solves

[Mag 32],

notes

to v a r i o u s totally

[Boo-Ro 66]

of

[Rei 50]

these

answers

cannot

in

in

such

examples

relator groups

discussed

positive

where

is

to

decision

uniform

even

:

the

word

problem

for

which have a solvable word problem,

where

reduced

methods

such

in the g r o u p c a s e

for one

[Rab 58] ),

situations

for a b i l i a n g r o u p s

The w o r d p r o b l e m

-

Two

lAd 55] or

we present

undecidability

proofs

the p r o b l e m s

to:

a) The h a l t i n g p r o b l e m

for T u r i n g m a c h i n e s

or

b) T h e

intersection

(i.e. where

is

L(G I)

problem n L(G 2)

L(G i)

is the

c) The q u e s t i o n

whether

for t y p e - 0 - 1 a n g u a g e s = ~

language

for

type-0-grammars

generated

by

G i,

Gi~

i=i,2)

or

nitely

long

a type-0-grammar

derivation.

allows

an infi-

II.

Finite

II.1.

,First

For

some

notion is

S e t s of R e d u c t i o n s

Concepts

set

E ~ s ~

true

there

in

are

as

E itself general

of

E

E,

relation a

the

As

means

indicated a

t

in

and

s

that

s ~ t

chapter

syntactic

recursively

because

On

the

the

word

I.l.

the

enumerable

enumerable.

the

consequence

~ = t~ . In p a r t i c u l a r ,

is

the

in

E I~ t ~

symmetry

point

of

avoids

the

to of

view loops

(tacitly)

different

occuring

in

as

set long

other

hand

problem

is

of

many

reductions not

E in

does

s. E s s e n t i a l l y

the the

this

of

right

side.

equality is

provided

an

can

in

a

disjoint

always

is a r e s t r i c t e d prove

all

nevertheless

which This

sign.

of

be

consequendecide

only

means, the

we

be

the will

applied

often

loose

computational

however,

symmetric

syntac-

reductions

can

From

advantage,

by t h e

have

This

a system

equations

that

if n e c e s s a r y .

necessarily

cases

assume

equations

them.

the v a r i a b l e s ,

does

system

left

always

equations

which

but

finite

from

of

will

a system

tic o p e r a t o r of

we

by r e n a m i n g

Informally,

be

describe

recursive

variables

achieved

ces

to

recursively

not

sequel set

sets

E.

E

terms

undecidable.

the

given

of

some

consequence

of

property

consequences

general In

ways

the

is

and

semantical

models

~ with

of

equations

t of

all

<E>

in

of

various

operator

is

E

since

equality

sign.

one

27

i0

Definition:

system

(i)

be (ii)

of

of

reductions

terms

denoted

A

term

t

~

s

which

by

v

is

t ~

an

substitution

of (iii)

u

(iv)

0(t) R~

R~

v

are

by

R

of

ordered

reductions

reduction

u

the

the

set

called

immediate

0

a

pairs

and

will

s.

(denoted

in u and

is

(t,s)~

term

v

is

result

in

u

by

0(s).

means

u

(t,s)~

v

v)

0(t) of

occurs

some

is

the

reflexive

is

the

transitive

hull

arises

reference

a term

iff

replacing

for

and

of

t ~

transitive

u

for as

a

by

some

subterm

some

occurence

s

R.

in

hull

o f R~.

+

(v)

If

A by

R~

no

more t

~

ambiguity

formal s

at

u

a ~

the

definition place v

A

of

(cf.

iff

an

is

in

common.

2.

Definition:

The

again

assumed

semantic

e__quational

two

consequences

of

of

R.

1 of

v

no

theory

that

to

R

is

chapter

= rep(u,

reductions

{s

omitted.

reduction

some substitution sbt(u,~) = 0(t)

(b)

It

R ~.

immediate

Definition for (a)

(t,s)~

of

have

I.l.)

of

u

is:

0:

~,

a

_= t I s ~ t ~ R )

0(s)).

variable

form

the

28

3.

Definition:

(i)

R

and

they (ii)

R

R' have

and

Derivation the

Systems from

called same

are

s R.~

of

the

is

theory

of

concepts

and

Z

egcuivalent

theory. eHuivalent

vice

iff

iff

s

R~

t

versa.

are

equationally

equivalent

true.

are

generalizations

formal

one

equational

systems

not

reductions

eHuationall

derivation t

equivalent

converse

language

4.

the R'

implies

but

are

languages.

defines

of

grammars

Analogous

for

a term

irreducible

with

to

known formal

t:

Definition:

(i)

(ii)

L(t,R)

= {s I t

L+(t,R)

= {s I t R ~ + s}

t

is

called

L+(t,R)

(iii)

(iv)

= 0;

otherwise

t is

respect

= Irr(R)

= {t I t i r r e d u c i b l e }

Red

= Red(R)

= {t I t r e d u c i b l e )

Irred(t,R)

Of

course,

any

of

reductions.

of

a

better

problems:

set

= Irr

Giving

equations up

computational

R

iff

n L(t,R)

= u(Irred(t,R)~teS)

of

to

reducible.

Irr

Irred(S,R)

two

R ~ s},

may

symmetry

for

be

set

regarded

this

behaviour

a

way

for

provides

S of t e r m s .

as the

a

system benefit

immediately

29

l)

Regarding

equations

infinitely

long

as

long

there

of

transitivity";

as

as

chains

reductions of

is no

not

avoid

or

even

loops

derivations

restriction

e.g.

may

s ~

"applying

t and

t

~

s may

the

law

both

be

reductions. 2)

Using

the

equations

information; that

e.g.

a certain

as

reductions

there

might

equation

s ~

may

be t

no

loose way

too

to

much

find

is

a consequence

are

introduced

out of

E.

For

this

reason

a system

5.

nice

reductions

properties

may

which

have.

Definition:

R

has

no

6.

of

some

the

finite

infinite

termination

chains

property

t n R ~ t n + I,

FT P

iff

t n • tn+l,

there

are

ne~.

Definition:

(i)

R

has

the

Church-Rosser

and

t I,

t2 £

L(t,R)

This

means

for

such

that

t I R~

t R~

pro~er__tt Z

we

have

t I,

t R~

t2

t 2 R~

s

s and

CR

L(tI,R)

iff

n L(t2,R)

there hold:

t

tI

for

t2

all ~

is s o m e

t 0. s

30 R

(ii)

has

the

t R'* t l ' and

t

weak

R'* t 2

t 2 R~

s

Church-Rosser

property

there

s such

is

some

WCR

that

iff for

t 1 tl.-~

s

hold: t

s

7. D e f i n i t i o n :

R

has

and

the

unique

Irred(t,R)

termination

contains

It is i m m e d i a t e

that

in the

of

presence

property

exactly

one

UTP

term

the t w o p r o p e r t i e s

iff

for

R

each

has

FTP

t.

UTP a n d CR c o i n c i d e

FTP.

8. D e f i n i t i o n :

R is c o m p l e t e

The s

finite

R~

also

iff R has

termination

property

is

well-founded.

allows

an i n d u c t i o n

t

FTP a n d

As

CR.

FTP

any

means

that

the

well-founded

principle

which

we

relation

relation

will

state

it

here

explicitely:

(IP)

Suppose If

for

that

P is a u n a r y any

t R~

t we s,

predicate

have:

then

If

P(t);

P(s) then

on the for P(t)

set of terms.

all for

s • t such all

t.

31

For

a complete

a

term

t

can

set

of

t

eventually

be

regarded

reductions

stops as

at

the

a

R

any

process

unique

normal

of

rewriting

irreducible

form

of

t with

term

t ;

respect

to

R.

There

are

above. not

various

First

imply

relations

we

see

that

between

even

the

in the

concepts

ground

introduced

case

WCR

does

CR.

9. E x a m p l e : 0,0,a,b,c,d R:=

{0

~

d ~

0,

are 0

constants ~

f(c),d

WCR

is

easily

the

absence

of

a,

~

0

~

f(b),

checked CR

and b,

0

~

b ~ and

because

f

is

a unary

c,

a

~

f(a),

the

following

L(a,R)

n L(b,R)

C

f(a)

b

f(c)

f(b)

f(f(a))

f(d)

f(f(c))

flflfla)))

c

~

symbol.

f(a),

c

~

d~

f(b)}.

0

./

function

f(f(b))

diagram = 0.

indicates

32

The

reason

existence

of

accident

10.

for

as

failure

infinite the

next

of

chains

CR

in t h e

of

reductions.

proposition

above

example This

is

is t h e not

an

shows.

Proposition:

Suppose

R has

Proof:

One

For

other

the

following

~

The

induction

all

terms

The

and

we

is

direction

WCR

iff

R has

WCR.

We

CR.

trivial.

we

assume

consider

the

P(x):

principle

s I, t

get

tI ~

finally

that:

t R~

--~ t 1 -*

assumptions

and

such

implication

R has

(IP)

=

will

L(tI,R)

show

n

L(t2,R)

that

P(t)

•

0)].

holds

for

t:

t R~ 2

WCR

Then

[(Vtl,t2EL(t,R))(t~tl,t~t 2

Assume tl~t,l,t

FTP.

predicate

P(t)

By

the

s 2,

sI, t ~ t 2 ~

u and

P(tl),

sl~t~s 2 and

t2 ~

P(t 2)

and

take

s2.

u for

P(u)

some

u.

provide

v I e L(sI,R)

n L(u,R),

v 2 ~ L(s2,R)

n L(u,R)

s

E

L(vI,R)

tl,t 2 s u c h

n L(v2,R)

that

33

t

"a/

vI

v 2

s

Hence

P(t)

Because or

follows

only

the

s 2 = t are

and

by

trivial

(IP)

cases

remaining

the

we

obtain

t ~

s I,

assertion

P(t)

t ~

CR

is

for

all

s 2 with

t.

sI = t

proved.

[]

Besides times tree

ii.

the

induction

useful. contains

an

says

that

infinite

an

(IP)

K6nig's

infinite

branch.

An

lemma

finitely

application

is

some-

branched is:

Proposition:

Assume

that

finitely Then

The

It

principle

proof

second

R has

many

each

is

FTP

s such

L(t,R)

is

immediate

assumption

is

and

that

for

each

that

t R~

s holds.

t there

are only

finite.

by

K6nig's

satisfied

for

lemma.

finite

R.

Note

that

the

34

There

are

perties

various

decision

introduced

problems

above.

We

connected

begin

with

with

some

the

pro-

undecidability

results.

12.

Theorem:

properties

for

(i)

The

finite

termination

(ii)

The

Church-Rosser

Undecidable

finite

systems

of

reductions

are:

Proof:

The

reduced

to

= < V N, VN

We

choose

problems

undecidable

G

Suppose

decision

some

V T,

problems

for

a o,

is

P>

= {a o ..... a n }

x ~ V: = V N

u VT

property

property

CR.

for

and

FTP

CR

will

be

type-0-1anguages.

a type-0-grammar

and

and

FTP.

VT

put

where

= {an+ 1 ..... an+m).

Vx

= {wx I w

~ V

};

the mapping ~:V

Let

Tx

algebra generated one Next

=


with by

~

Vx

~(w):=

Ao,°..,An+m,

n+m+i the

,

f>

constants

singleton

wx

be and

{x).

is

the one

Tx

is

a

a

bijection0

absolutely binary term

operation algebra

variable. we

define

two

~:

injective

V

~

mappings

Tx"

~x:

Vx

~

Tx

by X for ~(u)

and

=

I

Ai

for

u

= e u

f(Ai,~(w))

= ai for

u

= aiw,

w

free

• e

with

35

i

~x(U)

Finally

we

f(Ai~

~x(W))

= {<~(u),~(v)> u

= x

for

u

= aiw

<~x(~(u)),~x(~(v))>

{,



I

E T

I

e P)

0~i~n+m}

show

(*)

(a) We

u

put

R(G)

and

x for

u

Assume have

Case

the

p~ v

iff

u ~ v by following

~(u)

the

~(v).

production

rule

(i)

u

= r,

v

= s

(ii)

u

=Wlr,

v

= WlS

(iii)

u

= rw2~

v

= sw 2

(iv)

u

= wlrw2,

v

= WlSW 2

(i)

is

immediate.

because

~(r)

ponding

subterm

case

is

(iii)

a of we

.

cases:

For

~(wlr)

For

R(G)~

case

(ii)

<~(r),~(s)>

subterm ~(WlS); consider

of

similar the

we

~

~(wlr)

where

have

for

w I ~ e • w 2.

s • e

~(WlS)

and for

~(s)

is

the

corres-

s = e.

substitution

0(x)

= ~ ( w 2) a n d

get e(~x(~(r)))

which

and

8(~x(~(s)))

yields

~ ( r w 2) Case

= g ( r w 2)

(iv)

is

as

<~x(~(r)),~x(~(s))> (iii).

~

g(sw2).

= ~ ( s w 2)

36

Therefore

in all

(b)

other

The

injectivity ~(u)

direction

of

contains

cases

the

(*) it f o l l o w s

does

not

If

have

G 1 and

G 2 are

terminals

Then

is an

immediate

~,

a and

~(v)

@ and

holds.

consequence

@x

and

the

of t h e

fact

that

= e.

G has

hence

two

FTP

infinite

is

grammars

productions

derivations

iff

R(G)

undecidable.

(w.l.o.g.

(a o ..... a n , a o' ..... am'))

terminal G 1 and

u

that

FTP;

R(G)~

mappings

x iff

From

~(u)

then

a ~ a o,

with we

disjoint

add

a ~ a o' t o

a new

the

nonnon-

union

of

can

be

G 2.

L ( G I)

n

L ( G 2)

•

@

iff

the

following

square

completed.

A

oS \

\

\

R(G)

This

implies

Although one

is

FTP

be

discuss

FTP

/

~'

R(G)

that

CR

is a l s o

undecidable.

and

CR

are

general

interested

CR will

i I

in n e c e s s a r y

investigated here.

in

in

the

or

undecidable

sufficient

next

[]

section

properties

criteria. we

will

While shortly

37 13.

Definition:

R

is

t ~

14.

called s

for

compatible all

with

some

term

ordering

i

iff

s ~ t E R.

Proposition:

(i)

If

(ii)

R is c o m p a t i b l e

is w e l l - f o u n d e d

then

For e a c h

H with

system t ~

is a w e l l

Proof:

(i)

The

all

s,t s u c h

~ii)

If

t ~

s is d e f i n e d

of D e f i n i t i o n

4 in c h a p t e r

s

li)(b)

implies

is

term

conditions

e(t)

as

t

ordering

compatible

ordering

with

ensure

R,

t ~

s

t.

s R~

I.l.

R~

relation

s R~

s R~

ordering ~ which

FTP.

of a t e r m

that

term

FTP t h e

iff

founded

for

some

R has

s

also

t R~

with

t then

follows

0(s)

for

condition

from all

the

(i)(a)

fact

that

substitutions

0;

immediate.

[]

In

general

a

term

than

pairs

Term

orderings

theory. tion

If

s R~

an

is t h a t

leads

to

t

are

for

the t e r m s

contain

system

is

class to the

investigations

more

pairs

R compatible

independent

theory

each congruence

the

will

some

defined

equational

and that R reduces lem

ordering

of

given,

contains minimal

in c h a p t e r

any the

t i with

s 4-

equational ideal

a i-minimal ones. II.2..

This

situaterm prob-

38

II.2.

The Completion Algorithm

Incompleteness applications It d o e s power

not of

have

enough

order

cases

we

(and

it is i m p o r t a n t

seen

R of

to

obtain

introduce

II.l.

to

restricts

enlarge

to e s t i m a t e

it,

if

wanted

such a defect.

however,

its

k i n d of a d e f e c t :

in o r d e r

restrict

further some

as s o m e

to d e t e c t

results

in c h a p t e r

reductions

reductions

a system

undecidability

have In

a system

a n d c a n be r e g a r d e d

such

possible) ral

of

such

attempts.

FTP r e d u c e s

reductions

The

of

the and

geneAs

we

CR to WCR.

WCR

to

special

terminology.

i. D e f i n i t i o n :

(i)

(ii)

Left(R)

= {t I t ~

R~ht(R)

= {s I t ~ s E R, f o r

superpos(tl,t2!

=

s ~ R, for

some

s o m e t)

{0(tl) I 0 a

m.g.u.(t,t 2)

subterm (iii)

If S I, S 2 a r e superpos(Sl,S2!

(iv)

sets

of t e r m s

~(t I)

E

t I ~ s I,

superpos(tl,t2),

t of t I, t ~ Var)

I t I E s I,

then

the

~

pair

I)

(t2,s 2) a critical

pair

t 2 ~ s 2)

~--lmw 2 of R.

( W l , W 2)

Wl(=0(Sl))

o(t I )

is c a l l e d

some

t 2 ~ s 2 E R and

/ ~ ~ - - ~ ~ , s

for

then

=u(superpos(tl,t2)

If tl, t 2 E Left(R),

s}

with

39

Note

that

a finite

reduction

critical

pairs

The

decidability

next

algorithmic

and

each

system

R has only

s u p e r p o s ( t l , t 2)

result

is

finitely

many

finite.

is f u n d a m e n t a l

for many

further

considerations.

2. T h e o r e m :

If

R

is

finite

property

Proof:

CR

It

consider

is

and

has

FTP

then

the

Church-Rosser

decidable.

suffices

a situation

to of

determine

the

the

validity

of

WCR°

We

form

tl

A simple we

structural

can

some

restrict

we

finitely

many

We

ourselves

s E Left(R)

Finally

first

and

will

every

Now

assume

that

that

tI

substitution

(t,s)~

~

such

that

some

variables

we

unifiable

and

A then

can

shows

t

that

= 0(s)

for

ourselves

to

8.

in

for

assume

implies

is

some

|(t)

=

A(t I)

tI ~

(t,s)~ A(s I)

8(r)

and

r of

a

and

t I and

replacing

some some

~(t)

by

w.

substitutions

~ = 8 which have

s I in R

subterm

8(t I) g i v e s

standardizing

therefore

8 = A.~.

where

restrict

some

substitution

occurence

can

w

there

Neglecting

algebra

I.

Then

some

we

sI

8.

at

case

(t,s)~

substitution

~(s)

the

term

8 a substitution.

see

0 ( t I)

in t h e to

substitutions

remark

for

induction

means

m.g.u.

0

which that

=

separate

t and

~(t,r).

For

r are some

40 We have the c o m m u t a t i v e

t

for some term

l

diagram

~

i !i)°J

w'. Therefore

to standardizing

is a subterm This

means

pairs

it is sufficient

substitutions)

of terms t and r where

(t,s) ~ !

to consider

most general

(up

substitutions

t is a left side of a reduction

0

and r

of a left side of a reduction.

we

have

only

to test

the

finitely

many

critical

of R. []

According t I and

t 2 such

are both FTP

to the

and

that

is finite).

theorem t R~

irreducible

semantically. the

last

t I, t R~

a method

Of course,

R implies

it is a natural

tI ~ t2

idea

Although begin

there

are

only

with

this

process

reductions

give

rise

more

precise

assume

to iterate

finitely might

to n e w

we assume

that R is finite

not

to add

for algorithms.

tl~t 2 one

of

system R.

system

critical

stop

and c o m p a t i b l e

general

frame

many

a well-founded

is called an a l g o r i t h m

tl,t 2

enjoys

the procedure.

critical

The following

t,

(but enjoys

or t 2 ~ t I to the original

then one wants

find

the equation

This should be done in such a way that the new FTP again;

to

t 2, t I • t 2 and

in case R does not have CR

Therefore

reductions

we have

because

pairs.

pairs

the a d d e d

To m a k e

term ordering with

although

to

this

"4" and

"4" it is really a

41 3. D e f i n i t i o n :

The

(redundant)

Knuth-Bendix

completio n algorithm

(KBCA r)

is d e f i n e d by: (i) R or: = R (ii)

F o r m all c r i t i c a l p a i r s (Wl,W 2) of R~. r P n := {t I ~ t21 t 2 ~ t I, t l , t 2 E I r r e d ( w j , R ~ ) ,

(iii)

j=l,2, (iv)

Pnr

If R~,

= 0,

then

otherwise

the

put

(Wl,W 2) a c r i t i c a l

algorithm

stops

pair)

with

Rn+l:=r Rnr u Pnr a n d

result

return

to

(ii).

Here

we

bles. We

have

The

will

neglected

algorithm

substitutions

generates

give precise

notions

many

which

separate

unnecessary

varia-

reductions.

to deal w i t h this.

4. D e f i n i t i o n :

(i)

s ~ t E R is r e d u n d a n t

(ii)

a)

s = t

or

b)

t E Red(R)

or

c)

s E Red(R

R is r e d u n d a n t

Redundant

reductions

however•

be

steps

change

do

equational

replaced

theory

the

If s ~ t e R a n d

2)

If

E R,

(s and t are i d e n t i c a l )

iff R c o n t a i n s

cannot by

simply

"simpler"

set

of

Omit

s ~ s.

a redundant

be omitted.

reductions.

reducible

s

t R~ R"*

s~

t' then

or t ~ s' a c c o r d i n g w h e t h e r 3)

\ (s ~ t}).

terms

The and

reduction°

They

cane

following leave

the

invariant:

i)

S -* t

in R iff

then

replace

replace

s

s ~ t b y s ~ t' -* t

t ~ s' or s' ~t.

by

s'

-* t

42

These

steps

algorithm complete

the of

1')

s ~

If

Again

Starting dant

R-*

we

generate

be

that

with

t R ~ t' , t h e n

s R~

a complete

s ~ t E R,

will

•

s'

into

In

R

already

case

then

the

the

was

an

be

s ~ t by t

e

s ~ t

Irred(t',R).

s ~ t

an

algorithm.

carried

system

following

Observe,

out

parallel.

R r the

non-redun-

properties:

a)

s e L e f t ( R r)

b)

t e I r r ( R r)

c)

s Rr~

d)

s E Irr(R\{s

completion

non-redundant

i.e.

omit

into

reduction

R has

replace

Irr(R),

cannot

then

modify

only

t

steps

system

ways

take

incorporated

the

various

be s i m p l i f i e d :

and

s ~ t E R and

can

can

and t

in

redundancies.

2) w e

t e R

complete

If

Next

procedure

t'

this

however,

incorporated

removes

I) a n d

where

2')

be

which

Instead

If

can

t ~ t)).

algorithm

in o r d e r

to

reductions.

5. D e f i n i t i o n :

The two that

(iv')

non-redundant sequences

KB-completion

Rn,P n

(iv)

is

replaced

If

Pn

=

Rn+l

@"

in

the

algorithm

same

way

as

(KBCA) r R n,

P nr

defines except

by

stop

with

= ( s l ~ s 2 [ s2~sl, where and

result

R n,

otherwise

sjeIrred(tj,Rn(tl,t2)) tl~t2eRnUP

Rn(tl,t2)

n

or

put j=l,2

t2~tlERnUPn

= Rn\(tl~t2,t2~tl )

).

43 6. D e f i n i t i o n :

The

completion

each

neH

algorithm

and

each

tl,t 2 e Irred(wj,Rn),

The completion (a)

For

may

critical

(b)

is c a l l e d

terminate

pair

•

( W l , W 2) b u t

no

respect

=

2 )

all

comparable.

reasons:

has

Irred(w2,Rn). termination.

I r r e d ( w 2 , R n) terms

if f o r

(Wl,W

for t w o

(Wl,W 2) o n e

a successful

I r r e d ( W l , R n)

with

pair

t I . t 2, a r e

I r r e d ( W l , R n) This

sucessfully

critical j=l,2,

algorithm

each

continues

for

in t h e s e

to "~". T h i s

some sets

is c a l l e d

critical are

pair

comparable

a failure

termi-

nation.

Successful

termination

failure

termination

pletion

algorithm

nation

The

does

main

(Rn,ne~) general

not

produces continues

an

in

complete

incomplete

succesfully

sets

set.

then

while

If the

failure

com-

termi-

occur.

difference

is t h a t is not.

systems"

results

the

between first

For t h i s

in a d i f f e r e n t

the

sequences

is i n c r e a s i n g

reason

we

have

which

(R~,ne~) the

to d e f i n e

latter the

and in

"limit

way.

7. D e f i n i t i o n :

(i)

Rr

:=

U

(lirnl

(ii)

Rc

:=

u

(Rnl

{iii)

R~

:=

(s I ~

n ell) n E ~)

s21

sj

~

where and R ° is c a l l e d R

which

the

respect

j = 1,2,

t I ~ t 2 e R c or t 2 ~ t I E R c RC(tl,t2 ) = Rc\{tl~t2,t2~tl

limit to

Irred(tj,RC(tl,t2)),

system

"~".

of t h e o r i g i n a l

))

system

44

8. P r o p o s i t i o D !

Assume

t h a t the c o m p l e t i o n

algorithm

continues

ly. T h e n R r, R c and R ~ are all c o m p l e t e

succesful-

s y s t e m s and R ~ is

non-redundant.

Proof:

By

construction

completeness cing

cible terms, Hence

This

implies

from

t.

tained plete.

reductions

induction that

only

The in

gives

Rc

is

enjoy

not c h a n g e

Irr(R r)

finitely

many

the

definition

for

that

from

t

repla-

of

redu-

each

each

nE~.

term

t

s is d e r i v a b l e

in R ~ can

implies

The

r e m a i n the s a m e

Because

t such

this

set

= Irr(R n) for

derivable

R n already;

FTP.

B y the r e m a r k above,

complete.

irreducibles

some

By

does

systems

t h e r e f o r e the i r r e d u c i b l e t e r m s

too.

are

three

of R r is i m m e d i a t e .

redundant

there

all

that

be

ob-

R ~ is c o m -

R ~ is n o n - r e d u n d a n t .

[]

The

sets R n are the aRRroximations

will

see,

R ® can

be

the u n d e c i d a b i l i t y

infinite.

of the

This

word

or s t a ~ e s

of R ". A s w e

fact

follows

problem

(e.g. for

or groups).

O n t h e o t h e r hand,

R ~ depends

nal

R and the o r d e r i n g

"~". W e

system

related

In

the

simple. ground

questions

ground

case

Suppose terms.

in

we

Then

chapter

the have

also

completion a total

algorithm

term

this

and

is p a r t i c u l a r y

ordering

t E s u p e r p o s ( t l , t 2)

implies

-~. , °-~

"~"

on the

that

t is a

looks as f o l l o w s :

tI

2

discuss

II.4..

= s I ~. ° .~ w I

w

semigroups

o n l y on the o r i g i -

will

s u b t e r m of t I. The c r i t i c a l p a i r d i a g r a m

~

from

W 2

45

where

wI

and

w2

are

irreducible.

the

completion

algorithm

w2

~

both

w I

Because finitely

has

no

many

Later

on

we

consider

cases

such

pletion

reductions,

consider

algorithm

for deals

terminate

in g e n e r a l .

equations

are

the

we

axioms

These

~

wI

t I and

chains

i.e.

the

~

which

w2

w 2

there

or ~

are

completion

t l.

only algo-

group

and

will

a situation.

only This

words

give

words

situation.

with comes

present; written

the

fact

e.g.

the

law

example

it

complewill

that of

we

the com-

the

words

from

without

a (classical)

redefine

Although

ground

There

not

general

associati-

parenthesises.

which

arises

from

theory.

are E

In

by

will of

are

w I

a modified

such

implicitly

is r e p l a c e d

Finally

have

descending

or s e m i g r o u p

algorithm

vity

we

adds

reductions

(succesfully).

will

group

possibly

infinite

terminates

rithm

tion

; in

"~"

The only

order

to

= {e.x

define

= x,

a

x-l.x

= e,

KB-ordering

the

(x.y)-z

= x.(y.z)).

following

weights

are

choosen:

The

initial

does

(e.y, shows :

not

= 1

~2

= ~(')

= 0

~3

= ~(.-i)

= 0

stage

R

Ro

e I = ~(e)

for

= Ro

have

(x-l.x).y)

the

completion

= {e~x ~

CR, for

as

x,

e.g.

(x-l-x).y

x-l.x

the e

algorithm

~ e,

following

is

(x.y).z ~

x.(y-z)}.

critical

superpos((x.y).z,

pair

x-lox))

46 W e have: (x-l-x).y E s u p e r p o s ( ( x - y ) . z ,

e.y

x-l-x))

-l(x.y)

Y

The completion

algorithm

generates

as a c o m p l e t e

system

the

reductions:

e.x ~ x

e -i ~ e

x-l.x ~ e

(x-l) -I ~ x

(x.y).z ~ x.(y.z)

x.x -I ~ e

x-l(x.y)

x. (x-l.y)

~ y

x.e ~ x

Additional

remarks:

We

intend

do

not

concept pair

rithm.

and

These

independently) ce

of

to p r e s e n t

the

the

ideas

system various

have

of

been

the

general

algebraic

had m u c h

word

problem

of

the

treated

structures

influence

in was

is

in

[Ev 51]. done

the

a

of

the

critical

completion

(to

large

algoextent

An e a r l y a p p e a r a n [New

which

was

A breakthrough

for

1970

on the t e r m i n o l o g y .

rings a c o m p l e t e

account

reductions,

forms

Church-Rosser property

to

polynomial

an h i s t o r i c a l

by m a n y d i f f e r e n t authors.

applied

also

(x,y) -I ~ y - l . x - I

of a complete

notion

~ y

42]

in

[Kn-Be]

which

For the c a s e

of

s y s t e m c o r r e s p o n d s to a " G r 6 b n e r

47 base";

completion

thesis

[Bu 65].

tive

semigroups.

[La 81] The

and

We

this

area

and

related

More

We

mention

[Ka-Na-Si]. Additional pletion

first refer

studied

in

[Ba-La

81]

to

historical

is

have

for d i f f e r e n t

not

uniform.

topics

There are various to

also

of C R a n d W C R

are useful

nected

here

comments

can

Buchberger's for

commuta-

be

found

in

[Ba-Bu-La].

notions

which

was

is

A

in

[Kn-Be],

or

[La 75],

A good overview equations

procedures

of

have e.g.

The terminology

detailed

discussion

and

to e n s u r e

partial

of v a r i a t i o n s

purposes.

[Hu 80]

methodes

certain

a great number

of

this

are all

con-

[Hu-Op].

FTP w h i c h

total

well-founded

[La 79],

[Der-Ma],

is g i v e n in

also

been

[Pe-Sti].

in

orderings.

[Jou-Le-Re],

[Der 85].

"built

in"

in

the

com-

48

II.3.

T h e G r o u n d Case

In the case of finitely

generated

groups one works

in prin-

ciple w i t h two types of equations:

i)

The general

axioms of group theory;

2)

The specific

(ground) equations

for the group under

consideration.

Similar

situations

Boolean

algebras

situation

occur

etc..

frequently

The

considerably;

general

they allow

as

in

semigroups,

equations

simplify

the

e.g. to l e a v e the t e r m

algebra and to present terms by group words. It is of course desirable

to

avoid

the

explicit

equations

in t h e K B - c o m p l e t i o n

appearance

general means

general

equa-

tions into some form of a special c o m p l e t i o n a l g o r i t h m

which

one

works

has

on

the

promising

to

"built-in"

special

the

equations

if one has a c o m p l e t e

effect

only.

of

too.

the This

that

algorithm

of

the

This

system

is

in p a r t i c u l a r

for the general

equa-

tions.

Let

us f i r s t

special groups

of all

completion

leave

the

algorithm

w o r k i n g on words,

term

for

algebra

finitely

so that there

by m e a n s presented

of a semi-

is no longer need to

care about the law of associativity. As will is

more

although

be clear later, efficient it

and

is defined

this s e m i g r o u p c o m p l e t i o n easier parallel

to

handle

to the

and

implement

KB c o m p l e t i o n algo-

rithm.

Being precise we recall

algorithm

some formal definitions.

49

I.

Definition:

Let

Z

{ a I .....a n )

free

semigroup

free

semigroup

The

elements

are

called

The

identity

Two

2.

=

over

e

is

u:=

l~j~m

and

= v i for

ui

w

=

also

the

e)

X+

free

over

WlW2...w

called and

~ !

in

all

alphabet

denotes

monoid

(i.e.

the the

Z.

k

with

w i •

Z,

l~i~k,

~.

Ul...u k

are

X*

identity

over

e

finite and

X +,

words

l~i~k,

a Z

with w

words

be

i E

the

v:= Z

~R~Z

word.

V l . . . v m,

, denoted

ui,v j

by

u

•

= v

Z

for

iff

k=m

{ I ..... k } .

Definition:

Let A

Z be

set

a

finite

c

X

R

x

alphabet. X

of

ordered

pairs



of

words

is

i

called

a

reduction

system

over

over

its

elements,

the

reductions,

connected

to

reduction

Thue-system be

denoted

Nearly over

all

here

3.

u ~

algebras

over

X

(often

also

called will

v.

technical

term

systems

by

X*);

X

terms can

. For

be

defined

reasons

of

literally

for

completeness

sytems

reduction

they

are

reduction

of

listed

again.

Definition:

Let

R

be

a

reduction

sytem

over

Z

.

is

immediate

i

(a)

A

word

wI

E

Z

an

a

word

i

wo

(b)

E

X

by

u ~ v e

there

are

x,y

w o R~

wI

means

X wO

e

R

such u~v ~

(denoted that w I for

wo

by

w o u~v ~

= xuy

some

u ~

w I)

and

w I

v

E R.

iff

= xvy.

50

(c)

R~

is

the

(d)

w O R~ w I means

(e)

R~

is

the

i.e.

R~

Thus

two

iff

reflexive

is

the

words

v R~

w. w

by

congruent

modulo

R

class

of

w

of

R~,

generated

by

~.

modulo

is a g a i n

R

denoted

[w] R.

is c a l l e d

L(w,R)

=

{w).

every

w

E X

we

put

L(w,R):=

irreducible

(v I w R ~

with

= Irr(R):=

{w

e X

I w

is i r r e d u c i b l e }

Red

= Red(R): = {w

E X

I w

is r e d u c i b l e } .

Irred(w,R!

= L(w,R)

R

has

are

the

no

R has and R

each R

finite

and w

for

termination chains

unique

iff

iff

there

we

tn+ I,

property have

contains

FTP

t n • tn+ I,

CR iff

L(wI,R)

termination

Irred(w,R)

S ! Z

property

t n R~

Church-Rosser

W l , W 2 E L(w,R) the

to R

n Irr

Irred(w,R)

infinite the

has

FTP

u wES

v}.

respect

Irr

Irred(S,R ! =

(m)

are

R)

w

(i)

R~.

hull

called

A word

(k)

transitive

v,w6X

(g)

(j)

of

relation

For

(i)

hull

congruence

(f)

(h)

and

~ v(mod

congruence

transitive

w O R ~ W l or w I R~ Wo.

reflexive

Notation: The

and

n~.

for all

n L(w2,R)

wEX

• ~.

property

UTP

iff

exactly

one

term

R

has for

E X .

is c o m p l e t e

iff

R has

FTP

and

CR.

4. L e m m a :

Let

R be

(a)

The

a reduction

quotient

operation (b)

The [el,

system

[u]

quotient also

over

Z+/ •

Iv]

X

/

called

=

is

X

a

and



= R~

semigroup

.

under

the

[uv]. is

a

a monoid.

semigroup

with

identity

51 5. D e f i n i t i o n :

Let

R be

a reduction

We

call

/+/

system

resp.

I*/
monoid

presented

by

u

E

relators

~

v

R

its

over

R,

R

X

and

> the

its



= R~

~!~[2~

resp.

presentation,

and

the

.

the

elements

the

elements

ai ~

Z

its

generators. If

R

is

finite

we

say

X+/

resp.

X * / < R > is

finitei Z

presented. Notation:

X+/

=: S G w ( Z , R )

E * / I R > =: M w ( Z , R ) By

SGw(Z)

resp.

6.

resp.

the

free

Mw(Z)

we

monoid

also

denote

generated

by

the

free

semigroup

Z.

Definition:

Let

Z

(0:

Z ~

every

=

{a I .....a k} M\{0)

be

a

a

u

alphabet

function

( over

X

= UlU2...u n and

l
is

which

(i)

and

assigns

let a weight

to

vm

as

follows:

with

u i , v j ~ Z,

and

k
nl

Z ~(u i ) i=l

Z j=l

<

~(vj)

M

Z ~(u i ) i=l and or

or

Furthermore

defined

iff

n

(ii)

is t h e n

v = VlV2...

u ( v

n

or

finite

aiEZ.

A KB-0rdering For

be

we

have

=

Z j=l

~(Vj)

(a)

~ ( U I)

< ~ ( v I)

(b)

~(u I)

= ~ ( v I)

(c)

uI

e ( v

= vI

for

and

all

if

ul=a k

and

vl=a 1

u 2 . . . u n ( v2...v m

v E X +.

.

52 The

fact

X

has

,

that no

orderings algebra and

term

e,

not

meet

algebras, presented

reason,

(Definition

the

a

by

an

which

KB-ordering

over

the

the

KB-ordering

the

of

function.

(Lemma

monoid

of

identity

O-ary

corresponds

of

definition

later

special

algebras

the

since

discussed

13)

identity

a

term

For

that

we

will

12), for

KB-

semigroups

t o (.

Lemma:

Every

8.

term

other

word

does

is a l w a y s

define

7.

empty

weight, over

some

over

the

X

is w e l l - f o u n d e d

and

total.

Lemma:

Every

semigroup

SGw(Z,R')

has

a presentation

R

which

has

FTP.

Let

Proof: we

have

for

Hence

( be

some

every

u ~

R'

(

is t o t a l

v m u.

{ u ~ v I u ~ v e R',

u m v)

R2

:=

(u ~

v m u).

:= R 1 u

u ~

v

R

which

{v ~ is

termination

property

Since

R'

everything

u m v or

. Since

:=

R

and

E R'

X

RI

for

R

v

over

= R I u R 2, w h e r e

And

E

KB-ordering

is

induce

a

v I u ~ v ~ R',

u[

u ~ v

E R 2)

sufficient of of

R

have

condition

because

course

we

the

(

is same

umv

for

for

the

every finite

well-founded. congruence

relation

proved.

[]

Let

us now

sented

define

semigroups

a completion SGw(X,R).

algorithm

for

finitely

pre-

53 9. D e f i n i t i o n :

Given

a finitely

u ~ v for

The

(i)

all

presented

semigroup

u ~ v e R and

semigroup

some

com_pletion

SGw(X,R)

KB-ordering

algorithm

such

that

<.

S G C A r is d e f i n e d

by :

(i)

r R o := R

(2)

Form

all

critical

pairs

( W l , W 2) of

r R n,

i.e.

form CP n

:=

((WlrW2)l(B

u l 2 ~ r 2 e R nr s.t.

l l U ~ r I,

w I = rll 2, w 2 = llr 2) or

(B u l 2 v ~ r I,

s.t.

(3)

Pn

:= { (Vl'V2)l

w I = rl, w 2 = ur2v) }

Vl m v 2 a n d

v i e I r r e d ( w j, Rr),

i,j=l,2,

(4)

r 12~r ~ e R n

If Pn = 0, t h e n t h e a l g o r i t h m

( W l , W 2) 6 C P n}

stops

with

result

4; otherwise

R nr+ 1

put

:= R nr u P n

and

return

to

correspond

to

(2).

(ii)

By S G C A w e d e n o t e

The

critical

the

set

for

the

SGCA

of p a i r s

is

is to s h o w of t h e

which

not that

occurs

our the

KB-completion

resp.

these

in N i v a t ' s Hence

reduction

criteria

[Ni 70]

a proof

system

can

that be

the

easily

criteria.

intention. SGCA

S G C A r.

(rl,ur2 v)

of T h u e - s y s t e m s .

a complete

by c o n s i d e r i n g

this

non-redundant

( r l l 2 , 1 1 r 2)

completeness

generates

found But

pairs

the

can

be

algorithm.

What

we

viewed

will as

rather

do h e r e

a special

version

54 So

let

us

finitely

return

to

presented

subalgebra

of

ground

associativity

In

order

to

algebras can

of

also

a term

invariant congruence and

and be

remember

that

understood

algebra

as

which

the

is fac-

g e n e r a t e d by t h e

the

finitely

for

technical

many

a

law

equations

of

E.

be

i n t r o d u c e the

terms

(ass)

the p r e s e n t a t i o n

term

semigroup

t o r e d by the f u l l y of

the

precise

following

and

purposes

we

will

notations.

10. D e f i n i t i o n :

Let

~

= {a I, ....a n ) be

not

in Z.

Further of

(a)

let A

signature

By

=

For

TSG(Z)

:=

a finite

TM(Z)

finite

and A e

alphabet

and

=

e a

be

letter

algebras

<0 .....0,2>.

A(Var)

d e n o t e the t e r m

(b)

a

resp.

algebras

set

E of

TM(~)

:=

of A resp.

A e.

equations

over

Ae(Var)

TSG(Z)

we

resp.

w e d e n o t e by

SGT(Z,E)

= {w I w

E TSG(Z)/~ass,Em,

w ground term)

resp. MT(Z,E!

= {w I w

~ T M ( X ) / ~ a s s , f(x,e)=f(e,x)=x~, w g r o u n d term)

the f i n i t e l y p r e s e n t e d presented

(c)

For a free write

resp. the f i n i t e l y

monoid.

semigroup

SGT(Z)

MT(X,0).

semigroup

resp.

resp.

MT(X)

a free monoid instead

of

we

SGT(X,0)

simply resp.

55 A

completion

course,

algorithm

start

(KBCA)

for

SGT(Z,E)

(R(E)

:= R(E)

u {f(f(x,y),z)

:= {t ~

s I tEs

~

E E or

analogously

for

Ro(MT(/,E))

MT
:= R ( E )

with

the

ii.

carry

terms

over

E E and

some

~

s) 4)

system

~ f(x,f(y,z)),

x,

to w o r d s

t ~

KB-ordering

reduction

u {f(f(x,y),z) f(e,x)

to

of

f(x,f(y,z))}

set

for

In o r d e r

then,

with

Ro(SGT(/,E))

and

will

we

f(x,e) ~ x

)

define:

Definition:

Let

T and

(a)

The

Te

be

the

transition

defined

as

Obviously resp.

TW

between

TW

and

Te ~

TSG(Z)

:= e

(the

T W e ( a i)

:= a i

for

ai E

TWe(X)

:= x

for

x

: T ~

TW e are and

empty

is

TM(I).

recursively

of

word)

E Var

:= T W e ( t l ) - T W e ( t

restriction

MT(Z)

resp.

(Z u Var)

TWe(e)

T-W is t h e i.e.

T_We:

of

follows:

TWe(f(tl,t2))

(b)

carriers

2)

for

tl,t 2 ~ T e

T W e to T,

(ZuVar) +

isomorphisms Z .

between

SGT(Z)

and

~+

56 As

already

the

mentioned

KB-orderings

over

12.

~

for

term

little

algebras

discrepancy

and

the

between

KB-orderings

Lemma:

are

KB-orderings

KB-orderings TW(T) tI ~ of

~

over

t2

implies

"(" o v e r

{ a l , a 2 , a 3)

41

we have

be

for

by

we

have

which

with

the

to ( in t h e all

are

no

property

sense

ground

that terms

the

such

to

other that we

following

is d e t e r m i n e d

KB-ordering

by the

weights

over

same

T

= A(Var),

weights

that

as

(

al
and

A

=

and a total

ordering

f < a 2.

f(a3,a 3) ~i

a2

so t h a t

no ~ i

(.

hand,

if

~2

now demand

f(al'al) there

the

= 2.

f(al,a I) ~i

corresponds

Hence

T

there

a 2 m a l a I a n d a 3 a 3 m a 2.

the

{al,a2,a3,f)

except

correspond

instance

a KB-ordering

determined

Then

algebra

T W ( t I) ( T W ( t 2) f o r

= ~ ( a 3) = i, ~ ( a 2)

Of c o u r s e

Let

a term

X , such that

T.

Consider

~ ( a I)

( over

= (ZuVar) + w h i c h

Proof:

On

is a

~ .

There

of

there

~2 a2

is

a KB-ordering

defined

as

~i

f>a 2, t h e n w e h a v e

and

is no K B - o r d e r i n g

f ( a 3 , a 3) ~ which

~2 a2"

is e q u i v a l e n t

to (.

[]

Small

variations

algebras

suffice

of t h e d e f i n i t i o n to r e m o v e

of K B - o r d e r i n g s

the above

discussed

over

term

discrepancy.

57 13. Definition:

Let

~

be

weight

We

function

call

~SG

weight (a)

a KB-ordering ~ with

over ~(f)=0

a SGT-ordering

function

TSG(Z)

determined

and ~(ai)>0

over

TSG(X)

for all ai~

relative

if

tI i t2

ti£Z

and

implies for

tl

~SG

t2

if (i)

or

discrepancy

tity

a monoid

word, Note

can, that

algebras. t' E

t' is either rings

14.

is

e-free

~ M over

TM(X)

and

t l = f ( s l , s 2) and

s I ~ S G t2"

from

the fact that

is p r e s e n t e d

allowed

= ~ass, or

{i,j)={l,2}

t2eZ,

for every M

tjeZ

t I ~SG

be r e m o v e d

not

where

the

t2=f(Sl,S 2) and

MT(Z)

But since

It] M

Z.

tlEZ,

occuring

of course, this

~(tl)=w(t 2)

(ii)

The second e of

to

a

~ iff

tl ~ S G t2

(b)

by

in Z* by

by d e m a n d i n g for

t E TM(X) f(x,e)

identical

there

= f(e,x)

e,

we

can

the

iden-

the

~(e)

KB-orderings

sl

empty

= 0. over

is e x a c t l y = x~

such

define

term one that

MT-orde-

as follows.

Definition:

Let

~SG

weight aiEZ. We call weight

be

a

SG-ordering

function

~M

~

with

over ~(f)

a MT-ordering

function

TSG(Z)

determined

= 0 and ~(a i)

over

TM(Z)

by

the

> 0 for

all

relative

to

the

~ iff

(a)

for tl,t 2 e-free

(b)

e ~M t

(c)

tl ~M t2

for t ~ e if

we have and

tl' ~ M t2'

tl ~ M t2

iff

tl I S G t2

t ~ Var for t i' e

[ti] M.

58

since

for e v e r y

~(tl)

~SG

~(t2)

orderings

substitution and

t I ~M

~SG and ~ M

0 we

have

t2 i m p l i e s

are, of course,

t I ISG

t2 i m p l i e s

~(tl) ~ M

~(t 2) the

t er m orderings.

F u r t h e r m o r e we have

15. Lemma: For

every

KB-ordering

MT-ordering TSG(X)

~ over

resp.

Proof:

Let

weight

function

yields

over

TM(X)

resp.

Z*

resp.

there

Z+

a SGT-ordering

~SG

is a over

such that tl ~ M

careful

(

t2

implies

TWe(t I) ( TWe(t 2)

tl ISG t2

implies

TW(t I)

~M'

~SG

and

(

~0 w i t h ~(f)

inspection

of

the

be

(TW(t2).

determined

by

t he

same

= 0 and ~(a i) > 0 for a i E Z. A definitions

of

~M'

~SG

an d

then the lemma. []

We are now ready to prove: 16. Theorem: Given

a finitely

represented

semigroup

SGw(Z,R)

that u m v for all u ~ v G R and some K B - o r d e r i n g Let

E(R)

:= (t

-- s I t,s g r o u n d

and let ~SG be the S G - o r d e r i n g Of

course,

SGT(X,E(R))

of S G w ( Z , E ( R ) ) ,

i.e. w e h a v e

and t I ISG t2 implies

(.

t e r m s of TSG(X) a n d TW(t) ~ TW(s) E ~)

corresponding

is a t e r m

such

al ge br a

representation

TW(SGT(X,E(R))

TW(t I) ( T W ( t 2 ) .

to ~. = SGw(X,R)

59

Furthermore the

SGCA

the

reduction

after

completion chapter R~

:=

R wi

let

be

the

i steps system

after

generated as

~

The

i = 0:

First

Since

proof

of

is by

all

let

starting

on

compute

t ~

only

be KB-

Definition

s[

t,s

3

of

t ~

s ~ E(R)}.

ground

terms}.

i.

RI

.

superpos(Left(R~\{ass}),Left(R~))

can arise

(redundant)

by

with

s E R I +i"

induction

us

the by

f(x,f(y,z)),

T h e n w e h a v e for all i E R wi = {TW(t) ~ TW(s) I t ~

Proof:

by

defined

i steps

{f(f(x,y),z)

system generated w i t h R ow := R a n d R

starting

algorithm

II.2.

reduction

=

critical

pairs

from fass(t,x)

superpos(Left(ass),t)

=

if t = f ( t l , t 2 ) E L e f t ( R T k { a s s f(f(f(x,y),z),w)

where

The

fass(t,x)

KB-algorithm

CP~

:= I r r e d ( f ( t , x ) , a s s ) .

then

generates

=

{

=

Hence

we

the

= {(fass(t,x),fass(S,X)) u

pT

fass(t,x)

u

if t E L e f t ( a s s )

following

I t ~

sets:

s E R~\{ass}}

{(f(f(x,f(y,z)),w),f(f(x,y),f(z,w))}

"

fass(S,X)l

t "

S E RoT\{ass}}

po

obtain:

{TW(t)

~ TW(s) I t ~

s ~ RT,

t,s

ground

terms}

= {TW(t)

~ TW(s) I t ~

s E R~,

t,s

ground

terms}

= {TW(t)

~ TW(s) I t ~ s E E(R),

t

~SG s}

=

w

RO .

})

60 i ~ i+l:

We h a v e More

Let G i

:= {t ~ s I t ~ s E R~,

and Gxj

:= { f a s s ( t , x )

already

generally

Assertion:

Proof

seen the

R~

= (ass)

on

j = i:

trivial

j ~ j+l:

Obviously

on

superpos(Left(Gxj),Left(Gj)) ground

it has

be t h a t

to

at

Pj.

contribution

~ fass(S,X)l

if

to Pj. contributes

t ~ s e GPj} has

=:GxPj

to Pjo

no e f f e c t

on Pj.

we o b t a i n

Suppose

now

there

=

R~

u GPj

u GxPj

=

{ass}

u Gj+ 1 u Gxj+ 1

v ~ w ~ R ~ + I \ R ~. are

llU ~ rl,

ul 2 ~ r 2 ~ R~ such

that W

(a)

v E I r r e d ( r l l 2 , R ~)

and

w E Irred(llr2,Ri)

(b)

v e I r r e d ( l l r 2 , R ~)

and

w £ Irred(rll2,R~).

W.l.o.g. By the

where

reductions

contributes,

superpos(Left(Gxj),Left(ass))

R +I

or

hence

=

superpos(Left(Gxj),Left(Gxj))

(e)

Gxj;

Pi"

(c)

{fass(t,x)

Then

generates

superpos(Left(Gj),Left(R~))

Hence

holds:

j:

(b)

(d)

}

we have:

no e f f e c t

Let GPj

t ~ s E Gj

u Gxj

superpos(Left(ass),Left(R~))

all,

terms};

u G o u Gx o-

assertion

u Gj

ground

~ fass(S,X)l

R T = (ass}

following

by i n d u c t i o n

(a)

that

t~s

we a s s u m e induction

(a).

hypothesis

fass(ll~,U~)~rl t ~ E TW-I(t).

According

ease

~

and

to the p r e c e d i n g

fass(ll ~, fass(U~,X))

~

there

are

fass(U~,12

proposition

fass(rl~,X)

E

~)

there RT+ I.

r2

e Gi+ 1

is also

61

Hence

(fass(rl~,12~),fass(11~,r2~))

longing

is

the

critical

pair

be-

to

f a s s ( l l ~ , f a s s (u ~ ,1 2

)) E

superpos(fass(ll~,fass(U~,X)), fass(U~,12~))

By

a

second

conclude

Hence

then,

that

of

the

there

induction

hypothesis

can

are v~,w ~ such that

v ~ e Irred
)

w ~ E Irred(fass(ll~,r2~),R~+l

)

v ~ e w ~ ~ Gi+ 2

we

! R~+ 2 w h a t

was

and

to be p r o v e d .

T let t ~ s e G i + 2 \ R i + I.

Now Then p

application

there

E

is

(see the p r o o f

of the p r e c e d i n g

superpos(Left(GXi+l),Left(Gi+l))

RTi+1

such

proposition)

that

(a)

t E Irred(Pl,

)

and

s e I r r e d ( P 2 r R T + I)

(b)

t E I r r e d ( P 2 , R ~ + I)

and

s ~ I r r e d ( P l . ~ T + I)

where

(pl,P2)

Again

we

is the

stick

only

critical to c a s e

pair (a);

belonging

case

a

or

to p.

(b) c a n be p r o v e d

ana-

logously. To be m o r e

precise

there

fass(ll,fass(U,X)) such

are

however

~ fass(rl,x),

fass(U,12)

~ r 2 E RT+ 1

that

P=fass(ll,fass(U,12)),

Since by the

there

is

induction

TW(II)TW(u)

also

fass
hypothesis

~ T W ( r I)

Pl=fass(rl,12

and

~

),

P2=fass(ll,r2

r I e R~+ 1 we

can

).

conclude

that T W ( u ) T W ( I 2) ~ T W ( r 2) e R wi,

62 Hence

(TW(rl)TW(12),TW(II)TW(r2))

longing

to

A second

TW(p)

is

the

critical

pair

be-

= TW(II)TW(u)TW(12).

application

of the i n d u c t i o n

hypothesis

yields

then

of c o u r s e TW(t)

E I r r e d ( T W l P l ) , R ~)

Hence

TW(t)

i-th

step

~ TW(s) i.e.

is

and

TWls)

generated

by

TW(t) ~ TW(s)

~ Irred(TW(P2),R~).

the

SGCA r

during

the

E R wi+l"

[]

As

we h a v e

semigroups

seen

above,

is a c o m p l e t i o n

algorithm

w h i c h does no longer w o r k e x p l i c i t l y

of a s s o c i a t i v i t y , (a)

there

by m e a n s

omitting

for

w i t h the law

of

brackets,

i.e. t u r n i n g

to w o r d s

instead

of

terms and (b)

changing

the

ding

superpos-sets

the

simpler by

In the

tion algorithm group

axioms.

Again

instead

we will

treat

Analogously notations:

process

string

following

CR-test

of

we

will

terms

words

to

is

the p r o c e s s

replaced

overlapping,

by

which

of builthe

can

be

more done

matchings.

do even

for f i n i t e l y

of

so that

the

over

over

more

presented

the

free

and d e f i n e groups

group

(ZuZ-I) *, w h e r e

preceding

with

a comple'built-in'

generated

by

X

z-l={a-llaEZ).

terminology

we

fix

our

63 17.

Definition:

Let

be

a reduction

FR

:= {aa -I ~ e,

We

call

R,

R

its

The

free

group

and

will

be

assume

weight

to

~(ai)=~(aTl)±

is

algebra

and

group

completion axioms. the

SGCA

dure

To

and

will

for

called

be p r e c i s e

be

groups

group

R u F R ~*

presented u ~ v

by

~ R

its

generators.

X is of c o u r s e

rest

of

this

(ZuZ-I)*/

( over ~

paragraph

compatible

In p a r t i c u l a r ,

to

we

(ZuZ-I) *,

~\(0}

Definition

17

completion

Gw(X,R).

case

the

=

that

some

well-

suppose

determined

with

the

comby a

property

ai~Z.

to

on



elements

F T P by b e i n g

Thus

group

one

shown can

be

again

we

an

takes

in

SGCA for

can

the

even

finitely

pre-

more

advantage

following,

replaced

establish

with

leave

by

of great

a very

"symmetrizing".

we m u s t

starting

the

term

only.

achieve

which

the

algorithm

words

can

algorithm

As

the

X u Z -I

all

work

by

Gw(Z).

~:

a handy

groups

the

[ aeZ)

by Gw(Z).

a KB-ordering

for

sented

In

over

accordance

= RuFR

(ZuZ-I) *,

a E Z its

generated

R has

function

In

over

the

element

throughout

ordering

patibility

Ro

the

denoted

presentation

founded

a-la ~ e

representation, and

we

system

Gw(Z,R): = (ZuZ-I) * / < R u F R >

relators

Again the

R

some

notations.

efficient the

group

parts

simple

of

proce-

64

18.

Definition:

Let Q be a s u b s e t of R.

(i)

By

pAr(Q~R)

completion

we

algorithm

(i)

r.=R\Q Ro .

(2)

Form

the

lapping with

denote

following

for s e m i g r o u p s

following

the

some

the

left

left

critical

side

side

of

of

a

part

of

the

SGCAr:

pairs

by

reduction

a reduction

over-

from

of Q,

Rn i.e.

form CP n := { ( W l , W 2 ) I 3 l l U ~ r l E R ~, 3 llU~rlEQ,

ul2~r2EQ

or

ul2~r2eR ~

s.t. wl=rll 2 and w 2 = l l r 2 r or B u l 2 v ~ r I E R n, 12~r 2 E Q or B u l 2 v ~ r I e Q, s.t0

(3)

pn:=

w l = r I and w 2 = u r 2 v

{(Vl,V2) I v I m v 2,

If Pn=0,

RE, to

(ii)

Notice, SGCA r which

which

some

a l g o r i t h m stops w i t h put Rr n + l := Nnr u Pn and

result

return

(2).

we d e n o t e

particular, tests

the

that

the n o n - r e d u n d a n t

pAr(FR,RuFR)

CR-property

arise by o v e r l a p p i n g

FR w i t h

( W l , W 2 ) e C P n}

the

otherwise

By PA(Q,R)

in

then

}

viEIrred(wj,R~uQ) i,j=l,2,

(4)

12~r 2 •

only

for

is

pAr(Q,R) o

the

partial

critical

a left side of a r e d u c t i o n

left side of a r e d u c t i o n

of FR.

pairs not in

65

As

it w i l l

be p o s t u l a t e d

generated if

no

19.

by

the

in T h e o r e m

PA(FR,RuFR)

"undesirable"

is

reductions

22 t h e

just are

set of

Sym(R),

reductions

defined

below,

possible.

Definition:

For

1 ~

r ~ R we

Sym(l~r)

(i)

define

= {A ~

@

A@ -I or A-I~

i ~m~,

permutation

of

A

have

= aA'b

we

is a c y c l i c ir -I a~'(

and

for

Qb -I a n d

A'b ~ a -1 ) (ii)

In

Sym(R)

order

to

avoid

PA(FR,RuFR)

the

ditions

very

20.

are

u

=

(Sym(l

~

r) I 1 ~

"undesirable"

following

so

called

r ~ R).

reductions small

during

the

cancellation

con-

useful.

Definition:

Let

S be

positive

S

a

real

satisfies

wu,

wv

l,.i.

set

~ S

of

group

words,

i.e.

S c

(Xux-l) * a n d

A a

number.

the

small

implies

cancellation

lwl

~ A l w v I for

condition the

lenght

C'(A)

iff

function

66 21. D e f i n i t i o n :

(i)

A

group

word

czc!icall Z respect

w

reduced,

ments

of

words

S are

permutations (Let

if

bi ~

w

is

X u Z -I

is

called

irreducible

S

is s_ymmetrized

cyclically

reduced

if all

and all

The

of w or w -I are in S for any w6S.

W=bl°..b m

then

symmetrized

i.e.

the

cludes

of

closure

smallest

S,

ele-

cyclic

course

w-l=cm...Cl ,

c i = a j if b i = a j -I and c i = a j -I if b i = a 9, w h e r e

(iii)

with

to FR a n d b m is not the i n v e r s e of b I.

A set of g r o u p

(ii)

= bl...bm,

set

is d e n o t e d

of a set of g r o u p

that

is

with aieZ.)

words

symmetrized

and

S, in-

by Sym(S).

22. T h e o r e m :

Let

S

group If

result

The p r o o f

by of

groups:

satisfies

E R

the

~(a)=~(a-l)=l

) for

a finitely

presented

C'(I/2)-condition for

all

a£Z,

then

and

(

Sym(R)

is

de-

is the

PA(FR,RuFR).

can be f o u n d

carrying

Theorem

I l~r

Gw(Z,R).

Sym(S)

fined

16 by

:= { Ir -I

out

the

analoguously

to the p r o o f

of T h e o r e m

PA(FR,RuFR).

22 y i e l d s n o w the f o l l o w i n g c o m p l e t i o n

algorithm

for

67 23.

Definition:

Given for

a finitely

all

(i)

represented

l~r e R and s o m e

The group

(I)

R~

(2)

Form

group

GW(Z,R)

KB-ordering

completion

algorithm

such that

Imr

(.

GCA r is d e f i n e d

by:

(Wl,W 2) of R r,

i.e.

r I B l l U ~ r I, u l 2 ~ r 2 e R n,

s.t.

:= Sym(R)

all

critical

pairs

form CPn:=

{ ( W l ~ W 2)

wl=rll 2 and B ul2v~r I,

w2=llr 2

or r s.t. 12~r 2 E R n,

wl=r I and

(3)

Pn

:= {(Vl~V2)/

Vl~CV 2 and

r), vi£Irred(wj,R n

i,j=l,2, (4)

If Pn=0, R~,

(ii)

Again

Additional As

we call

to

at

use

examples

the

a l g o r i t h m stops w i t h put R n + I .._ - R nr u Sym(P n)

GCA the n o n - r e d u n d a n t

the There

additional we

end

incorporate

t i o n procedure. to

otherwise

(Wl,W2)ECP n ) result

GCA r.

remarks:

mentioned

methods

then

w2=ur2v

of

additional

are also

laws

mention

chapter

[Ni

axioms

a number

in t h e 70]

I.l., there

ground

for

into

are the

of d i f f e r e n t case.

monoids,

As

various unificamethods

important

[Ba-La

81]

for

88 abelian mial)

semigroups

rings,

and

see e.g.

case

of B o o l e a n

This

seems

the

work

of

Buchberger

[Bu 84]. Of p a r t i c u l a r

algebras

because

first to be h a n d l e d

in

of

its

on

(polyno-

interest

relevance

[Hs 8 2 ] , c f . a l s o

to

is the logic.

[Ka-Na85].

69

II.4.

First Analysis of_the CompletionAlgorithm

In c h a p t e r

II.2.

stages

or

selves

to t h e

semigroup

case

case.

Successful

ordering

The

R ~ is a l s o t

tain

different

i.

which

is

this

the

which

with

into

reduces

semantically

The

next

our-

continues

the

group

assume

on the g r o u n d two

its

a

and

well-

terms.

cases:

complete

system

R n.

R ® is i n f i n i t e .

each term equivalent

constructions

systems.

to we

and

restrict

algorithm

section

a finite

system

we

is p a i d

is total

splits

and

section

completion

this

algorithm

of

R~

to

and

proposition

t to the q - m i n i m a l t.

We

need

refers

to

to

want

ob-

compare

notions

in

II.l..

Proposition:

Two

complete

the

same

Proof: s S~ t E

limit

complete

term

In

Throughout

Termination

(II)

the c o m p l e t i o n

attention

continuation

(I)

chapter

where

Special

term

complete

defined

approximations.

successfully.

founded

we

equationally

irreducibles

W.l.o.g. t.

We

Irr(R),

can hence

it

are

suffices

also

assume

t E Irr(S).

equivalent

systems

derivation

equivalent.

to

show

that We

R

also

that

s ~

R and S w i t h

t E R

implies

is n o n - r e d u n d a n t . have

some

r such

Then that

70

s

t

r

holds;

t E Irr(S)

then

gives

t = r and we

get

s S~

t.

[]

2. P r o p o s i t i 0 n :

Suppose

R and

S are

(i)

complete

(ii)

non-redundant

(iii)

either

a)

or

derivation

b) e q u a t i o n a l l y same

Then

R

Proof:

= S up

By

the

to a r e n a m i n g

previous

proposition

systems

assume

that

S are derivation

We

will

Assume s'

show

Irr(S)

t

E

for

By assumption

R; all

we

then proper

also

s S~ tl and

obtain

dundant.

have

s R~

and

the

and

R ! S u p to s o m e

s ~

~ Red(R)

=

with

the

of v a r i a b l e s .

equivalent

Irr(R)

equivalent

irreducibles.

vation

R and

equivalent

same

the

that

irreducibles

equivalent

Red(R) renaming

fact

deriwe

can

and

= Red(S)

holds.

of v a r i a b l e s .

s E R~d(R)

and

subterms

s' of

by

non-redundancy

s.

have S~

t I which

...

S ~ tn

gives

t

= t = t I because

R

is n o n - r e -

71

Therefore

The

we

same

have

u ~ v ~ S

argument

and

a

substitution

%1~

~ v

s

~t

provides

so ~

to

e R

and

a

0 such

that

substitution

T

for which

So~

holds.

This

renaming

implies

so

=

~t O

s,

to

= s;

therefore

0 and

T are

substitutions.

£3

3.

Corollary:

Under have

The

the

assumptions same

argument

where

A

the

we

deal

typical

ground

also

works

with

words

application

system

from

system

of

"below"

reductions

of

and

previous

proposition

R and

S

reductions.

for

the

instead

is

and

the

the

from R~

group of

and

case

terms.

construction

"above". is t h e

semigroup

of

Suppose limit

R

the is

system.

a

limit finite

72

We

also

have

E

= E(R)

= {s ~ t I s ~ t e R}

and

= {s

~

R'

~ t I t ~

(which

is

s,

E I~ s

t}

(s ~ t I t ~ s, s R ~

t

] in the

semigroup

By r e m o v i n g dant R~;

redundancies

complete hence

system

we

have

In p a r t i c u l a r ,

case for

where a

with

determined

may

however

is

of

the

a second

result

equivalent

redundancies

For the g r o u p a n d words

only

non-redundant

the

non-redunto

"above".

removing

(ground)

by

get

derivation

R ~ from

system.

presentation

is u n i q u e l y orderings

deal

R' w e

which

process

determined

we

given

R~

obtained

the

in a u n i q u e l y

from

case).

chosen

we

results

semigroup state

complete

ordering;

in e s s e n t i a l l y

that

system

different

different

limit

there

three

systems. Running

through

possibilities

As der

we

Clearly

For

2)

There

3)

All

limit are

{t

are

finite

limit

orderings

presentation

systems

and

systems

algebras

and

and A are

n-ary

as t h e

the

limit

systems.

infinite.

c a n occur. ordering

First 4.

For

we

consi-

a finite

A we put

~ t a ~-minimal

T(A)

R:

infinite

are

fix

are

finite.

see, all p o s s i b i l i t i e s

A with carrier

each

defined

Mf

All

finite

T(A):=

Knuth-Bendix

for a g i v e n

i)

will

algebra

all

term

denoting

in o n e - o n e

operation

f

the

some

element

in A].

correspondence.

multipication

table

is

set of e q u a t i o n s

= { f ( t I .... t n ) ~ t

[ t I .... t n , t E T ( A ) ,

A I~ f(t I ..... tn)~t};

73

We p u t u

M=

The

(Mfl

f an o p e r a t i o n ) .

multiplication

table

of

a constant

consists

of

just

one

equation.

If

RM

then

all

= {s ~ t I s = t E M,

right

sides

of

sides

are

irreducible

sides.

RM

is c o m p a t i b l e

FTP. B e c a u s e which

the

steps

( 2')

not

apply

fore

the

non-redundant

also

a subsystem

Next

isolate

are

(3)

right

irreducible;

coincide

with

ordering to the

UTP.

in R M

and

because

already

the

t reduces

redundancies

(i'),

they

with

have

does

we

equations

iff

term

A I~ t ~ t' w e

Removing the

each

s t t}

and

chapter

sides

complete

are

right

therefore term

RM

t' for

simple.

II.2.

From

step

irreducible.

system

has

is c o m p l e t e .

is p a r t i c u l a r l y in

the

minimal

Therefore

left

belonging

(i')

There-

to

RM

is

of R M.

a case

where

the c o m p l e t i o n

algorithm

termi-

nates.

4.

Proposition:

Suppose group

G with

algorithm rings

Proof: If

IGI

at m o s t

is

a

finitely

an

initial

terminates

and there

Consider

are

the

because

system

(i.e. only

the

finite

R o.

R W is

Then

finite)

finitely

complete

= n t h e n the m a x i m u m n-i

presented

limit

many

group the

system

KB-orde-

systems.

R ~.

l e n g t h of an i r r e d u c i b l e

irreducibles

semi-

completion

for all

such

or

are closed

under

word

is

taking

74 subwords. Therefore are w o r d s finitely bound

w with many

IR MI

is

length

there finite

tables.

redundant complete

that there

Hence

there

tables

are o n l y

is a u n i f o r m

KB-orderings.

is a u n i f o r m and

in the m u l t i p l i c a t i o n

< n which means

different

< k for all

Therefore R®(~)

the e n t r i e s

there

bound

are

only

on

IR~I

too,

finitely

hence

each

such

non-

many

systems.

[]

These considerations equations

with

An example

fact

This

is

finite,

in

even

when

general

occur.

idempotent

idempotency

is t h a t

useful

each

semigroups;

is x 2 = x.

finitely

particular

the g e n e -

generated

the

free

idempotent

ones

(see

for

are

in-

[Gr-Re]).

is n o t

finitely square,

describing

here

semigroup example

variables

sometimes

is g i v e n by the

ral e q u a t i o n A key

are

quite

many

obvious

words

on

i.e. a s u b w o r d

n

because

for

letters

which

of the

An immediate consequence

form

n > 2 there do

not

contain

a

uu.

is

5. P r o p o s i t i o n :

Each

finitely

finite complete

A w a y to c o n s t r u c t We

start

This

with

gives

describe

the

a

the

generated

idempotent

semigroup

has

a

system.

the s y s t e m

is as follows:

reductions

system

which

semigroup.

is

AA ~ A for complete

each but

generator does

not

A. yet

75 We have two

kinds

1)

Ordinary

2)

Adding

of a d d i n g r e d u c t i o n s :

s t e p s of the S G C A reductions

through

all

maximal

length

present

system.

The t e r m i n a t i o n

words

form

length

words

uu ~ u where l~k,

where

v e Left(R),

Completeness

b)

E a c h w o r d of l e n g t h n, n the m a x i m a l

In p a r t i c u l a r finitely

gives

generated

discussed

at

is

the

R being

the

l e n g t h of w o r d s

is reducible.

of the s e m i g r o u p

this

k

u runs

are

a)

By the f i n i t e n e s s

the a l g o r i t h m

a solution

idempotent

several

places

to the

semigroups; in

the

stops°

word

this

problem

question

literature

(cf.

in was

[Gr-Re]

[Ger]).

A disadvantage lar t h e the

of t h i s m e t h o d

complete

tors

has

more

fact The

next

system;

this

332

this

was

big,

The

to

the

In p a r t i c u -

the

fact

increases

semigroup

many

is

due

semigroup

388

procedure

with

that

sharply

4 genera-

elements. use

of

c a r r i e d out

in

some

conditional

[Sie-Sz].

We will

below.

example

will

lead to d i f f e r e n t Fibonacci-groups

defined

is q u i t e

of g e n e r a t o r s .

efficient

discuss

is its c o m p l e x i t y .

idempotent

e.g. a l r e a d y

reduction

The

system

s i z e of t h e f r e e

with the number

A

of

of

conditions

v 6 Left(R),

or

of t h e

show

that

limit F(2,n)

different

systems on

orderings

An_IA n = A 1 AnA 1

= A2

in

for f i n i t e algebras.

n generators

A 1 .....A n a r e

by the e q u a t i o n s A i A i ÷ 1 = Ai+ 2

may

for l ~ i ~ n - 2 ,

7G

If

we

the

For

Al

= A,

equations

for

the

The

first

system

just

For

the

But

reduce

ABBABA

= B

BABABBA

= A

we

~(A)

= 5

~(B)

= 1

choose

the

remaining

constants

to

the

weights

is:

A

~ B3

A-I

~ B-3

B-6

~ B5

B6

~ B -5

gives

the

immediate

information

that

the

~(A)

= ~(B)

= i.

~ A-IB-I

F(2,5)

ZII.

second

system note

remove

F(2,5)

system

The complete

This

A 2 = B and

ordering

complete

This is

put

that

ordering

system

is

take

is:

B-2

~ B A -I

B-IA-I

B2

~ A B -I

A - 2 B -I ~ A B

A B A -I ~ B

A2B

~ A - I B -I

A-IB

~ B A -I

A3

~ B A -I

BA-IB

~ A-I

BA-2

~ A2

BA

~ AB

A 2 B -I

~ A -2

B-IA

~ A B -I

A-3

~ A B -I

much

the

we

less

useful

commutativity

for

an

immediate

is e x p r e s s e d

overview.

directly.

77

We

now

turn

as

finite,

to

depending

There

are

which

we

limit

system

ciently One

various will

many

of t h e

is

ways

shown

the

termination

new

ordering

was

of

are

other

word

If w e

are

to

we

that

know

the

suffi-

old

in

result

completion

algorithm

presented

Already

can

the

several

~

the

[Kn-Be]

it

in a l o s s

of

algorithm.

one

by

The

changing

the

have

the

to

be

re-

(

terms

(Xl{X2(,..(Xn_iXn)...)))-i

equal of

reductions

(xy)-l.

words to

in t h e i r tn

would

,

congruence contain

n~2

classes

all

xk

but

(any more

)-i).

take sn

then

prove

= i.

that

:=

minimal

occurrences

II.2. w h e r e

the

systems

elements.

theory.

from

to

the completion

ordering

for

to c o n s i d e r

s

well

e.g.

tn

which

as

limit

suffices

where

group the

)-i)

obtain

suggests

infinite

inverse:

y-lx-i

This

often

order

irreducible

obtained

be

infinite

In

in c h a p t e r

property

the

we

it

or

changing

consequence

versed,

later.

pure

~((

a

describe

examples

for

that

R ® can

ordering.

to

infinite

given

was

As

the

reductions

system

for

where

investigate

was

complete

on

classical

terminates

weight

situations

for

all

minimal

system

n

:= X n + l ( X l ( X 2 ( . . . ( X n X n + l ) . . . ) ) )

sn ~ tn

too.

This

R compatible

Sn

holds means

with

R~

this

tn

and

that

all we

ordering:

proper have

for

-I~

n~2

subterms any

of s n

complete

78 with

s E

Irr(R)

Therefore have

sn

for

This

the

of

s

deal

s

E

subterms

Left(R)

~ must

be

a

irreducible

that

example

with

the

are

words

following

proper

and

some

renaming tn

s of

substitution

of

would

be

s n.

variables a

0

we

because

substitution

in-

too.

implies

Another

all

some

= 0(s);

otherwise stance

for

limit

the

now

system

idempotent instead

conditional

R"

and

must

be

infinite.

semigroups;

of

terms.

complete

note

In

system

was

that

[Sie-Sz] given

we the

for

this

theory: (i)

xx

(ii)

~

xyz

x

~

(Here

xz

provided

(u)

denotes

variables

We

view

system

is

We

and

is

of

to

it

{x}

system

of

as

in

suffices

of

reductions.

sn

:=

XlX 2

...

XnXlX n

rn

:=

XlX 2

...

XnX n

tn

:=

xI

u, in

an

to

{xy)

i.e.

The

order

to

consider

= {z}

the

set

infinite

non-redundant show

that

a

certain

it

...

...

Xn_iXnXn_l

x2x 1

x2x 1 -..

xI

of

sn

get sn ~

Furthermore

we

have

(a)

Each

(b)

sn

(c)

The

(d)

tn

for

proper

is

tn .

all

n~2:

subword

reducible

only is

rn ~

only

subword

irreducible.

by

is

irreducible

renamings

reducible

in

rn

of is

sn ~ XnX n

is in-

put ~,.

of

u.)

ordinary

redundant).

set

and

occurring

exhibit;

again

= {z}

content

constants

course

difficult

infinite

finite

the

conditional

(which

system still

this

and

that

rn

79

The

technique

restrict

of

ourselves

substitutions Suppose of the is

in

xI

half

are

is

only in

shows

It f o l l o w s sn ~ t n Note

y.

{x)

the

weight

of

the

constants. get

The

Now

would

now

the

all

algorithm

with

contain

be

in

x,

we

a

and

of

xI

all

reduction i > i,

in

hence

x1

x i and

y contains

have

that

xi~

xy

we

variables.

by

some

to

that

(d) t h a t

z; we

because see

some

that

other

xk

a contradiction.

the

works

special

reductions

generates

the our

minimu~

because weight

considerations

of

from

which

complete

of

xm

does

is

idempotent

not

semi-

system.

arbitrary because x for

with

infinite

by t h e

otherwise

every

n ~ m

also

ordering.

and

variable violate

compatible arguments

with from

Any the xi FTP. the

above.~

case. u

= yaw ~(a)

an

replaces

system

ground

u

generated

reducing

xn ~

semigroups

system.

case

length

= < a I ..... anl

that

with

KB-orderings

of i d e m p o t e n t

a finite

hence

i.e.

all

is a l w a y s

finitely

be

to t h e

for

this

general

to

and

such

reduced

occurrences

complete

complete

weights

to

hence

theory

any

once,

is

of

and

KB-orderings

reduction

exactly

z has

Combining

a

G

sn

sees

renamings

x starts

means

substitution

Suppose

to of

variables

the

has

turn

be

The

a finite

KB-orderings

We

have

argument

admits

Therefore

one

same

be o m i t t e d ,

group

give

First

remaining

general

consider

in

x I has

specific

special

one.

s n.

Each

reduction

a typical

(a) a n d

for

have (2)

the

• (z);

from

this

(i)

cases

and

This

cannot

we

xz

two

that

above

~

of

is

subword

Therefore

not

which

a

xyz

xyz.

there

that

form

first

to

all

involved

e.g.

the

proof

~

e> and

> ~(v)

where a

is

+ ~(w).

some not

a

= a i occurs

in vw.

Then

the

We

choose

completion

80 R ® = {a ~ v - l w -1, a -1 ~ wv}.

We

observe

here

"Freiheitssatz" that

for one

defining

of

relation

there

Magnus

relator

word denoting Notice

that

groups

occurs

the

identity

connection

e.g.

each

generator

in

[Ly-Sch]).

each

to This

occurring

non-trivial

the says

in the reduced

element. c h o i c e of w e i g h t s

may

lead to an

limit s y t e m R m.

F i n a l l y w e will give an e x a m p l e

w h e r e the l i m i t

inifinte

for all K B - o r d e r i n g s .

Consider

the so c a l l e d G r e e n d l i n g e r

G = < A, B, C

For each

a

(cf.

also

that a less skillful

infinite

is

weight

I

function

group

ABC ~ CBA

~ABC

E { ~

s y s t e m R W is

>

I~ is a w e i g h t

function

s.t. ~ ( A ) < ~ ( B ) < ~ ( C ) } the c o m p l e t i o n

algorithm

GCA generates:

R W = {CBA ~ ABC, C-IB-IA-I

A - I c B ~ BCA -I,

~ A - I B - I c -I, B - I c - I A ~ A C - I B -I,

c-lAB ~ BAC -I, A ( B C ) n A -I ~

is

easily

functions

seen

WACB'

that

~BAC'

the

~BCA'

B ( C A - I ) n B -I ~

(A-Ic) n,

(CB) n, B ( A C - I ) n B -I ~

C( A - I B - I ) n C - I

It

B - I A - I c ~ CA-IB -I

~

(B-IA-I)n

limit ~CAB'

systems

I

nE~

for

W C B A are also

(C-IA) n, }.

all

weight

infinite.

81 Additional

For

remarks:

practical

purposes

it

is

important

to

have

algorithms

w h i c h c o m p u t e the c o m p l e t e s y s t e m R ® (if it is finite) fast. An i n t e r e s t i n g w a y to avoid the i n v e s t i g a t i o n of u n n e c e s s a r y critical

pairs

is d e s c r i b e d

however,

that such m e t h o d s

of the a l g o r i t h m

in

[K~ 86].

do not

It should be added,

always

improve

the

speed

: One a l w a y s has to test w h e t h e r the m e t h o d

a p p l i e s and these tests also c o n s u m e s o m e t i m e and space. Other

heuristics

and

strategies

are e.g. f i r s t to o v e r l a p

short w o r d s and s i m i l a r devices; each i m p l e m e n t a t i o n usually contains

such

methods.

n e g a t i v e aspects; nearly unknown.

T h e y all

theoretical

have

results

positive in this

as

well

direction

as are

82

The

II.5.

Special

Word

Cancellation

The oldest problem It

for

allows

relator power

of

~u I >

success

algorithm

on

important

of

Dehn's

that

E S

In t h i s

v

if

is

u v -I

the

and

Small

s o l v e s the w o r d

Dehn's

= e

length

found

Group

in this

function

I...I. T h e

over

in the b o o k

Theory" context

which

algorithm.

is a d e f i n i n g

investigated

c a n be

conditions

are

several by L y n d o n

([Ly-Sch]). is the c o n c e p t

sufficient

for

of the

algorithm. conditions

Sc(I

A ~

we

already

lwl

met

are

the

so

~+.

u I-l) * s a t i s f i e s

implies

section

Groups

which

groups

was

notion

C'(A)-conditions,

Remember

of

"Combinatorial

cancellation

wu,wv

u by

Iv I f o r

cancellation

called

classes

a good overview

The most

Small

popular algorithm

replace

this

Schupp

small

most

to

for

Theory

certain

and

decades; and

and

Problem

the

C'(E)-condition

iff

< A • lwvl.

we need some

more

conditions,

the

so c a l l e d

T(p)-conditions.

I. D e f i n i t i o n :

Let S

S ¢ (I u I-l) * and p E N

satifies

w I ....W n ~ S for

iSi~n,

the

T(p)-conditi0n,

where

iff

w i is not t h e

there

w 2 w 3 ..... W n W 1

be given.

is a t

which

least

for

literally

all

n,

3~n
inverse

one of the products

is f r e e l y

reduced.

and

of w i + 1 W l W 2,

8B The

following

theorem

Dehn's a l g o r i t h m

{of

and proved

Greend!inger) in

is

central

for

[Ly-Sch].

2. T h e o r e m :

Suppose

G

=
t e d group, clically are

in

= {a I .....a n } I S>

S symmetrized

reduced S

for

satisfies

(i.e. all

a n d all

any

and

Then every freely

in G c o n t a i n s

These

conditions

C'(A)

and

word

more

are

S

elements

satisfies

C'(i/6)

or

wE(Zuz-l) * for w h i c h w

than

sharp

half

among

of the

some

element

conditions

of S.

of

type

in s e c t i o n

II.3. the K n u t h - B e n d i x

completion

algorithm,

in p a r t i c u l a r the g r o u p c o m p l e t i o n

algorithm

generates

reduction

which

Dehn's a l g o r i t h m

for take

GCA.

over

t i o n theory: and

But

the

does

systems

namely

as a s u b s y s t e m .

conditions

the

for

Dehn's

there

it

This

some

results

get

GCA,

contain

means

that

all

are

also

sufficient

algorithm

has

combinatorial

Sym(R)

suffi-

disadvantage

simply

of

cancella-

O n e d o e s not get an i n s i g h t

not

= e

T(p).

As d e m o n s t r a t e d

cient

S

T(4).

reduced

holds

and

presen-

of S are cyp e r m u t a t i o n s of s or s -i

cyclic

s E S)

C'(I/4)

is a f i n i t e l y

the

the

small

to

into t h e p r o c e d u r e s

any

information

on

impact

new

sufficient

conditions

of

stronger

algorithms. For

this

systems

reason were

conditions the

small

section algorithm the

developed

by Hans

are of n o n - n u m e r i c a l cancellation

(II.5.1.) as

a

special

case,

(cf.

k i n d but

conditions.

contains

C'(1/6)-groups.

B~cken

most e.g.

of

some

Hence the

for

reduction

[B~79]).

These

of t h e m

imply

the

results

Greendlinger's

following on

Dehn's

results

on

84

II.5.1.

Superpos

Solvability

In

this

section

Gw(X/R) and

Z

also =

by

R

every

element

is

finitely where

Furthermore

we

t is a c y c l i c

supposed of

Furthermore

we

ring

(Z

presented

S

=

assume

(ir-l[

group

l~r

e

R},

that

permutation

(

on

= 1

for

obtained

to

be

irreducible

Sym(S)

is

freely

fix

of

s or

s -I

for

some

sES)

u

z-l) *

l~i~n. by

Hence

ordering

lexicographically

First

study

we

generated

the

with

by

effect respect

the

(

is

if

of

respect

to

FR,

our

discussion

a

G-orde-

weight

function

~

where

just

words

with

reduced.

throughout

~.--I a n d

is

a

C'(1/2).

Since

be

denote

,

:= (tl

satisfies

~(a i)

Word P r o b l e m

of the

we

{a I ....,an}.

Sym(S)

- C h a i n s and C r i t e r i a for the

- Deduction

the

ordering

according they

Sym(R)

have

to the

u FR

=:

which

their

same

can

length

length.

R G where

Sym(R)

to ~.

3. L e m m a :

IIrred(t,RG) I = 1 for

Proof :

We

take

every

subword

t = 1112131415

such 1213 ~

Then

C'(I/2 ) i m p l i e s

13r~i12

=

1314r21

t of

some

s e Sym(S).

that rl,

=

1314 ~

r 2 e R G.

131415vli12 •

85 T h e r e f o r e w e obtain:

rl I = 1415vi I, r21

= 15vlll 2

and 1112131415

iii~iv-li~ii~ii4i5

lii2i~il~iv-ii~ii5

vi

[]

As an i m m e d i a t e c o n s e q u e n c e

Suppose

Sym(S)

satisfies

pair

Sym(R)

then

of

that wl=rll 2 and cal

pair

Our next aim which

The

there

is

(Wl,W 2)

is to d e s c r i b e

a

critical

l l U ~ r I, u l 2 ~ r 2 e S y m ( R )

in p a r t i c u l a r ,

such that

is

u l 2 v ~ r I,

there

such

is no c r i t i -

12 ~ r 2 E Sym(R)°

a k i n d of a " s t a n d a r d

deduction"

w e w i l l call s u p e r p o s - d e d u c t i o n - c h a i n .

idea runs as f o l l o w s :

Starting reduced side

C'(I/2 ) a n d

w2=llr2~

(rl,ur2v)

we remark:

at as

of

another procedure

an

element

of

far as p o s s i b l e .

the

irreducible

reduction

a critical Now

word

at the just

is a p p l i c a b l e ,

is a p p l i e d to the o t h e r

More formally:

as

and

pair,

this

word

left or at the much

is a d d e d

so on. T h e n

is

right that

the

same

w o r d of the c r i t i c a l

pair.

86 4. D e f i n i t i o n :

Let A

(Wl,W 2) be a c r i t i c a l sequence

p a i r of

Sym(R).

(En,I n .....E o , I o ~ W l , W 2 )

deduction-chain

o_ff l e n g t h

(i)

If n = 0 t h e n

(2)

If n

called

either

Ek,

0
or

If

En

to t h e

is of t h e

Ek = SpEk_ 1

(b)

Sp...s I

(e)

S p I k _ 1 is r e d u c i b l e

(d)

Ik = I r r e d ( S p I k _ l , R G )

(~)

Ek = Ek_it q

(S)

tl...t q

(¥)

I k _ i t q is r e d u c i b l e

(6)

I k = I r r e d ( I k _ i t q , R G)

=

particular

5. E x a m p l e :

SIot we

Let

for

put

is

conditions:

and irreducible

n>0,

we

with

with

irreducible

Eo(W2):=



form

p+q=k

following

(a)

is

superpos-

I o = I r r e d ( w l , R G)

E k = S p . . . S l I o t l . . . t q, subject

a

n if:

E O = w I,

> 0 then each

is

respect

with

with

respect

En(W2):=

put

to R G

respect

respect

to R G

to R G

to R G

sw2t;

in

w 2.

= Gw(Z/R)

be

a finitely

presented

group.

Assume

and

Then chain

IkU k ~ r k ~ Sym(R)

for

kE{0,1,2),

uil i ~ r i e S y m ( R )

for

iE{4,5}

uol 3 ~ r 3 ~ Sym(R).

we of

have

for

length

instance 4 for

the

following

(Wl,W 2) =

superpos-deduction-

( r o l 3 , 1 o r 3)

i

0

I

k,-,

,,~

I

~'~

N

N

I

~

~

k-,,

I-~

I

II

~

o

~

v

~

(-~

~

~

0

~

I~

I

~

~

~

I'J

,I~

~

I~

0

II

F"~

i

I

N

< o

II

~

~

o

II

II

N

0F,,.'

F~

N I- ~

0

o

0

II

0

0

o

II

I"

NI I

88

Next

we

dition

will for

obtain

the

a first

solvability

(more of

technical)

the

word

sufficient

con-

problem.

6. T h e o r e m :

Suppose

G

Sym(S) For

=


satisfies

all

critical

L ( E m , R G)

e ~

u R~ G

Proof:

Let

pairs

( W l , W 2)

of

Sym(R)

and

all

( E m , I m ..... E o , I o , W l , W 2 ) ,

n L ( E m ( W 2 ) , R G)

=

Obviously suffices

e

u

is s o m e

v

= 0

for

R

e be

~ G

word

can

to

freely

v FR ~

tllSltlt21s2t2

v

I r r e d ( u , R G)

implies

be

and

super-

m~0,

0 < k < m

such

assume

prove

u such that

to

e by

I r r e d ( v , R G)

indirectly

We put

w

and

sieS

R G.

of

assume

such

minimality

that

of

l~i~k.

u e L(v,R G)

it

D {e}.

t R G ~ w I, t RG ~ w 2,

w there

w

= Vlllal2v 2

wI

= Vlrll2V2

w2

= Vlllr2v 2 with

Since

I r r e d ( v , R G)

= m i n { t I tEL(V,RG),

the

for

= {e).

Irred(Wl,RG)={e}

Because

= {e}.

reduced.

°..tk-I skt k,

reduced

We p r o c e e d

Sym(R)

R={s~elseS})

Irred(ImlEm(W2),RG).

Then

There

(=G(Z/R),

C'(I/2 ).

pos-deduction-chains that

I S>

and

e~Irred(w2,RG))

exist

I r r e d ( v i , R G)

lla~r I, a l 2 ~ r 2 in

= v i for

i=i,2.

89 Because chain the

I r r e d ( W l , R G)

=

{e)

there

is

a

superpos-deduction-

(Em,I m .....E l , I i , E o ~ ! o , r l l 2 , 1 1 r 2 ) , m > 0 ,

reduction

of

wI

and

Im

From

the definition

to

e

where

Em

which

describes

= vlIrred(llr2,RG)V 2

= e.

and

of w we get:

L(rll2,RG)

n

L ( I I r 2 , R G)

=

L ( E k , R G)

n

L ( E k ( l l r 2 ) , R G)

= 0

The hypothesis

of the t h e o r e m

for

0
implies:

m

e E Irred(ImlEm(llr2),RG

which

)

yields:

e E Irred(e-iEm(llr2),R

G)

e E Irred(Em(11r2),RG) e E I r r e d ( w 2 , R G)

a contradiction

to t h e

;

assumption.

[]

The

disadvantage

theorem obtain

lies

in

a more

restrict

the

of its

the

sufficient

condition

non-constructive

algorithmic conditions

version (d)

and

we

(6)

nature. have in

in

the

in t h e

last

In o r d e r

to

particular

to

definition

of

a

superpos-deduction-chain.

7. D e f i n i t i o n :

R G satisfies ( W l , W 2) implies

of

the condition Sym(R)

K1

I r r e d ( w i , R G)

I r r e d ( W l , R G)

=

iff f o r •

some

critical

Irred(wi,FR)

Irred(w2,RG).

for

pair i=i,2

90 Obviously

we have:

On the other violated

If R is complete,

hand

if R s a t i s f i e s

K1 the

CR-property

KI. can

be

only if Irred(wi,RuFR)

Hence

then R satisfies

if R s a t i s f i e s

K1

superpos-deduction-chain s atisfying

L(wI,R)

= Irred(wi,FR).

and for

(Em,I m ..... E o , I o , W l , W 2 )

some

n L(w2,R)

pair

(Wl,W 2) of

= ~ then

Io = Irred(Wl,R)

The following

critical

is a

lemmas are useful

= Irred(Wl,FR).

for the fu rt he r discussion.

8. Lemma : Suppose pair

R G satisfies

occuring

from

K1 a n d

( W l , W 2) be a c r i t i c a l

L 1 ~ R I, L 2 ~ R 2 e Sym(R)

Irred(wi,RG)=Irred(wi,FR) for

let

such

that

and I r r e d ( W l , F R ) * I r r e d ( w 2 , F R )

i=i,2.

Then we present:

with

Proof:

L 1 = llv~lu

R 1 = rlv 2

L 2 = uv~ll2

R 2 = vlr 2

Irred(Wl,RG)=rll2 ,

I r r e d ( w 2, RG )= ll r2 , r l l l l ~ 1 2 r 2 I.

We look at

llviluvll2 r l v 2 v ~ l ~ ~2l

~llv[ivlr

r~lt2 tl~r2

2

91 and a s s u m e

rlll I = 12r21o

T h e n we have:

Case

a)

either

a)

r[ 1 = 12t

and

r21

= tl 1

or

b)

12

and

11

= tr21

rl I = 12t From and

= tilt

r I = t - l l ~ I, r 2 = l~it -I

implies

rll 2 = t = l l ~ l 1 2

+ t-i

llr 2 = lll[It -I ~ t-i w e get

Case

b)

is h a n d l e d

a contradiction.

in the same manner. []

9. Lemma: If for

(lu,vr)

to RG,

w e have

Proof:

Because

E Sym(R)

v-ll

is i r r e d u c i b l e

with

respect

iv i ~ lu{.

v-ll

is i r r e d u c i b l e

one o b v i o u s l y

(i)

Ilu{

=

ill+lu{

(2)

{v-ll{

=

{vl+ll [ K Irl+lul

Hence

we have:

lll+lul-(Ivl+lll)

which

implies

lvl

~ Ivl+lrl

~

obtains:

= IvrI = lru-li

Ivl+Irl-(Irl+lul),

< jui.

[]

92 I0.

Definition:

Suppose pair

RG

satisfies

of S y m ( R )

in t h e

K1

which

notation

(i)

assume

D

is

of L e m m a

a

be

a critical

the CR-property,

such that

= rll 2 • llr 2 = I r r e d ( w 2 , R G) where

block

for

rl=rllD 1 and

( w l , w 2)

superpos-deduction-chain that

(Wl,W 2)

8 w e have:

DE{DI,Dr),

called

let

violates

I r r e d ( W l , R G) Further

and

iff

12=Drl2r.

there

is

a

( E l , I i , W l , I o , W l , W 2) s u c h

either D = D1

(i) (ii)

(D 1 is a l s o

called

a left b l o c k )

E 1 = Srll 2

(iii)

L ( E I , R G)

(iv)

n L ( s l l r 2 , R G)

=

I1 = Irred(srll,RG)Dll2

or D = Dr

(i) (ii)

L ( E I , R G)

(iv)

(3)

n L ( l l r 2 t , R G)

block)

= O

I 1 = r i D r I r r e d ( 1 2 r t , R G)

R_G s a t i s f i e s

K2

critical

which

Let

a right

E 1 = rll2t

(iii)

(2)

(D r is a l s o c a l l e d

pair

D 1 be

a left

the critical

iff

there violates

block

pair

is a b l o c k

and

(Wl,W2).

the

D for

every

CR-property.

D r be

a right

D I D r is c a l l e d

block

for

invincib-

le if (i) (ii)

D I D r is i r r e d u c i b l e For

D 1 = Dl'a,

with

aD r is

respect

not

some (iii)

For

D r = aDr',

Dla

is n o t some

(4)

R_G s a t i s f i e s if all

K iff

left b l o c k s

invincible.

RG

satisfies

D 1 resp.

all

a

to R G

subword

of

1 ~ Left(Sym(R)) a subword

of

i E Left(Sym(R)).

K1

right

and blocks

K2

and

D r are

93

Now we

ii.

see at once:

Lemma:

Suppose

w

l~i
= DID2...Dk,

Dj

is a r i g h t

D m D m + 1 is i n v i n c i b l e

12.

D i is a left block

for

for

l<m
block

for

(Wj_l,Wj), then

w

(wi,wi+l),

n+l<j~k,

and

is i r r e d u c i b l e .

Lemma:

Let

R G

satisfy

condition

K

(En,I n .....E l , I i , r o L o , I o , r o L o , l o R o ) tion-chain Ik

of

length

n~

~ I r r r e d ( E k ( W 2 ) , R G)

Then

In

are

where

for

is En

blocks

for

0
D o1D or a n d

a

=

suppose

superpos-deduc-

Sp...SlIotl...t q

and

l
nl~r r = S p D p1 . .i'. . ~ il~o~o...Dq_iT q,

D IiD i-l' 1

and

and

n=p+q,

0<j
D jrD jr+ 1 are

and

invincible

DI,Djr

where the

for

products

0
and

0Sj
Proof:

The

proof

pos-deduction-chain

by

induction

is

left

to

on the

the

length

of

the

super-

reader.

[]

Now

we

can

condition

13.

replace

the

hypothesis

of

Theorem

6

by

the

K.

Theorem:

Let

R G

have

property

~Em,I m .....E l , I i , W l , I o t W l , W chain Im

for

some

critical

~ I r r e d ( E m ( W 2 ) , R G)

Then

2)

pair

and

K is

a

assume

superpos-deduction-

( W l , W 2) of S y m ( R )

L ( w I , R G)

e E Irred(ImlEm(W2),RG).

and

n L ( w 2 , R G)

such = ~.

that

94

In the n o t a t i o n

Proof:

of L e m m a

12 we have

I m = S p D p-l'" 1 "D10D r ' ' ' D r ~- 1 T - ' The proof

is by induction

First we consider: Since

RG

according

on m.

m = 0.

satisfies

in p a r t i c u l a r

to L e m m a

8

condition

K1

there

flu ~ rl, ul 2 ~ r 2 e Sym(R)

rll 2 = w I = I r r e d ( W l , R G) whenever

m=p+q.

and

(Wl,W 2) is a critical

are

such that

llr 2 = w 2 = I r r e d ( w 2 , R G)

pair

which

violates

the

CR-

property. Hence we have IolEo(W2) Assuming lul

a

rllll

= w i l w 2 = 121r11ilr2 •

is i r r e d u c i b l e

1/21rllll

which

contradicts

satisfies

C'(I/2 ). T h e r e f o r e

Lemma

3)

and

wanted

to show.

hence

we o b t a i n

rllll

l~ir~lllr2

~

the is

u -I ~ r l l l l , i.e. fact

that

reducible

121u-lr2

~*

Sym(S)

to e,

u -I

(by

what

we

I

For the induction

step assume

e E I r r e d ( I m t l E m _ l ( W 2 ) , R G ).

A g a i n we have two cases: (a)

Em(W 2) = SpEm_l(W2). Then we have: I~IEm(W2 ) = Tql(D[g_l )-I...(D Ip_l )-ISplspEm_l(w2) As a c o n s e q u e n c e of Lemma 12 we have: -i and Ip-lUp-I ~ Vp-lWp-i D p-i 1 Sp=ipVp-i E Sym(R) ipWp_ 1 ~ Sp This and C'(i/2)

implies

E Sym(R)

Spllp ~ Wp{ I-

Thus we have: r )-i .. . ( D ~ _ l ) - i s ~ l l p v ~ l l E m _ l ( W 2 ) I~iEm(W2 ) = Tql(Dq_!

_~

TqI(D~_I)-I... (D Ip_l)-lwpllvp{IEm_l(w2)

: Im11Eml

95 The

induction

hypothesis

implies e

(b)

Em(W 2) = Em_l(W2)Sq, The proof is as in case

E

Irred(ImllEm_l(W2),RG).

(a). []

14.

Corollary:

If R G s a t i s f i e s R G solves

Proof:

of

numerical

tions.

C'(i/2 ) then

the

reduction

system

the w o r d problem.

Theorem

The p r o p e r t y

K and

6

K can

and

Theorem

be t e s t e d

character

like

13.

effectively the

small

although

it

cancellation

is not condi-

96

II.5.2.

The

Small

Cancellation

Condition

This

section

tion

K and

particular word

section;

15.

small

the

The

connection

cancellation

some classical

problem.

and

the

K

establishes

the

Conditions

results

notation

the next L e m m a

is

between

conditions.

the

We

condi-

derive

for the s o l v a b i l i t y the

is c o n n e c t e d

same

as

in

with Lemma

in

of the

the

last

8.

Lemma:

Suppose

RG

satisfies

L I = llVlU,

Let

C'(A)

L 2 = uv212,

w I = rlv~iv212 ,

with

R 1 = rlv21,

w 2 = llVlVllr2

ILiRii T h e n the f o l l o w i n g

A ~ 1/4 .

and

R 2 = vllr2 ,

assume

IL2RIi.

holds:

(i) If ivluv21 a ~ILIR[IJ

then I r r e d ( W l , R G )= Irred(w2,F~3).

(2)

If

IVlUV2J < lILIR~iI

and r[lil

= 12r21

I r r e d ( w l , R G)

(3)

If

IVlUV21K rll 2 and

Proof:

Concerning

Therefore

XIL2R~ll

and

and

we have e i t h e r

(2)

we

= Irred(w2,RG).

ril I m i2r~l

llr 2 are i r r e d u c i b l e

(I)

then

with

observe

then

respect

r[ili

to R G.

= i2r{i.

97

The

only

Case

11 = t r 2 1

b)

r[ 1 =12t

and

r21

:

In o r d e r

If

and

:

b)

that

12 = r l l t

reductions

a)

Case

a)

to

we

Irred(wl,FR)

= rll 2 = r l r l l t

~ t

Irred(w2,FR)

= llr 2 = t r 2 1 r 2

~ t

Irred(Wl,FR)

= rll 2 = t - l l ~ l 1 2

Irred(w2,FR)

= 11r 2 = l l l [ i t -I ~ t -I

(l,r)

write

rI

possibilities (a)

(b)

Both

cases

selves cause

to

(3) w e

consider

e Sym{R)

reduces

=

rl'l'

and

12

only

~ t-i

one

case

and

assume

rll 2.

=

i"12",

then

we

have

two

:

II'l a 1 / 4 1 1 r - l j ~ a l l r - l l [1"[ a 1 / 4 [ l r - 1 [ a A [ l r - l [ . are handled (a). W e

have

L 2 is f r e e l y

Hence

= tl I

are:

prove

some

or

in the

same

way

or

and

l"r -I = v 2 1 L { i r l ' ,

we but

restrict v21

our-

= e

be-

reduced.

l"r -I = L l l r l ' = u - l v l l l l l r l ' .

Therefore

L 2 is not

freely

reduced,

a contradiction.

[]

16.

Lemma:

Assume and Then

R G satisfies

l l V l U ~ rlv~ I, rlrl 2 •

reducible,

i.e.

K1

uv212

~ vllr2

I r r e d ( z - l l l r 2 , R G) rlrV21u-lvll

~

e RG

and

implies z-ll I.

r I = Zrlrz-ll I

is

ir-

98

Pr0of: ' From

Suppose

z-i11

C'(I/2 ) and L e m m a

is

reducible.

3 we

get: z-lll

and

~ rlrv~lu-lv~ 1

r l r V 2 1 u - l v l l r 2 ~ rlrl 2

which

is a c o n t r a d i c t i o n . []

17.

Theorem:

Suppose C'(I/6)

Proof:

then

By

Lemma the

G

8 we

=


R G has

Lemma have

the

15

for

If

property

RG

every

S>.

satisfies

K.

satisfies critical

Sym(S)

KI.

pair

S9

according

(Wl,W 2)

violating

CR-property:

There

are

L1 ~

L 1 = llVlU, with

In o r d e r

R I,

L2 ~

R 2 E Sym(R)

L 2 = uv212,

w I = rlv21v212

I r r e d ( w l , R G)

= rll 2

to s h o w

K2

such

that

R 1 = rlv~ I, R 2 = v [ i r 2 and

w 2 = llVlVllr

and

I r r e d ( w 2 , R G)

it is s u f f i c i e n t

2, = llr 2.

to c o n s i d e r

the c a s e s

E 1 = 13z-lzv3v41Dll2

,

r I = zv3v~IDl,

13v 3 ~ r3v 4 E R G

B) E 1 = r l D r V 4 1 V 3 z z - l l 3

,

12 = D r V ~ i v 3 z ,

v313

A)

to

Case

A)

We d i v i d e

a)

~V3V41~ Assuming

the p r o o f

into

~ v4r 3 E R G.

parts:

< 1/6 min( l l 3 v 3 v 4 1 r 3 1 I , ~LIRIII) the

contrary

C'(I/6 ) yields:

v3v ir 113 : v3v ID1v lulvillilz

99

and we have W.l.o.g.

z = e

because

we a s s u m e

(the c a s e s

13z-i

is i r r e d u c i b l e .

13 = v l l l l I.

13 = u r l v l l l l I , w h e r e 13 = v r ~ l l I

,

where

and 13 = ll~ , where can be t r e a t e d in the same way.)

u = UlUr, v I = VllVlr 11 = lllllr

T h e n we obtain:

E1 = 13v3v41Dl12

~ r3v4v41Dl12

= uv2D~iv4v~iDll2

~ uv212 vllr 2

and

This

1311r 2 = v ~ l l ~ l l l r 2 is a c o n t r a d i c t i o n

to

L ( E I , R G) n L ( s l l r 2 , R G)

b~

IDll

= 6.

i/6JLiR~ll

~

Again we proceed First

~ v ~ i r 2.

indirectly.

of all we h a v e a c c o r d i n g

t:= v 3 v 4 1 D l V 2 1 u - l v l l Putting

together

to L e m m a

= rlrV21u-lvl I

~

Iv3v411

< 1/61LIR~II,

IDII

< 1/61LIR[I I

16

z-ll I

and

fret.rill < 1/61LiR{it we o b t a i n

ttl < i/2 ILIRilJ. Therefore shows

(b).

t

is

irreducible;

this

contradiction

100 Next we consider

E 1 = 13z-lzv3v41Dll 2 ~ r3v4v41Dll 2 r3dll 2 .

c)

r3Dll 2 is i r r e d u c i b l e

Assuming

or

that

this

is not

such

that

(i)

D 1 is a s u b w o r d

(ii)

with

respect

to R G.

the c a s e

there

is L~R £ R G

either

L = r 3 r d I,

of L,

where

i.eo

L = r3rDll21

DI=DIIDIr,

r3=r31r3r

and

12=12112r. In b o t h

cases

we

have:

(~}

IDll

~ Z/61LIRI11

or

(8)

Ir3rl

~ 1/61LR-ll

or

(~)

11211

a 1/61LR-I I

Case

(i):

By C ' ( I / 6 ) w e h a v e

Dlv~lu-lv[ll[izv3v~

v3v~ 1 = e hence

11

= r3r

reduced; Thus

is

and

D1

= e

same

that

r I = zv3v~iDl

In c a s e

(8) we h a v e

fact

that

(v 3 • e by c o n s t r u c t i o n ) ,

because way

is r e d u c i b l e

r3rDll21R-i

(~):

1 = DIIrR-ll I = Dll21R-ir3r

impossible

in the

in c a s e

13v3v41r31 we

which

get

is

i r = 121

contradicts

freely = e.

the

fact

contradicts

the

is i r r e d u c i b l e .

by C ' ( i / 6 ) :

= r3rV4v~ll~ir31, r I = zv3v4-1dl

is

which freely

reduced.

101

Finally

in c a s e

121R-ir3rDl

Hence

v2=e

because But

contradiction =

yields:

= 12112rr21vluv2 .

ly r e d u c e d .

LIRI I

<~) C ' ( i / 6 )

R 1 = rlv21

since

to t h e

llVlUrll

u ~ e by c o n s t r u c t i o n fact

=

= zv3v41DlV21

is f r e e -

there

is a

freely

re-

that

llVlUDlV4v3lz-i

is

duced.

Case

(ii)

So w e

have

that

DI

are

(Wl,W2).

which

implies

in c a s e

(A).

K2.

property

D I D r is

irreducible

is a block.

as

RG satisfies

to v e r i f y

analogously.

r 3 D l l 2 is

(A)

arguments

Thus

for

be s h o w n

that

in c a s e

Cas___eeB'. T h e

In o r d e r

can

K consider

irreducible,

two

blocks

because

D 1 and D r

r l l 2 is

irredu-

cible. Suppose i.e. We

D1

= Dl'a

and

aD r

is

a

subword

of

some

LELeft(RG),

L ~ R E R G. have

C'(I/6)

~aDrl implies

Therefore But fact So tion

> IDrl

rI

v2

v2 = e to t h e

Analogously subword

Drt2R-itla

ends with

= SDl'a

that

of

a !/61L2R21

ends

and

u

fact

that

it some

is s h o w n

a and

so v 2 1 a

is f r e e l y

ends

Put

L = tlaDrt2;

then

= Drtr~ivluv2 .

with

R 1 = rlv2 i

I.

with

LIR[I

that

a.

starts

; this

a -I.

contradicts

the

reduced. Again

= llVlUrll

for

with

we

have

a contradic-

is f r e e l y

D r = b D r'

Dlb

reduced.

is

not

a

L E Left(RG).

[]

I02 18. Theorem:

Suppose

G =
then R G has

Proof:

By

to L e m m a the

the p r o p e r t y

Lemma

15

are

R G satisfies

=

L 1 ~ R I,

llVlU,

critical

I r r e d ( W l , R G) K2

L2 ~ R2

L 2 = uv212,

w I = rlv~Iv212,

C'(i/6 )

KI.

So again

pair

according

(Wl,W 2) v i o l a t i n g

we have

and

to look

(essentially)

(B)

E l = r l D r V 4 1 V 3 z z - l l 3,

of

steps

such

that

R 2 = v~ir2 ,

,

I r r e d ( w 2 , R G)

occur.

= llr 2.

at two cases:

rl=zv3v41D 1 ,

13v3~r3v4

• RG

12=DrV41V3Z,

v313~v4r3

E RG

(B) is omitted,

essential

Sym(R)

R 1 = rlv~l,

= rll 2

El=13z-lzv3v41Dll2

The proof

•

w 2 = llVlV[ir 2,

(A)

a)

satisfies

CR-property:

LI

all

If S y m ( S )

K.

8 we have for every

There

For

S>.

we consider We proceed

only

case

in several

(A) w h e r e

steps.

Iv3v41Dl [ ~ 1/41LIRII[ Assuming Lemma

the

opposite

inequality

we

have

according

16

v v iDlV lulvil rlrVlU-lVil zi11 :

(where and

Iv3v~lDl I leads

+

Iv~lu-lv[ll

to a c o n t r a d i c t i o n .

<

Z/21LIR[II

Zrlr

=:rl);

to

103

As

in T h e o r e m

b)

17 o n e o b t a i n s :

Iv3v411

< i/4-min(l!3v3v41r31,~LIRlll

Therefore

c)

D 1 ~ e and w e h a v e

for r I = zv3v41Dl

E 1 = 13~izv3v~iDll2

~ r3Dll 2

r3Dll 2

is i r r e d u c i b l e

respect

Assume

the

contrary,

respect

to

L ~ R e R G.

C'(I/4 )

implies

with i.e.

L = t l D l t 2,

< 1/4

min(ILR-II,ILIRIII),

It21

< 1/4

min(ILR-II,II2R211),

Putting

and

tl,t 2 ~ e

e

we

have

This

Therefore

By

for

i~j,

Dl,tl,v3v41

RG

one

C'(i/4), i.e.

shows

satisfies

that

Sym(13v3v41r31)

D1

e Left(RG),

is

i,je(l,2,3}

T(4),

for

i.e.

and

in a d d i t i o n

(c) holds.

(wl,w2).

D r is a r i g h t

to c o n s i d e r

irreducible.

= Dl'a i.e.

Sym LIRiI

block

for

(wl,w2).

K2.

K it r e m a i n s DID r

and

• e.

condition

D 1 is a l e f t b l o c k

property

cible, Lo

T i • Tj

contradicts

Analogously Hence

•

: v4v iz-iL1v2Dyt have

with

S y m ( L R -I)

T 2 := t l l r 3 1 1 3 v 3 v 4 1

we

reducible

12 = t 2 1 2 r .

T 1 := D l t 2 R - i t I

and

t o R G-

r 3 D l l 2 is

~tll

r 3 = r31t I

For

)

Lo

and

aD r

= s l a d r s 2.

two

Assume is

blocks

DID r a

is

subword

D 1 and not

D r-

irreduof

some

104

Putting

:= DrS2RolSl a S 2 := a - l ( D l ' ) - i v 4 v ~ i z - l l l v l U V 2

E

Sym(LoRo I )

e

Sym(RILI 1 )

S 3 := v~lu-iR212rD~ 1

E

Sym(R2L21)

S1

and we have

S i • Sj

for i~j,

i,je{l,2,3}

which

implies

uv21 ~ e, D r • e and leads to a contradiction

to

b • e,

T(4).

Therefore DID r is invincible. []

T h e last t w o t h e o r e m s

are s l i g h t

results on Dehn's a l g o r i t h m reductions

(L,R) with

conditions).

Greendlinger Greendlinger

used

To as

of c l a s s i c a l

(Dehn's a l g o r i t h m does not allow

ILl = IRI). We want to show that also

in our case the condition C'(A)

extensions

C'(I/6 ) is still

prove

it,

we

take

a counter example

t o o k as a c o u n t e r e x a m p l e

sharp the

(among the same

[Gre 60].

The

is r e d u c i b l e

group word to

with respect to RG; but another word will do.

19. Example: G = , S = {ABCA-IB-Ic-I], Sym(S)

Sym(R)

= {ABCA-IB-Ic -I, BCA-IB -Ic-IA,

CA-IB-Ic-IAB,

A-IB-Ic-IABc,

B-Ic-IABCA -I, C-IABCA-IB -I,

CBAC-IB-IA-I,

BAC-!B-IA-Ic,

C-IB-IA-ICBA,

B-IA-ICBAC -I, A-IcBAC-IB-I }

= {(CBA, ABC), (C-IB-IA-I,

AC-IB-IA-IcB,

(A-IcB, BCA-I), (B-IA -Ic, CA-IB-I), A-IB-Ic-I), (B-Ic-IA, AC-IB-I),

(c-lAB, BAC-I), (ABCA -I, CB), (BCA-IB -I, A-Ic), (CA-IB-Ic-I, B-IA-I), (BAC-IB-I,c-IA)}.

e

105 G has

The

none

of the

word

w

the

pairs that

respect

to

of Dehn's a l g o r i t h m relations

exist

and

there

"little"

proof

have

the

is o n l y

are

condition. of

with

RG,

but

it

is

the

-I

is

identity

Remarks:

study

that

T(4) or K.

in G.

Additional

The

C'(i/6),

= ABABABCBCA-IcA-IB-IA-IB-Ic-IB-!C-IB-Ic

irreducible element

properties

small

pieces

comes

[De ii]. The fact

means

cancellation

that

critical

condition

ensures

overlapping";

in a n u m e r i c a l

The number and

common

"little

taken

goes back to

the

sense

measures

like

the

C'(i/6)

1/6 arises from the geometrical ultimately

from

the

Euler

of

method

polyeder

formula. In

the

approach

different and

the

origin. fact

difficulties results

presented

of

dissertation

Essentially

that

violating

(compare

II.5.

here,

are

[BO 79].

e.g.

the

number

it arises the the

essentially

from

C'(i/2) proof

1/6

of

contained

a very

has

3 • 1/6

causes

= 1/2

immediate

Theorem in Hans

17).

The

B~cken's

106

II.6.

Relations between the Todd-Coxeter

A

famous

method

Todd-Coxeter tion. U c G The

to

G

of

In t h e

latter

case

U = {e}.

Then

with

the

short

We

informal

relators

u

aI

is

below

a2

presented

(left

the

enumeragroup

resp.

is finite.

also

able

to

the

output

the

case

can

and

of t h e

the

and

the

trivial

equations

Todd-Coxeter For

this

XuZ -I.

occurs With

in

each

an_ 1

coset We

subgroup giving

between in t h i s

such light

first

a

is given.

one such

is a s s o c i a t e d :

....

the

as

purpose

TC-algorithm

right)

algorithm.

interpreted

deriving the

construct

of

of

be

generator ai E

coset

is

subgroup.

procedure.

each

of

[G:U]

to

description

= al...a n,

as s h o w n

and the

groups

iff

is to c o m p a r e

that

sequel)

generates

algorithm

elements

presented

is a f i n i t e l y

from

here

completion

assume

table

the

aim

the

generated

one

table

ourselves

Our

in

terminates

restrict

to g r o u p

finitely

algorithm

U and

multiplication

names.

(TC

is a f i n i t e l y

cosets

names

study

=

Todd-Coxeter

Procedure

Alqorithm

method

Suppose

Completion

an

of

the

relator

a

107

The

ai

slots for

are

markers

will

(not

be

filled

in

of t h e

the

by

necessarily

Insertions entry

for

internal

symbols

gn"

different)

the

table

dividing

n e ~

elements

are

; the

of

encodings

lines.

The

gn a r e

names

G.

for

equations;

an

form ai

gj stands Such

for

equations

equations

tional

are

lists:

collapse The

the

the

gj'ai

recorded

= gk

for

definition

and

gk.ai

convenience

list,

the

1 = gj.

in

bonus

three list

addi-

and

the

list.

algorithm

determined

as

it

will

because

slots

are

(i)

Initial

it

be d e s c r i b e d

leaves

now

open

how

write

gl

is n o t

certain

completely tables

and

leftmost

and

selected.

step_ t

Select

some

rightmost

table

slot

and

of

the

first

aI

in

the

row:

•..

an

gl All

(2)

lists

are

Iteration (a)

If t h e r e that

create

step: is

gk

is a

If

(a) d o e s

(if

it

new

symbol

leftmost

some not

new

rightmost (b)

empty.

gk in t h e in

row

tables

any

row

in

and

write

and

the gk

some

table

such

leftmost

slot

than

leftmost

and

in

the

slot. not

exists) gk or

apply

then

with

not

not

yet

rightmost

select all

used

a row

slots

and free

put

in s o m e

filled. it

slot

Create

either of

table

the

a

in t h e row:

108 (i )

a

gi

resp.

, I

(ii)

a

gk

Add

the

the

definition

gk

equation

resp.

gi "a = g k

gj

gj-a -I = g k

to

list.

If the r o w is c o m p l e t e l y

f i l l e d w e h a v e the f o l l o w i n g

situation:

a

The

two

case

equations

(i)) resp.

(ii)) have

If

gk.b

= gj

gi "a = gk

the

now

union

two

gi "x = gk' w i t h

(~)

gk = gk'

(~)

all

of

-g'j'b-ll = g k

gk.a

= gi

bonus

of

(for c a s e

and

the

bonus

form

then a colla~

with

gk

in the

list

gi "x = gk'

takes

is a d d e d to the c o l l a p s e

rows

(for

list.

definition

equations k'
and

and

to be a d d e d to the

finally

contains

b

place:

list;

leftmost

slot are

omitted; (~)

all o c c u r r e n c e s finition

(c) If

neither

(a)

of g k in the tables,

and b o n u s

nor

(b)

is

list are

replaced

applicable

the

the deby

gk'"

algorithm

stops.

The

basic

fact a b o u t

this

algorithm

is

(cf. e.g.

[C-M

80]):

t09

i. Proposition,,! .....

The T o d d - C o x e t e r

When

the

algorithm

algorithm

collaps,

l~i~n,

stops,

denote

t h e y are the e n t r i e s columns occurs

contain

terminates

then

the

the

not

n elements

in the first

permutations

in each c o l u m n

gi

and

last column;

of the gi" This

exactly

each

gi

gi.x -I

for

some

j,k is a r e c o r d e d

For the c o m p a r i s o n v e n i e n t to The

with

introduce

symbols

gn are

becomes

the

(2b)(i)

resp.

and

construct

empty

x e Z

equation.

The

group

1

by words.

word

the

The

words

ties

(they

are

collapse

steps

equations

into

this

of G.

it is con-

initial

symbol

introduced

in

gl

steps

are

ua

equations

equations

From

or

modification:

replaced

ua-i

are

b e c a u s e the m u t i p l i c a t i o n

All

gi

a

elements

introduced

each

table

algorithm

a

u

table

gi "x = gk

the m u t i p l i c a t i o n

a notational

(2b)(ii)

a

all o t h e r

means,

either

the c o m p l e t i o n

e;

by

once.

for

one can e f f e c t i v e l y

eliminated

of G. In e a c h

Furthermore = gj

iff G is finite.

from

between

words

are e q u a t i o n s the

definition

introduced does

represented

not

reductions:

this change

way

1

by

u

words

Ul~ ....Un;

is c o n c a t e n a t i o n

between

words.

list

are

trivial

and

the

applications

this).

Now

the

we

identi-

transfer

of the

t10

2. D e f i n i t i o n :

The ux

Todd ~

v

bonus

- Coxeter

system

where

E

x

X

RTC

u

Z -I

now

the

example

Z = {A},

A

only

Both

this

u-x

reductions

= v

is

from

the

of

leave reason

we

A

A

A

A2

1

A

A2

1

A

A2

1

A

A2

equation

reading each

A 3 = i:

1

nontrivial

ways

would

and

the

list.

Consider

The

contains

this

word

in

is

as

a

the

introduce

the

bonus

reduction

inverses

the

equation

of

following

(A3~I A

A 3 = i.

resp.

I~A 3)

irreducible.

extension

For

into

the

algorithm:

After

each

nition right slot

of

however

c

w

are

(in

this

word

not

we

this

insert

case

equation

has

I-A -I also

the

corresponding

= A2); to

the

appear

defi-

word

on

in t h e

the

first

row.

from

fail

Suppose

of

some

any

to

even

equation

equation side

Given RTC

bonus

w

such

left

the

case

to have

a word

to

right that

finite

ubvcw

introduced;

that

w

~ u i in G

reduces RTC

is

as

the

above

complete.

application

of

w

is

In

to

u i.

fact,

It it

may

termination:

is

assume

introduced

as s o m e

furthermore

that

gk before later

b

b and

= ubvcw

!11

and

c

= ubvcw

the

collapse

but

in t h e

and

list.

final

no

other

Then

the

tables

equation

collapse

we have

with

eliminates

not

have

FTP by

violate

Responsible

b ~ ubvcw,

ubvcw

c ~ ubvcw

e RTC

i.e. R T C

does

now

modify

the

word

reversing

the

reductions

would

on the o t h e r

CR. for

of a c o l l a p s e

Suppose

1

FTP.

Restoring hand

that

b and c

,

ubvcw

says

both

in

c ,,

which

appear

entries

b

1

ubvcw

this

effect

those words

the algorithm

is the

which

rule

are

which

omits

in c a s e

introduced

last.

We

by a l l o w i n g

strategies

which

will

select

to be o m i t t e d .

~ is any

well-founded

term

ordering

on the words~

3. D e f i n i t i o n :

The

i

- strateg Z

If u

~ v appears

omitted

This

simplifies

because given

resp.

by

one the

N o w we h a v e

can

says:

in t h e

replaced

the

collaps

handling

and

a better

and

u ~ v then

v is

by u.

immediately

bonus

list

of

insert

bonus the

the d e f i n i t i o n

situation:

equation

minimum equation.

somewhat

of the

words

112

4. P r o p o s i t i o n :

If

the

rates

TC-algorithm the

(i)

The

words

u I .....u n,

u i are

the

elements, taking (ii)

RTC

Proof:

from

RTC

for

to

determined

particular

u k RTC ~

v

uk

If

now

in G t h e

is

again

= u i.

some

w

letter is

hence

words the

gene-

the

group

denoting

ui

are

closed

by

we

letter)

equation

under

Irr(RTC)

(i) a n d

in

G

(ii)

are

holds, an

=

As

(namely

to a u n i q u e l y

ui

w

= u i.

for

some

u i,

{u I .....Un}.

v ~ Uk,

v RTC ~

w

FTP.

RTC

implies

v RTC ~

u k = v,

of

reduces

this

obtain

shows

"~" it e n j o y s

application

irreducible

This

Uk=U i

with

some

u i satisfying

that

assertions

and

system.

word

u i. If w

now

says

- strategy

then,

is c o m p a t i b l e

any

right

Assume

This

in

~

- minimal

is a c o m p l e t e

above, left

i

the

subwords.

Because

shown

follows

holds,

then

there

ui°

i.e.

we

immediate

have

i=k.

Now

the

consequence.

[]

The

system

RTC,

although

reductions.

Converting

(cf.

II.2.)

sides

Hence

chapter are

already

there

coincides algorithm. methods:

is

some

There

into

may

contain

a minimal simple

redundant

complete because

system the

right

irreducible:

has

the

it

is p a r t i c u l a r l y

One

with

complete,

only

to d e l e t e

non-redundant system are

R~(~)

several

some

reductions.

complete generated differences

R T C ° c RTC; by

the

this

completion

between

these

113

a)

The

TC-method

saves b)

has

a

the c r i t i c a l

simpler

pair

termination

not

latter

later

will

than

the

continue

it

terminates

at

test.

O n the o t h e r h a n d the c o m p l e t i o n least

criterion,

algorithm

TC-algorithm;

if

it h a s

in

already

general

some

the

complete

systems. c)

If

the

TC-algorithm

Generating

a complete

elements.

The

task:

This

the c o n s e q u e n c e for any

generate

The

TC-method,

duction

last

dividual

(I)

systems

remark steps

two

jobs:

enumerating

the

group

is g e n e r a t e d

and then

that

the T C - a l g o r i t h m

group

considered violate

splits this

chapter

redundant

the

IV.l.).

(from s o m e

does

point

not

on it

reductions). as

all

an

algorithm

standard

to

principles

generate for

re-

like FTP or CR and still be s u c c e s s f u l .

(d) m a k e s

it d e s i r a b l e

to

compare

also

in-

of b o t h methods.

Discusslon

consider

done

(cf.

infinite

only

may

of a t r a n s l a t i o n

[Completion

We

enumerated

may

and

system

are

has

it h a s

m e t h o d of r e d u c t i o n s y s t e m s

elements

reductions,

The

system

First a complete

terminate

d)

terminates

of

[TC-steps]

into

steps]:

a defining

The bonus equation

relation

r = al...a n = i.

in the f i r s t r o w of t h i s t a b l e

just m e a n s

to s p l i t t h e r e l a t i o n : -i a I

Rows

with

some

....

a k

=

a n

-i ....

w o r d u in the f i r s t

relation

u.r.u-l=l

ak+

slot

1

•

split

the c o n j u g a t e

114

They

are

rithm

In

recorded

only

case

in

of

a

(by

the

case

with

the

wu

~

v

This

=

at

can

the

•••

ak+l...al_ bonus

we

I.

equation

u

is

have

.°.

1

W

. . . . .

r

that

collapse

aI

with

symmetrizing)

in

a

the

ak

1

We

assume

•••

wur

completion

subword

of

following

...

WU

the

aI

r.

situation:

"""

an

1

wur

ws

the

= w s a ] -I

I

•••

row and

algo-

is we

W

.....

closed

have

at

a1

a collapse

ak.

also

critical

be

obtained

either

by

a

direct

reduction

or

by

pair

wur

vr

from The

wu other

are

the

and

cases

they

ur ~ are

of

the

addition, critical

similar. steps

actually

pair.

apply

respects

completion

the

-1

sal-l.

for

TC-algorithm

order

the

v

candidates

Whether

In

~

wsa 1

This in

implies

the

all

completion

depends the

that

again

~-strategy,

on

TC-steps

algorithm. the

fact

that

"4"

being

the

algorithm.

completion

algorithm

will

reduce

further

115

Discussion

(II)

of a t r a n s l a t i o n

of

CCompletion

steps]

into

[TC-steps] :

Because be

symmetrizing

simulated

also

see that

is

the

result

of

conjugation

this

can

inductively

one

can

by T C - s t e p s .

Proceeding

superposition

can be simulated:

Assume aI

...

ak

ak+l ...................

U

]

ua k = vak~ I

,,,

,,,

v

and bj

bI

are

given

such

pair

then

This

applies,

that

similar

of o t h e r

reductions

of an

ua k

and

however,

Applications

the c a s e

I

as a b o v e

gorithm.

in p a r t i c u l a r

W

wbj

to the

reductions

cannot

responsible infinite

I z give

we o b t a i n

only of

bj+ 1

rise

to a c r i t i c a l

this by TC-steps.

redundant to

left

completion or

right

be

immediately

imitated.

for

the a b s e n c e

of

group.

al-

sides

This

is

termination

in

III.

Infinite Sets of Reductions

III.l.

Regular Systems

III.l.1.

Regular System- as Special

In

the

following

nonterminating

paragraphs cases

answer questions

of

we

the

will

KB

How

can

we

handle

of

frequently,

many

as

algorithm the

for

for

generating there

the

an

finitely

on to

for i n s t a n c e the algo-

system

computable

of

all

an

is

noe~

newly

algorithm

such that

occurs

show.

generated

information

systems

most of

Fortuna-

very

often

is a v a i l a b l e .

completion R n for for all

algorithm

all

hEN.

stages

in R n h a v e a s m a l l e r

generated

gene-

implementation

groups

a nonterminating reduction

system

algorithm

with

w a y that c e r t a i n

stepwise

sides

KB c o m p l e t i o n

presented

infinite

example

is s o m e

the

algorithms

reduction

KB c o m p l e t i o n

the left s i d e s of all r e d u c t i o n s than

look

if the KB c o m p l e t i o n

infinite

experiments

resulting

in such a r e g u l a r Consider

closer

runs f o r e v e r

Nontermination

tely

(concerning

is a v a i l a b l e

r a t e d by a n o n t e r m i n a t i n g

the

a

like

word problem)

-

have

completion

W h a t k i n d of i n f o r m a t i o n

rithm

Infinite Systems

reductions

If

n~n o

length

(i.e. re-

117

ductions

from

solvable

in the f o l l o w i n g

Given

R n + I \ R n) then,

words

u

than

We

call

above

both

v one

the

generates

the

reductions

left sides h a v e a g r e a t e r

systems

with

up

length

the property

described

as

exploding. that

the

completion

algorithm

small-cancellation-presentation explodes

(see

chapter

which

infinite

starting

with

a

of a g r o u p e i t h e r t e r m i n a t e s

II.5.).

S p e c i a l c a s e s of e x p l o s i o n by

is

u and v.

reduction

Remember

or

the w o r d p r o b l e m

way:

and

to a s t a g e w h e r e

of course,

are the r e g u l a r

systems

can

be

reduction

handled

in

a

systems finitary

manner.

I. D e f i n i t i o n :

A regular are

reduction

regular

scheme

expressions

is a 3 - t u p l e over

the

(L,01,R) if L,

alphabet

Z with

R

the

following property

(I)

1 ~ X*(L\{I,e})E*

and

if

the

following

(2)

(3)

If

there

ordering

04:

Z

~

E

for all

is a d e t e r m i n i s t i c

is a b i j e c t i o n

all

w

is

GSM-mapping

with

properties.

O4

04

i E L;

from

L to R a n d u n d e f i n e d for

e Z*\L.

is c o m p a t i b l e

no

need

to

'~' w e

will

omit

with

the

(I.e.

04(1) ~

emphasize the

term

the

i n d e x and

ordering

'~'.

1 for all

underlying write

1 ~ L).

term

(L,~,R).

118 2. Example: Take

the

infinite

reduction

system

{abna-l~bnln~l)

and

put

L:= a b * a -I • R:= b * Consider the following GSM

b

Obviously tions

condition

(2)

induced

and

by

reduction

the

(i) is f u l f i l l e d

(3)

are

above

fulfilled

GSM.

Hence

for

L and

the

the

condi-

GSM-mapping

(ab*a-l,0,b *)

is a

0

regular

scheme.

We have seen that a regular but an e v e n t u a l l y

infinite R

where

for

L as w e l l

GSM-mapping for

every

are

regular

reduction

{I

~

regular

regular

0 -I) p r e s e r v e s

L'cL a n d

system

0(I) I 1 E

as R = 0(L) are

0 (resp.

too.

:=

r e d u c t i o n s c h e m e is n o t h i n g e l s e

L}

expressions

the r e g u l a r i t y

R'cR t h e

sets

and the

such that

0(L') a n d

0 - 1 ( R ')

119

Moreover is

condition

irreducible

whereas

the

property Notice as

a

Let

with

states

respect

condition

for

that to

every

all

reduction

other

(3) g u a r a n t e e s

the

l~0(1)eR

reductions finite

of

R,

termination

R.

also,

that a usual

reduction

following

(2)

scheme

reduction (l,d,r)

i ~ r can be i n t e r p r e t e d with

0

induced

by

the

GSM.

i := I i 1 2 . . . i n e Z

121e

inlr

3. D e f i n i t i o n :

We the

call same

a

finite

set

underlying

of

regular

term

reduction

ordering

schemes

a regular

with

reduction

system.

Before thing the

so

studying about

their

called

investigated

regular

reduction

relationship

conditional by

Brand,

to

systems other

reduction

Darringer

and

let us reduction

systems, Joyner

(see

say

some-

systems,

which [BDJ

were 78]).

!20 We

start

with

represents each of

the

the

ground

that

infinitely

word

reductions

a r e of the

remark

u.

is

many

A more

the

a general

use

ground

of

P

is

tention true.

although logic. tions that

a predicate that

usual also

certain

unconditional

culties.

(with

conditional

u2~u

for

to d e n o t e

sets

reductions.

where

notions be m a d e

aspects case

These

schemes

some and

cannot

is m o r e

[BDJ

P

in

s or

is a p p l i c a b l e

P belongs

here.

(cf.

if

variables

reduction

it is in g e n e r a l

can

duction

the

language

The

way

x2~x,

form

is

The

say

reductions

sophisticated

s ~ t

Here

reduction,

kind

to of

concepts It be

general

handled

but

is q u i t e

for

Our

be added,

case

provides

easy of

if

inP is

arbitrary

systems

as

the

only

first order

should

78]).

t);

predicate of

reduc-

however, as

in

the

regular

re-

similar

diffi-

121

III.l.2.

Applications of Regular Reduction Systems to Sets of

In o r d e r need by

an

to w o r k

with

operator

applying

First

4.

Words

of

regular

which

reduces

a regular

all

let us

reduction regular

reduction

recall

the

systems

we

sets

regular

to

obviously sets

scheme.

following

technical

terms.

Definition:

Let

L and

(i)

(ii)

T be

regular

expressions

Irr(L)

:= {t e X

I no

Irr(T,L)

:= I r r ( L )

N T

Red(L)

:= {t

E Z

over

Z

.

1 e L is a s u b t e r m

I t contains

at

least

of

one

t}

leL

as a s u b t e r m } Red(T,L)

Notice

that

Irr(T,L) cible respect

for

is

the

elements to

a

regular

set of

n T

of T

reduction

all

and

reducible

we

in

have

with

scheme

respect

Red(T,L)

(L,d,R)

Furthermore chapter

:= R e d ( L )

is

elements

accordance

the

(L,o,R)

irredu-

to

the of

(L,O,R)

set

of

all

set

with

T.

with

Definition

4

of

II.l.: Irr

=

Irr(R)

Red

= Red(R)

=

Irr(Z

,L)

= Red(Z*,L)

and for

L:=

{i~

l~r

e R}.

122 5. L e m m a :

Red(T,L)

and

Irr(T,L)

are

regular

whenever

L and

T are

regular.

Proof:

Obviously

we

have

Red(T,L)

= Z LZ

Irr(T,L)

=

nT

and

T\Red(T,L).

[]

6.

Definition:

A

regular

tion

expression

scheme

(i)

T

(L,~,R)

T to

is

reducible

a regular

n + Z i=l

= Irr(T,L)

where

by

a

regular

expression

reduc-

S iff

UiLiV i

U i, L i, V i a r e and

regular

Li [ L

expressions

for

all

l~i~n

and n

(ii)

S

= Irr(T,L)

+

UiRiV i i=l where

Formally

In t h e

we w r i t e

following

we

T

will

(L,o,R)~

apply

R i = ~ ( L i) f o r

S

or

reductions

all

(L,o,R)(T)

only

from

right. A formal

definition

of

this

procedure

is g i v e n

below.

l~i~n.

= S

left

to

123

7. D e f i n i t i o n :

A

regular

duction

expression

scheme

S

is the

result

(L,d,R) a p p l i e d

to

a

of

the

regular

regular

re-

expression

f r o m the left iff n

(i)

T = Irr(T,L)

where t

+

Z i=l

UiViL i

= uiliv i e

i i is the

are

UiLiV i

regular,

UiLiV i

first

L i c L and for e a c h

(i.e. u i e U i,

IEL w h i c h

occurs

lieL i,

v i e V i)

in t for l~i~n.

and n

(ii)

S = Irr(T,L)

÷

Z i=l

UiRiV i

w h e r e R i = d(L i)

Formally

we

write

T

(L,d,R)~ 1 S

a n d call

such a reduction

also

or

for l~i~n.

I(L,d,R)(T)

= S

leftmost.

8. A g r e e m e n t :

In

the

tions,

The

following Hence

following

scheme

we

we omit

theorem

will the

shows

(L,d,R) c a n be a p p l i e d

consider index

that

only

leftmost

reduc-

'I'.

every

to e v e r y

regular

regular

reduction

expression.

9. T h e o r e m :

For tion that

every

regular

scheme

expression

(L,d,R) t h e r e

(L,d,R)(T)

= S.

T

and

every

regular

is a r e g u l a r e x p r e s s i o n

reducS

such

124

Since

Proof: that

Red(T,L)

T

= Irr(T,L)

can

be

u Red(T,L)

written

it

suffices

to

show

as

n Red(T,L)

where

U i,

L i,

For

that

10.

Lemma:

For

V i are

purpose

every

regular

=

X i=l

regular

we p r o v e

regular

UiLiV i

and

the

following

reduction

expressions

I i,

for

Li ! L

lemma.

scheme

L i with

l~iKn.

Li !

(L,0,R)

there

are

Irr(L),

IKi~n,

and

n

n

Z Li = L i=l In p a r t i c u l a r

we

in

that

such

a

1 e L as

Proof: A

such

way

can

Of

course

t e

Irr(L)L

is t h e

case

for

choose

every

we

the

t E

can

have

however

instance

=

Z i=l

regular

IiLiZ

expressions

IiL i c o n t a i n s

= tllll2t 2 with

Red(L) be

exactly

8 ~

•

= Irr(L)LZ

of

whenever

(Irr(L))

t

Red(L)

Ii one

a subterm.

word

i.e.

that

the

there

form

is an

. t = t l l t 2. T h i s

l=lll2eL

with

12D(L)

1112 , 12t 2 e L and

tll I e

Irr(L).

n

So

in o r d e r

to

write

Red(L)

as

the

sum

Z IiLiX

with

the

i=l property word

that

of

for

L which

words

from

actly

all

e

t

occurs

Irr(L)L ta

every

= ijljwj i n t,

having

IrrlL)L

such

not

e

we

must

this

that

IjLjZ

, lj

is

eleminate

property.

t contains

the all

Those an

first those

are

i e L.

ex-

125

Put Q:=

{ta E I r r l L l L

I t = t l l t 2 for

Then we obtain according

Q =

IEL,

to a t h e o r e m

z (Da(Irr(L)L)

tl,t2EZ

, aEZ}.

of B r o z o w s k i :

n T LZ

)a

aEZ

and h e n c e Q is as the f i n i t e

s u m of r e g u l a r

sets r e g u l a r .

Therefore Red(L)

It

remains

l~i~n

to

= (Irr(L)L\Q)Z.

show

that

there

are

IicIrr(L)

and

LicL

for

such that n

Irr(L)L\Q

Since

Q

c_ Irr(L)L

=

we

Z i=l

have

IiL i.

according

to

Lemma

3 of

chapter

1.2. k

Q =

Z IiL 5 j=l J

with for

Ij:=IjD(Irr(L)L) characteristic

B e c a u s e of L e m m a

c

J

Irr(L)

11 .....I k

4 of c h a p t e r

and

E

where

n=k+l

= Irr(L)

and

1.2.

L i : = L \ L i _ 1 for

2
~

L

we h a v e k

\ I IjLj j=l

Ii:= Irr{L)

IjS(Irr(L)L)nL

L.

k

Irr(L)LiQ

Lj:=

Z

j--i

k \ Z Ij,

n I~(L\L~) J

J

Ii:=Ii_ I,

=:

Z

IiL i

i-i

LI:=L

and

j=l

[]

126

Continuation

of the p r o o f

of T h e o r e m

9:

n

Since

Red(T,L)

is the

= TnRed(L)

partition

have to s h o w

of

=

Red(L)

that there

n

X (TNIiLiZ*), i=i according

where

to the

Z IiLiZ* i=l

last

is also for Ti:= T N I i L i Z

lemma,

we

a disjoint

partition n-I Z j-i

Ti =

such

that

l<j
Ij,Lj,Uj

with ijETj,

occurs

are

the

regular

property

ljeLj,

and

that

ujEUj)

IjLjUj

Ij c_ Irr(L),

for every

lj is the

Lj ! L

t=ijljuj

first

word

for

E IjLjUj

of

L which

in t.

According

to L e m m a

3 of c h a p t e r

1.2. w e have b e c a u s e

of

T i c_ IiLiZ ni Ti =

where

Bk,U k are

regular

A second

application

for

i yields

l~n

the

Z BkU k k=l

and B k c_ IiL i

of L e m m a desired

for

3 of c h a p t e r partition

of

l
hence

of

Red(T, L). []

ii. Corollary: Let and

T be

regular

and

(L,~,R)

S: = (s E Z ~ B t E T

a regular

with

reduction

t = tll~t2 , where

the first w o r d of L w h i c h such

T h e n the

set

that

S is regular.

t

i~(i)~

s}.

occurs

scheme 1 ~ is in t,

127

12.

Definition:

Let

T,S be r e g u l a r

expressions,

tion

scheme

and R r e g a r e g u l a r

(a)

W e call T r e d u c i b l e (formally: iff t h e r e Rreg

(b)

The

is a r e g u l a r

n-fold

S

=

or

or

system.

T R r e g ~ S)

reduction

scheme

(L,o,R) of

(L,~,R)(T).

application

(L,~,R)n(T)

reduction

reduc-

to S by R r e g

S:= Rreg(T)

such that

(L,O,R) a r e g u l a r

of

(L,o,R)

T (L,d,R)~ n S)

is

to

T

(denoted

defined

by

recursively

as follows: (L,O,R)n(T)

:= ( L ~ 0 , R ) ( ( L , 0 , R ) n - I ( T ) )

Analogously

we define: n-i

llrneg(T) := Rreg(Rreg)) (c)

(L,0,R)*(T)

:=

u nE~

(L,0,R)n(T)

is the set of all w o r d s by the *

Rreg

(d)

regu]ar

(T)

:=

reduction

by the

regular

(denoted

by

(L,0,R)

and

u RreglT) nEN

set of all

n-fold

scheme

n

is the

The

w h i c h c a n be r e d u c e d f r o m T

words which

reduction

parallel

can be r e d u c e d

system

Rreg.

application

P(L,0,R)n(T))

is

from T

of

defined

(L,0,R)

recursively

as follows: k P(L,0,R)n(T)

:= Irr(T,L)

+

I i R i ( L , 0 , R ) ( U i) i=l

if

P(L,O,R)n-I(T)

= Irr(T,L)

+

to T

k X liRiU i • i=l

I28

Analogously

p.n __reg(T)

if

(e)

we define:

:=

Irr(T,Rreg)

k Z IiRiRreg(U i ) i=l

+

PRn-l(T)-reg = Irr(T'Rreg)

P(L,d,R)*(T)

where

:=

Tn:=

+

k Z IiRiU i. i=l

u P(L,O,R)n(Tn ) neR

{tlllt2.~.Intn+leTl

lieL,tielrr(T,L) I i is the

first

of L w h i c h

and word

occurs

in

ti...tn+l,l~i~n}. is the set of w o r d s tET

having

subterms

(f)

n disjoint i i are

Irred(T,L) is the ble

:= (L,o,R)

which

Analogously

words

13.

subterms

!iEL,

l~i~n,

word these

0(li).

(T) n Irr{L)

with

can

by

for every

respect

be deduced

to

(L,d,R)

from

T by

irreduci(L,O,R).

we define:

Irred(T,Rreg ! is the

arise w h e n

replaced

set of all

words

which

:= Rreg(T)

set of all which

with

n Irr(Rreg) respect

can be d e d u c e d

to R r e g

irreducible

from T by Rreg.

Definition:

Let

T

be

duction We

a

regular

scheme

call

(L,d,R)

expression,

and Rreg respo

a regular

Rreg

(L,O,R)

reduction

T-founded

iff

such that

resp.

a regular

(L,d,R)n(T) = (L,o,R)n-!(T) n n-i Rreg(T) = Rreg(T).

re-

system.

there

is a ne~

t29

14o

Proposition:

For

every

regular

regular

reduction

Irred(T,Rreg)

Proof:

Since

some

:

n~

are

expression system

Rreg

and

Rreg

Irred~T,L)

This

GSM-mapping

If

Irred(T,Rreg)

proposition

the

maps

regular

reduction

regular

sets

to

reduction

Irred(T,Rreg)

T was

regular.

are

sets

T-founded

Irred(T,L)

and

T-founded

we

have

for

= R~eg(T)

the

resp.

every

= (L,o,R)n(T)

from

Rreg

the

are

follows

system

and

regular.

(L,OrR)

resp.

T

fact regular

scheme

(L,0,R)

is n o t T - f o u n d e d , generally

that

no

a deterministic

sets.

resp. the

longer

the

sets regular

regular

Irred(T,L) provided

[]

15.

Then

Example:

one

Let

has

T

:= {ab)*

and

Irr((ab)

,ba)

= e ÷ ab

Red((ab)

,ba)

= abab(ab)

(L,~,R)

:= ba ~ ab.

Thus (ha ~ a b ) ( ( a b ) * )

= e + ab

+ a2b2(ab) *

(ba ~ a b 2 ) ( ( a b ) *)

= e + ab

+ a2b 2 + a2bab2(ab)

(ba ~ a b ) n ( ( a b ) *) = e + a b

+ a 2 b 2 +.o.+ a k + i b k + !

+ ak+ibk+l-mkabmk+l(ab)

for

n=(k(k+l))/2+m k and

*

0~mk~k+l.

130 Hence

we

have

for

l~mk~n+l:

(ba ~ a b ) * ( ( a b ) * )

Z anb n +

= e +

• anbn-mkabmk+l(ab)

n=l

Therefore

Irred((ab)*,ba~ab)

but

no

longer

16.

Example:

n=2

=

{anb n

(abc)

and

I nE~}

is

context-free

regular.

Let

T

:=

(L,¢,R)

be

defined

as

follows: L and

:= ba

~(ba)

Analogously

+ ca

:= ab,

to the

last

0(ca)

is n o t

even

Nevertheless Irred(T,L) scheme not

In the

and

order

to

following by

expression

right

resp.

in

many

the

cases

+ bc

:= bc.

}

even

regular

regular

when

T

characterize

these

cases

lemma.

means

It

says

that

parallelly regular. that,

t is

reduced

is

as

regular

it

reduction

scheme

is

useful

set

every

of

word that

running often

for

Rreg

is

regular.

the

(Remember

while

the

expressions

reductions

of c o u r s e

t,

+ ac

we h a v e

provided

is a g a i n

through

0(cb)

= {anbncn I n e H

Irred(T,Rreg)

reducing

parallel

:= ab

context-free.

gain

(L,~,R)

T-founded,

arises

in

we

R

:= ac,

example

Irred(T,L)

which

+ cb,

of

a

producing

once as

words

from

possible).

to

know which

regular a word left

to

131

17.

Lemma:

Let

T

tion

be

a

regular

scheme

Then

and

Rreg

P(L,¢,R)*(T)

Proof:

In o r d e r

construct

a GSM

expression

to

0,

(L,~,R)

a regular

and

show

whose

reduction

PRreg*(T)

that

a

are

reduc-

system.

regular.

P(L,0,R)*(T)

GSM-mapping

regular

o has

is the

regular

we

following

property 0(T)

for

all

regular

The

second

Let

Mir r

T. T h i s

proposition

accepting

ministic

GSM

W.l.o.g.

we a s s u m e

Q~

from

Q

:= { q E Q \ F

first proposition.

c a n be p r o v e d

Irr(L)

be

and

o

=

analogously.

a

finite

deterministic

(Q,I,Z,~,qo,F)

the

deter-

(L,0,R).

Q n QIrr

= {qo )"

I BwEI*

states

s.t.

which

peF whenever

lead

in o n e

~(q,w)

:= Q~ u Fir r GSM

and

the

:= Fir r u {qo }.

O c a n be d e f i n e d

0

9 are

F

:=

following

as f o l l o w s :

(Q,I,I,f,qo,F) subsets

= (p,v)}

s t e p to s o m e

and put

The desired

where

the

we define

t h e s e t of all state

implies

= (Qirr,~irr,qo,Firr)

automaton

Further

= P(L,0,R)*(T)

of Q x z x Q x

halting

132

(i)

~(qo,a):=

(ii)

~(q,a):=

{ ( ~ i r r ( q o , a ) , a ) , ~(qo,a)} for all aEZ s.t.

((Oirr(q,a)~a),

~Irr(qo,a)eFirr;

~(qo,a))

if q ~ F i r r \ { q o} and ~ i r r ( q , a ) e F I r r ~ aEZ; (iii)

~(q,a):=

~(q,a) if q £ Q ~ \ { q o } and p~F

for ~(q,a)=(p,v);

~(q,a) := (s,v)

{iv)

if q e Q ~ \ { s }

and peF for ~ ( q ~ a ) = ( p , v ) . []

18. Theorem:

Let

(L,d,R)

Then

Irred(T,L)

T

whenever

(Remember: resp.

Proof:

with

end

Let

is

B(L)

B(T) parts

N c(R)

resp. of T

for

every

scheme. regular

expression

= 0 = e(L) N S(R).

c(T)

is the

set

(see D e f i n i t i o n

= Irr(T,L)

of

initial

parts

5 of c h a p t e r

1.2.))

n + Z liRiU i i=l

and Ri=0(L i) for l ~ i ~ .

B(L)

o c(R)

Irr(T,L)

Hence

reduction

regular

(L, 0, R)(T)

Iiclrr(L)

The c o n d i t i o n

be a regular

it can e a s i l y

+

n Z IiR i i=l

n B(R)

c

implies

Irr(L)

be shown by i n d u c t i o n

Irred(T,L)

Now the p r o p o s i t i o n

= ~ = c(L)

that

= P(L,d,R)*(T).

follows

obviously

by the

last

lemma.

[]

133

19.

Theorem:

Let

T

be

duction Then

Proof: Hence

a

regular

(L,o,R)(T) have

Irred(T,L)

=

Together

is r e g u l a r

c T

implies

(L,o,R)*(T)

(L, ~, R)*(T)

with

Irred(T,L)

Irred(T,L)

which

of

and

(L,6,R)

a

regular

re-

scheme.

Irred(T,L)

we

expression

course

what

Irr(L)

c T.

c T for

all

ne~.

implies:

¢

D Irr(T,L)

=

(L,o,R)(T)

(L,~,R)n(T)

c T,

n

whenever

T

we

n

Irr(L)

= Irr(T,L)

obtain

Irr(T,L),

[]

is regular.

Conclusion:

For e a c h

free

monoid

(L,0,R)

with

R

a

regular

T and each

c T

regular

Irred(T,L)

is

reduction

scheme

regular.

20. Theorem:

Let

T

he

reduction

(i)

If

(ii)

If

=

Irred(T,L)

(L,~,R)(T)

then

and

(L,0,R)

a

regular

scheme:

(L,~,R)(T)

then

expression

£+TS =

=

Irred(T,L)

with

IS

I+ST

IS*cIrr(L)

(This

with

is

of

S Tclrr(L)

and

I=Irr(T,L)

course

regular)

and

I=Irr(T,L)

= S I (This

is a g a i n

of

course

regular)

t34

Proof: teT is

(i)

such not

Since

that

(L,O,R)(T)=I+TS

there

is

(L,~,Rln(tI~(L,~,R)n-l(t),

irreducible

with

respect

to

for

i.e.

every

n~

(L,0,R)n-llt)

(L,0,R).

Let Tn

Then

:= { t E T

we

T

I (L,0, R ) n ( t )

= (L, O , R ) n + l ( t )

and ~

(L,o,R)

= Z

have:

=

X

Tn

and

(T)

n=O

i.

(L,o,R)n-l(t)).

(L,0,R)n(t)

(L,o,R)n(Tn).

n=O

(L,0,R)n(T)

Statement:

=

n-i I IS i + T S n

for

all

neH.

i=0 Proof Assume

by

induction

Statement

on

n:

1 is t r u e

for

(L,e,R)k+I(T)

n=k.

Then

we

have

=

(L,O,R) ((L,e,R)k(T))

=

(L,e,R)(

k-i X

ISi

+ T S k)

i=0 k-i Z

IS i +

(I + T S ) S k

i=0 k IS i + T S k + l i=0

2.Statement:

( L , 0 , R ) n ( T n)

Proof

by

induction

First

we

have

on

= IS n

for

all

neM.

n:

( L , 0 , R ) ( T I)

=

((L,o,R)(T)

N Irr(L))

\ I

=

((I

N Irr(L))

\ I

=

(I

= IS

+ TS)

+ Red(T,S)S)

N Irr(L)

a

135

Assume

statement

2 to be t r u e

(L,~,R)k+I(Tk+I)

f o r n=k.

= TS k+l

n Irr(L)

k \ ~ IS i i=0

n Irr(L)

=

have:

k \ X (L,~,R)i(Ti) i=l

= ((L,~,R)k+l(T)nIrr(L))

k = ( Z IS i + TS k+l) i=0

Hence

Then we

IS k+l.

we obtain: e

(L,o,R)

*

(T)

=

Ii

(L,o,IR)n(T

n)

=

n=0

analogously

(ii)

to

Z

£S n

=

IS

n=0

(i)

[]

21.

Theorem:

Let

(L,0,R)

expression

be

a

regular

reduction

and T n defined

as

schemer

in t h e p r o o f

T

of

a

regular

Theorem

20.

T h e n w e have: k

If

(L, 0,R)n(Tn )

=

11

n

i=0

then

Proof :

Irr(T,L)

is

Irr(T,L)

for

UiSiVi , regular

all

n>mEH

Ui,S i, V i, l
regular.

=

Z

(L,O,R) (T n)

n=O m-t

11 ( L , O , R ) i ( T i ) i=0

and

(L,~,R)J(Tj)

+ j=m

0

~" m

c~ ~

f~

IA

0

~

~" ~

0

r.~ ~'N

~

II

H Q

<~

t~. r~ F,,~t

M

+

~

8

M

<

~.

t_J. t~ M o

II

+

a

H

H 0

M

8

II

II

8

I I-~

II

Ii

I"'

t,~,

M

+

M

i1

8

I

i.J. ii

~i

i~ • r~ l.~ t,,j. <

M

+

a

8

CO O~

137

III.l.3.

Working

The U n d e c i d a b i l i t y of the C h u r c h - R o s s e r

with

property

regular

reduction

is of course

Unfortunately

of great

systems

the

Property

Church-Rosser

interest.

we have the following

theorem:

22. Theorem:

It is undecidable system

Proof:

whether

an infinite

regular

reduction

is complete.

The

([O'Du 83]),

following

proof

who proved

the

ness of infinite reduction

R D := {wij ~ ail

As OWDunlaing

aieZ,

is b o r r o w e d

undecidability

from of the

complete-

systems R D of the form:

wijER i, R i is regular,

we proceed

O'Dunlaing

l~i<m,

je~}.

from the fact that for every Turing

m a c h i n e TM there is a finite reduction system R T M corresponding to TM

Proceeding reduction iff T M

(see M. Davis

from R T M system

IDa 58],

pages

we can construct

R(w)

88 - 93).

a well-founded

(w any w o r d of X*),

which

regular

is c o m p l e t e

stops on input w.

The u n s o l v a b i l i t y mediately

the

of

the

halting

undecidability

of

problem the

for TM

yields

Church-Rosser

im-

property

of R(w).

In

particular,

one

can

(eventually

by

adding

some

dummy

138

symbols)

construct

ding

the

to

following

Let (a)

E be

Turing

the

input is

a

are

regular

of

of T M

t RTM~

t e CONF

is a Any

subterm

s are

is c o m p l e t e .

(d)

For

every

(HALT

Let

~3Z

the

but

Then

define

wEZ

, w~HALT

R(W)

Z

way

iff

corresponding

and

such

there

w

e

input

h RTM ~

be

TM

that

the

a

map

f:

to

the

CONF

~

set

set

of

that

f(s)

overlap

f-l(set

disjunct we

e

on

:=

with

w

word

stops

a

TM ~

f(t)

some

1 6 Left(RTM)

some

s e CONF

only

if

1

if a n d

only

if

identical.

RTM

TM

such

correspon-

t.

can

(c)

ducible

CONF

overlap

of

t e CONF

t and

s

can

in

RTM

of T M

set

configurations

Any

TM

system

fulfilled:

alphabet

configurations

(b)

reduction

machine

properties

There of

a finite

w

CONF,

a

such

w

iff t h e r e

of

halting

respect

that

the

is s o m e

to R T M

Turing

irre-

machine

h e HALT

configurations

of

TM))

.

alphabet

isomorph

is a w i t h

of

RTM

to ~ w h e r e

regular

and g:

• an a l p h a b e t ~ ~ •

reduction

which

is

is a b i j e c t i o n .

system

R(w)

for

some

as f o l l o w i n g :

:= R T M

u R T M ~ U {I ~ g(1) I 1 e Mw},

where and

Mw:= RTM~:=

HALTu{tEKONF[ {g(1)

~

t~w

g(r) I i ~

, t irreducible) r E RTM}

139

R(w)

Proposition:

is

stops

Proof:

Because

of

iff t h e r e

is s o m e

But

is

this

Turing

on

the

input

true

TM

iff

Turing

(a)-(c)

such that

iff

stops

the

mashine

TM

w.

properties

h e Mw

only

machine

incomplete

R(w)

incomplete

Irred(h,RTM)

Irred(h,RTM)

on i n p u t

is

= w

~ M w.

, i.e.

iff

the

w.

[]

Additional

In

remarks:

[O'Du

83]

studied

where

right

sides

shown

that

regular as for

a different the

systems

limit the

of

KB-orderings

infinite

cases

group

the

are

regular

form

one

in

a

for

have

a

regular

such

chapter

van

III.l

groups

complete

the

regular

n=2j

for

some

j>l.

2)

k~5,

m=2

and

n=2j

for

some

j>l.

3)

k , m e {2,3},

k~m

and n~6.

4)

k , m E {3,4},

k~m

and

n=2j

for

5)

k=2,

m=4

and

n~4.

6)

k=3,

m=3

and n~3.

7)

k=4,

m=2

and

for

some

j~2.

however. which

In c h a p t e r R ® is n o t

It

was

kind

of

arise

instance, defined

system.

by The

following:

and

word

For

D(n,m,k)

m=5

undecidable

the

word. The

been

and

naturally

k=2,

n=2j

has

set

systems.

algorithm.

Dyck

exactly

systems

single-letter

completion

the

= 1

of

i)

with

for

only

undecidable

systems

= (AB) n

system,

of

sides

studied

Ak = Bm

Groups

left

consist CR w a s

kind

problem IV.5 w e

regular

cannot

will for

some

have

j~2.

a complete

see an e x a m p l e

any

KB-ordering.

of a

140

III.l.4.

Although finite

A Possible

the

Church-Rosser

regular

reduction

the finite some

property the

property

systems

property,

which

are

Test

is

Rreg,

undecidable even

if they

it is p o s s i b l e

sufficient

for

the

for

in-

possess

to p o s t u l a t e

Church-Rosser

of Rreg.

following

usual

will

termination

criteria

In the

Chureh-Rosser

test

decide

we for

whether

will

show

that

under

all

w

Rreg

is c o m p l e t e

certain

conditions

E Superpos(Left(Rreg),Left(Rreg)) or not.

23. Definition:

For regular

expression

Superpos(Ll,!2 !

LI,L 2 we define:

:= {weZ

I B ueZ

, llUeL I, ul2eL 2 such that

24.

W=llU]2}.

Theorem:

(i)

Superpos

(ii)

LI,L2° For t w o

(LI,L 2) is regular

regular

(L2,02,R 2)

reduction

Superpos(LI,L

into the f o l l o w i n g

Superpos (LI ,L 2 ) such

that

l~i~n,

=

regular

schemes 2)

splits

expressions

(LI,01,R I) and in p a r t i c u l a r

sums: n X i=l

LliEi

Lli!L I, Eic~(L2),

l~j~k.

for

=

k Z j =i

BjL2j

BjcB(L I) and L2j[L 2 for

t41

Proof:

(i)

Obviously

we have:

S u p e r p o s ( L i , L 2) = LlC(l 2) n ~ ( L I ) L 2. Hence p r o p o s i t i o n ty of initial (ii)

Since

(i) f o l l o w s

parts

resp.

immediately

f r o m the regulari-

end parts.

Superpos(L!,L 2) c_ LI~(L2),

Theorem

3 of

c ha pt er

1.2. y i e l d s : n

Q: = S u p e r p o s ( L I,L 2) = The

second

part

n

Z IIiE(Q)IIiD(Q) i=l

of the p r o p o s i t i o n

=

Z LilE2i i=l

can be proved

analogous-

ly. []

25.

Definition:

Let

(LI,~I,R I)

and

s c h e m e s and

(L2,~2,R 2)

n Z i=l

S u p e r p o s ( L i , L 2) =

be given as in T h e o r e m

We

call

schemes

(TI,T 2)

a

LIiE i

=

regular k Z j =i

pair

for

(L2,~2,R 2) if

n

and

reduction

BjL2j

22.

critical

(LI,OI,R I) and

be

T 1 :=

X i=l

~i (LIi)E i

T 2 :=

k Z j=l

BjO2(L2j).

th e

reduction

142

26, D e f i n i t i o n :

A

regular

to R

Notice

expression

if Irred(v,R)

that

separated.

As

following

deciding

n Irred(w,R)

in p a r t i c u l a r

course,

the

T is c a l l e d

the

theorem

Church-Rosser

every

s_eparated w i t h

= ~ for all

set T w i t h

shows

there

property

for

is

respect

v,wET,

v~w.

~T I = 1 is,

a

finite

a regular

of

test

reduction

system Rreg with

the

property

be r e p r e s e n t e d separated

27.

= { ( L i , O i , R i ) I l~i~n)

that

every

as a f i n i t e

with

respect

Superpos(Li,Lj), sum

of r e g u l a r

l~i,j~n

can

which

are

sets

to Rreg.

Theorem:

Let

Rreg

= {(Li,0i,Ri) ~ !~i~n}

be

a regular

reduction

system. If

for

all

regular

separated

sets

T

! Superp°s(Li,Lj),

iSi,j~n Irred(Ti,Rreg)

where

(TI,T 2) is the

= Irred(T2,Rreg)

critical

pair

belonging

to

T,

then

w e have IIrred(w,Rreg) I = 1

Proof:

Assume

there

is s o m e w E Z

for all

weZ

.

such that

IIrred(w,Rreg) I > i. W.l.o.g. underlying

we

consider term

ordering

w

to 4.

be

minimal

according

to

the

143

Then

we

have

w

=

(L2,~2,R 2) E R r e g Let

T be

IIUI2ET

l l u l 2,

such

a regular

i.e.

separated

subset

let

(TI,T 2) be the

Furthermore

let

WleIrred(rll2,Rreg)

i.e.

r I = 01(llU) we

have

Since wI

such

that

T

is

therefore

In the

which

to T.

w2EIrred(llr2,Rreg),

w 2 E Irred(T2,Rreg).

we

implies

we

then

have

that

w e Irred(l~,,reg)

separated

belonging

there

with

is

rh =

some

02(u'l ~.

N Irred(l~u'l~,Rreg).

have

llul 2 = l~u'l~

and

w I E Irred(llr2,Rreg).

same way one

The m i n i m a l i t y

shows

of w and w

[Irred(llr2,Rreg) hence

pair

r 2 = ~2(u12),

we have w I E I r r e d ( 1 1 u l 2 , R r e g )

Since

(LI,01,RI),

Superpos(Li,L 2) w i t h

and

= Irred(T2,Rreg)

Irred(T2,Rreg)

l~u'l~ET

of

critical

w I E Irred(Ti,Rreg),

Irred(TI,Rreg)

e

Hence

and

are

llU E L 1 and ul 2 ~ L 2.

that

and

where

there

w I is identical

w 2 E Irred(rll2,Rreg).

~ llr 2, w

~ rll 2

implies

I = IIrred(rll2,Rreg)

then

I = 1

with w 2.

[]

If I r r e d (TI,T 2) viously, true,

(TI,Rreg) belonging Rreg

is

~ Irred(T2,Rreg) to

some

T

incomplete.

as the f o l l o w i n g

example

for

the

critical

= Superpos(Li,L The

reversal

shows.

2)

pair

then,

however

ob-

is not

144

Example:

Let

Rreg:=

{(Li,0i,Ri)~

l~i~4}

be

given

as

fol lows : L1

:= c a

+ bd,

R1

:= x + y,

L2

:= a b

+ dc,

R2

:=y+x,

L3

:= x ( a

+ b

+ c

+ d),

R3

:=

(a + b

+ c

+ d)x,

L4

:= y ( a

+ b

+ c

+ d

R4

:=

(a

+ c

+ d

+ x),

+ x)y,

of(ca)

=

x,

Ol(bd)

= y

o2(ab)

=

y,

o2(dc)

= x

o3(xa)

= ax,

~3(xb)

= bx,

o3(xc)

= cx,

03(xd)

= dx

o4(Ya)

= ay,

04(Yb)

= by,

04(YC)

= cy,

o4(Yd)

= dy,

04(YX)

= xy

Then

we

have

for

instance

Superpos(L

I , L 2)

and

Irred(Ti,Rreg) = xb

+ yc

longing

to

Testing

all

=

and

the

such

that

yc

cy

+ bx

=: T

bx

where

is

the

(TI,T 2)

critical

with

pair

be-

T.

other

critical

Nevertheless

bdc

Irred(T2,Rreg),

T 2 = cy

T

:= S u p e r p o s ( L i , L

I r r e d ( T I,

for

+

= cab

~b

T1

+ b

there

Rreg)

pairs

is

=

~Irred(w,Rreg)l

w > i.

also

Irred(T2,Rreg)

(TI,T 2)

some

j) y i e l d s

belonging

e

Z

, for

to

T.

instance,

W

= cab,

145

There the

are however

property

incomplete

that

Irred(Ti,Rreg)

=

the

to

assumptions

reduction listed

of being

systems

:= S u p e r p o s ( L i , L j) for

the

we

with have

critical

pair

separated

is in fact

necessary

of the KB-completion-test.

under

system

T

reduction

T.

condition

for the correctness

The

all

Irred(T2,Rreg)

(TI,T 2) belonging Hence

for

regular

which

the

is decidable

in the following

completeness

of

a

regular

by the K B - c o m p l e t i o n - t e s t

are

corollary.

28. Corollary:

Let Rreg

:= {(Li,di,R±) I l
be a r e g u l a r

reduction

T = Superpos(Li,Lj),

l~i,j
be d e c o m -

posed into the sum of finitely

many with

system. If v e r y

separated

sets

for every

T k one can decide

critical

then it is decidable

that generally

separated

and that

Nevertheless satisfy

the

there

=

whether

Irred(T,Rreg) are many

assumptions

Irred(T,Rreg)

pair

(Tlk,T2k)

belonging

to

Irred(T2k,Rreg) Rreg

is complete.

it is not decidable

tions for the regularity of

to Rreg

whether

Irred(Tlk,Rreg)

Notice

respect

m Z Tk k=l

T =

such that

can

of

can not always

regular

reduction

Corollary

28.

and in particular

were already

whether

given

in

a set T is

be computed. systems

Sufficient

which condi-

the c o m p u t a b i l i t y

section

III.l.2.

146

III~2.

Forward

The

free

the

single

the

-

abelian

Backward

group

equation

completion

Systems

generated

AB

= BA.

by A a n d

For the w e i g h t s

algorithm

generates

an

B is d e f i n e d

by

~(A) = ~(B)

= 1

infinite

set

of

reductions:

BA ~ AB, and

B - I A -I ~ A - I B -I, B - I A ~ A B -I, A - I B ~ BA -I

B ( A - I ) n B -I ~

Similarly

the

(A-I) n, A B n A -I ~ B n

completion

algorithm

Greendlinger

group which

be r e g a r d e d

as a g e n e r a l i z e d

II.4. For

we

recall:

the

weights

algorithm

~(A)

1

the

also

seen

do n o t

tions

which

tion.

If

from

B ( A C - I ) n B -I ~

that

is n o

the

fact

that

redundant would

right

to

be left

ones.

in s o m e

(C-IA) n for

complete

n~l

system

sets are regular. A l t h o u g h

s u f f i c e to d e s c r i b e

one

to f i n i t e

finite

(A-Ic) n

group.)

the c o m p l e t e

become

symmetry

there

for this

reflect

e.g.

(B-IA-I) n,

From

completion

(CB) n, B ( C A - I ) n B -I ~

regular expressions

duce

=

A ( B C ) n A -I ~

In b o t h e x a m p l e s

collapse

= ~(C)

ABC),

in a n y K B - o r d e r i n g

tions

of c o m m u t a t i v i t y ) .

set of r e d u c t i o n s :

~

the

= CBA ( w h i c h can

an i n f i n i t e

C ( A - I B - I ) n c -I ~

they

ABC

for

generates

SYM(CBA

(We h a v e

= ~(B)

n ~ i.

explodes

is d e f i n e d by law

for

able both

This

the i n f i n i t e s y s t e m s

they

after to

contain

some

infinite

restricted

many

trivial

apply

suggests

the

the

only

manipula-

a few

systems idea

reduc-

reducwould

to

reintro-

and controlled

manner.

147

This

has

to

reduction

be

system

it a p o w e r f u l The

notion

short)

such

not

for

a way

loose

decision

partly

simulate

this

the

concepts

features

of

which

a

makes

problems. system

idea.

an i n f i n i t e

concepts

that

those

a forward-backward

captures

general

in

does

tool

of

sometimes The

done

A

((f+b)-system

in

(f+b)-system

can

finite

ordinary

will

be

defined

( R f + R b)

is

an

reduction for

system.

arbitary

term

algebras.

i.

Definition:

An

(f+b)-system

ordinary term

Rf

is

reduction

orderings

called

system;

both

Rf ~ and

Rb~

2.

the are we

~f

and

and

assumed

f~ a n d

with

( R f , R b)

of

well-founded

resp°.

system

in g e n e r a l

write

compatible

~b

forward

pair

Rb

to be

is

the

finite.

backward Instead

of

b ~ in short.

Definition:

(i)

A repetition u

in


(ii)

the

vation

system

derivation R

such

un

= v and

An

(f+b)-derivation

Notation:

usual

free

I l~i~n>

derivation

AS

systems

ordered

u

from

f+b ~

= Rf

that

u i • u k for

of

v means

u to v. B e c a u s e

u Rb

is

a

v from

a term

sequence

u i R ~ ui+ 1 f o r

l~i~n,

uI

= u,

i • k.

of v f r o m

v from

u f+b ~

of a t e r m

u such

u is a that

repitition

v ~f

free

u.

v

the

existence

repititions

of are

some always

(f+b)-deriredundant

148

this We

is

equivalent

refer

Note

to

also

Having

u R~

v and

(f+b)-derivations

that

these

defined

notions

to

of

u

as

concepts

f÷b ~

v,

irreducible

v ~f to

are

we

and

derivations

not

can

u. in

(Rf+Rb).

symmetric.

formally

reducible

take

terms

over

from

the

chapter

II.l.. Furthermore,

the

will

in t h e

be

used

concept same

of

the

way

as

Church-Rosser in c h a p t e r

property

II.l.,

i.e.

CR the

square U

uI

can

be c o m p l e t e d

It

is h o w e v e r

of

the

for

not

reduction

steps

repitition we

WCR are

and

are

The

u,

u I and

to t a k e

not

but

free. first

given

useful

properties

(f+b)-derivations

later;

uI

FTP.

on

over

This

arbitrary subject

substitution

focus

u 2.

the is

formal

due

to

iterations

to

the

for

WCR

are

no

equivalents

the of

condition will

be

fact

that

elementary of

being

discussed

FTP.

3. D e f i n i t i o n :

( R f + R b) has free

weak

derivations

FTP in R

iff

there

= Rf

u R b.

infinite

repitition

149

If

( R f + R b) e n j o y s

term

u the

{v[v

can be

weak

sets

FTP t h e n

L(u,Rf+Rb):=

reached

from

by K 6 n i g ' s

{v I u f+b ~

u by a r e p i t i t i o n

lemma

for each

v) u {u)

and

free d e r i v a t i o n )

are both finite. An i m m e d i a t e

consequence

is

4. P r o P o s i t i o n :

Suppose

(i)

(Rf+R b) has w e a k

Irreducibility

(ii)

For each

FTP.

Then

is a d e c i d a b l e

term

the

sets

that

L ( u , R f + R b)

systems An

weak

next

contains

CR

+ weak

exactly

consequence

examples

beginning

will

equivalent of this

one

FTP, term.

the e q u i v a l e n c e is t h a t

from

UTP As

again for

u.

means

ordinary

of CR and UTP.

a complete

(f+b)-system

The f r e e a b e l i a n

and d e f i n e

preceding

to

cases;

the

finite

(f+b)-systems

infinite

systems

which

given

at

are the

g r o u p on two g e n e r @ t o r s A and B:

B - I A - I ~ A - I B -I, B - I A ~ A B -I, A - I B ~ B A -I }

R b to c o n t a i n

orderings

in b o t h

show

section.

We put Rf = {BA~AB,

The

as

and c o m p u t a b l e

the w o r d problem.

derivation

i)

defined

FTP g u a r a n t e e s

immediate

solves

Our

is

of terms.

L ( u , R f + R b ) n I r r = I r r e d ( u , R f + R b)

and L ( u , R f + R b) are f i n i t e

Completeness

property

are we

B for ~f;

the

the

reversed

KB-orderings

choose

the

reductions.

with

weights

lexicographic

in ~ b B p r e c e d e s

A.

~(A)=~(B)=I

ordering

with

A

150

2)

The G r e e n d l i n g e r

We put

Rf = {

group:

CBA ~ ABC,

A - I c B ~ BCA -I,

C - I B - I A -I ~ A - I B - 1 C - I '

B - I c - I A ~ A C - I B -I,

c - l A B ~ BAC -I, R b again contains The o r d e r i n g s and

reductions.

are t a k e n as in the a b o v e e x a m p l e

with A~fB~fC

C ~ b B ~ b A.

Both

(f+b)-systems

infinite They

ordinary

also

enjoy

many

words

Because system ordinary

The

FTP

to

above.

because

right

and

left

sides

have

t h e r e are only f i n i t e l y

see

KB-ordering

that

there

(f+b)-systems

is are

no

finite

more

complete

powerful

than

systems.

have

example

from

and for e a c h w e i g h t

any

formation

both

systems

equivalent

h a v i n g that weight.

for we

have CR and are d e r i v a t i o n

weak

the s a m e w e i g h t s

In

the r e v e r s e d

B - I A - I c ~ C A - I B -I };

of

FTP

(Rf+R b)

may

from

loose

two

even

systems

weak

Rf

FTP,

and R b w h i c h

however.

As

an

we c o n s i d e r

Rf = {A ~ BB)

with weights

~f(A)

= 3, ~f(B)

= i;

R b = {B ~ AA}

with weights

~b(A)

= i, ~b(B)

= 3.

(Rf+R b) we have the i n f i n i t e A

which means In general

that

~

BB

~

repetition

AAAA

~

free d e r i v a t i o n

...

FTP fails.

we have the f o l l o w i n g

undecidably

result.

5. Theorem: It is u n d e c i d a b l e tion

free

whether

derivations.

a s y s t e m R has i n f i n i t e

repeti-

151

Proof:

This f o l l o w s

II.l.. The tion

existence

in R a m o u n t s

in R(G) w h i c h

from the proof of T h e o r e m 12 of chapter of

an

infinite

to an infinite

is an u n d e c i d a b l e

repetition

repetition

free

deriva-

free derivation

property.

[]

As

a consequence

(f+b)-systems. FTP

we

get

Therefore

the u n d e c i d a b i l i t y a positive

of

criterion

weak

FTP

for

which

ensures

is some KB-ordering

with a

is useful.

6. Proposition:

(Rf+R b) has w e a k FTP if there weight

Proof:

function

~

such

that ~(s)k~0(t) for each

In a d e r i v a t i o n the w e i g h t never

s~tERfuR b.

increases and there

are only finitely m a n y words for each weight.

[]

The

criterion

satisfied tions may

in

with

however

in the

the

last

examples

non-descreasing for

a given

also strictly w e i g h t

proposition from

above

weights

descreasing

This is possible because

in

where

have

KB-ordering

is

particular

certain

been

reduc-

reversed.

be possible

It

to reverse

reductions.

some other K B - o r d e r i n g may be used.

The existence of such an ordering

is decidable.

7. Proposition:

Suppose of

R

a weight

is

a

finite

function

~0

system. for

which

Then

the

~(s)

> ~(t)

existence holds

for

152

each can

s ~ t e R be

is

decidable;

in t h e

effectively

constructed,

Proof:

Suppose

generators

reduction

s ~ t

the

induces

an

Z

k(s) o~(a k)

where

k(u)

is t h e

If w e

introduce

integer

k(s)

condition

Presburger metic,

linear

>

exponential. and

A its

This method INF k a n d

of

a I .....a n , of

Z k=l

the

such

Then

form

is

xk w e h a v e

~

Xk~l

in

which

k(t)

xk

domain

of

is

decidable,

a

it

[Ble]

linear programming

superSUP-INF

and

problem

[Sho]). numbers

that

in a real

solution

for

x k)

SUP k = Max(x I x occurs

in a real

solution

for

x k)

our

of

arith-

is

Bledsoe's

(cf.

~.

formula

Presburger

is

Shostak

inequality

integers

in f a c t

method

by

each

the

problem

efficient;

efficient

for the

SUP k s u c h

is

very

refinement

gives

solve

programming

not

more

to

n Z k=l the

of a k in u.

INF k = M i n ( x I x o c c u r s

Because

each

k(t) - ~ ( a k)

occurrences

xk

arithmetic

however,

method

number

unknowns

n Z k=l

This

case

n

k=l

extra

are

inequality

n

with

positive

problem

has

real

solutions

it

is

solvable

iff

solution

is

constructed

by

reducing

the

problem

in t h i s

solutions

iff

I N F ~ SUP.

it

In the

substituting

has

integer

latter

SUP 1

for

case x

a

and

way.

[]

We

note

actual

that

for

solution

deciding

of t h e

weak

linear

FTP

we

need

programming

not

to

problem.

know

the

153

I I I , 3 , ..... The

Property

Church-Rosser

of

Forward-Backward

that

for

forward-backward-

will

consider

Systems

In

this

chapter

systems of

forward-backward

systems

even

that

for

paragraph

function

o

that

we

~ ~(t)

(ii)

s ~

t £ R

and

~(s)

= ~(t)

t

v

the

u

I

u

~-~

u

following

and

special

we

R

=

v

and

~(u)

> ~(v)

v

if

u R~

v

and

~(u)

= o(v)

I ~ v

if

u

or

u

~-~

that

the

and

s ~ R

notatlon:

u R~

describe

prove

+ R b,

t ~

if

follows,

will

Rf

~ v

it

kind

have:

~(s)

use

a

undecidable.

=

both ~

is

below),

t e R

1 . and

u

CR

this

weight

(see

s ~

~ r

in

class

throughout some

(ii)

~-~

this

We

(i)

will

From

s

for

assume

and

show

decidable.

We

We

will

is n o t

that

CR

we

v

~

is

transitive

~-~

v.

symmetric.

and

reflexive

Let

~-~*,

hulls

~-~*

of

~,

~ ~.

R~

:= {s ~ t e R I o ( s )

R H

:=

{s

denotes

a

directions, iff

v

is

~

t

new

I s ~

kind

> O(t))

t e R

of

rules,

and

o(s)

which

= ~(t)}.

can

be

applied

i.e.:

obtained

from

u

by

applying

s ~

t or

t ~

s.

154 We

put

M

denoted We

= X*/~-~*.

by

[u]

u'

Let M ~

The

equivalence

class

of

a

word

w

is

~-~

v'

[w].

write

and

The

M--~

[v]

iff

Bu',v'

u

~-~

u'

and

v

~-~ v' denote

rules

the t r a n s i t i v e

Is] M ~

on t h e m o n o i d

M

It], which

and

s ~ t E R~ only has

reflexive

hull

of M ~.

give a reduction

forward

system

RM

rules.

i. Lemma:

The

following

Proof: cise

to

literally

are e q u i v a l e n t :

The

equivalence

the

reader.

the s a m e

The

of

(i) and

(i)

R

has

CR

(ii)

R M has

CR

(iii)

R M has W C R

(ii)

equivalence

as the one

for t e r m

of

is

left (ii)

algebras

as and and

an

exer-

(iii) the

is

free

monoids.

[]

This

gives

testing

Now

we

notion:

WCR

come

a criterion

to

check

the

completeness

of

R

by

for R M.

to a c e n t r a l

but

somewhat

complicated

technical

!55 2. D e f i n i t i o n :

For

w',w",IEZ

defined There

the

narrow

is

some

such

for

r
all

following

finite

that

double

~(0,0)=l,

there

are

(2)

[(2a)

(in

cause

and

tv'

= u~(r,r)v

and

sv'

= ~(r+l,r+l)

[(Bv'

possible Take Then

Hence

for

words

~(g,V),

~(n,n)=w" u,vEZ

such

or

= e)

and that

the

• e) {Bu'

= u~(r,r)v

and

u's

= ~(r+l,r+l)

and

= ~(r,r)

and

u'sv'

= ~(r+l,r+l)

to c a t c h

direction need

not

enlarge

have

the

want

u'

= e)

of

this

so

~-~

= ~(r,r+l) ]

to

to

be

the

]].

apply

a

This

meaning

the

1 and

of

:= l

@(0,I)

:= U o ~ ( 0 , 0 )

~(i,i)

:= S o V ~

~(0,1)

~

~(i,I)

be i.

to a p p l y

= Uo~ ,

~(0,0)

rule

might

subword

word

to

= ~(r,r+l)

intuitive

t o ~ So).

example

or

= e

u'tv'

we

(u = e

E X*)

and

want

]

E Z*)

u't

[(Bu',v'

v'

= ~(r,r+l)

and

to

and

(v = e

= v'v

w e put:

we

and

t

Suppose

tO

is

= u~(V,r)v

= uu'

(2c)

the

w"

[(Bu ' • e ) ( B v ~ E z* )

u = v

tion.

of

g(0,n)=w',

t

(2b)

we

~I-+

holds:

(¥U ~ r) ~ ( g , r + l )

First

sequence

t~sER~_~

(i)

or

w~

as:

0~gNV~n

or

derivation

• e, ~

t o to

impossible, But the

it

So k-~ t o •

1

be-

might

be

rule.

a subword

= toV ~

by

defini-

(Vo:=e)

of

I.

156

The

following

figure

shows

the

meaning

of the c o n d i t i o n s

2a-2c:

~(0,5)

= w'

(2a)

-

~12,2)

6(2,5)

l !

,

(2c)

~

(2b)

~(3,3)

~(3,5)

(2b)(2c)

~{4,4}

~(4,5)

(2a) W(5,5)

= w"

Note

that

rule

t~s,

a

next

words

3.

in

order

to

t must

subword

The

~(5,5)

of

overlap

from

@(r,r)

with

@(r,r)

to

@(r+l,r+l)

but

t need

by

not

the

to

be

@(r,r).

definition

w' a n d

come

w"

discusses

from

possibilities

of

extending

the

above:

Definition:

wI

~Ii-12-- # w 2 iff

(1)

uw~v=w I

there

and

uw~v

are

u,v,w[,w h e Z

= w2

and

wI

such

that

~Ii- ~ w ~ and

(2a)

(12

or

(2b)

(Bu'~

Z* )(u

= e

or

(2c)

(Bv'e

Z* )(v

= e

or

(2d)

(Bu',v'~

(2)

= w 2)

X* )(u

and

w 2 = u'l 2)

and

w 2 = 12v')

= v = e

and

w 2 = u'12v')

157

In t h i s

case

w I = uw[v also

The

we

call

= u~(0,n)v

a narrow

the

derivation

~

u~(l,n)v

~-~

.o.

~

u~(n,n)v

= w2

derivation.

following

figures

show

the

meaning

of

the

conditions

(2a-2d):

~(0,0)

o,o,o,l

wI

wI JL

,,,,,,,,

u

( n, n ) =w~

v

~(n,n)=w~

w2

v

w2

12

12

(2a}

~(0,0)

(2b)

wI

[~(0,0)

wI

~(n,n)

w2

,o-

U

~(n,n)

w2

(2c)

Note, subword

that

12

must

of ~ ( n , n ) .

overlap

(2d)

~(n,n),

but

1 2 need

not

be

a

158

4. L e m m a :

If

UlllV 1

~

statements

o n e of

the

following

three

holds:

There

(i)

then

u212v2

are

v~,

u~,

v i, v[,

w

such that

[

+

*

(il) (i2)

U~llV ~

and

(i3)

v 2 = v~wv~

are

u~,

u~,

v~,

uI

and

(ii2)

U~llV ~

and

(ii3)

u 2 = u[wu~

are

~-~

ul,

u~,

u[u~

u i, v[,

uI

~-~

and

(iii2)

U{llV ~

and

(iii3)

u 2 = u[u~

Let U l l l V 1 = w o

vlv[

w E Z

and

such

that

v [ u ~ 1 2 v 2 ~-~

v1

~Ii-~ w

(iiil)

Proof:

vi

and

~Ii- ~ w

(iil)

There

(iii)

U212v~u~

and

There

(ii)

Ul ~- ~

v i, u~,

u~ u~

v~ E X

and

v{v~

such

~

that

v1

~ii-12-- ~ u ~ 1 2 v ~ !

and

~

wI

.

v2v I = v 2

~-~

...

~-~ W n _ 1 ~-~ w n = u212v 2 .

We

write

He

will

u ~ v if u is a s u b w o r d try

to

find

a double

conditions

in D e f i n i t i o n

satisfies

~(~,r)

E w~

We a s s u m e

~(v+r)

E w v for

Of c o u r s e ,

there

is at m o s t

~(~,A+I)

If

we

cannot

of D e f i n i t i o n

define

E w~

of v.

sequence

~ which

2 for a n a r r o w for

satisfies

derivation

the

and which

~v~n.

~r~A. one possibility

for

~(V,A+I)

to d e f i n e

V~A+I.

properly,

2 one of t h e f o l l o w i n g

two

then cases

by

condition

takes

place:

2

159

(a)

For

some

w,

w',

w"

6

w A = w~(~,A)w'tw"

(b)

For

some

w,

w',

w"

this

We

both

case

cases

we

will

have

make

w o .....w n

by

For

purpose

this

some

following

sequences

some

we

t

only

~ g 4:

wp

global

change

more

appropriate

~-~s

~ R H : holds.

~-~s

E RH: holds.

consider

case

(a).

In

= w~(~,A)w'tw",

and

replace

the

sequence

sequence

W _ l , W o .....Wn-l"

put w-i

The

t

wA+ 1 = w'sw"~(A,A)w

similar

for

a

and

e Z and

are

some

wA+ 1 = ww(A,A)w'sw"

and

w A = w'tw"~(~,~)w

Because

and

= Wo

figure

= w~(0,~)w'tw"

w~

:= w ~ ( ~ , h ) w ' s w "

for

0~A,

w~

:= w ~ + 1

for

A~n-I

shows

(wp)0~p~ n and

the

relationship

between

the

(w~)-l~n-I

11 11 ~(0,0)

wo

w'

t

3(0,0) 3(0,0)

w'

w'

t

S

w '

S

W"

~o

wI i

wA

~

w ~+I

~

w~+2

~(~,x)

w' t

w

w

~(k,~)

w' t

w"

....

¢(~,~)

s ' w"

W I

i

Wn- 1 wn

If

~

is a g a i n

the process Note,

that

go t h r o u g h

of

not

properly

changing

before

we

(w~)A+l~v~n,

expandable

the

had

(w~)),

we

iterate

sequence.

gone

but

(in

from

after

the

(w~)

to

change

(w~), we

we

only

had

to

have

to

160

go through tually

(wv)£+IKgKn_I.

after

tending

of

~

W-k+l

~-J

=

(n-k)-times

and

''"

~-~ Wo

~-J

ex-

situation:

''"

~-J Wn-k

= Wn

~iii~i

~g = ~ ( g , n - k ) v {

for 0KgKn-k

where

u212v2

f

the

sequence

stop even-

fi[611iVlVi-'-"

:

~(O,n-k)

This

will

w. Then we will have the f o l l o w i n g

Wo = W-k

wo

k changes

Hence the process

u I ~-J

UlU I-"-'

and

v I ~-J

VlV I-'-''

= Wn = Wn-k = u { ~ ( n - k , n - k ) v {

looks as follows:

uI

vI

11

:Wo:W_l:W_ k

L

|

j,,

=Wo

ii=~(0,0) .i

| 1

7

1 I

-

r

-[ |

~(n-k,n-k)

=Wn- k =Wn =u212v 2

161

We c o n s i d e r

(I)

(II)

{III)

the

u212

Then

i)

holds.

12v 2 E v~,

Then

ii)

holds.

v[

{ v2

u~ ~ u 2

and

(iii)

Hence

uI

:= u 2

w'

!

This

w' ~Ii--12-~ w"

will

~ 12v2

and

and

U lIU l.'--"-' - 'U l'U

hold.

to the above

v2 ~ vi

u2u~

:

We put

ui:u 2 •

and

w ' -- u 2' ~w ( 0 , n - k ~v~:ui~iv I,

Then we have:

(iii) will

and

case.

and hence ~ ( n - k , n - k )

Then ul

Hence

UlM

E ~(n-k,n-k)v i

is s y m m e t r i c

u2 ~ ui

T h e n u{ : u 2 u ~

u I' := u 2 ' ~ { w" := u 2 w ( n - k , n - k )

--I

:= U l l l V 1

(iii)

v 2 E ~i

12 E ~ ( n - k , n - k ) .

holds,

Then we have

(VI}

and hence

u 2 E u{ ~ u212 E u l ~ ( n - k , n - k ) . We put

(v}

six cases:

E ul.

Then

(IV)

following

~ 12 .

~i : v~v2

U l'- U 2'-' U I,

vi=v 2,

w ~ ~11-12-~

V l2-'V-l,V ,

w" : 12 .

w"

hold. []

162

The

following

theorem

tells

us the c r i t i c a l

pairs

for R M.

5. T h e o r e m :

R M has W C R

iff

for all

ul,v~,u½,v~

e Z

we have:

U~llV ~

~ii-12- ~

u~12v ~

[u~rlv{]

Proof:

=:

l l ~ r I,

We p u t

implies M--~

a:=

12~r 2 e R ~

[w]

for

some w e X

and

[u'l 1 1 v'] 1

a n d f o r all

[u~r2v~]

= [ u ~ 1 2 v 2]

M--~

[w].

and consider

[u'rlVl]l

a

[w]

f

f

f J f

[u~r2v~]

By W C R

~:

Let

Then we

there

must

be s o m e

a, b I, b 2 e M s u c h

for

some

11 ~

r I,

wee

which

that

completes

a M ~ bl'

12 ~ r 2 e R ~

and

have: a

= [UlllV I] =

[u212v2 ]

b I = [UlrlVl] b 2 = [u2r2v2]. Hence,

in p a r t i c u l a r

UlllVl

~-~

u212v20

the d i a g r a m .

a M~b

2•

Ul,Vl,U2,V2

e

163

According

to L e m m a

(i)

{ I

uI

v½U{llVlV ~

we h a v e

Hence

w

PM

:= u 2 r 2 v i u i r l v I

(iii)

u~ ~

ulu I,

with

u 2 = uiu ~

assumption

to

will

is a n a l o g o u s

and

b2 =

[u2r2v~UillVl]

work.


vl,

viv I

there

[u'rliV']l M ~

Furthermore

t h r e e cases:

v~UillV I

b I = [u212v~ulrlVl]

(ii)

By o u r

the following

u212v~u i

v2 { { Hence

4 we consider

and

is w'

[w']

and

uillvl

v~v{

e Z

H~-I2-+ uil2v i

= V2

such

[u~v2v~]

holds.

that:

M~

[w']

we h a v e

bI =

[UlrlVl]

= [UlUlrlVlVl]

[u~w'vl]

M~

and

b a -- [u2r2v 2] -- cu[u~r2v~v [] Hence

w:=

u~w'v~

will

.*

[ui*vi]

work.

[] Because

there

! ([ U irlvl]

•

algorithm not will

be

! [ u 2! r 2 v 2] ), t h e

which

decidable, assign

input.

might

tests

infinitely above

WCR.

We

whether

R has

a reduction

system

many

theorem will CR.

does

indeed In

critical

order

to a n y T u r i n g

pairs

not

give

us an

show•

that

it

to

do

machine

is

this,

we

and

its

164

6. D e f i n i t i o n :

Let of

T

be

its

state, Let

a Turing

states~ A

machine

qo

the

put

Z

by u , w , v , l , r , t , s , . ,

For

R

state,

alphabet

qh

the

= R (T'T)

(RI)

bib

(R3,r)

ar>B

(R4)

u<

every

words

the

~-~ b I

from

A +

following

halting

the

by

rules

from

O,T . . . .

for

T E A+:



(R2)

b2B

~-~ Bb 2

for

0~rKm

~-~ arB>

~-~

words

(R2')

~-~ Bb I

instruction

contain

and

•

introduce



set

a o = u ~ A u Q.

Z := A u Q u {B,<,>,bl,b 2} a n d d e n o t e

(RI')

R will

= {qo .....qh } t h e

symbols.

We

For

Q

initial

= {a I .....a m } its

<,>,bl,b 2 n e w

with

I ' b2


l =

((q~,ar),(q~,av,0))

following

e

T.

reduction:

(R5,~,A)

q~ara A

~-~ a v q ~ a A

for

0&A~m,

if

0 = R

(R6,%,A)

aAq~a r

H

for

0~A~m,

if

@ = L

(R7,%)

q~a r

if

0 = N

If t h e

state

q~aAa~

~-~ q ~ a U

of T is .

• .. B B . . . B a i l a i 2

.0 all

a i

l+lail+2

.

"" a i k B . . .

B

...

f q~ then this

state

"''B
will

2

be d e s c r i b e d ...a i

lq~ail+l

by w o r d s a

i f + 2"

..

of t h e f o r m

ai k

B...B>B...

165

7. Lemma:

If 1

.

=
''al k

>, k>k'

and

u

~i-~ v~

then u = B...BIB0..B v

Proof:

and

= B . . . B < a j l ' " a j n , q ~ a 3n,+ ....l

Let ~ be the d o u b l e

We claim,

that

for

V~V and

sequence

for

B...B
This

is e a s i l y

v = ~(n,n)

have

shown the

by

from

n>n' .

Definition

2.

i~>I'~:

q~

ail,v+l,~...ail~,~>B...B,

induction

desired

a j n>B...B,

on

V;

hence

u = ~(0,n)

and

form.

[]

8. Lemma: Assume

11 = < a i l . . . a i k ' q~ a i k , + l . . . a i k

>,

12 = < a j l ' ' ° a J n, qn a j n , + l ' ' ' a j n > ' Then

wI

~ii--12--~ w 2

Proof:

Consider

By L e m m a

7 we have

for

w i : B...BIIB i~ =
some

Because hence

we

u=v=e,

cannot

implies

w = uwlv ,

. .

.... B ahp '

overlap

w[ = Wl,

q~

: uw~v

w2

w~ ahp

12 w i t h

wh = w 2 and

,

n'
w I ~Ii-- ~ w 2.

and

: B...BI~B...B +i • "'ahp > ,

w~ it f o l l o w s finally

k'
w i ~iI- ~ w~

p>p'

12

.

= 11 and

w I ~ii- ~ w 2. []

166 9. Lemma:

The

following

(a)

The

Turing

blanc

Proof:

machine

(a):

T with

input

~-~

u2V2]°

T will

stop with

a

Let

p be

the

amount



by

of

space

needed

by

machine. ul:=

B p =:v I

u l < q o T >u 2 (R5),(R6),(R7)

tape

we

we g e t

~--~

and

the

Again

by

(a) ~

we get



~-j

T

will

stop

with

blanc

Assume

assume

is m i n i m a l .

in t h a t

of

length

wO

~-~*

n-2.

Wq

(R4) case

The

and

some u 2 , v 2 e { u }

w n = u2v 2

wO

w n.

~-~ w I

that

we

H*

with

(R5),

(R6),

(R7),

These

words

a r e of t h e

~-~

the

is not

same Wq

for

w o = u l < q o T > V I,

w.l.o.g,

Then

holds

u 2 < q h u > v 2.

and

because

that



(R3),(R4)

(b):

may

fact

(R3)(R4).

see t h a t

~

We

) [ulV 1

tape.

(b) ~

Putting

By

equivalent:

B ( U l , U 2 , V l , V 2 e {B}

(b)

the

are

...~-~

length

used

n of

the

both

backward

able

to

would

be

holds

for

(R3).

w n with

(R3),

find

Hence

form:

Wp

= B...BB...B

Wq

= B...BB...B

and

.

forward

a derivation we

(R4) a n d

0fq~Kq~n.

derivation

may Wp

assume 'j lJ* W q

.

167 In

the

(R7)

last

is

state

step

applied

Wq_ l

forward

Wq

one

because

of

the

T cannot

rules

(R5),

enter

the

(R6),

halting

backwards.

If o n e

of t h e s e

rules

w o u l d be a f o r w a r d cation.

But

then,

a derivation

of

would

because

length

in the d e r i v a t i o n

(R5),

(R6),

(R7) one

describes

the

t h a t the m a c h i n e

be a p p l i e d

application

Hence

vation

~-~

backwards

then t h e r e

f o l l o w e d by a b a c k w a r d

T is d e t e r m i n i s t i c ,

we

appli-

would

find

n-2. Wp

only

~-~

W p + 1 ~-~ -.- ~-~ W q the r u l e s

applied

forward.

computation

of

stops with blanc

Hence

T with

input

this

deri-

T. W e

see

tape.

[]

N o w we will

p r o v e the m a i n

r e s u l t of this

section.

i0. T h e o r e m :

R(T,T)

has

CR iff the T u r i n g

not s t o p w i t h

Proof:

T with

input T does

tape.

We h a v e to show: T(T)

I)

blank

machine

If T(T) aus {B}

stops with blanc tape

stops

with

w e h a v e by

blanc a:=

tape

iff

then

[ulVl]

M

R M has not WCR.

for

Ul,Vl,U2,V 2

= [u2V2]

~[UlblVl ]

a~ ~

~ ~ [

some

~b2v2l

and

168 Hence

2)

We

RM does

assume

will

Let

that

T(T)

use Theorem

ll~r I,

Then and

not h a v e WCR.

does

5 in o r d e r

12~r 2 £ R ~

11

blanc

t h a t R M has

tape.

We

WCR.

U l l l V 1 ~ii--12-~ u 2 1 2 v 2 .

Ul,U2,Vl,V2E(B)

by L e m m a

7

cases:

= .

U l V 1 = u 2 v 2,

w:=

blUlVl

= blU2V 2

[UlllVl]

M~

[UlblVl]

= [blUlVl]

= [w]

[u212v2]

M~

[U2blV2]

= [blU2V2]

= [w]

and

11 = 12 = . This

(III.)

to s h o w

and

the f o l l o w i n g

= 12

Then

(II.)

with

8.

We consider

(I.)

stop

and

ll,12E{,} Lemma

not

c a n be t r e a t e d

like

(I.).

11 • 12 Without

loss of g e n e r a l i t y 11 =

Then It

we

have

follows

blanc

tape.

and

Lemma

Hence

this

assume:

12 = < q h B >

ulV 1 from

we m a y

~ 9,

case

u 2 < q n B > V 2. that is

T(~)

stops

with

impossible.

[]

Ii.

Theorem:

CR

Proof:

is u n d e c i d a b l e

This

for

(f+b)-systems.

is an i m m e d i a t e

consequence

of T h e o r e m

10.

169

Additional

The

remarks:

undecidability

comes

from

the

of

fact

the

that

CR-property

finite

ordinary

systems;

in g e n e r a l

a finite

test

for

As

there

are

CR-test the

also

can

narrow

be

here

WCR.

in t h e

sufficient

carried

derivations

out

one case

criteria

successfully.

form

a

(f+b)-systems

(f÷b)-systems

infinite

set

for

regular

correspond

cannot of

An (or

restrict

regular

which

example

to

systems

ensure

even

to

that

a

is w h e r e

finite)

set.

IV.

Aut-~ta

IV.I.

General Aspects

In t h i s Z be

We

chapter

the

In

chapter

and

a

Obviously Irr(R)

consider

left

III.l.2.

is

regular

we

groups

system

set

and

semigroups;

let

R of

reductions

is one

~.

observed

regular.

its

only

alphabet.

is a r e g u l a r

is a g a i n

Irr(R)

their

we

that

Left(R)

Irr(R)

Reductions

underlying

recall

where

and

that

In f a c t ,

we

for have

such

R

the

Red(R)

set

= Z .~.Z

complement.

Red(R)

is

is c l o s e d

closed

under

corresponding

under

taking

automata

forming

subwords. we

need

superwords

In o r d e r

some

graph

to

and study

theoretic

notions. Suppose

A=(Q,Z,6,qo,F)

is

a

finite

deterministic

automaton.

I. D e f i n i t i o n :

(i)

The and

word

~raph

labelled

(ii)

the

vertices

(b)

for

p,q ~ Q

For

E = F(A)

wEL(F)

iff

w

is

the

following

directed

graph:

(a)

labelled

F(A)

by

the

of

F(A)

are

\ F there a E Z iff

language

labels

a path

the

states

is an

edge

6(p,a)

= q.

L(ff) e Z in F(A)

in Q from

p to q

is d e f i n e d starting

\ F;

by

in qo"

171

The closure A

which

property

accepts

particular,

of Red(R)

Red(R)

for

each

at q o w i t h the s a m e

now

that

path

in

implies

Irr(R) £(A)

for an a u t o m a t o n

= L(F(A))

there

holds.

is a p a t h

In

starting

labels.

2. D e f i n i t i o n :

If

AO

£(R)

is

the

minimal

= F(A o) is the

Somewhat

word

the g r a p h

£(A)

determines

path

in F(A) c o r r e s p o n d s

T(A o)

is c a l l e d

ciate

to e a c h

T(A o)

the

a

labelled

word

tree

the

word

as

£(A o)

is

then

call for o t h e r A a c c e p g r a p h for R.

tree

to a path

node

Red(R)

g r a p h for R.

F(A) a l s o a w o r d

Each

this

accepting

inconsistently we sometimes

t i n g Red(R)

to

automaton

for

T(A)

such

that

in T(A) a n d v i c e R.

In T(A o)

corresponding

we

can

each

versa. asso-

to a p a t h f r o m qo

node.

as

well

A o and t h e r e f o r e

also

completely

r(A o) is h o w e v e r

determined

by

Irr(R).

not d e t e r m i n e d

by the

s e m i g r o u p d e s c r i b e d by R~ As w e w i l l

see b e l o w

complete

by the c o m p l e t i o n

algo-

r i t h m f r o m the s a m e g r o u p p r e s e n t a t i o n u s i n g d i f f e r e n t

orde-

rings

systems

for

Red(R')

are

The cycle tant is:

which

role

not

R a n d R' g e n e r a t e d

the

automata

accepting

Red(R)

and

isomorphic.

structure for

minimal

t h e r e are t w o

the

of

the

decision

word

graph

problems.

F will A

p l a y an i m p o r -

simple

observation

172

3. P r o p o s i t i o n :

The f o l l o w i n g

The word

if

Irr(R)

(ii)

The

{iii)

The w o r d tree

elements R

pondence whole

{i)

g r a p h and the

reducible table

are e q u i v a l e n t :

was

to

word

word

but

algebraic

finite. graph

tree

also

complete

the

is

has

is finite.

represent

a part

structure

not o n l y the ir-

of the

(i.e. Irr(R)

semigroup

no cycles.

elements).

is

multiplication

in o n e - o n e

For

finite

can be r e p r e s e n t e d

corres-

Irr(R)

the

by a graph:

4. D e f i n i t i o n :

The

reduction

which

G(R)

has in a d d i t i o n

following

is an

ponding

to

node

the

word

labelled

ding

to

directly

are

R

is

the

labelled

graph

to the n o d e s and e d g e s of T(A o) the

end

edge

we

for

l a b e l l e d edges:

If p

Next

g_~h

with

T(A o)

w

and

a from

(a leave

node)

a e Z then p to

the

corres-

there

node

is an

correspon-

Irred(wa,R).

interested

from

of

the

in w a y s

reduction

to construct

system

which

we

these assume

graphs is

left

regular. The s i m p l e s t ting

the

accepting

way

left

is to c o n s t r u c t

sides

Red(R).

I i = ail...aik i,

of

R)

In c a s e

is f i n i t e

s i t i o n g r a p h of A:

a

(from an a u t o m a t o n

non-deterministic

where this

R

= {i i ~ ril

leads to t h e

B accep-

automaton

A

1 ~ i ~ m},

following

tran-

173

a12

..

am~

O

am2

The

transformation

yields This

the

will

We

recall if

now

for

lent

finitely Now is

many

given

fixes such

we

v • T

was

let

that

uD(L)

Q

2)

if

is

have

automaton

however.

Therefore

T

uD(T)

= wD(T)

and

if

of

a E

v

v

by

a;

proper

W.l.o.g.

directed

nodes

Z,

to

be

the

set Q of

each

for

each

non-redundant

= Left(R)

equivalent labelled

words

ua

e Q,

of

non

we

can

and

u the

w

are

are

regular

T.

set R of of

left

left

only

reductions sides

of

proper is s o m e

assume

1 e

graph

R.

pre-

u there

labelled

set

equiva-

there

equivalent

prefix

veQ

Q.

G(L):

G(L);

a proper then

set

a regular

a word

u and

= vD(L).

u E Q,

for

e T};

for

set

E T;

of

= {vluv

following

the

uv

prefix

uD(T)

a maximal

L; h e n c e

I)

minimal

method.

is a p r o p e r

regular L

of

the

refined

non-equivalent

choose

define

u

to

a left

and

and

inefficient,

a more

1 we

respect

suppose

Next

We

of

with

rather

a word

some

derivation

is

describe

that

a deterministic

graph.

construction

we

T

word

to

prefix

there

is

an

of

L and

edge

from

ua

is

u to

174

3)

if

u E Q,

prefix

of

L,

then

obtained

from

right;

is

v

an edge

For

a E Z and

some

w

let

ua

by

E Irr(R) v be

u to

we

w

put

to

but

the

omitting

equivalent

from

~ Q

ua

some

labelled

ua

not

a proper

first

proper

letters

from

w

~ Q and

prefix left

we

to

require

by a.

Q' = Q u { w }

and

add

the

following

edges : a)

For

each

edge For

b)

This the

5.

w

labelled

an

Q and

and

two

by

edge

of

the

such

f r o m w to w

G'(L).

automaton

final

that

ua

~ Red(R)

an

a.

graph

an

automata

state

a E Z

extended

graph

state

define

~

a E Z an

transition

initial

we

labelled

In

order

have

to

b y a.

to

obtain

specify

the

states.

A 1 and

A2

which

have

both

IEQ' as

by p u t t i n g

(i)

the

set

of

final

states

of A 1 is

(ii)

the

s e t of

final

states

of A 2 is Q.

{w};

Proposition:

A 1 is

(i) (ii)

Proof: and

minimal

automaton

accepting

Red(R);

A 2 is t h e m i n i m a l

automaton

accepting

Irr(R);

G(L)

(iii)

We

each

defines

initial We

u to

u

It

the

is t h e w o r d

is

easy

to

graph

verify

for

R.

the

equivalence

= uD(L)

implies

vaD(L)

= uaD(L)-

E Irr(R)

which

is

of

(i),

(ii)

(iii). will

First Second proper

prove

we we

(i).

observe notice

prefix

of

that that L we

vD(L) for have

each avD(L)

av

= vD(L).

not

a

t75

Using If

induction

the

then

word

vD(L)

For

uaD(L)

v

is

length

the

of the p a t h

label

of

the

this

path

yields:

from

1

to

uEQ

= uD(L).

Furthermore dant.

on the

each

uEQ

is i r r e d u c i b l e

irreducible

= {i).

This

u and

implies

because

a E Z we

for

a

R was

have

word

w

ua

non-redun-

E Red(R)

labelled

from

iff 1 to

u: wa E Irr(R)

Therefore this

is

the the

invariant

language

automaton

equivalence

iff

ua e Irr(R).

accepted

by

corresponding relation

of

A1 to

is the

Red(B)

it

Red(R).

Because

canonical is

the

right

minimal

one.

[]

We

will

illustrate

the c o m p l e t e

by

some

examples.

First

we

consider

system

R

The

this

nondeterministic

°

=

{B 3 ~

i,

automaton

BA

~

AB,

A 4 ~

accepting

i)o

Red(R)

is

....

B

I J. u

-QL

,

v

A

which

gives

as t h e m i n i m a l

A

deterministic

,oY automaton

for Red(R)

176

A

A

/ B

The w o r d g r a p h

B

B /

is:

F r o m t h i s we get f i n a l l y the w o r d t r e e and r e d u c t i o n

graph:

177

/ /

(9

/

I

/"

1"" "~@/ / \

~

@

@\

/

/ J

\

~"

.@

"v"

/

\

[

\ \\ \ /

B/

B

\ /B

I

<~;>,,

. f---~©

A\ \

/

\ \

/

\

% %.

A',

%

\

The

dotted

in o r d e r The

arrows

to

proper

get

are

the

prefixes

those

arrows

reduction of the

added

to

the

word

graph

graph.

left

sides

of R are

Q = {i, B, B 2, A, A 2, A 3)

which the

are

mutually

minimal

graph.

non-equivalent.

automaton

accepting

This

leads

Red(R)

and

directly to

the

to

word

178

Since

the

complete

system

depends

on the ordering

used the

word graph also depends on the ordering.

The example

Fibonacci

in fact may obtain

group F(2,5) will

show that we

of the

two non-isomorphic graphs in this way. From

chapter

II.4.

we

recall

the

defining

equations

for

F(2,5) ABBABA = B The c o m p l e t e

and

BABABBA = A

system for the weights ~(A) = 5, ~(B) = 1 leads

to the word graph B

B

B

~

B-I

but the weights ~(A)

= ~(B)

B-I

B

A

Jm

B-I

B-I

= 1 induce the word graph

I A B -I

B

A B-I

B

A A

v

A-I B-I v

A-i

179

It is of graph

interest

and

the

to k n o w

word

on the s e m i g r o u p

tree

are

products:

with

word

then

the

obtained

by

connecting

reduction, be

left

Because

If

R

of

regular

systems

semigroups

G i for

l&i~n,

product

union

with

word

~(Gil

of

some

regular

is a n

the

subsemigroups.

left

disjoint nodes

operations

W e c o n s i d e r the f o r m a -

and

direct

left

This

the

new

and

l~i~n)

F(ai)'s

initial

u ~

v

is

is and

node.

one

new

R of R u {u ~ v} n e e d not to immediate

problem

consequence

for

finitely

of

the

presented

semigroups.

of

Irr(R)

c

Irr(R)

the

by c e r t a i n p a t h s

c a s e R_ is i t s e l f

If

is

the

of the w o r d

to a l g e b r a i c

complete

t h e n the c o m p l e t i o n

represented

a)

the

t h e i r initial

regular.

and

for

taking

undecidability groups

quotients

F ( A i) d e f i n i n g

graph

Quotients:

related

If R i are

graphs word

the c o n s t r u c t i o n s

under consideration.

t i o n of d i r e c t p r o d u c t s ,

Direct

how

Irr(R)

in the w o r d

left r e g u l a r

is

finite

irreducible

words

graph

of

R

are

r(A) of R°

In

w e observe:

then

the

automaton

B for

R

is a

m

subautomaton

of the a u t o m a t o n

is of c o u r s e b)

In g e n e r a l

c)

always

the

for R (take

cribing

a free

If t h e r e

for

in B (and is

e.g. R as t h e

u

some

labelling n>l

replaced the

simpler

structure

Subal~ebras:

then

by

n). T h e r e f o r e cycle

B for R has

G' a r e

words

trivial

If G' [ G

represented

by

a cycle

the

whole

a finite

formation

nodes

than

system

des-

in A such t h a t cycle

path,

is a

of q u o t i e n t s

subsemigroup

are for G' • G not c l o s e d

leads

words

under

in

on to a

automata.

of G a n d

s y s t e m R t h e n the

certain

vanishes

depending

in the c o r r e s p o n d i n g

g i v e n by a left r e g u l a r c o m p l e t e of

more

monoid).

is a w o r d

u n E Red(R)

(and in t h i s c a s e

left regular).

automaton

the one

A for R

elements

Irr(R);

the f o r m a t i o n

G is

these of sub-

180 words all

(otherwise

of G).

denotes exists)

If for

a

word

G'

contain

the g e n e r a t o r s

u e Irr(R)

and

then

word

a cycle

the

labelled

by

some

and hence

n ~ 1 the

graph

for

word

un

(if

it

G'

u n.

remarks:

construction

given

in

given

to s y s t e m s

be f o u n d

would

some in

contains

Additional

The

they

[Gil

in

of

79]

[Ho

the

word

although

graph

there

of r e d u c t i o n s . 83].

using

was

A more

prefixes

no e x p l i c i t detailed

was

first

connection

analysis

can

181

IV,2.

For

The Complexity of Reduction A l g o r i t ~ s

a finite

over

a finite

Irred(w,R) In

complete

this

this

reduction

alphabet

is

Z and

system a word

R w

= {]i ~ ri over

~ l~i~n}

E the

word

in

unique.

section

we

deal

with

the

complexity

of c o m p u t i n g

word.

If IRR(R)

is finite,

the

of w).

length

from

left

to

root.

The

reduction

of the

final

can

done

be

This

method

reduction deals a word

node

is a s i m p l e

the

through

graph

word

the

Irred(w,R).

w and

go

reduction

is d e s c r i b e d

is

linear

algorithm

letter

graph,

in c h a p t e r

to

letter

starting

IV.l..

It is e a s y

by

(in

see

The

at

label

that

this

in O ( l w I) steps.

will

graph

with

We take

right

the

there

not is

this

work

for

infinite,

case.

is i r r e d u c i b l e

infinite

too.

or

rest

of

because

the

this

section

we want

to d e c i d e

whether

be a c o m p l e t e

reduction

F i r s t of all

The

Irr(R)

not.

i. D e f i n i t i o n :

Let

R

=

rev(w)

rev(abc) The with

{i i ~

ril

denotes =

search

l~i~n)

the

reversal

of

the

word

w,

for

system. example

cba. tree

a common

of

root

R and

consists r i as

of

leaves.

the

paths

of

r e v ( l i)

182 We

will

plete

illustrate

this

example

and

consider

the

com-

system R

The

by an

search

tree

for

=

{B 3 ~

i,

BA

~

AB,

A 4 ~

i}.

R is

A

A

/

I

~

B

A

The

search

can

decide

tree in

irreducible;

For no

w

w to

~

left

O(lwl) the

m a x ( l l i l ) . l w I. reduce

f o r R is u n i q u e

If

w

steps

side

over

T.

This

explains

and

a

given

complexity

is

with

reducible the

a E T we

1 ~ {lil

the

whether

R is c o m p l e t e .

worst-case

w' = v r i u s i n g

IRR(R)

because

following

search

have

l
w'

with

word

w

is

= vr i,

So w e

E Z

given then

we

is by can

tree.

wa

E wa

algorithm.

IRR(R) = xl,

iff

there

x being

is

a word

183 2. R e d u c t i o n

Input

Algorithm:

of

system

the

R

algorithm

= {I i ~ ril

is

a finite

l~i~n}

and

complete

a word

w

reduction

= WlW2...Wm_lW m

.

from

Z .

Output

First

of

the

of all f r o m

tialization, of t w o

$~

algorithm

We

stacks

At t h e

Let

and

pointer pointer" tion

of

Start

left

the the

and left

is the

be

for the

the

stack

R

for

describe

to d e c i d e

beginning

L

generate

the

the

whether

word

pointers

"working left

top"

stack)

Irred(w,R).

search-tree

array

a stack

s 2 the stack

word

algorithm

in a o n e - d i m e n s i o n a l

the

of t h e r i g h t

we

now

is a m a r k

sI describes

R

is the u n i q u e

right

for

to

simulation

letters.

is e m p t y

or not.

a n d the c o n t e n t s

w I at the top.

the

(that

of

ini-

stack.

is e m p t y with

by

for

top

means

remember

position, M

M

is a " m e m o r y

the

last

posi-

L pointer.

situation:

w11w2 f

f

L

R

wolq

M

I n the

sequel

we c a n

describe

the

stacks

as f o l l o w s :

s2

I: : Isl I

I $ f

f

t

L

M

R

a

184

At

the

beginning

sI

following

situation

Situation

i:

(u e

is

empty

and

we

consider

the

Irr(R))

u

I I $

•

.

I I -I

o

,..............X..

•

.

.

I I

y

t

t

L

R

z

................. I

" , ",, "

I $

M

If

s2

is

empty,

then

the

algorithm

terminates

and

{u}=Irred(w,R).

If

s2

we

get

is

not

directly

mation

READ,

to

top

the

empty

Situation

then

from

because

of

we

consider

situation the

top

i.

situation

We

letter

call of

2,

this

s2

is

which

transfor-

transferred

s I.

2: v

l~I

i Ixl~l

Izl

t

t

L

R

...... I~I

M

Now

we

test

whether

{ r e v ( l i ) I l~i~n}.

there

is

some

can

be

done

This

therefore

we

will

call

~)

e

IRR(R)

we

If

ny

position, ~)

If

ny

e

and {ill

we

this

set get

l~i~n}

v

for

using

operation

the a

then

we

the

rev(v)

search

e

tree,

SEARCH.

pointers

situation

which

of

consider

L and type

M

to

the

i.

situation

3.

y

t85 Situation

3: 1i iii

I I I

I'

a

From

situation

of t y p e

i,

we

,b

3 we call

I I z

...... i¢i

t

t

t

L

M

R

get t h e

following

situation

the t r a n s f o r m a t i o n

t

t

B

R

which

is

REDUCE,

M

The

operations

are

repeated

until

the

algorithm

termi-

nates. It

is

easy

finite tion

to

number

see of

that steps,

the

algorithm

therefore

terminates

the

following

after

a

proposi-

holds.

3. p r o p o s i t i o n :

Let R be a f i n i t e alphabet word

It

in

remains

algorithm. special There not

Z.

The

reduction

system

reduction

algorithm

computes

Irred(w,R)

to

for

determine

We will

every

the

over a finite the

unique

w ~ Z .

complexity

of

see t h a t t h e c o m p l e x i t y

the

reduction

is l i n e a r

in s o m e

cases.

also exist

linear

plexity

complete

but

simple

examples

polynomial,

is e x p o n e n t i a l

for w h i c h

Unfortunately

as w e

will

see

the c o m p l e x i t y

in g e n e r a l

later.

is

the c o m -

186 4. P r o p o s i t i o n ;

Let

R

over

= {I i ~ the

The

alphabet

reduction

in O ( ] w [ )

Proof:

are

each

after

&

at

SEARCH

step

because by

tested

T(w)

have

> [ril the

and

lli[

are

each

for

unique

system

all

i.

Irred(w,R)

r:=

max{]ri[ ] l~i~n}.

at

a READ most

application

r letters. if t h e

because

for

all

> ]ri[.

There

in at m o s t

word

step

lw]+ of

An u is

and

r. lw ] R E A D -

REDUCE

upper

there-

the

bound

irreducible,

stack for

a

this

1 steps.

number

= TREDUCE(W)

(i+

reduction

w e Z .

follows

is g i v e n

r. Iw]

[lil

there

most

the

complete

REDUCE-operations

for at

if T d e n o t e s

=

a

computes

every

lw]

step.

SEARCH-operation

Hence,

which

r i we

REDUCE

s 2 increases

be

Z for

for

most

ii ~

operations

can

be

i: = max{ [lil I l ~ i ~ n }

reductions

After

l~i~n}

algorithm

steps

Let

There

ri]

of

letter-operations

+ TREAD(W)

we

have

+ TSEARCH(W)

+ ([w[+r'lw ] ) + l(]wl+r'Iw])

2-r+l+l'r)lwl

e

O(lwl).

[]

5. D e f i n i t i o n :

A

system

is c a l l e d

R

= {I i ~ monadic.

ril

l
with

r i E Xu{e}

for

all

i

187

The only possible reduction are

at

system

most

complete.

length preserving are of t h e f o r m

IZI-I

From

b ~ a

reductions

this

we

get

reductions

of

with

this

directly

for a m o n a d i c a,b E Z. T h e r e

form

the

because

following

R

is

corolla-

ries.

6. C o r o l l a r y :

Let R be a f i n i t e the

alphabet

unique

word

Z. in

complete The

monadic

reduction

Irred(w,R)

in

reduction

algorithm

system

over

computes

the

O([w I) steps

for

every

,

w E ~ .

7. Cot_ollary:

Let

R

over

= {i i ~ the

ri~

alphabet

or r i E Xu{e} unique

l~i~n]

word

Z.

then in

be

If the

a complete

for

all

i

reduction

Irred(w,R)

we

reduction

system

have

>

algorithm

in O ( l w I )

lli~

Iril

computes

steps

for

the

every

.

w ~ X .

8. E x a m p l e : over

We

Z = {a,b)

Hence

IRR(~)

If w e

consider which

the

reduction

system

R

= (ba ~ ab]

is c o m p l e t e .

= (aibJ I i~9 E H).

reduce the word

w = b n a n,

n~l,

we must apply exactly

n 2 = I w 1 2 / 4 E O(lwl 2) r e d u c t i o n s

to get t h e i r r e d u c i b l e w o r d

Irred(w,R)

there

= anb n.

the p o s i t i o n

Next

we

is not

on w h i c h

consider always

In

this

is

no

restriction

to

w e a p p l y the reduction.

a more

complex

a polynomial

with a finite complete

case

example

algorithm

reduction

to

show

that

to r e d u c e a w o r d

system.

there w~X

188

9. E x a m p l e : order It

We

system

we

l's

followed

reduce

reductions

the

i:

0*

~

*I

2:

i*

~

0#

3:

#0

~

0#

4:

#i

~

i#

5:

#S

~

*$

6:

##

~

#

7:

#*

~

*

by

a

every

is

applicable,

=

the

of

in*$

we

Z

= {*,0,i,#,$}

with

set forms

a complete

reduc-

e Z*, a

$.

need

in*$

f(n)

to

denotes

denotes

get

a word

the

from

in*$

the

above

of

number the

n of

word

*ins.

reduction

words

following

* and

which

At

we

that

a word

irred(in*$,R)

count

alphabet

R.

Now

for

the

* < 0 < 1 < # < $.

is o b v i o u s

tion

consider

step

because the

number

form of

exactly the

one

of

reduction

in*$.

This

reductions

process

is

the

which

we

is

reductions unambiguous

reason need.

why

we

can

In p a r t i c u l a r ,

have

We

stress

is

also

the

fact

= 2n+2

that

the

non-redundant.

complete

systems

that

the

in

f(n)

are

group is

complete

Exponential

somewhat case

nential

system

known.

further

applications.

- n - 4.

no

system examples

easier complete,

We

in

finally

to get.

this of

We

redundant

also

non-redundant give

two

example

remark expo-

examples

of

189

10. E x a m p l e : over

the

Then

Let

R

alphabet

there

is

be

a

finite

complete

reduction

system

Z.

a

simple

algorithm

for

solving

the

word

problem, For

Wl,W 2 E

Z

u2=Irred(w2,R) equality. order

So

of

we and

first

then

the

we

word

complexity

compute

Ul=Irred(Wl,R)

test

words

the

problem is

the

can

same

be

as

and

u I and

solved,

for

the

u 2 for and

the

reduction

algorithm.

ii. E x a m p l e : over

the

Let

alphabet

If R d e s c r i b e s we w

can E

the

Irred(wv,R)

Let

w=

and

W--I

which of

we

be

finite

complete

reduction

system

X.

a -I

representation

• Z then

we

word

E

unique =

a

a group

identify

Z

R

v

can

and compute

IRR(R)

for

if

for

for which

all

a&Z

every

word

we

have

io

WlW2...Wn_lWn,

wiEZ , then we have Irred(ww-l,R) = 1 -i = wn-lWn_l-l...w2-1Wl . So w e h a v e v = I r r e d ( w - l , R ) can

complexity

compute is a g a i n

by

the

the

reduction

same.

algorithm.

The

order

190

IV.3.

T h e C y c l e S t r u c t u r e and the G r o w t h F u n c t i o n

In c h a p t e r

IV.I.

left-regular the

we

complete

non-existence

section graph

we will

in

more

contains.

In

aspect

of

G:

words

will

words

may

Suppose length

system

investigate detail

some

lead

for a s e m i g r o u p

finiteness in t h e

word

the c y c l e which

sense

the

cardinality

to

new

denote

terminology

the

which

is

the

of the

an

element.

allows

to

by

We

word

about

G

it

asymptotic

formation

in G a l t h o u g h

same

a

E. In t h i s

information

that

elements

graph

structure

see

of G m e a n s

G with

is c h a r a c t e r i z e d

and

Infinity

a

that

of a c y c l e

sometimes

introduce elements

observed

of

new

different will

express

first

how

many

w e get by a g i v e n n u m b e r of words.

G is a s e m i g r o u p

generated

iT(g) of g e G w i t h

respect

by a finite

set

T. T h e

to T is d e f i n e d as usual.

i. D e f i n i t i o n :

The g r o w t h defined

function

analogous

algebras.

R ~

~

of G w i t h

respect

to T is

by: YT(n):=

An

YT:

card({g

definition

can

E G~

IT(g)

~ n})

be

given

for

more

general

191

2.

Examples:

a)

The

group

growth

integers

function

For

b)

of

the

~T(n)

free

= 2n

abelian

get

For

the

free

monoid

By

definition,

but of

on

~ turn

Let

3.

the

X:

the

~ ~ H

by

T = {i}

has

the

generated

by

m

elements

we

21 • (m p). (pn) •

m generators

we obtain

~ m n.

growth

generating out

groups

over

~T(n)

generated

+ i.

m = X p=0

YT(n)

c)

{Z,+)

function

set

T

too.

to b e

independent

be a n y

function.

~ depends Certain

of T,

not

only

important

on

G

aspects

however.

Definition:

(i)

¥

is

and of (ii)

¥

in

we

general

n ~ n o,

no

adding

a

all ~,

is

all

Because

Ro!znomial

iff

n ~ ~.

denoted

The by

are

for

some

a and

all

head

coefficient

iff

interested

some

d is c a l l e d

¥(n)

~ c n,

fixed

certain

integer. summand;

n~n o n.

in a s y m p t o t i c

that

constant

¥{n)~c-n d

such

for

c,d the

e

degree

for

some

c>l

and

we

will

~°

require some

least

~ c-n d

deg(¥).

exponential n e

¥(n)

In of

we

could

such

only

inequalities This

in

e.g.

instead

ask

for

a polynomial

a dominating

aspects

hold

general of

for

all

amounts

to

requiring

¥(n)

~

c.n d

case

c

is

polynomial.

+

a

for

called

the

192 We n o w need a few basic

facts

about these

concepts.

4. P r o p o s i t i o n :

Suppose group

two

finite

sets

S and

T generate

G and

assume

bl-nP ~ ~T(n) ~ b2.nq

the for

same

semi-

0~p~q

and

b l , b 2 > 0. T h e n t h e r e are s o m e Cl,C 2 > 0 such that cl.nP

we

Proof:

choose

k and

~ Ys(n)

~ c2.nq

This

implies

Hence

On

~s(n)

the c h o i c e

the

other

we

¥s(in)

Putting

d = bl.l-P

Ys(n) for s u i t a b l e

~

YT(kn)

~

for all

xES

is(Y) ~ 1

for all

yET.

b2"kqnq.

of c 2 = b2.kq will

hand

IT(X) ~ k

1 such that and

holds.

work.

have ~

and

YT(n)

>

n = a.l

~s(a-l)

bl.nP

+ b,

~

= (bl.l-P).(in)P.

O
we get

d-(a01) p= d . ( n - b ) P

~ c I.

c I.

[]

5. Corollary: The

properties

independent the

degree

semigroup.

of

being

polynomial

of the g e n e r a t i n g of the

growth

set.

function

resp.

exponential

In the p o l y n o m i a l is an i n v a r i a n t

are case

of the

193

H e n c e t h e r e are t h r e e p o s s i b i l i t i e s

a)

G is p o l y n o m i a l

b)

G is e x p o n e n t i a l

c)

G is p r o p e r l y

in b e t w e e n

exponential

We

have

seen

reported

in

examples

course,

passing

and the

b).

An

example

for

c) w a s

context this will

however.

quotients

polynomial;

a) a n d

in our c o m p u t a t i o n a l

t u r n to be i m p o s s i b l e ,

Of

the p o l y n o m i a l

case.

for

[vdD-Wi];

for G

of p o l y n o m i a l to

the

semigroups

quotient

does

not

are

again

increase

the

degree. The

situation

involved.

with

We will

respect

consider

to

subgroups

is a

little

more

the g r o u p case.

6. P r o p o s i t i o n :

Suppose subgroup

(i)

G

U. T h e n

The index

(ii)

Proof:

(i) ~

take

modulo

there

[G:U]

group

generated

degree

(ii):

a finite U with

For e a c h

by

S with

a

are e q u i v a l e n t :

is f i n i t e

U is a f i n i t e l y g e n e r a t e d p o l y n o m i a l same

We

is a p o l y n o m i a l

group with

the

as G.

We set

start R of

with

some

standard

representatives

of

arguments. the

cosets

1 e R.

g e G w e h a v e g = g .Ug,

g

e R,

Ug e U;

further-

more (I) (2)

ugl...g n = ugl(g2...gn)*. Ug-I r = (Ug(g-i r ),)-I

o..'U

gn-lgn

*"

Ugn

and holds.

194

From

this

form

U s r , s e S u S -1,

Hence

we

see

that

u is f i n i t e l y

assertion

is a l w a y s

(ii) ~

c>O,

of t h e

by T = {Usr ~ s e S,

[G:U]

from

r e R} a n d

holds.

• YT(n)

the

fact

that

d e g ( ¥ T) ~ deg(¥ s)

true.

(i):

Because

~

follows

of e l e m e n t s

r E R.

generated

¥s(n)

The

u e U is a p r o d u c t

We

of

take

Yx(n)

no e ~

finite

~ ~y(n)

generating

and

the

sets

xcy

assumption

of

there

U resp. are

G.

some

such that yy(n)

Furthermore

for

all

/ ~x(n)

£>0,

~

c

for all

k e N there

are

n ~ no .

infinitely

many

m

such that (,)

Yx(k+m)

/ ~x(m)

~

i+~

Otherwise

~x(k+m)

/ Yx(m)

>

l+c

Yx(r)

/ Yx(N)

~

(i+~) 1

large

f o r all

sufficiently

n.

Hence I =

holds~

[r-N/k],

hence

for

If t h e r e that

which

G,

gl ..... gt

e G,

ly(g i) < k f o r

t

Now

(*)

and

finally

an

exponential

growth

fixed

N and

for

U and

too.

are

each

gives

for

<

~y(k+m)

implies

gi

each

/ Yx(m)

t < (l+e)-c

[G:U]

~ gj

mod

i, t h e n

<

for

U

for

for m>no:

c'(~x(k+m)

each

i • j such

£>0,

/ ~x(m))

therefore

t ~ c

< c.

[]

If U h a s shows

an

that

infinite the degree

index

in G t h e n

strictly

the

decreases.

next

proposition

195

~° P r o p o s i t i o n :

Suppose

G

infinite.

Proof: and that

We

is

polynomial

Then

take

U has

finite

representatives

r~R

ly(r) is m i n i m a l

T h e n we h a v e

Because

from

the

sets left

~ n}

Un

:= {g E U~

K n}

Rn

:= G n N R.

is

u

infinite

[G:U]

is

cosets

Ix(u)

we

get

modulo

G

U such

w e p u t for nEN:

card(R k

l~k~n)

u R0.U n

\ Rk_ I) > i;

from

n ~ ~xlk). k=0

>

then

Xx(n)

holds.

and

x c y for U resp.

{(R k \ R k _ l ) . U n _ k l

Yxln)

deg(~x)=m,

Yx(n)w ~ c.(~)

generating

:= {g e G I ly(g)

D

d+l

d e g r e e ~ d.

in rU. F u r t h e r m o r e

this we obtain

If

degree

Gn

Gn

[G:U]

with

This

~ c-n m

for

some

c>0,

therefore

gives n

~y(n)

and

the

right

degree m+l

hand

>

c • • i=0

term

which proves

is

(~);

essentially

a polynomial

of

the a s s e r t i o n .

[]

T h e d e f i n i t i o n of the g r o w t h of

length.

ordering

Alternative]y ~

and

define

function

replacing

concepts

introduced

function

one in

length above.

an by

can

r e f e r r e d to the n o t i o n

refer

analogue weight

and

to

some

other

KB-

way

the

~-growth

all

the

remaining

196

The

fact

little

that

is

all

KB-orderings

expressed

underlying

in

the

differ

next

asymptotically

proposition.

very

Z denotes

the

belonging

to

alphabet°

8. Proposition:

Suppose the

X1 and

same

semigroup

and 42.

(ii)

put

Then

deg(¥ I)

Suppose

e I and

the

~2"

words

iff

= deg(¥2)

weight

iff

Now

if ¥I is b o u n d e d

hold

¥2 is b o u n d e d

and

in this

s.

belonging

> e(n)

e I (u)

=

k.602(u)

~2 (n)

<

¥i

to 41 and 42

~

p(k.n)

by a p o l y n o m i a l

therefore

~

is e x p o n e n t i a l .

p(k).p(n),

of the

for some e x p o n e n t i a l

X1

(k.n).

p then

~l(n)

symmetric

41

~2 is exponential.

by a p o l y n o m i a l

¥2(n)

and

KB-orderings

¥2 is p o l y n o m i a l

functions

u we have

therefore

~2(n)

two

resp..

and

If

and

k:= max(~l(a) I a e X).

for all

hence

functions

presentation

¥i is e x p o n e n t i a l

Proof:

growth

Then

case

We

the

¥i is p o l y n o m i a l

(i)

are

~2 are

function

a The

same degree.

e then

e(n/k ) rest

follows

by t h e

argument. []

197

Next

w e w a n t t o g e n e r a l i z e the n o t i o n of the g r o w t h

to a r b i t r a r y p a r t i a l plete

o r d e r i n g s 4. T h i s can be d o n e

(not n e c e s s a r i l y

is c o m p a t i b l e

a left

regular)

w i t h 4. In p a r t i c u l a r ,

a g i v e n p r e s e n t a t i o n the c o m p l e t i o n

system

this

function if a c o m -

exists

which

is the c a s e

algorithm

if for

continues

suc-

cessfully.

9. D e f i n i t i o n :

(i)

s u p p o s e R is a s y s t e m of r e d u c t i o n s . The

(ii)

R-growt h

~R(n)

:=

If

is

4

¥

= YR: ~ ~ ~

is d e f i n e d

by

c a r d ( { u e Irr(R) i l ( u ) ~ n } ) . a

partial

equations class

function

ordering

defining

of

words

4-minimal

word

a

and

semigroup

equivalent then

the

E G

such

modulo i-growth

is

a

set

that

E

of

each

contains

function

a

X = ¥~

is d e f i n e d by X~(n)

:=

card({u i u minimal

If R is a c o m p l e t e is a K B - o r d e r i n g one f r o m

above,

it f o l l o w s all

n.

Now

we

word

assume

graph

£. W e

r is a d i r e c t e d

l(u)~n}).

w i t h ~ t h e n YR = ~4" If

then

is c o m p a t i b l e

our

notation

with the original

the d e f i n i t i o n

that

[u] E,

system compatible

Compared

from

in

R

will

is

left

that

the

growth function

~ (n) ~ ¥(n) is true for

regular

and h e n c e

u s e F in o r d e r to c o m p u t e

graph with

with

a starting

possesses XR"

a

Suppose

node.

i0. D e f i n i t i o n :

(i)

£ is p o l y n o m i a l

iff no t w o

cycles

of

F have a node

in c o m m o n . (ii)

F is e_xponential

iff F is not p o l y n o m i a l .

t98

Exponential intersect

graphs

in s o m e

are

those

where

at

least

two

cycles

node~

11. D e f i n i t i o n :

A minimal

(i)

path

and v i s i t s

in

maximal For

(iii)

the

starting

n u m b e r of c y c l e s on a m i n i m a l

of

g r a p h the p a t h

minimal

paths

node

once.

g r a p h the c y c l e n u m b e r

a polynomial

number

from

e a c h n o d e at m o s t

For a p o l y n o m i a l

(ii)

F starts

which

c(F) is the

path.

number

p(F)

intersect

t

is the cycles,

t=c(F).

Note

that

vious

each path

determines

some

minimal

path

in an ob-

manner.

12. D e f i n i t i o n :

If the

F is l a b e l l e d label

then

a cycle

word

is a w o r d

which

is

of s o m e c y c l e of F.

13. P r o p o s i t i o n :

Suppose system

R

is

with

a

left

regular

(not n e c e s s a r i l y

complete)

w o r d g r a p h F.

Then:

(i)

X(R)

is

polynomial

c a s e deg(¥(R)) nomial

(ii)

¥(R)

iff

F is p o l y n o m i a l .

= c(F) and for

the h e a d

term

is e x p o n e n t i a l

some

In t h i s

dominating

is less or equal

to p(F).

iff F is e x p o n e n t i a l .

poly-

199

Proof: from

Assume the

first

starting

that

node

in

i)

The

minimal

path

2)

The

number

of

sects A word

w

where

on

= WlZlJl

F is u n i q u e l y M determined

runs

through

a path

w2z2J2

Wl,..Wk+ 1 is t h e

l ( z i ) = n i.

polynomial.

Then

each

determined

path

P

by

by P. each

cycle

which

inter-

M.

labelled

w

F is

Note

P can

°.. W k Z k J k

word

that

then

on

are

represented

as

Wk+ 1

M and

w2,,.w k

be

z i are

cycle

non-empty

words

because

with F

is

polynomial. With

k*l r =

we

Z i=l

get

Let

k E i=l

yM*(n)

P which 1(w)

~

~s(n)

(resp.

has

M

YM(n))

as

n).

Then

=

n Z

its

we

~M

ty

r = 0 we

~M n ~

N

number

- r.

of

words

on

some

path

path

*

s u c h t h a t l(w) = n (resp. n~r (n) ~ ( ) a n d for r • 0

(p~r)

N

(n-~+l)

one

cycle

~

(~)

N

n k.

p=l

have

exactly

and the

same

inequali-

holds.

The

number

maximal C

be t h e

have

(p)

niJ i = l(w)

minimal

p=l

For

l(w i )

= P(F)

Hence

c of

number be

¥(R)

the

minimal of

cycles

path

paths

intersecting

number

is p o l y n o m i a l

is f i n i t e ;

of

with

let

some

t

minimal

F.

degree

at

most

= c(F)

t.

be

the

path

and

200

On

the

other

hand

each

solution

for the

Ji of

k (*)

gives

rise

to

struction

of

shows

that

words

in

Choosing

Z i=l

some F

word

from

the

different

niJ i

w

in

~

s - r

L(F)

minimal

with

l(w)

automaton

solutions

of

(*)

< s. T h e

accepting

produce

conIrr(R)

different

L(F). m

= max(nil

sufficiently

large

n:

¥(n)

~

l~i~k)

¥(sm

and

s

+ r)

( (n-r)/m

Z

=

[(n-r)/m]

we

get

for

(~)

- 1

)...°.(

(n-r)/m

- k

)

> k'

( (n-r)/(m-k)

- I

)k

a.(n-b) k

for

some

a>0,

b>0

for

some

d>0.

> d.n k

Therefore

we

sufficiently

Now

we

two

words.

large.

assume

such that of

get

two

that

¥(n) Hence

let

D.n t

deg(¥)

with

= c(F)

F is e x p o n e n t i a l .

vertices

cycles;

~

start

w I and

t as

above

n

holds.

We choose

a node

at q w h i c h

are

w 2 be

corresponding

the

and

initial

q in F

segments cycle

201 For the

free

SGcL(F). has

semigroup

Therefore

exponential

Because

the

exhaustive

SG g e n e r a t e d

a subset

of

L(F)

by

w I and

and

hence

w 2 we have L(F)

itself

growth.

cases

of

T being

and e x c l u s i v e

polynomial

or

the p r o p o s i t i o n

exponential

are

is proved.

[]

Additional

The

general

cycle tion The

a

connection

structure 13 w a s

of

first

generalization

is of nate

remarks:

interest for

group

growth

the

function.

the g r o w t h

graph in

slower

completion In c h a p t e r than

as

[Gil

of the o r d i n a r y

if the

a

word

observed

KB-orderings. with

between

functions

described 79];

growth

ordinarily

we

the

in

Proposi-

see also

[Ho 82].

f u n c t i o n to a ¥(R)

algorithm IV.4.

and

does

not

termi-

will

investigate

growing

generalized

202

E f f e c t i v e A s p e c t s of G r o m o v ' s Theo,rem.,

IV.4.

It is a n a t u r a l with a first

q u e s t i o n to c h a r a c t e r i z e

polynomial

the c l a s s of g r o u p s

g r o w t h f u n c t i o n by a l g e b r a i c methods.

s t e p in t h i s d i r e c t i o n

w a s d o n e by J. A. W o l f

in

The [Wo].

i. T h e o r e m :

A

finitely

growth

generated

nilpotent

group

has

a polynomial

function.

This

implies

that also finite extensions

grow

polynomially.

also

the

converse

the c e l e b r a t e d

J. M i l n o r in true.

of n i l p o t e n t g r o u p s

conjectured

The

proof

of

in

[Mi 68]

this

that

conjecture

is

r e s u l t of M. G r o m o v :

2. T h e o r e m :

Let G be a f i n i t e l y g e n e r a t e d mial

growth

finite

The

proof

function

iff

G

group. has

a

T h e n G has a p o l y n o -

nilpotent

subgroup

of

index.

uses

deep

analytic

methods

(e.g.

Hilbert's

5 th

problem). There

is a s t r o n g

in p a r t i c u l a r under

in

additional

interest

in s i m p l i f i c a t i o n s

removing

its

assumptions).

of t h i s proof,

nonconstructive Also

one

wants

parts, to

find

(even the

203

nilpotent to

subgroup

exhibit

the

the

growth

proved

by

"explicitly",

significance

function H.

Bass

for

in

of

the

to

determine

its

index

and

the

numerical

magnitudes

of

group.

An

important

step

was

[Ba 75]:

3. T h e o r e m :

Suppose

and

the

rk

nilpotent

group

has

the

lower

G

~

D rn(G)

= {e}

is

= FI(G)

the

rk(G)/rk+l(G).

rank

Then

degree

elementary

been

of

G has

the

free

a polynomial

series

abelian

group

in

growth

function

of

n-i d =

An

...

central

given

~ k=l

proof

k.r k .

for

in

[vdD-Wi]°

an

infinite

the

case

of

linear

growth

has

also

4. T h e o r e m :

If

G

is

function then

G

For

the

the

reader

be

The

~ has

group

recalled

e.g.

some

that

X(n)

subgroup

theoretic to

here,

commutator

groups

such

finitely

[Ka

generated - X(n-l)

U

~ Z

concepts

- Me];

some

group

with

growth

~ n for

some

n ~ ~,

satisfying

mentioned of

the

[G:U]

above

main

~ n4/2.

we

refer

notions

will

and

sub-

however.

of

U,V c_ G t h e

a,b

E G

notation

is

[a,b] [U,V]

= a-lb-lab will

mean

the

for

group

gene-

204

rated of

by

G

all

[a,b]

with

Fk+I(G)

potent

iff

will

word

graph

system

in

Fn(G)

The

with

lower

central

series

...

of

some

G/Fk+I(G)

and

G

is

nil

with

the

n e ~.

of

Gromov's

theorem

group.

section

ordering

the

for

aspects

respect

We c o n s i d e r

D

center

= {i}

the

this

term

D F2(G)

the

connect of

Throughout

G

on to

linear

is

the

finitely words

presented,

and

R~

"4"

= R~(~)

the

some limit

"4"case.

Proposition:

If

R~

is

linear

left

regular

the

word

the

(ii)

and

Proof:

(i)

subgroup

U c G of

G

cyclic

of and

Let

finite

graph

cycles

[G:U]

¢

and

the

R~-growth

function

is

then

(i)

U

bEV.

[Fk(G),G].

is

now

partial

= FI(G)

=

Fk(G)/Fk+I(G)

5.

a~U,

is: G

We

with

~

represent

the

label finite

index

generated

R ~ has

at

path

number

a cycle index. which

by u k for

most

a subgroup

(p(F)/2)-lu i, where

p(r)

u

F of

of

in F; Then

is

k.

is

cycles;

U ~ Z of

index

a cycle

word

F.

u generates

there

normal

some

u

two

is in

some G;

U

a cyclic subgroup is

again

205

Now

suppose

vnEu

for in

1>0

l<0.

By

labels

some

either or

v

the

n~l,

i.e.

given

Proposition

We

growth

10(i)

want

IV°3.; not

to

this

to

be

additional large

of

apply

the

or

factor

the

chapter

p(F).n

that

cycle

is

of a

u

u -1,

such

that

generator

we

v

is

depending

on

for

d;

u

Proposition

the

finite

have

generator to

is

i. T h e r e f o r e

of

we

in t h e

Adding

lul

some

G/U

+ d

V

u

Because

IV.3.

corollary

case.

too.

vn=u I for

subgroup

requires the

of

<

the

cycle

cycle

~ ( R ~)

the

some

is 5 of

of

G

for

2n+l. chapter

which

generators

obtain

some

needs

gives

an

sufficiently

n

[G:U]

c(F)

+ d

2n

+ 1

c(r)

~

. lul

~

• ]u I . 2

[1

6.

Example:

In

the

graph

G

=
KB-orderings

with

is : b

b-1

(ab) 2 =

weights

(ab-l) 2 = i>

~(a)

> ~0(b)

= 1 the

word

206 There

are

word

minimal

b has

l e n g t h one.

The

estimate

iff

nZ8

from

we

assume H

group

always

is

index

of

(cf.

Gromov's

limit

has

the

of H are

G

has

free

a nilpotent

generated

a central subgroup

Hi/Hi+ 1 = Z

same

represented

cycle

structure

nilpotent

series

with

H of

finite

for

l
h a v e n cycles,

Malcev-basis

as

F itself

l a b e l l e d by c e r t a i n

u = Ultl(u)

elements

by a set of p a t h s

in

because

d e g r e e of growth.

E a c h u d e s c r i b i n g an e l e m e n t

The

hence

s y s t e m R ~ is left r e g u l a r w i t h the w o r d g r a p h F

a n d G h a v e the s a m e we

= 4.

group G with polynomial

(i.e. has

a torsion

cycle

= 2048.

finitely

... ~ Hn+ 1 = {i},

t h e n the e l e m e n t s

Hence

84/2

theorem;

polycyclic and has

~

A

8/2

the

b e c a u s e of ¥(n)-¥(n-l) ~ n

infinite

index.

~

and

[Ka - Me]):

= H1 D H2 D

F which

[G:U]

a cycle

[G:U]

4 gives

of an

finite

factors)

If the

Therefore

Theorem

case

subgroup

cyclic

intersecting

the i n e q u a l i t y

For the g e n e r a l growth

paths

denoted

of H and

the the

u i,

l~i~n.

in H is of t h e f o r m

untn(U)

by

words

ui

ti(u)

are

tuples

~ Z

traditionally (tl(u) .....tn(U))

called are

a

the

Malcev-coordinates.

The

word

table, the word

graph

namely

above

represents

for t h o s e

situation

graph directly,

a

part

of

the

multiplication

u , v ~ I r r ( R ~) for w h i c h

I r r e d ( u v , R ~) however:

can

be

uveIrr(R~).

computed

from

In the

207 7. T h e o r e m :

For

u = U l t l (u)

v = U l t l (v)

... u n t n ( U )

... U n t n ( V )

we h a v e I r r e d ( u v , R m)

t (uv) ... u n n

= Ultl(uv)

where ti(uv)

=

[a p o l y n o m i a l +

For

the

tial

proof

idea

we

is

commutators

the

coordinates

using

refer

to

the

case

of

+

to

[ui,ug] of

more

of

Q in

(tj(u),tj(v)lj
ti(v)

[Ka-Me],

determine

commutator

In t h e

ti(u)

over

Thm.

first

the

the

cycle

general

17,

25.

The

essen-

Malcev-coordinates words

products

and

in

to

these

of

express terms

by

identities.

quadratic

growth

we can

make

a more

specific

statement.

8. P r o p o s i t i o n :

If G is n i l p o t e n t for

k~2

Proof: that 9~k

The for

C =

[G,G]

finiteness

each

with

From

and

of q u a d r a t i c

k such

then

r I = 2, r k = 0

is finite.

of

the

that

r(Fj(G)/Fj+I(G))

the Bass formula

growth

lower

Fk(G) a

is

central infinite

series there

gives

is

some

i.

we obtain

d=2=rl+2r 2

which

rules

out

rl=l~ If

rl=2

then

otherwise

Now

assume

r2=0;

r3

=

rl=0.

in t h i s

case

r(F2(G)/F3(G))

Then

[G:C]

•

is

C has

to be

finite

because

0.

finite

and

hence

C

is

of

208 quadratic

growth.

r2

On

the

=

Comparing

r(C/[C,G])

other

hand

fore

C/[C,G]

is

group

which

= i,

[C,C] the

the growth

of G a n d C w e

r(C/[C,C])

c_ [C,G]

quotient

and of

is a c o n t r a d i c t i o n .

both

= r2(C)

are

C/[C,C]

Hence

get

= 2.

finite;

There-

by a f i n i t e

rl=0

is

sub-

impossible.

[]

9. P r o p o s i t i o n :

A group

G of

finite

quadratic

We

take

C

[H,H]

is

finite.

that

[H:H]

such

Therefore

has

an

abelian

subgroup

of

index.

Proof: =

growth

H c G,

[G:H]

On

the

< ®

other

and hand

< ~ a n d H_ is t o r s i o n

C n H = {I)

and

hence

H

nilpotent.

there

is

some

Then H

¢ H

free,

H is a b e l i a n .

[]

This

result

degree

is c e r t a i n l y

~ 3. W e

have,

not

true

however,

if t h e

the

growth

following

function

has

conjecture:

! 0 . Conjecture:

If

G

has

complete abelian

The

polynomial system

section

for

subgroup

conjecture which

is we

growth

of

some

finite

supported

also

think

and

KB-ordering,

a

left-regular

then

G

has

an

index.

by can

admits

be

the

results

extended.

of

the

next

I

I

23-

i

i

D

© 2r

2J"

12r

j-

i

~v

©

ir

i

O~

rQ ©

~.w

g~"

",~ V

t~

~r" II

I'q

" " II II 4=y

rqo-,

O.-..

oE2

p~

.="o

~<~

:D ~D

210

IV.5.

A

Relation

Completion

If

G

=

<X,R>

between

a

finitely

well-founded

term

cribes

some

infinitary

aspect

is

IV.3. s h o w s regular.

We

will

cannot

now be

This

use

to

the

aspect

of

function

such

the

group

in

certain

case

of

[a,b]

the

words

There

= a b a - l b -I d e n o t e s known

N = a n b m c k,

is in f a c t

system

Another

the

aspects

in o r d e r

free

We

"i"

some

R~(~)

des-

infinitary

material

these

cases.

and

in

chapter

in c a s e

to p r o v e

will

that

restrict

nilpotent

group

R~

R"

this

of

class

commutator

of a

has t h e p r e s e n t a t i o n

a n d b. It is w e l l the

G.

between

arguments

group

limit

and

G = < a , b I [a, [b,a]]

where

and the

Function

presented then

relations

regular

discussion two.

ordering

growth

some

is

Growth

Algorithm

is

the

the

that

=

as u s u a l

the

the elements

n , m , k E Z,

a finite

[b, [b,a]]

complete

= i>

of G c o r r e s p o n d

where

c =

system

to

[a,b].

R such

that

N = Irr(R).

I. D e f i n i t i o n : R

= (ca ~

ac,

c a -I ~ a - l c ,

c-la

cb ~ bc,

cb -I ~ b-lc,

c - l b ~ bc -I,

ba ~ a b c -I,

b - l a ~ ab-lc,

~ a c -I

•

c - l a -1 ~ a - l c -I

r

c - l b -I ~ b - l c -I,

ba -I ~ a-lbc, b-la-i

~

a-lb-lc-l).

211

F r o m this d e f i n i t i o n Irr(R)

= N

holds.

it f o l l o w s

Also,

R has

by i m m e d i a t e FTP and

UTP;

i n s p e c t i o n that hence

R

is c o m -

plete. It is s t r a i g h t f o r w a r d ordering. partial

R

is,

term

that

R

however,

ordering

is not c o m p a t i b l e

compatible

with

w i t h any KB-

the

following

"E":

2. D e f i n i t i o n :

For

u = Xl...Xn,

x i E {a,a-l,b,b-l,c,c-l},

we put

Kx(u)

:= c a r d ( { i I x i E {x,x-l)}),

V(u)

:= c a r d ( f ( i , j ) I i<j,

xiE{b,b-l),

xjE{a,a-l}});

V'(u)

:= c a r d ( f ( i , j ) {

xiEfc,c-l),

xjEfb,b-l}})

and f i n a l l y

u E v

i<j,

x E {a,b,c};

iff

(Ka(U),Kb(U),V(u),Kc
It

is

"~" d e n o t e s

easy

to

compatible

check

a word

procedure The

for

cycle

a,b

and

deg(¥(R)) the

In

this

way

c

taking

its

each.

word

= A(R)

Hence

being

normal

the

graph

F(R)

equations

of

form

a

is

standard

process".

cycle

we

is

R.

cycle

the R-growth

which

generate

accepting

three

5-tuples.

ordering

"collecting

namely

on

the d e f i n i n g

for "K" w i l l

to

A

ordering

is a t e r m

the name

structure,

= 3, ¥(R)

Irr(R) on

a path,

number

is

function;

have

the

has

3

a

one and

i.e. for

cycle

number

= 3.

contrast

degree

"E"

algorithm

automaton

corresponding

c(r(R))

that

known under

minimal

simple

lexicographic

w i t h R. In fact~

G, the c o m p l e t i o n Reducing

the

to this

the

classical

4. An easy w a y to see this

(cf. c h a p t e r

IV.4.):

growth

function

is to use the Bass

of

G has

formula

212

We

have

G1

central This

Now

= G,

series;

G 2 = <[a,b]> G I / G 2 is f r e e

gives

we

investigate

for

number

of

G 3 = (e)

abelian

for

of

rank

of

the

the

2 and

lower

G2/G3~N.

d = 1.2 + 2.1 = 4

will

gorithm

and

the

behaviour

a KB-ordering.

constants

We

in t h e

know

that

completion

this

depends

al-

on the

alphabet.

We put

and

take

as

E

= ~o u E 1

~o

= (a,b),

defining

relations

[a, [b,a]]

additional

equations

of

fix

an

R~=R~(~)

arbitrary

the

=

the

c i = w i,

We

~i

= {Cl ..... Cn}

for

[b, [b,a]]

= 1

form

l~i~n,

KB-ordering

corresponding

G besides

limit

for

some

"4"

on

wie(ZouZgl)*.

Z

and

denote

by

system.

3. P r o p o s i t i o n :

I r r ( R ~)

Below

we

provide need

Let

~=~(a),

will

some

some

~:

is n o t

~o

regular.

give

a proof

comments

for

technical

~

~

S=~(b).

be

for

the

the

case

general

E1

= ~ and

situation.

will

First

we

lemmas.

the

weight

function

defining

"~"

and

213

In

addition

to

the

notions

defining

"K"

we

put

for

U=Xl..°x n

+

Kx(U)

:= c a r d

({i I x i = x})

Kx(U)

:= c a r d

({i 1 x i = x-l})

+

Kx(U)

:= Kx(U) - Kx(U)

V+(u)

:= c a r d ( { ( i , j ) I i<j,

V-(u)

for

:= c a r d ( { ( i , j ) l

V(u)

:= V + ( u )

In o r d e r

either

xi=b

or

xi=b-i

either

xi=b

or

xi=b-i

i<j,

and

xj=a

and

xj=a-l}) x j = a -i

and

and

xj=a})

- V-(u).

to c o m p a r e

M(r)

x E Zo;

the

two

orderings

:= m a x ( V ( u ) I ~(u)

~ r and

we define

Ka(U)

for

= Kb(U)

r>0

= 0).

4. L e m m a :

Assume

r ~

2(~+B)

satisfying Then

u has

Proof: This

We

obtain Next

Suppose K~(u)

u'~u,

M(r)

[a,b]. u

show this

an a

each

of

u'

that is can

~

1

[a,b]

the

assume

= u(mod

contradiction.

s G)

with case, > 0. which

i-minimal

Ka(U) for

and

hence

[a,b]M(r)(mod

not

the

= aSb-ta-Sbt

of t h e

n starts

is

~ r and

a , a - l , b , b -I

Because =

~(u)

u

= ab-la-lb

permutation

> 0 we

yields

u

shows

also

we

form

w

that

is a c y c l i c word

the

suppose

= M(r),

consider

example

It f o l l o w s

tor

V(u)

and

get

V(u)

= Kb(U) some

= 0.

s>0,

~(u)~r,

t*0.

V(w)=l.

~ i.

occurs

in u.

M(r)-th

power

is

the

in

word

Therefore

u

of t h e c o m m u t a center

of

G

of

a.

we

G). some

positive

i.e. Then

u some

starts

power

= vaSz. cyclic with

aS;

Because

of

permutation we

obtain

214

A

similar

If

u

argument

= a S v a k,

Therefore

shows

then

u

has

the

u now

that

n ends

= as+kv(mod

with

G) a n d

some

as+kv

power

We

have

to

form

which

have

With

p = ti

the

u

For We

m=p+q get

There

= a s l b t l a s2

show

k=2.

and

q

(i)

we

are

t i and

tj

for

= tj

we

get

with

consider

v

w I = UlbmVu2

~(w2)~r

and

by

= av I = v2a.

and

a direct

w 2 = UlVbmu2 • computation

- V(u)

=

V(bmv)

- v(bPvbq),

V ( w 2)

- V(u)

=

V ( v b m)

- v(bPvbq),

V ( w I)

- V(u)

= m. K a ( V )

- P'Ka(V)

=

V ( w 2)

- V(u)

= 0.Ka(V)

- P. K a ( V )

= -P.Ka(V).

three

cases:

q. K a ( V ) = 0

gives

either

which

bmv

V(u) ~

= V ( w 2)

and

wI i u

vb m ~ bPvb q

and

w2 ~ u

(ii)

q°Ka(V)>0

yields

(iii)

q. K a ( V ) < 0

implies

k=2

= V ( w I)

q. K a ( V ) ,

bPvb q

contradicts

again

i<j

sign.

or

immediately

there

k

Z s i = ~ t i = 0. i=l i=l

V ( w I)

Th~s

Therefore

with

= ulbPvbqu 2

~(Wl)Kr,

are

o .o b t k

Otherwise

same

b:

~ u.

k U

of

the

minimality

V(Wl)>V(w)

P. K a ( V ) < 0

of

which

and

= M(r).

ho]ds

u.

is

impossible.

V ( w 2)

> V(u);

a contradiction.

is

from

true; Ka(U)

the

rest

= Kb(U)

of

= 0.

the

assertion

follows

215

5.

Lemma:

(i)

(ii)

M(4~Bn)

= ~Sn 2

If

the

u

is

~(u)

Proof:

~

(i) B y

V(u)

=

From

~(u)

4~Sn

Lemma

= 2~s

f(x)

This

and

n>0.

word

satisfying

Ka(U)=Kb(U)=0,

4

then

u

= a -8 n b

u

= aSb-ta-Sb t

= max

~ 4~B

(s.t I s,t

its

maximum

gives

the

=

V(u) u has

= M(r),

the

form

-~na-S n b - ~ n .

holds

on

at

follows

E R,

the

x = Sn

solution

~Sn 2

this

it

E ~,

(x.y I x , y

= ~/s.x.(2Sn-x)

M(4~Sn)

From

i-minimal

+ 2Bt

max

f has

all

which

gives

s.t.

M(4~Sn)

If

for

x

~s

+ St

x,y~0,

real

~ 2~8n)

~x

+ SY

= 2~n)

interval

and =

that

M'

Sn,

y

= M ~.

[0,2Sn]

= f(Sn)

= ~Sn 2

=

~,

~n

e

then holds.

therefore

holds.

(ii)

follows

directly.

[]

Now

we

come

to

the

Proposition

3:

Proof

of

Then

some

I r r ( R ®) class

finite

=

of

back

L(A).

words

Suppose

automation A

pumping

discussed

A

I r r ( R ~) will

lemma in

Lemma

is

accept

argument 5

regular.

will

Irr(R~), applied

lead

to

i.e.

to

a

the

contra-

diction: Let

Q(z~u)

denote

let

zo

the

be

initial

ciently

large

for

positive

all

the

such

state

reached

state

that

integers

the

of

A.

from We

pumping

state

z by

input

choose

s I ..... s 4

lemma

guarantees

m I ..... m 4 t h e

equality

u;

suffithat

216

Q(zo,alslml

b-s2m2

a-S3m3

=

Taking

mI =

b-S2

the

a-S3

special

bS4)

~.n

~0n

~ , sI

m2 = ~ , s2

m3 = ~ , s3

Lemma

5; t h i s

Taking

word

t

u

such

= a ~n

b -~n

shows

that

that

x

a -~n

b ~n

z I is an

>

y Sy2

- - , Sl

m2 = ~ , s2

m3 = - - , s3

word

w

=

= Kb(W)

= 0

m4

a sy2

= s4

I r r ( R ~)

because

of

state.

= s2.s 4

~y2

another

of

accepting

= t.sl.s 2

holds.

zI)

~0n

e

Sy2

to

(=

choice

S'n

a

leads

Q(zo,aSl

n = Sl. S 2 . S 3 . S 4

gives

mI =

bs4m4)

and

choosing ~y2

m4 s4

b -~x2

a -Sy2

b ~x2

with

Zl=Q(Zo,W). We the

have

Ka(W)

minimal

hence

w

~

word

and

representing

Irr(R~);

w

V(w)

= ~ S ( x y ) 2. B y

is

Lemma

a S X Y b - ~ X Y a - ~ X Y b ~xy

5

• w;

a contradiction. []

The

essential

discussion orderings finitely tions

the

work,

The

part

of

this

of

the

two

"E"

and

"4".

In

many

new

arbitrary simple only

sophisticated

use

proof

growth

of

not

discuss

here.

The

use

nonstandard

functions Gromov's

of

was

first

theorem

model

the cycles

axis,

connected

pumping

in t h e

in word

in

for

in

the

[vdD-Wi]. become

infinitesimal

this

the

The

where

does

case which

of

simplified In

the

as a b b r e v i a -

context

essentially

paths.

with

argument

analysis

the

implicit

situation

occur

lemma

have

an

associated

can

nonstandard

models

in

general

constants

introduced

given

by

more

authors

of

consists

functions

the

the

the

methods

proof

not

use

some

we

will

growth

proof

of

nonstandard a real

half-

assumption

that

217

R ~ is regular then finally leads to a topological bility

between

~3 (arising

from

incompata-

"K") and ~4 ( a r i s i n g

from

V. D e c i d i n g

Algebraic

Properties

of F i n i t e l y

Presented

Monoids

F. Otto F a c h b e r e i c h Informatik Universit~t Kaiserslautern P o s t f a c h 3049 6750 K a i s e r s l a u t e r n

In this chapter ing a l g e b r a i c turn out, general.

we are going

properties

However,

general,

of finitely

all the p r o b l e m s

word p r o b l e m

we are

we have already

and the finiteness

become

decidable

through

tems.

So a f t e r e s t a b l i s h i n g we will

presented

sight

which

Presentations

they are r e s t r i c t e d involving

I. Then

i~ addition

is d e f i n e d

a monoid

[w] R (w 6 E*) (u,v c E*), monoid

MR,

R. As we have

the elements

and the

operation

[e] R serving as the

of the free m o n o i d

for f i n i t e l y

restricted

classes

gain a d d i t i o n a l

and by R a r e d u c t i o n

relation

on Z* c o n t a i n i n g

mentioned

in-

Transformations

to the r e d u c t i o n

gruence

sys-

systems.

~+

defines

in

that are

reduction

problems

certain

of these

on Z*, w h i c h

in

as the

to m o n o i d s

results

decision

through

and Tietze

it will

problems

complete

In this way we will

power

As

concern-

that are also u n d e c i d a b l e

Again we denote by Z a finite alphabet,

ecs/ivalence r e l a t i o n

monoids.

seen that certain

are given

systems.

problems

in are u n d e c i d a b l e

the u n d e c i d a b i l i t y

into the c o m p u t a t i o n a l

Monoid

presented

problem

presentations

reduction

decision

interested

show h o w to solve these

monoids

of complete

V.I.

finite

when

given

above,

to i n v e s t i g a t e

~,

R also

induces

this

identity.

Thus,

con-

congruence

are the c o n g r u e n c e

of which is given by[u] R o

E* g e n e r a t e d

an

as the smallest

seen already,

of which

system on

classes

[V]R = [uv]R

M R is the factor

by Z m o d u l o

the c o n g r u e n c e

7 Whenever ordered

pair

a monoid (E;R)

M happens

is c a l l e d a

ing the set of g e n e r a t o r s of this presentation.

to be isomorphic (monoid)

and R b e i n g

that are f i n i t e l y presented,

monoids

that are given

both

the

sets ~ and

the

through

R are

i.e.,

(M T

MR),

of M with

the E be-

relations

only be d e a l i n g

with

we will only be c o n s i d e r i n g

presentations

finite.

MR

set of d e f i n i n g

In the f o l l o w i n g we will

monoids

to

presentation

of the form

(E;R),

where

219

Let oroblem

(Z;R)

be a finite

for this p r e s e n t a t i o n

INSTANCE:

Two words

QUESTION:

v hold

So the w o r d o r o b l e m word p r o b l e m

there

([Da 58]). scribe

{b,c}, fined

systems

~(a)

finite p r e s e n t a t i o n s

ed ? In o r d e r

this o b s e r v a t i o n

to answer the n o t i o n

1.2. Let

Then p r e s e n t a t i o n application

(3

if

= E, and

may de-

shows.

Then

Z2 =

the m a p p i n g

an i s o m o r p h i s m

(ZI;R I) and

raises

(E2;R2)

~ de-

from the

are dif-

the f o l l o w i n g

question:

depend on the a c t u a l l y

this question, of e l e m e n t a r y

(Z;R) and

(E';R')

(Z';R')

but also Tietze

Does

chosen

of the m o n o i d p r e s e n t for future

reference,

transformation

be two finite

is said to be o b t a i n a b l e

(Z';R')

Tietze

satisfies

R' = R U {(u,v)},

transformation

for finite

condition

presentations. from

(Z;R)

of type

by an

i for some

(i) given below.

w h e r e u , v 6 Z* satisfy

v.

,

R' = R-{(u,v) }, where

Z' = E U {a} for

word p r o b l e m

([Ti 08]).

{ R, but u ~ +

Z' = Z, and

and hence,

presentations

example

or is it a p r o p e r t y

of an e l e m e n t a r y

i 6 {1,2,3,4},

(u,v)

finite

there exist

same monoid.

of the w o r d p r o b l e m

presentations

Definition

(2

w o r d problem,

= b induces i.e.,

of the

finite m o n o i d p r e s e n t a t i o n ,

we i n t r o d u c e

does

is n o t h i n ~ but the

~I = { ( a 3 ' e ) ' ( b 2 ' e ) ' ( a b ' b a ) } '

MR2,

the m o n o i d

the d e c i d a b i l i t y

i.e.,

with u n d e c i d a b l e

different

= cb and ~(b)

ferent

Obviously,

of M,

(Z;R)

R 2 = {(cbcbcb,e), (b2,e),(bcb,c)}.

onto

the word

problem:

R. As is w e l l - k n o w n

as the f o l l o w i n g

MRI

(I) Z'

system

presentations

1.1. Let E I = {a,b},

through

decision

same e l e m e n t

with u n d e c i d a b l e

monoid

monoid

the

for the p r e s e n t a t i o n

On the other hand,

and

M. Then

?

finite m o n o i d

the same monoid,

Examole

is the f o l l o w i n g

for the r e d u c t i o n

reduction exist

of a m o n o i d

u,v 6 E*.

Do u and v r e p r e s e n t u ~+

finite

presentation

some letter

(u,v)

6 R satisfies

a { E, and

u ~

R' = R U {(u,a) } for

v. some

word u 6 Z*. (4

There

exist a letter

(u,a)

6 R. Let ~: Z* ~

by ~(a) E'

The

= u and ~(b)

= Z-{a},

and

following

transformations

a 6 E and a word (Z-{a})*

u 6

denote

!emma

shows

to a finite

b 6 Z-{a}.

induced

Then

£ R-{(u,a)}}.

that by a p p l y i n g

presentation

such that

the h o m o m o r p h i s m

= b for all letters

R' = {(~(l),~(r)) I (/,r)

(Z-{a})*

(E;R)

elementary of a m o n o i d

Tietze M we only

220 get further p r e s e n t a t i o n s of M. The proof of this lemma is straiqhtforward, and t h e r e f o r e Lemma

1.3. Let

(E';R')

(E;R)

it is left to the reader.

and

is obtainable

Tietze transformation. monoid,

i.e.,

(E';~') be two finite p r e s e n t a t i o n s

from

such that

(Z;R) by an a p p l i c a t i o n of an e l e m e n t a r y

Then these two p r e s e n t a t i o n s define the same

MR''

~{R

Observe that the e l e m e n t a r y Tietze t r a n s f o r m a t i o n s of type 2 are inverses of each other,

I and

that the inverse of an e l e m e n t a r y Tietze

t r a n s f o r m a t i o n of type 3 is one of type 4, and that the effect of an e l e m e n t a r y Tietze t r a n s f o r m a t i o n of type 4 on a finite monoid presentation can be r e v e r s e d by a finite number of a p p l i c a t i o n s of elementary Tietze t r a n s f o r m a t i o n s of types

I to 3. We will use this o b s e r v a -

tion in the proof of the f o l l o w i n g theorem,

which describes the basic

reason for c o n s i d e r i n g Tietze transformations. T h e o r e m 1.4. Let same monoid.

(E;R)

and

(E';R') be two finite p r e s e n t a t i o n s of the

Then there exists a finite sequence of e l e m e n t a r y Tietze

t r a n s f o r m a t i o n s that transforms

(E;R)

into

(E';R') .

Proof. W i t h o u t loss of g e n e r a l i t y we may assume that the sets E and X' of g e n e r a t o r s are disjoint. monoid,

we have

M R ~ MR,.

Since

Thus,

(E;R)

and

(Z';R') define the same

for each a 6 E, there exists a word

u a 6 E'* such that a and u a describe the same element of this monoid. Also, for each b 6 E ~, there exists a word v b 6 E* such that b and v b describe the same element. Using these words the p r e s e n t a t i o n

(E;R)

is

t r a n s f o r m e d by a finite sequence of e l e m e n t a r y Tietze t r a n s f o r m a t i o n s as follows: (a)

(E;R) ~

(E U E';R U {(Vb,b)Ib 6 E'}) by

IE~i e l e m e n t a r y Tietze

t r a n s f o r m a t i o n s of type 3. (b)

Let R ° = R U {(Vb,b) ib 6 E'}, and let g denote the i s o m o r p h i s m

from MR~

onto M R that is induced by m a p p i n g b onto,v b for all b 6 E'

Then for all

(/,r)

6 R ~, g(1)

~+

(E U E';R o) ~

(E U E';R ° U R') by

tions of type

1.

(c)

implying i + ~

r. Thus, o IR' I e l e m e n t a r y Tietze t r a n s f o r m a -

Since for each a 6 E, a, u a, and g(u a) all define the same ele-

ment of the monoid

MR,

we have a +~+ g(u a) ~

(Z U E';R o U R') ~ (X U Z~;R U R by

g(r)

u a. Thus, we obtain

U {(Vb,b)Ib E Z ~} U {{Ua,a)!a E E})

IEl e l e m e n t a r y Tietze transformations of type I. Let E" = E U E'

and R" = R U R ~ U {(Vb,b) Ib E E'} u {(Ua,a) la E E ] . been t r a n s f o r m e d into Tietze transformations.

Then

(E;R)

has

(E";R") by a finite sequence of e l e m e n t a r y

221

(d)

In an a n a l o g o u s

so by the remark into

(E';R')

tions.

manner

proceeding

by a finite

the vb

a uniform

process

same m o n o i d (b 6 E')

the t h e o r e m

sequence

into each other,

to the

since

in general.

isomorphism

(Z";R"),

Tietze

transforma-

Two finite

Do these p r e s e n t a t i o n s

presentations

On the other hand, is o b t a i n e d

plication

(E;R)

and

describe

of

(a £ E) and it does not give

(E';R').

the same monoid,

it can be seen easily,

for

(E';RI),

(E;~)

Tietze also

conclude

the f o l l o w i n g

Corollary

1.5. Let

(E;R)

same monoid.

that

i.e.,

induces

Then

and

for

the d e c i d a b i l i t y

finite p r e s e n t a t i o n s .

(E~;R ') be two finite

(E';R')

for

(E;R)

1.4 we can

im-

of the w o r d p r o b l e m

In particular,

there

of

if and

is an invariant

presented

of

the d e c i d a b i l i t y

for a f i n i t e l y

finitely

presentations

is d e c i d a b l e

is decidable.

we can speak a b o u t

undecidability

exist

to

to the word pro-

from T h e o r e m

of the word p r o b l e m

Hence,

ap-

result.

the word p r o b l e m

if the word p r o b l e m

by a single

then a solution

a solution

Hence,

if a p r e s e n t a t i o n

(E;R)

transformation,

and vice versa.

mediately

Thus,

ua

In particular,

from a finite p r e s e n t a t i o n

of an e l e m e n t a r y

the word p r o b l e m for

the words

does not

presentations

M R ~ MR, hold ?

does

(E';R')

construction

two finite

problem:

INSTANCE:

only

into

can be t r a n s f o r m e d

of e l e m e n t a r y

out that the above

QUESTION:

the

(Z";R")

for t r a n s f o r m i n g

are not known

a so l u t i o n

blem

can be t r a n s f o r m e d

[]

It should be p o i n t e d yield

(E~;R')

presented

monoids

or

monoid

M.

with u n d e c i d -

able w o r d problem.

V.2.

Markov

Properties

Given a m o n o i d

M through

like to d e t e r m i n e specifically INSTANCE: I. QUESTION:

of E i n i t e l y

Presented

some finite

presentation,

some of the algebraic

interested

in the f o l l o w i n g

A finite p r e s e n t a t i o n Is the m o n o i d trivial,

i.e.,

one w o u l d

properties decision

often

of M. Here, problems:

(E;R) .

M R given t h r o u g h does

Monoids

this p r e s e n t a t i o n

M R ~ {e} hold

?

we are

222

2. QUESTION:

Is the m o n o i d

M R finite

3. QUESTION:

Is the m o n o i d

MR commutative

4. QUESTION:

Is the m o n o i d

M R cancellative

5. QUESTION:

Is M R a free m o n o i d

6. QUESTION:

Is the m o n o i d

? ? ?

?

M R a group ?

7. QUESTION:

Does

the m o n o i d

M R contain

any n o n - t r i v i a l

idempotents?

8. QUESTION:

Does the m o n o i d

M R contain

any n o n - t r i v i a l

elements

finite 9. QUESTION:

Does

In this ([Ma 51],

c.f.,

the d e c i s i o n will

then

izations

we will

e.g.,

problems

leave

are p r e s e n t e d

M R a free g r o u p ? learn about

[Mo 52]),

2.1.

ever y m o n o i d

complete

(a) A p r o p e r t y

that

(b) A p r o p e r t y

P of f i n i t e l y

if it satisfies

systems

(O)

P is invariant. There

exists

a finitely

property

P, and w h i c h

finitely

presented

There

exists

(c) Finally, hereditary

This

for monoids

or certain

that

special-

a property

presented

Whenever

above

monoid

submonoids

a finitely

if

property

P it-

is a M a r k o v property,

M I which

does not have

to a s u b m o n o i d

having property

P of finitely

of any

P.

monoid M 2 having property presented

presented

monoids

P.

is c a l l e d

m o n o i d M has P, all

of M also have P. property

of finitely

presented

or not P is a M a r k o v property,

can be r e l a x e d

There exists

monoid

isomorphic

presented

a finitely

of whether

invariant

possessing

monoids

presented

P is an h e r e d i t a r y

for c h e c k i n g

(I) g i v e n

is c a l l e d

three conditions:

is not

a finitely

if w h e n e v e r

to a m o n o i d

presented

the f o l l o w i n g

(I)

(1')

of M a r k o v

in general.

these p r o b l e m s

reduction

P of m o n o i d s

is isomorphic

this property.

then

are u n d e c i d a b l e

us the task of solving

self p o s s e s s e s

fini t e l y

result

w h i c h can be used to show that all

stated above

by finite

a fundamental

thereof.

Definition

(2)

M R contain an element of infinite

?

Is the m o n o i d section

of

?

the m o n o i d

order 10. QUESTION:

order

to the f o l l o w i n g

presented

monoids,

condition

condition:

m o n o i d M I not h a v i n g p r o p e r t y

P. Before exam p l e s

stating

and p r o v i n g

of M a r k o v properties.

Markov's

r e s u l t we want

to give

some

223

Definition Then

2.2.

In w h a t

the properties

(a) PI(M)

~

follows

PI,P2,.°.,PIo

M is t r i v i a l ,

(b) P2(M)

.: > M is f i n i t e ;

(c) P3(M)

<

are

i.e.,

> M is c o m m u t a t i v e ,

m I o m 2 = m 2 o m I, w h e r e ~ (d) P4(M)

let M be a f i n i t e l y

= ml o

m3

implies

monoid.

as f o l l o w s :

M Z {e};

i.e.,

f o r all m l , m 2 6 M, we h a v e

denotes

.~ ~- M is c a n c e l l a t i v e ,

ml ~ m2

defined

presented

the o p e r a t i o n

i.e.,

of M;

f o r all m l , m 2 , m 3 C M,

m 2 = m3,

and m I o m 3 = m 2 o m 3 implies

m I = m2; (e) Ps(M)

~

M is a f r e e m o n o i d ,

alphabet (f) P6(M) m'

6 M such ~

there (h) P8(M)

M ~ E* f o r

some

i.e.,

t h a t m o m'

= e M,

~. M d o e s n o t c o n t a i n

for a l l where

m 6 M,

there

eM denotes

a non-trivial

the

m # e M a n d m o m = m;

<

a non-trivial

: M does not contain i.e.,

is an e l e m e n t identity

idemmotent,

is n o m 6 M s a t i s f y i n g

order,

finite

E;

----~- M is a g r o u p ,

(g) P7(M)

i.e.,

element

of M;

i.e.,

of f i n i t e

there

is n o m 6 M - { e M} s u c h t h a t t h e r e e x i s t k > I k+n n i and n > O satisfying m = m , where m s t a n d s for m o m o . . . 0

m

(i-times); (i) P9(M)

<

there

:. M d o e s n o t

(j) PIo(M)

~--~ M is a f r e e

Obviously, be

seen

they

all

easily,

satisfy

However,

contain

an e l e m e n t

is n o m 6 M s a t i s f y i n g

properties

and

remaining

properties

following

examples.

(a) L e t

E = {a,b},

2.3.

M = E*

is a f r e e m o n o i d ,

a free monoid.

This

of w o r d s

that generates


(b) L e t

E = {a,a}, by

submonoid


the

So Markov

let

is the

{a}*,

and

can b e

are

they

order,

6 ~ with

i.e., i # j;

invariant.

As can

let A = { a b , a b a , b a b } . its

seen

contain

group b y a.

and hence,

are Markov

submonoid from

the

Then

the

generated

fact

that each

by set

A.

F I of r a n k This

and

properties.

and PIO are not hereditary

R = {(aa,e),(aa,e)}. free

of M R g e n e r a t e d

free monoid

group,

and

(E;R)

i.e.,

P5,P6,

while

A is n o t

presented

PI,P2,...,PIo

(2),

the

monoid

i,j

PI to P4 a n d P7 to P9 are h e r e d i t a r y ,

(I')

shown by the Examnles

infinite

group.

the p r o p e r t i e s

conditions

of

m i % m j for all

Then

the monoid

I. N o w

submonoid

it is n o t a g r o u p ,

consider

MR the

is i s o m o r p h i c let alone

to

a free

a in o r d e r

to p r o v e

properties,

tion

2.1.

with

condition

Obviously, (I).

that properties

we must they

check all

P5,P6,

conditions

satisfy

(I)

condition

and PIO and

are

in f a c t

(2) of D e f i n i -

(2), w h i c h

leaves

us

as

224

Lemma 2.4. There exists a finitely p r e s e n t e d m o n o i d M I that is not isomorphic to a submonoid of any finitely g e n e r a t e d free monoid. Proof. Each finitely g e n e r a t e d submonoid of a free m o n o i d has a decidable w o r d problem.

Now let M I be a finitely p r e s e n t e d m o n o i d with an

u n d e c i d a b l e word problem.

Since the u n d e c i d a b i l i t y of the word p r o b l e m

is an invariant property,

we conclude that M I is not isomorphic to a

submonoid of any finitely g e n e r a t e d free monoid,

a

So p r o p e r t y P5 is also a Markov property. Lemma 2.5. There exists a finitely p r e s e n t e d m o n o i d M I that is not isomorphic to a s u b m o n o i d of any group. Proof. Let M I be given through the p r e s e n t a t i o n Then M I is not cancellative, m o n o i d of any group,

and hence,

({a,b,c};{(ab,ac)}).

it is not isomorphic to a sub-

a

This implies that also p r o p e r t i e s P6 and PIO are M a r k o v properties. We can summarize the results o b t a i n e d so far as follows. T h e o r e m 2.6. Properties PI to PIO are Markov p r o p e r t i e s of finitely p r e s e n t e d monoids. N o w the main result of M a r k o v

[Ma 51] states the f o l l o w i n g un-

d e c i d a b i l i t y result. Theorem 2.7

([Ma 51]). Let P be a M a r k o v property.

Then the following

p r o b l e m is undecidable:

INSTANCE:

A finite p r e s e n t a t i o n

QUESTION:

Does the m o n o i d M R given through this p r e s e n t a t i o n have

(Z;R).

property P ? Because of T h e o r e m 2.6 this immediately gives the following. C o r o l l a r y 2.8. The d e c i s i o n p r o b l e m s

I to 10 are all u n d e c i d a b l e

in

general. Since a m o n o i d M is trivial ({a};{(a,e)}),

if and only if it can be p r e s e n t e d by

the u n d e c i d a b i l i t y of p r o b l e m I induces the u n d e c i d -

ability of the i s o m o r p h i s m problem. Hence,

there is no hope for ex-

tending T h e o r e m 1.4 to a solution for this problem. It remains to prove T h e o r e m 2.7. For doing so we need the following lemma the proof of which is e s s e n t i a l l y taken from [Bo 84].

225

Lemma

2.9.

letters,

Let

and

R on Z a n d

I be a f i n i t e

let

F = E U {c,d}°

two w o r d s

reduction

system

alphabet,

R

u,v on

U,V

Then

6 E*r

one

let c,d given

~ Z be two a d d i t i o n a l

a finite

reduction

can e f f e c t i v e l y

Z and a homomorphism

h:

system

construct

a finite

Z* ~ F* such

that

either (i)

u ~+

v and the

congruence

~

is t r i v i a l ,

or

utv

(ii)

u ~

v and h i n d u c e s

an e m b e d d i n g

of

M R in M R UtV

Here,

the

congruence


,

-~ is c a l l e d

trivial

if it s a t i s f i e s

~,V

x ~

y for all w o r d s

x,y

6 r*.

utv Proof.

Let

R be a r e w r i t i n g

a rewriting a 6 r} and

system

a homomorphism

R ~ Ru, v, we h a v e

system

Ru, v by t a k i n g

h(/)

h:

on E,

and

let u , v

6 E*.

Ru, v = R U { (cud,e)}

E* ~ F* by a ~ a for all

= I ~

r = h(r)

We d e f i n e

U { (acvd,cvd) 1 a 6 E.

for all

(/,r)

MR i n t o

MR

Since

E R, w h i c h

UrV

implies

that

h induces

a homomorphism

from

. Obviously, U~V

R

a n d h c a n be c o n s t r u c t e d

U,V

remains

to c h e c k

that

effectively

conditions

from

(i) and

(ii)

R a n d u,v. are

Thus,

it

satisfied.

e

First lowing

assume

chain

that

u + ~ + v. T h e n

of c o n g r u e n c e s :

for e a c h

a 6 F, we h a v e

acud

acvd

a ~

*

~

U,V

~------+ c u d ~

e,

UrV

i.e.,

Now

assume

UrV

~

is in fact

UtV

fol-

cvd

~,V

the c o n g r u e n c e

the

~

trivial.

UiV

that

u ~

v. We m u s t

prove

that

the h o m o m o r p h i s m

an embedding,

i.e.,

it

for all w o r d s

x,y

A

h:

induced

MR ~ MR

by h is

is

1-to-1.

To

U,V

this

en~

,

it s u f f i c e s

to p r o v e

that

6 E*,

x ~

y U,V

implies

x +~+ y,

Let

h(x)

R I = R U {(cud,e)}.

= x a n d h(y) As an

I. For

Proof.

all x , y

If z 6 F*

function

as left

occurrences and whenever

intermediate

C Z*,

if x ~

y,

satisfies

z ~

and right

parantheses,

of c a n d

d are

z = ZlZ2,

x,

correctly

then

Iz]l

>

then

then

the n u m b e r

Let x = x ° +~ length

step we p r o v e

of o c c u r r e n c e s x] + ~ I

of y f r o m x in

is a c t u a l l y

an

R I.

x +~

the

fol-

respectively,

Izll

i.e.,

. Here,

of c and

d in z

and

further,

Izl c

Iz

=

for a 6 F,

the

Id ,

lwl a

d of the

"'" + ~ I xn

letter

a in w.

= y be a d e r i v a t i o n

If x i 6 E* for all

R-derivation

y.

occurrences

balanced,

c denotes

tion

= y.

claim.

lowing Claim

since

implying

that

i, t h e n

x ~+

the

y.

of m i n i m a l above

deriva-

226 So a s s u m e

t h a t for

Then there exists

some

i 6 {1,2,...,n-!},

an index p E {1,2, .... n-l}

Ixil

= IxiJ > O. c d such t h a t

< ~Xll < ... < IXp I = IXp+ I ~ + 1, i.e., Xp is the f i r s t c c c c in the a b o v e d e r i v a t i o n to w h i c h the rule cud ~ e is a p p l i e d .

O = IXol word Thus,

Xp = w c u d z

+ ~ I wz = Xp+ I for some w o r d s w,z

6 F*.

Since

x = x E E*, a n d since the r u l e s of R do n o t c o n t a i n any o c c u r r e n c e s o of the l e t t e r s c a n d d, the p a r t i c u l a r o c c u r r e n c e s of c a n d d t h a t are c a n c e l l e d together

in the

step f r o m Xp to Xp+ I m u s t h a v e b e e n

at some e a r l i e r

index q 6 { O , I , . . . , p - I } +~I W l C U d Z l

= Xq+1'

step q 6 { O , ! , . . . ~ p - 1 } .

wl +~I w and

cud

+ ~ I XP = w c u d z is not t o u c h e d

shorter

z I ~I

z. So the a b o v e

+ ~ 1 wz = Xp+1 original

+~1

xq=

+ ~ I wz = Xp+ 1 + ~ I during

derivation

WlZ I

the d e r i v a t i o n

Therefore,

has

WlCUdZ I

of Xp f r o m x q+1' x = x O ~I

"'" + ~ I Xn = Y' thus c o n t r a d i c t i n g

derivation.

derivation

"'" + ~ I Xn = y" S i n c e the f a c t o r

of y f r o m x as follows:

i 6 {O,1, .... n}.

introduced t h e r e e x i s t an

and w o r d s W l , Z 1 6 F* such t h a t Xq = WlZ I

the following form: x : x ° Xq+1

Hence,

we c o n c l u d e

we get a

"'" + ~ I Xq = WlZ 1 the c h o i c e of the

t h a t x i 6 E* for all

o

N o w the p r o o f

of L e m m a

2.9 is c o m p l e t e d by p r o v i n g

the f o l l o w i n g

claim. W

Claim

*

2. For all x,y 6 E*,

Proof.

if x ÷ .... * y, t h e n x +--+ y. Ru,v R1 xI ~ ... ~ x = y be a d e r i v a t i o n

Let x = x ° ~

UfV U,V l e n g t h of y f r o m x in Ru, v. If this d e r i v a t i o n c o n s i s t s of , a p p l i c a t i o n s of r u l e s of R I only, t h e n it w i t n e s s e s x + ~ I y' and we

of m i n i m a l

are done. So a s s u m e as well,

Ru, v

a n d let p E { 1 , 2 , . . . , n - I }

Xp c o n t a i n s implies

that rules from

c v d as a factor.

that the i n i t i a l

i.e.,

length.

Xp = wcvdz,

of Xp f r o m x. A c c o r d i n g

we c a n c o n c l u d e

the rule c u d ~ e is n o t a p p l i e d this m e a n s

"'" ~ I

of y f r o m x g i v e n above, Hence,

i n d e x such t h a t in Ru, v - R I

s e g m e n t of l e n g t h p of the a b o v e

it is an R 1 - d e r i v a t i o n

minimal

be the s m a l l e s t

N o w the f o r m of the r u l e s

is of the f o r m x = x O + ~ 1 xl + ~ I

the d e r i v a t i o n

- R I = {(acvd,cvd)la 6 F} are u s e d

a l s o this

derivation

w h e r e w,z

6 F*,

to the c h o i c e of

R1-derivation

f r o m the p r o o f of C l a i m

d u r i n g this d e r i v a t i o n .

is of I, t h a t

Since x 6 X*,

t h a t t h e r e e x i s t an index q 6 {O,1,2 .... ,p-J} and w o r d s

227 w

w1,zl

6 r* such that Xq = WlZ I +~] WlCUdZ I = Xq+ I, w I +~I w, u ~ i

and z I ~ . z. Since u,v E E*, C l a i m I induces u + ~ ] ing our assumption u ~-'~v. H e n c e , we c o n c l u d e that m i n i m a l length of y from x in R of R . .

ufv

v,

v, thus c o n t r a d i c t a derivation

of

only contains a p p l i c a t i o n s of rules

oD

i

Definition

2.10. Let M I and M 2 be two monoids that are given through

presentations

(EI;RI)

and

(E2;R2), respectively,

where El ~ E2 = ~"

Then the free product M I * M 2 of the monoids M I and M 2 is the m o n o i d which is defined through the p r e s e n t a t i o n

MR

(E;R), where E = E I U E2

and R = R I U R 2 . w

As can be seen easily,

we have u ~

w

v if and only if u +~+ v

1 for all u,v E El, i = 1,2. This y i e l d s the f o l l o w i n g fundamental p r o p e r t y of the free product. T h e o r e m 2.11. Let

(EI;RI) and

El ~ E2 = ~' and let

(E2;R2) be two p r e s e n t a t i o n s

(E;R) denote the p r e s e n t a t i o n

Then for i = 1,2, the identity m a p p i n g ding of the m o n o i d In particular,

MRi

in the m o n o i d

such that

(E I U ~2;RI

U R2).

id.: E~ ~ E9 induces an embedl l l

M R = MRI*MR2.

this induces the f o l l o w i n g c o r o l l a r y we will make

use of in the proof of T h e o r e m 2.7. C o r o l l a r y 2.12. Let M I and M 2 be two finitely p r e s e n t e d monoids that the word p r o b l e m for the free product M I * M 2 is decidable. the w o r d p r o b l e m s Actually,

such Then

for M I and M 2 are decidable.

also the reverse implication holds,

i.e., two f i n i t e l y

p r e s e n t e d m o n o i d s M I and M 2 have decidable w o r d p r o b l e m s if their free p r o d u c t M I * M 2 has a d e c i d a b l e w o r d problem. of this fact in the case of groups see e.g.

if and only For a proof

[Ma-Ka-So].

N o w f i n a l l y we are p r e p a r e d to prove M a r k o v ~ s result. Proof of T h e o r e m 2.7. Let P be a M a r k o v property.

Then there are two

f i n i t e l y p r e s e n t e d monoids M I and M 2 such that (I) M I does not have p r o p e r t y P, and M I is not isomorphic to a subm o n o i d of any finitely p r e s e n t e d m o n o i d h a v i n g p r o p e r t y P, and (2) M 2 does have p r o p e r t y P. In addition,

let M 3 be a finitely p r e s e n t e d m o n o i d with an un-

d e c i d a b l e w o r d problem,

and for i = 1,2,3,

let

(Ei;R i) be a finite

p r e s e n t a t i o n of the m o n o i d M i. W i t h o u t loss of g e n e r a l i t y we may as-

228

sume that the sets of generators

E1~ E2r and E 3 are pairwise disjoint.

Now the free product M R of M I and M 3 is given through the presentation 2.11

(E;R), where Z = E I U E 3 and R = R I U R 3, A c c o r d i n g to Theorem

M R does not have p r o p e r t y

M I and M 3 are e m b e d d e d in M R . Hence,

P due to the choice of M I, and its word p r o b l e m is u n d e c i d a b l e due to the choice of M 3 and C o r o l l a r y 2.12. On the other hand,

the w o r d pro-

blem for M R is e f f e c t i v e l y reducible to the p r o b l e m of d e c i d i n g property P as we will see in the following.

Hence,

the latter p r o b l e m is

in fact undecidable. Let c and d be two new letters,

that are not c o n t a i n e d in any of

the E~ c o n s i d e r e d so far, and let F = E U {c,d}. We will describe an l

e f f e c t i v e process that, given two words u,v 6 E*, yields a finite oresentation

(E4;R 4) satisfying the following equivalence:

(*) The m o n o i d

MR4 p r e s e n t e d by (E4;R 4) has p r o p e r t y P if and only if

w U

~

V.

So let u,v 6 E*. Using the c o n s t r u c t i o n of Lemma finite p r e s e n t a t i o n

MR

is trivial,

2.9 we obtain a

(P;R ,v) such that either u +~+ v and the m o n o i d

or u ~

v and the monoid

M R is e m b e d d e d in M R

UrV

Further,

define E4 = F U E2 and R 4 = Ru, v U R 2. Then the finite

presentation J~R

. ~,V

(E4;R4)

describes the free product MR4 of the monoids

and M 2. Obviously,

the p r e s e n t a t i o n

(E4;R 4) can be c o n s t r u c t e d

UrV

e f f e c t i v e l y from u and v, since the p r e s e n t a t i o n s

(E;R)

and

(£2;R2)

are given in advance. It remains to verify that the monoid

MR4 p r e s e n t e d by (E4;R4) does w

indeed satisfy e q u i v a l e n c e monoid

MR

is trivial,

(*). So assume first that u +~+ v. Then the

i.e.,

UrV

MR

~ {e}, and hence,

MR4 =

UrV

MR

*M 2 ~ M 2. Now M 2 h a v i n g p r o p e r t y P, and P b e i n g an invariant u,v p r o p e r t y imply that MR4 does have p r o p e r t y P. If on the other h a n d we have u ~

v, Lemma 2.9 and T h e o r e m 2.11 y i e l d the f o l l o w i n g chain

of embeddings:

MI

~ MR ~ MR

~ MR4. Hence, by the choice of M I MR4

U,V

does not have p r o p e r t y P. Thus, e q u i v a l e n c e

(*) is satisfied,

i.e.,

the word p r o b l e m for M R is indeed e f f e c t i v e l y r e d u c i b l e to the p r o b l e m of deciding p r o p e r t y P. This c o m p l e t e s the proof of T h e o r e m 2.7.

D

V . ~ A u t o m a t a for R e d u c t i o n System s The d e c i s i o n problems c o n c e r n i n g algebraic p r o p e r t i e s of finitely p r e s e n t e d monoids that we stated in the p r e v i o u s section are all un-

229 decidable when

in general.

However~

we will prove

they are only c o n s i d e r e d

stricted scribe

classes

the c l a s s e s

present

mentioned

Recall

petty. *

that we will

that y i e l d

that are a s s o c i a t e d

that a finite

there

reduction

termination

If R is a finite

complete

is a unique

system

property

and we for r e c o g n i z systems.

the d e c i d a b i l i t y

of ~ from u is u n i f o r m for finite

R on E is c a l l e d

system on E, then w o r d ~ 6 Irr(R) effectively.

in R, we c o n c l u d e

complete

complete

and the C h u r c h - R o s s e r

reduction

A

word p r o b l e m

use,

automata

with reduction

irreducible

u, and given u, u can be d e t e r m i n e d

putation

re-

section we de-

above.

A

u +~

by c e r t a i n

In this

will be very useful when p r o v i n g

if it has the finite

w o r d u 6 E*,

systems

they can be solved

presented

systems.

constructions

languages

constructions

results

for m o n o i d s

reduction

of r e d u c t i o n

some t e c h n i c a l

ing c e r t a i n Thes e

of finite

that

reduction

that

systems

pro-

for each

such that

Since

the com-

the u n i f o r m

is e f f e c t i v e l y

de-

cidable: INSTANCE:

A finite

complete

reduction

system

R on ~, and two words

u,v 6 X*. QUESTION:

Are u and v c o n g r u e n t

In general, tion

system

systems nomial

it is u n d e c i d a b l e

is complete.

are considered,

we will

if

sometimes

llt > make

restricted." So recall

right(R)

modulo

recall

R-{(/,r)}.

system

Systems

R on Z, there

tion~

given

determined duction whenever

that

exists

for each

this

6 R. In what

Thus,

R on E is c a l l e d m o n a d i c c E U {e}, where it is c a l l e d

seems a p p r o p r i a t e .

special

I nor r can be r e d u c e d

finite

complete

"reduced"

complete

reduction to R w i t h

systems the

system

are c a l l e d equivalent,

same congruence. system

d e a l i n g with

our a t t e n t i o n

by

reduction

and e q u i v a l e n t

equivalent

when

we can r e s t r i c t

follows

that are even more

R on Z is c a l l e d non-

finite

defining

in p o l y -

= {e}.

system

two r e d u c t i o n

same a l p h a b e t

effectively.

system

6 R, n e i t h e r

a unique

reduc-

length-reducing

of this type are also c a l l e d

R the n o n - r e d u n d a n t

systems

systems

6 R}, and that

R' is n o n - r e d u n d a n t

IRR(R v) = Irr(R) . Here they are on the

(/~r)

?

system R on Z is c a l l e d

(/,r)

right(R)

right(R)

that a r e d u c t i o n

It is known

R' on ~ such that

(/,r)

with

if for each rule

some authors.

rule

and satisfies

6 E*:

if only

v hold

finite

can be a n s w e r e d

a reduction

that a r e d u c t i o n

= {r 6 E*I~l

Finally

hand,

use of r e d u c t i o n

if it is l e n g t h - r e d u c i n g

redundant

Here

does u + ~

or not a given

this q u e s t i o n

IrJ for each

if it is l e n g t h - r e d u c i n g

R, i.e.,

whether

On the other then

time ([Ka-Kr-McN-Na]).

length-reducing

modulo

In a d d i -

R' of R can be

finite

complete

to n o n - r e d u n d a n t

reones

if

230

In the f o l l o w i n g ognizing

certain

we describe

languages

The first one has already only m e n t i o n e d Theorem

3.1.

following

here

associated

There e x i s t s

OUTPUT:

A deterministic Irr(R)

next c o n s t r u c t i o n

finite

Theorem

reduction

systems.

and t h e r e f o r e

it is

construction

that

solves

the

3.1

system

3.2.

following

R on Z.

state a c c e p t o r words modulo imolies

set of irreducible deals with

~u 6 S: u ~ v} of a regular reduction

system

of i r r e d u c i b l e

R on Z, the

the

recognizing

the

set

R.

that for each finite words

is a regular

set of d e s c e n d a n t s

set S ~ Z* with r e s p e c t

reduction

language.

L(S,R)

The

= {v 6 Z*I

to a finite monadic

R on Z.

There

exists

an e f f e c t i v e

construction

that

solves

the

task: A finite m o n a d i c

OUTPUT:

before,

rec-

of completeness.

an e f f e c t i v e

reduction

In particular,

INPUT:

with finite

of a u t o m a t a

task: A finite

Theorem

constructions

been p r e s e n t e d

for reasons

INPUT:

system

three

reduction

acceptor

A with m states

A finite

state

acceptor

system

R on Z, and a finite

recognizing

the

subset

A* w i t h m states

state

S of Z*.

that r e c o g n i z e s

the

set L(S,R). Proof.

Let

R be a finite

r 6 Z U {e}, let D(r) all l e f t - h a n d

reduction

= {l 6 Z*l{l,r)

system on Z. For

E R},

i.e.,

D(r)

sides of rules of R that have r i g h t - h a n d

let S be a regular ministic

monadic

finite

subset

of Z* that

state acceptorr A =

is r e c o g n i z e d

(Q,Z,6,qo,F),

is the

by the n o n d e t e r -

where

Q =

{qo,ql ..... qm_1 } is the finite se__~to~f >tate______~s, 6: Q×Z ~ P(Q) transition of final is the

function,

states

To obtain

an a c c e p t o r transitions

is as follows: some

subsets

suppose

states qi,qj

from qi to qj with transition

P(Q)

denotes

state,

the power

as usual

([Ho-U1]).

for L(S,R)

we m o d i f y

if possible.

that

for

The

set of Q,

function

the a c c e p t e r

a 6 Z,

i.e.,

set P(Q)

6: Q×Z ~ P(Q)

idea of adding

some letter

is the

and F ~ Q is the

A by ad-

transitions

some w o r d I £ D(a),

6 Q, qj 6 6(qi,l) . Then we must add a t r a n s i t i o n

label a,

is to capture

from qi to qj with

initial

of Q. The t r a n s i t i o n

to Q×I* ~ P(Q)

ding c e r t a i n

and

of A. Here

set of all

is e x t e n d e d

qo 6 Q is the

set of

side r. Further,

if qj ¢ 6(qi,a).

the notion

th&t

label a is e q u i v a l e n t

The

since

intent

of adding

this

I ~ a, a t r a n s i t i o n

to a sequence

of t r a n s i t i o n s

231 from qi to qj with label I. £ £ D(e),

Further,

and some states qi,qj

suppose that for some word

E Q, qj E 6(qi,/).

and each qk 6 Q, if qk E 6(qj,a),

qi to qk with label a, if qk { 6(qi'a)" transition

Then for each a 6 Z

then we must add a transition

The intent of adding this

is to capture the notion that since la ~ a, a transition

from qi to qk with label a is equivalent from qi to qk with label la. qj is a final state,

to a sequence of transitions

In addition,

if this situation occurs and

then qi also becomes

a final state.

This whole

process must now be iterated until no further transitions and no additional

final states can be introduced.

IEI'm 2 iterations

suffice.

It is clear that this basic c o n s t r u c t i o n for

a subset of L(S,R).

transitions

On the other hand,

is not difficult

will lead to an accemtor

since the process of adding

to show that the resulting

construction

the set L(S,R).

outlined above

can be added

This means that

is iterated until no further transitions

actually recognizes

P:

from

finite

can be added,

it

state acceptor

Below a formal description

of the

is given.

begin INPUT:

A finite monadic

reduction

system R on E, and a finite

state acceptor A = (Q,E,6,qo,F); (1)

z + I;

(2)

while

[3)

begin for all qi,qj

z <

IZI.IQI 2 do

(4)

if qj E

6 Q and all r 6 E U {e} do

U 6(qi,l) lED (r) b e~in if r = e then

(5) (6)

then

begin for all a E E and all qk 6 Q d__o

(7)

if qk 6 6(qj,a)

(8) 9)

6(qi,a) ÷ 6(qi,a) U {qk}; if ~. E F and qi ~ F then F ÷ F U __{qi ]

and qk { 6(qi~a)

then

end 10)

else

if qj { 6(qi,r)

11)

6(qi~r)

then

+ 6(qi,r)

U {qj}

end; 12)

z ÷ z+l end end. D

SO for a finite monadic S ~ Z*, the set L(S,R) ification

reduction

system R on Z and a regular

is also regular,

of S, one can derive a formal

fact, this process can be performed easily from its formal description.

and given

R and a formal

specification

in polynomial

of L(S,R).

set specIn

time as can be seen

232 Book, certain

Jantzen,

infinite

on a proof

monadic

together

Thue

due to Berstel,

In Book and Otto

there

and W r a t h a l l

with

[Bo-Ot]

duction

some remarks

the degree system

Our

final

systems.

stated T h e o r e m

proof

appeared

on h o w to e x t e n d

is based

in Book

for a special

case

time b o u n d depends

construction

stems from Book,

deals with

complete

Jantzen

is taken

3.3.

the c o n g r u e n c e

There exists

task:

INPUT:

A finite m o n a d i c finite

on the re-

class

[S]R =

congruence Proof.

Given

finite

state acceptor

well-known

techniques

ministic

finite

set S I = L(S,R) Since

complete

state a c c e p t o r

solves

the

R on Z, and a

subset

S of Z*. the

subset

system

3.2.

the set L(S,R)

by

Using the c o n s t r u c t i o n

of

state a c c e p t o r

AI =

R on Z and a

S of Z*, we can con-

C that recognizes

R. By c o m b i n i n g

theory

we n o w obtain

(QI,~,61,qo,FI)

the

A* and C using a deter-

recognizing

the

n Irr(R).

the m e m b e r s h i p

system R is complete,

problem

descendant for

A I . It remains pushdo~

from R and A I.

a word u 6 Z* is in

modulo

R is in S I. Thus,

[S] R we can p r o c e e d to d e t e r m i n e

w, and then w 6 S I is t e s t e d u s i n g

by a d e t e r m i n i s t i c tible

that

B that r e c o g n i z e s

A* for r e c o g n i z i n g

words modulo

if its irreducible

state a c c e p t o r

system the

reduction

the

of T h e o r e m

Given a word u 6 Z*, u is reduced scen d a n t

reduction

automaton

from automata

the r e d u c t i o n

if and only

the proof

IS] R.

3.1 we also get a finite of i r r e d u c i b l e

set and

that also

However,

construction

A recognizing

pushdown

state a c c e p t o r

set Irr(R)

solving

complete

A recognizing

the c o n s t r u c t i o n

a result

[Bo-Ja-Wr].

an e f f e c t i v e

a finite m o n a d i c

struct a finite

Theorem

class

system,

is a

[Bo 84].

state a c c e p t o r

A deterministic

applying

reduction

and W r a t h a l l

from

following

OUTPUT:

A.

R under c o n s i d e r a t i o n .

R is a finite monadic

Theorem

presented

state a c c e p t o r

U [u] R of a regular set S ~ Z*. It implies that this class u6S d e t e r m i n i s t i c c o n t e x t - f r e e language, w h e n e v e r S is a regular

given b e l o w

[Bo 83]. is given

it. The a l g o r i t h m

in the size of the finite

of the p o l y n o m i a l

3.2 for

The proof given here

a sketch of which

a detailed

is also p o l y n o m i a l

However,

[Bo-Ja-Wr]

to p r o v e

automaton

[S] R for

as follows:

its irreducible

the d e t e r m i n i s t i c

definite

that this task can be p e r f o r m e d B that

is e f f e c t i v e l y

construc-

233

Let # be a letter that is not already c o n t a i n e d set Q I × ( Z U {#}) is to be the set of p u s h d o w n (qo,#)

in ~. Then the

symbols of B, where

serves as the initial symbol on the p u s h d o w n store. The transi-

tions of B are o r g a n i z e d store c o n t a i n s the bottom,

in such a m a n n e r that w h e n e v e r the p u s h d o ~

(qo,#), (ql,al) ~(q2,a2) ..... (qm,am), where

(qo,#)

and q l , . . . , q m 6 QI' a 1 ' ' ' ' ' a m 6 Z, then we have

61(qi'ai+1 ) = qi+1

for all i, 0 < i < m. To facilitate this b e h a v i o r

the first c o m p o n e n t of the t o p - m o s t symbol on the pushdown

store at

each given m o m e n t is also part of the actual state of B, i.e., top-most

is on

symbol of the p u s h d o w n

store is

if the

(q,a), then q is also stored

as part of B's actual state. Now we say that a letter a' 6 Z is pushed onto the p u s h d o w n store to mean that the symbol

(61(q,a'),a')

is p u s h e d

onto the store, where q 6 QI is the state symbol of A I that is part of B's actual state at this very moment.

Using this c o n v e n t i o n the be-

h a v i o r of B can be d e s c r i b e d by the f o l l o w i n g two p r o c e d u r e s READ and SEARCH. READ:

I n i t i a l l y a READ o p e r a t i o n

is attempted.

If the input tape is non-empty, onto the p u s h d o w n store,

read an input symbol, p u s h it

and go to SEARCH; otherwise,

accept

if and only if the state symbol of A I that is part of B's actual SEARCH:

state b e l o n g s to F I.

Pop from the p u s h d o w n store the longest string I occurs as l e f t - h a n d side of a rule detected,

then restore the p u s h d o w n

c o n d i t i o n and go to READ.

(/,r)

(if any)

that

in R. If no such I is

store to its p r e v i o u s

If such an I is detected,

the c o r r e s p o n d i n g r onto the p u s h d o w n store

then push

(we assume an

o r d e r i n g of rules with 1 on the l e f t - h a n d side and choose the first such r). Go to SEARCH. Obviously,

the above c o n s t r u c t i o n yields a d e t e r m i n i s t i c p u s h d o w n

a u t o m a t o n B. Now given input u 6 Z*, B finally halts with empty input tape,

its p u s h d o w n

store c o n t a i n i n g

(qo,#),(ql,al),...,(qm,am),

w = a l a 2 . . . a m is an irreducible w o r d modulo mined from u by a l e f t - m o s t reduction,

and since R is complete,

the unique irreducible d e s c e n d a n t of u modulo and only if w 6 S I = L(S,R)

where

R. Actually, w is deterw is

R. Hence, u C [SI R if

N Irr(R), which a c c o r d i n g to the construc-

tion of B holds if and only if qm 6 F I, since qm = 61(qo'W)" Hence, B accepts on input u if and only if u 6 [Si R . This c o m p l e t e s the proof of T h e o r e m 3.3.

234

V.4.

Deciding

Algebraic

properties

Complete

Reduction

Systems

As we saw all undecidable

the p r o b l e m s

in general.

through

a presentation

system,

then

problems

mentioned

where

before

the r e m a i n i n g

problems

to m o n o i d s

when

m 6 M, there

? Here we will

in this

while

will be shown to b e c o m e presented

by finite

reduction

normal

in solving

I to 3 and

in the next

decidable

monadic

form

the decision

see that p r o b l e m s

situation,

2 are

M is given

complete

is a unique

help

by Finite

of Section

a monoid

R is a finite

does this p r o p e r t y

decidable

restricted

On the other hand, (K;R),

5 to 6 b e c o m e

Presented

listed at the b e g i n n i n g

for each element

u m 6 Irr(R) . How much

of M o n o i d s

section

when b e i n g

complete

reduction

systems. Let

R be a finite

the f o l l o w i n g

three

complete

statements

reduction

system on ~. Then o b v i o u s l y

about the m o n o i d

M R presented

by

(Z;R)

hold: (I) M R is trivial (2) M R is finite

if and only if and only

(3) M R is c o m m u t a t i v e Given Irr(R) this

if a ~ if the

if and only

R, a d e t e r m i n i s t i c

can be c o n s t r u c t e d

if ab ~ +

finite

Tog e t h e r Theorem

these 4.1.

INSTANCE:

complete

reduction

observations

The f o l l o w i n g A finite duction

induce

from which

In addition, systems

(Z;R), w h e r e

2. QUESTION:

Is the m o n o i d

M R finite

3. QUESTION:

Is the m o n o i d

M R commutative

In c o n t r a s t

through

to the r e s u l t s

that the p r o b l e m

a finite p r e s e n t a t i o n

decidable, reduction

even

Now we want or not a m o n o i d free monoid,

and

that c o n t a i n

the u n i f o r m w o r d prodecidable.

result.

R is a c o m p l e t e

re-

through

? ?

just given N a r e n d r a n

(Z;R)

whether

to p r e s e n t a t i o n s

complete

whether

or not

M R given

or not remains involving

rules only

reduction

un-

complete

[Na-O'Dun].

5 and 6. Is it decidable

a finite

is it d e c i d a b l e

and O ' D u n l a i n g the monoid

is c a n c e l l a t i v e

length-reducing

to turn to p r o b l e m s given

set of

?

of d e c i d i n g

if it is r e s t r i c t e d

systems

the

system on Z.

M R trivial

have o b s e r v e d

recognizing

are decidable:

Is the m o n o i d

I. QUESTION:

E Z.

the c a r d i n a l i t y

is e f f e c t i v e l y

the f o l l o w i n g

problems

presentation

is finite.

ba for all a,b

state a c c e p t o r

effectively,

set can easily be determined.

blem for finite

e for all a 6 Z. set Irr(R)

whether

system

it ~s a group

is a

? So far

235

only p a r t i a l results c o n c e r n i n g these problems were known,

in [Ot-]]

it is shown that the former p r o b l e m can be solved in linear space for all p r e s e n t a t i o n s

i n v o l v i n g finite l e n g t h - r e d u c i n g c o m p l e t e r e d u c t i o n

systems, and Book

[Bo 82] proves that the latter p r o b l e m is solvable

in p o l y n o m i a l time for all p r e s e n t a t i o n s c o m p l e t e r e d u c t i o n systems, all p r e s e n t a t i o n s and complete.

i n v o l v i n g finite monadic

and that it is solvable in real time for

involving finite reduction

systems that are special

In the f o l l o w i n g we want to prove that these p r o b l e m s

become d e c i d a b l e when they are r e s t r i c t e d to m o n o i d s given through finite complete r e d u c t i o n

systems.

Recall that a finitely p r e s e n t e d

m o n o i d M is free if and only if it has a p r e s e n t a t i o n of the form (F;~)

for some finite a l p h a b e t F, i.e.,

if and only if it is iso-

morphic to the m o n o i d F*. So let R be a fixed finite complete r e d u c t i o n system on a finite a l p h a b e t E. We are interested in finding out w h e t h e r or not the m o n o i d M d e f i n e d by the p r e s e n t a t i o n

(E;R)

t r a n s f o r m the given p r e s e n t a t i o n p r o p e r t i e s of this normal

is free. For doing so we first

into a certain normal form. U s i n g the

form we can then c h a r a c t e r i z e a set of free

g e n e r a t o r s of the m o n o i d M, p r o v i d e d M is indeed a free monoid. Finally, we show how this c h a r a c t e r i z a t i o n can be turned into an a l g o r i t h m for testing whether or not M is a c t u a l l y free. Since for each finite complete r e d u c t i o n alent one that is finite, complete,

system,

there is an equiv-

and n o n - r e d u n d a n t and that can be

d e t e r m i n e d e f f e c t i v e l y from the former,

the first step in our normal-

ization of the given p r e s e n t a t i o n

c o n s i s t s in c o m p u t i n g a finite

(Z;R)

complete n o n - r e d u n d a n t r e d u c t i o n system R

on E such that R and R O

equivalent.

Then the o r d e r e d pair

tion of the m o n o i d M. Further,

are O

(E;R o) is another finite p r e s e n t a -

R° b e i n g c o m p l e t e

implies that the

empty w o r d e does not occur as l e f t - h a n d side of a rule of R , i.e., + o C left(R o) _ N o w assume that the r e d u c t i o n system R form

(a,w)

c o n t a i n s a rule of the o for some a 6 Z. Since R° is n o n - r e d u n d a n t and complete,

this means that the letter a does n e i t h e r occur in w nor in any other rule of R o. Hence, a p p l i c a t i o n of an e l e m e n t a r y Tietze t r a n s f o r m a t i o n of type 4 results in the p r e s e n t a t i o n

(E-{a};Ro-{(a,w)}) . Obviously,

Ro-{(a,w) } is a finite complete n o n - r e d u n d a n t reduction Z-{a}, and by Lemma 1.3 the p r e s e n t a t i o n

system on

(Z-{a};Ro-{ (a,w) } ) still de-

fines the monoid M. Iterating this process we finally obtain a finite presentation

(Z1;RI)

of M such that R I is a n o n - r e d u n d a n t c o m p l e t e re-

duction system s a t i s f y i n g left(R I) Q ZI = ~" Thus, the left-hand side of

each rule of R I is a word of length at least 2. The p r e s e n t a t i o n

236

(EI;R I) Lemma

4.2.

(E;R) ates

is a n o r m a l

Proof.

then

Let

~%1R g i v e n

there

exists

E = {al,a2,...,an}, that

the m o n o i d

of r a n k m for

a.

the

same

word

element

M R ~ F*,

j = 1,2, .... m. that

exists

of

v.3 6 Z* such

Since

a subset

and

I

presentation of E that

o

of the f o r m

freely

gener-

element

of

present

the

u.

(E;R)

there that

exists

that

for e a c h bj

Then

system it is

an a l p h a b e t

M R = E*/+~

6 r* such

1

reduction

is free.

~ F*.

So for

a. a n d u. 1

F =

represent

1

6 F, t h e r e

exists

a

a n d v. r e p r e s e n t the same e l e m e n t of M R. 3 3 no b i~ 6 F r e p r e s e n t s the i d e n t i t y of M R, and so vj~ # e, Further,

hand,

the w o r d s

an o c c u r r e n c e

MR a s a i £ same

vj

6 E* can be c h o s e n

that

Hence,

element

if a i + ~

e,

,k}

e.

some

letters

ul£ represents

the

same

,

and the

of M R. Since

a i +~

i = 1,2,...,n.

v. = a. a . . . . a. for 3 11 12 ik

I 6 ~I,2, .

bj

in such, a w a y

of a l e t t e r , la' 6 E w i t h

u i = e if and o n l y

,aik 6 E. F o r e a c h . . . . .

ai2, .

R be a f i n i t e by

b

Let b. 6 F, and a s s u m e 3 all,

m such

a word

M R. A n a l o g o u s l y ,

that

no vj c o n t a i n s

On the o t h e r

let

some m < n. Hence,

6 E, t h e r e

1

(E;R).

by a f i n i t e

JgR p r e s e n t e d

{ b l , b 2 , . . . , b m} of c a r d i n a l i t y each

presentation

monoid.

on E such free

If the m o n o i d

is free, this

f o r m of the

word UilUi2

M R ~ F*,

. U i k c F* r e -

this

implies

that

b_3 = ui u . . . . u . By the c h o i c e of v:,j we h a v e u i l # e for all l I 12 Ik i m p l y i n g t h a t k = I, i.e., v 6 ~. 3 Take ~ = {a 6 E l E b 6 F: a = v }. T h e n ~ = {vjlj = 1,2 ..... m} o 3 3 o a s u b s e t of E that f r e e l y g e n e r a t e s M R. m Assume of N1 t h a t

that

is a u n i q u e same

the m o n o i d

M is free.

generates

¢~1. H e n c e ,

freely word

element

of

u a 6 ~*o such

M. D e f i n e

that

4.3.

Proof.

to(U) . ~*] that

For

Since

all

to(V).

Since

we p e r f o r m e d ,

that

u a 4= e f o r

ing

from Lemma

all

each 4.3.

each

we have

letter

i.e.,

the

a subset

E

o

a C ~1" t h e r e

a and Ua d e f i n e

the

following.

w 6 Y~, u ~

C ~Oe,

if a n d o n l y

[,[I > 2 f o r

tion

exists

letter

u + ~ I v if a n d o n l y

W for

B u t to{U),~0(V)

to(u) + ~ ] ~(v)

there

each

a homomornhism_ to: 5-I -~ ~*o b y t a k i n g

u,v 6 ~,

t0(W) ~

for

a +~I ua'

to(a) = u a for all a 6 E I . T h e n we h a v e Lemma

Then

is

rule

a n d ~o f r e e l y

if to(u) = [~(v).

(£,r)

if %0(u) = <0(v) .

v if

and only

generates

This means

a 6 I~1 • H e n c e ,

~ implying

D

6 R1 a c c o r d i n g

~I c_ Irr(R1).

if

to

the

normaliza-

in p a r t i c u l a r ,

we c a n c o n c l u d e

the

follow-

237

C o r o l l a r y 4.4. For each letter a 6 E I and each word u 6 E$, u

a, The

then

either

following

of the set ~

o

u = a or

two

lemmas

lui a will

if

O. help

us

to

derive

a characterization

that freely generates the m o n o i d M. +

Lemma 4.5. right(R I) ~ X I, i.e., form

R I does not contain any rule of the

(£,e).

Proof.

Assume

Then I ~ I hand~

to

the

contrary

e, and hence ~(£)

we h a v e

Finally,

I~(/)

I >

1/I

that

(/,e)

= ~(e)

6 R1 f o r

~ 2 contradicting

let ~2 = E1-Eo"

some word

g C ~.

= e by Lemma 4.3. On the other the

equality

just

stated.

Then this subset of ~I can be d e s c r i b e d

as follows. Lemma 4.6. Z2 = right(R1) word 1 6 Z~ such that

N ZI,

i.e.,

for each a 6 I2, there exists a

(/,a) 6 RI, and furthermore,

these are the only

rules of R I with r i g h t - h a n d sides of length I. w

Proof.

Since Z I ~ Irr(R1) , we can conclude that u a ~la for each letter +

a 6 Z I. For a 6 Z2' the fact that u a 6 Z+o then implies that u a ~I a, which yields a E r i g h t ( R I) by Lemma 4.5. Thus, N o w assume that

(/,b)

Z2 ~ right(R1)

6 R I for some letter b 6 Z O. Then

and since u a 6 Z+o for each letter a 6 Z I, this gives while

N ZI. Ili > 2,

l~(1) I >

Ill > 2,

l~(b) I = Ibl = I, thus c o n t r a d i c t i n g Lemma 4.3. Hence 12 =

right(R I) n Z I . Thus,

[]

Z ° = Z1-(right(R1)

from R I. Hence,

~ ZI),

i.e.,

Z ° can easily be o b t a i n e d

the m o n o i d M is free if and only if it is freely

g e n e r a t e d by the set Z ° = Z1-(right(R1)

n ZI).

It remains to show that

it is e f f e c t i v e l y d e c i d a b l e w h e t h e r or not the set Zo freely q e n e r a t e s the m o n o i d M. To this end observe that Lemma 4.6 does not only c h a r a c t e r i z e the letters that we m u s t try to get rid off, but t o g e t h e r with C o r o l l a r y 4.4 it also shows that these letters can be e l i m i n a t e d by e l e m e n t a r y Tietze t r a n s f o r m a t i o n s of type 4 p r o v i d e d the m o n o i d M is free. So we are going to p r e s e n t an a l g o r i t h m that p e r f o r m s the following task:

If the m o n o i d M is free,

the p r e s e n t a t i o n

(ZI;RI)

then this a l g o r i t h m will t r a n s f o r m

into a p r e s e n t a t i o n that clearly indicates

the fact that M is a free monoid.

On the other hand,

if M is not free,

then this a l g o r i t h m will realize this fact at some stage of ~ts computation,

and it will then reject the input.

238

A l g o r i t h m 4.7. INPUT:

A finite alphabet reduction

(I)

begi n

E1~ and a finite complete non-redundant

system R I on E I such that Z I c Irr(R1);

if e 6 right(R I) then reject;

(2)

E 2 + right(R I) n /I;

(3)

while E 2 # ~ d_£o

(4)

begin choose a letter a 6 E 2 together with a rule (/,a) 6 RI; if f contains an occurrence

(5)

of the letter a then

rejej~; (6)

E 2 + E2-{a};

(7)

Z I ÷ Z1-{a};

(8)

R I + RI-{ (£,a) }; Substitute each occurrence

(9)

of a in each rule of R 1

by the word l end; (10)

if R I ~ {(w,w) !w 6 E*} then accept else reject end.

Lemma 4.8. Let R I be a finite complete non-redundant on E I such that E I ~ Irr(R1).

reduction

system

Then algorithm 4.7 accepts on input

(EI;R I) if and only if the monoid M presented by

(ZI;RI)

is a free

monoid. Proof.

If M is a free monoid,

freely generates

then there is a subset E ° of E I that

M, and by Lenmla 4.5 R I contains no rule with right-

hand side e, i.e., e ~ right(R1). in line

(I). Let E2 := EI-Zo"

Hence,

algorithm 4.7 does not reject

By Lemma 4.6 we have E 2 = right(R I) N

~I" For a 6 ~2' we thus have at least one rule with right-hand so let I E E~ such that Ill a = O by Corollary line

15). In lines

type 4 is performed (l,a)

(l,a)

6 R I. Then

4.4. Hence,

(6) to

which

in

(9) an elementary Tietze transformation

that results

from the presentation

the form

i~i > 2 implying that

a l g o r i t h m 4.7 does not reject in deleting

(EI;RI).

Also,

E 2 giving the set E~. Let (E';R') denote the presentation tion yields.

side a.

of

the letter a and the rule

a is deleted

from the set

of M this Tietze transforma-

Then for each b 6 El, R' contains

at least one rule of

(v,b) with v 6 E'*. Now v 6 E'* = (EoUE i)* ~ I T and v +~I b,

imply that

Ivl b = O by Corollary

4.4. Thus,

the loop

is p e r f o r m e d until E 2 = ~ is reached without rejecting When this loop is left, moment is of the form

the p r e s e n t a t i o n

(Eo;R')

(3) - (9)

in line

(5).

of M we have at this

for some reduction

system R' _c E*×E*.o o

239 But M is f r e e l y g e n e r a t e d by Zo' w h i c h implies that R' ~ {(w,w) I w 6 Z~},

i.e., a l g o r i t h m 4.7 accepts.

N o w assume to the c o n t r a r y that a l g o r i t h m 4.7 accepts on input (ZI;RI). Then the p r e s e n t a t i o n tation of the form

(Zo;R')

(ZI;RI)

is t r a n s f o r m e d into a presen-

for some subset I o of Z 1 and some r e d u c t i o n

system R' 5 {(w,w) lw 6 Z~}° But d u r i n g this p r o c e s s only Tietze transf o r m a t i o n s are a p p l i e d to p r e s e n t a t i o n s of M starting with Hence,

(Zo;R')

(ZI;RI).

is itself a p r e s e n t a t i o n of M, and so M is a free

m o n o i d that is freely g e n e r a t e d by Z O.

D

Hence, we have shown the f o l l o w i n g result. T h e o r e m 4.9. The f o l l o w i n g p r o b l e m is decidable: INSTANCE:

A finite p r e s e n t a t i o n

(Z;R), where

R is a c o m p l e t e reduc-

tion system on Z. QUESTION:

Is the m o n o i d M R given through this p r e s e n t a t i o n a free monoid ?

Before turning to the next d e c i s i o n p r o b l e m we give an example of a n o n - t r i v i a l p r e s e n t a t i o n of a free monoid. 2 Example 4.10. Let Z = { a , b , C , d l , d 2 , d 3 , f } and R = {(abC,dl),(adlC,d2), (ad~c,d3),(d2cb,f),(d3b,ad2f)}._ Then R is a finite n o n - r e d u n d a n t reduction

system on Z, and it can be c h e c k e d easily that R is n o e t h e r i a n

and confluent. Now Z 2 := right(R)

N X = {dl,d2,d3,f},

and I ° := ~-Z 2 = {a,b,c}.

On input (I;R) a l g o r i t h m 4.7 yields the p r e s e n t a t i o n (Zo; 3 22 3 3 22 3 {(a bcabc ~ bcabc b,a bcabc a bcabc b)}), thus p r o v i n g that the m o n o i d M R p r e s e n t e d by

(Z;R)

is a free monoid.

[]

O b s e r v e that we do not only decide w h e t h e r or not a m o n o i d given by a p r e s e n t a t i o n

(Z;R), where R is a finite complete r e d u c t i o n system

on Z, is free, but in the a f f i r m a t i v e we also determine the rank of this free m o n o i d by d i s p l a y i n g a set of free generators. Finally,

we want to show that the p r o b l e m of d e c i d i n g of whether

or not a m o n o i d given through a finite p r e s e n t a t i o n

is a group also

becomes d e c i d a b l e when it is r e s t r i c t e d to p r e s e n t a t i o n s finite c o m p l e t e r e d u c t i o n

systems.

involving

So let R be a finite complete re-

duction system on Z. Then the m o n o i d M given t h r o u g h the p r e s e n t a t i o n (Z;R)

is a group if and only if, for each w o r d w C Z*, there exists a

word w' 6 Z* such that ww' +~+ e. Obviously,

this is e q u i v a l e n t to

saying that for each letter a 6 I, there exists a word u a 6 Z* such

240

that

au a +

determine

e.

In the

reasonable

R is a c o m p l e t e these

show how

for t h e s e

words.

system,

we can t h e n

actually

satisfy

the

step we d e f i n e

of Z i n d u c t i v e l y 11

candidates

we will

reduction

candidates

As a f i r s t

following

above

a sequence

to e f f e c t i v e l y Using

easily

the f a c t

check

that

whether

congruences.

Z1,E2,...,Ei,...

of

subsets

by t a k i n g

:= {a £ I I B u , v

6 I*:

(uav,e)

6 R}

and El+ I := {a 6 I-Zi!gu,v. £ Z* (w,uav) 6 R} U I.. ! Then

we h a v e

viously,

I] ~ 12 ~

if E i = Ii+ I for

finite,

we c o n c l u d e

finite

number

that

... ~ I i ~ Ii+ I ~ some

the

of d i f f e r e n t

E l , E 2 , . . . , E n, w h e r e

Bw £ I~:l (uav,w)

i ~ I, then

above

chain

sets only.

Thus,

U j~1

... E

Ei =

of

n is the c a r d i n a l i t y

6 R or

U

X.J c - I. Oh-

Z..

j>1

Since

J

inclusions

E is

contains

it s u f f i c e s

of I, and this

a

to c o n s t r u c t can be done

effectively. Now

for e a c h

letter

If a 6 1 I, t h e n such

rule,

(uav,w)

,Ua

11

u

one

...u aim

6 R for

u

ai2 ai]

word

The m o n o i d

if a n d o n l y

Z = E n, w h e r e

(ii)

au a + ~ + e for all

if the

If c o n d i t i o n s

I, then Again

chosen.

N o w we take

Thus,

in this

through

way,

for e a c h

effectively.

the p r e s e n t a t i o n

two c o n d i t i o n s

the c a r d i n a l i t y

letter

are

(I;R)

is a

satisfied:

of Z;

a 6 In .

(i) a n d

it r e m a i n s

(ii)

are

to p r o v e

satisfied,

then

M is o b v i o u s l y

the c o n v e r s e

implication.

there

a w o r d w 6 E*

So a s s u m e

M is a group.

a w + ~ + e.

sequence Now

been

that

following

n denotes

Let a 6 I. S i n c e that

i >

u , v 6 I* and w 6 I~. 1 Then words

u a is d e t e r m i n e d

M given

(i)

that

already

u. N o t i c e

group

a group.

some w o r d s

some

Let w = a. a . . . . a. 6 Z~. 11 12 im l

6 I* h a v e

a corresponding

4.11.

Proof.

If a 6 l i + 1 - E i for

Im

:= vu

a C E n,

:= vu.

(w,uav)

,...,Ua.

a

ua

such rule.

12

Lemma

(uav,e) take

6 R or

we c h o o s e Ua.

and

a £ I , we d e t e r m i n e a w o r d u 6 E* as f o l l o w s . n a 6 R for some w o r d s u , v 6 Z*. We c h o o s e one

M is a group,

R is c o m p l e t e ,

of r e d u c t i o n s

(vl,e)

6 R yields

of the that

and

exists

so aw ~ e,

i.e.,

f o r m a w = V m ~ Vm_ I ~

v I 6 Z~.

For

each

i >

there

such

exists

a

4.. ~ v I ~ e.

I, vi+ I ~ v i i m p l y i n g

241

that

vi+ I = x l y a n d v. = x r y for some w o r d s x , y 6 E* a n d a r u l e (/,r) l of R. If v i E E i,* we h a v e r 6 Z[, w h i c h in t u r n g i v e s i 6 E*i+1. Hence, if v i E E * i, t h e n vi+1 6 ~*i+I " Bv- i n d u c t i o n this i m p- l i e s a w = v m 6 E*m" B u t ~I ~ ~2 ~ "'" E ~n j c ~, a n d so we a c t u a l l y see t h a t =

a w 6 En*' i.e.,

a 6 In.

In a d d i t i o n , so,

we p r o v e

by

.

au a ~ +

~lZj

-

Thus,

condition

we m u s t

show

induction

on

e holds.

that

Then

is s a t i s f i e d .

condition

i, that

Let a E El"

(i)

(ii)

for e a c h

u a = vu,

also holds.

To do

I and each

a 6 E.,

i >

where

u,v

1

6 E* are w o r d s

w

such

that

implies u

(uav,e)

aUa

= vxu,

is a rule

= avu

where

~

e,

u,v

of

since

R. T h i s

gives

M is a group.

{ ~* a n d x = u

...u a l•m

a

al1'ai2'''''aim

6 Ei such

that

(all ai2 . .a.lm,Uav) .

E . R. Hence, .

ailal2

Uai I +~

a i m ua'lm

u a v + ~ + e, w h i c h If a 6 El+l-El, u

ai2

e by

ai I

ailai2

induction

then

with

(uav,a ilal2 . . . . aim) u a .v x ~ +

in t u r n

E R or .almX

=

hypothesis.

Since

M is a

w

group, Lemma

this

implies

4.11.

Since

~+

e,

thus

completing

the p r o o f

of

[]

the

fectively, duction

au a = avxu

set

a n d the w o r d s u (a 6 E ) can be c o m D u t e d efn a n since the u n i f o r m w o r d p r o b l e m for finite c o m p l e t e re-

and

systems

E

is d e c i d a b l e ,

we h a v e

thus

derived

the

following

result. Theorem

4.12.

INSTANCE:

The

following

A finite duction

QUESTION: Given

time.

an a l p h a b e t system

As

a 6 El'

can be where

ponential plexity

time

shown

Example

seen

easily

problem

n ~

= 3n.

Further,

let

R is a c o m p l e t e

this

presentation

re-

n 6 ~ a n d a finite,

induction

£ R}+I.

or n o t for

on

ua

i,

a group complete

in p o l y n o m i a l

lu a I ~ i

So it m a y

for all

take

up to ex-

(a 6 En ) . F i n a l l y ,

au a ~

R which

e holds

the c o m -

depends

may be a r b i t r a r i l y

[Ba-Ot].

I be an

{al,bl,Cl,a2,b2,c2,...,an,bn,C JZi

through

the w o r d s

of w h e t h e r

and Otto

where

set En can be d e t e r m i n e d by

to d e t e r m i n e

Let

(E;R),

E of c a r d i n a l i t y

of the w o r d

4.13.

M R given

~ = m a x { l / I , Irl i (/,r)

by B a u e r

is d e c i d a b l e :

on ~.

R on E, the

of c h e c k i n g

complexity as

presentation system

Is the m o n o i d

reduction

problem

integer,

and

let

n} be an a l p h a b e t

Z = of c a r d i n a l i t y

R = {(alblCl,e),(a2b2c2,alCl)

,

on the high

?

242

( a 3 b 3 c 3 , a 2 c 2 ) , . . . , ( a n b n C n , a n _ I C n _ 1 ) } . T h e n R is a f i n i t e c o m p l e t e d u c t i o n s y s t e m on X. In a d d i t i o n , R is n o n - r e d u n d a n t . For X we get the f o l l o w i n g

sequence

XI,X2,...,Xi,...

re-

of s u b s e t s

of X: X I = { a l , b l , C l } , X 2 = { a 2 , b 2 , c 2} U X 1 , . . . , Z n = { a n , b n , C n} U Xn_ I = X. In p a r t i c u l a r , c o n d i t i o n (i) of L e m m a 4.1I is s a t i s f i e d . Further, ual

we get the f o l l o w i n g

= blC I, Ubl

= cla I, ucl

words

u a for a 6 Xn:

= alb I,

Ua 2 = b 2 C 2 U c l Ua I , Ub 2 = C 2 U c l U a Ia 2 ' u c 2 = u c Iu al a~b~, z z

Ua n

= bnC u u , ub = CnU u uc = u u a b . n Cn_ 1 an_ I n Cn-1 an-1 an' n Cn_ I an_ I n n Hence,

i.e.,

for e a c h i £ { I , 2 , . . . , n } ,

some of the c a n d i d a t e s

exponential tion

(ii)

Thus,

length.

of L e m m a

the m o n o i d

We h a v e

However 4.11

inverses

is n o t s a t i s f i e d ,

of the l e t t e r s

presentations.

Thus,

a

for c o n s t r u c t i n g

restricted

(Z;R)

Lemma

u

decidable, involving

the c a n d i d a t e s

(a 6 ~n ) , the p r o p e r t y In fact,

= blCla I He.

is not a group,

5 and 6 become

that c o n d i -

when

complete u a for the

of R of b e i n g c o m -

4.11 h o l d s

for all f i n i t e

in o r d e r to be a b l e to c h e c k w h e t h e r or n o t the

M R p r e s e n t e d by

following

(X;R)

very easily

since blUbl

to f i n i t e p r e s e n t a t i o n s However,

p l e t e w a s n o t u s e d at all.

monoid

by

seen t h a t b o t h p r o b l e m s

systems.

I = lUb i = lUc I = 2i+I-2, l 1 1 i n v e r s e s of l e t t e r s of X h a v e

it can be c h e c k e d

MR presented

they are r e s t r i c t e d reduction

for the

lUa

(I;R)

is a group,

version

it is s u f f i c i e n t

of the w o r d p r o b l e m

for

(X;R)

t h a t the is d e c i d -

able:

INSTANCE:

A w o r d w 6 Z*.

QUESTION:

Does w + ~ + e h o l d

w

Also

?

the r e s u l t c o n c e r n i n g

presentations, stated above

problem

for w h i c h the r e s t r i c t e d is d e c i d a b l e .

Thus,

as s p e c i a l c a s e s of t h e f o l l o w i n g Theorem

5 can be c a r r i e d version

of the w o r d p r o b l e m

the r e s u l t s o b t a i n e d general

over to f i n i t e

can be v i e w e d

result.

4.14 (lOt 86]) . Let C be a c l a s s of f i n i t e p r e s e n t a t i o n s

that there exists

an a l g o r i t h m

A(C) ((Z;R) ,w)

A(C)

satisfying if w ~

e

if w ~

e

=

such

243

for all p r e s e n t a t i o n s

(X;R)

from C and all words w 6 X *. Then it is

d e c i d a b l e w h e t h e r or not a given p r e s e n t a t i o n

(X;R)

from C defines a

free m o n o i d or a group.

V.5.

D e c i d i n g A l g e b r a i c P r o p e r t i e s of Monoids P r e s e n t e d by Finite M o n a d i c C o m p l e t e R e d u c t i o n Systems

In [Bo 83] Book has e s t a b l i s h e d a syntactic class of formulae,

called

linear sentences over X, such that each r e d u c t i o n system R on X induces an i n t e r p r e t a t i o n of these sentences. Hence,

these sentences

can be used to d e s c r i b e certain p r o p e r t i e s of the c o n g r u e n c e g e n e r a t e d by R. The m a i n r e s u l t of

[Bo 83] states that it is d e c i d a b l e w h e t h e r

or not a given linear sentence is true under the i n t e r p r e t a t i o n induced by a given finite monadic complete r e d u c t i o n system. Thus, a finite m o n a d i c c o m p l e t e r e d u c t i o n

for

system all those d e c i s i o n p r o b l e m s

can be solved by using the technique of linear sentences that are exp r e s s i b l e by these

sentences.

right d i v i s i b i l i t y problems,

P r o b l e m s of this kind are the left and the power problem,

b l e m for a finitely g e n e r a t e d submonoid, for finite sets,

etc. U n f o r t u n a t e l y

perties of cancellativity,

set p r o b l e m

it is not known w h e t h e r the pro-

idempotency,

e x p r e s s e d by linear sentences.

the m e m b e r s h i p pro-

the independent

Hence,

and t o r s i o n - f r e e n e s s can be

in order to solve our r e m a i n i n g

five d e c i s i o n p r o b l e m s at least for monoids p r e s e n t e d by finite monadic complete reduction Definition

systems,

we must devise d i f f e r e n t techniques.

5.1. Let M be a m o n o i d p r e s e n t e d by

left-cancellative It is c a l l e d

if, for all x,y,z 6 X*, xy ~

.right-cancellative

implies x + ~

y. Finally,

teft-cancellative

(X;R). M is called

if,

for

all

x,y,z

xz implies ~ ~

£ X*, xz ~

M is called cancellative,

z.

yz

if it is b o t h

and right-cancellative.

In what follows we will be dealing only with the p r o p e r t y of leftcancellativity,

but by symmetry a c c o r d i n g r e s u l t s can e a s i l y be ob-

tained for the p r o p e r t y of r i g h t - c a n c e l l a t i v i t y . are taken f r o m N a r e n d r a n and O ~ D u n l a i n g

The results p r e s e n t e d

[Na-O'Dun].

Lenuma 5.2. Let M be a m o n o i d p r e s e n t e d by

(X;R). Then the f o l l o w i n g

two s t a t e m e n t s are equivalent: (i) (ii)

M is l e f t - c a n c e l l a t i v e ; for all letters a 6 X and all words x,y 6 X*, ax + ~

ay implies

x~y. Proof.

Obviously,

(i) implies

(ii). To prove the reverse implication

244

assume

that

the m o n o i d

a word

of m i n i m a l

xy ~

xz a n d y

letter

a 6 E and first

choice

of x, a n d

(ii)

Corollary

5.3.

all

Let

second

either.

a n d let x 6 ~+ be

y,z

6 ~*

satisfying

x as x = au for

uy +

y ~

a(uy)

the c h o i c e

some

uz or uy ~

z contradicting

we h a v e

= xy ~

xz

of x. Thus,

uz.

the

IxI

= a(uz) = I,

i.e.,

[]

immediately

R be a c o m p l e t e (E;R)

exist

Now either

case

contradicting

by

condition

letters

there

I, we can w r i t e

uy + ~ + uz a n d

5.2 we can

M presented

following For

in the

satisfied

Lemma

that

[xl >

we h a v e

uz a g a i n

From

such

left-cancellative,

some w o r d u 6 ~+.

case

is not

monoid

length z. If

In the

a n d uy ~

M is n o t

conclude

reduction

the

following.

system

is l e f t - c a n c e l l a t i v e

on E. T h e n

if and

only

the

if the

holds:

a £ E a n d all

words

x,y

6 Irr(R),

ax + ~ + ay

implies

x = y. In p a r t i c u l a r , reduction contain Lemma

system

a rule

5.4.

there

of the

Let

If the m o n o i d exist

this

means

R presents form

that whenever

(ax,ay)

by

(Z;R)

Yl

(ii) (iii)

ay I +~+ ay 2, no rule of R can be a p p l i e d

Proof.

Assume

that

(x,y)

letter

M is n o t

of w o r d s

be a p a i r

Clearly, conditions

from

such

reduction

this

then

Finally, rule

means (ii)

Z.

that

of

ay I a n d ay 2,

the

that

and

that x ~

let

S consist

y but

ax + ~

set S is n o n - e m p t y .

[yl[

+ [y21

have

z I and

z 2 are

be f a c t o r e d

rz I + ~

t h a t Yl a n d Y2 are are

that

satisfied. (iii) aYl

R. T h e n

is m i n i m a l ,

irreducible,

If e i t h e r

irreducible

satisfying

ay for

and

let a

is s a t i s f i e d

1 = ax,

Yl

= xz1'

and hence,

ay I or ay 2 is a l s o

as well,

a n d ay 2 can be r e d u c e d

rz 2, w h i l e

as r = sbt,

of

So let

a n d Y2

and we are

by a p p l y i n g = xz2'

ay I = axz I = /z I ~ , r z I a n d ay 2 = axz 2 = /z 2 ~ rz 2. Since

(tzl,tz 2)

on

then

f r o m E.

condition

assume

(/,r)

we a l s o

minimal

R cannot

system

left-cancellative,

to b o t h

such

5.2,

S such

letter

(i) a n d

irreducible,

same

complete

left-cancellative,

f r o m E*

a 6 E. By L e m m a

be a c o r r e s p o n d i n g

done.

complete

then

# y.2 '

all p a i r s

(yl,Y2)

monoid,

a 6 E.

is n o t

a 6 E and y 1 , y 2 6 Irr(R)

(i)

some

where

R be a l e n g t h - r e d u c i n g M presented

a non-redundant

a left-cancellative

with

where

tz I

6 S. On the o t h e r

as

suffixes

z I # z 2,

of Yl

i.e.,

zI ~

s,t £ E* a n d b £ E,

Itzi[

<

[rzi[

~

~

[xzi[

aY2'

respectively,

z 2. Hence, such

tz 2 a n d btz I + ~ + btz 2. T h i s hand,

~Yl

and Y2'

the

implying

that

{tl

r can is

implies =

lyiI

for

245

i = 1,2, and thus,

(tzl,tz 2) £ S c o n t r a d i c t s the choice of

Hence~ no rule of R can be a p p l i e d to b o t h ay I and ay 2. Obviously,

the c o n d i t i o n given

(yl,Y2 } 6 S.

u

in Lemma 5.4 is not only necessary,

but it is also sufficient for M b e i n g not l e f t - c a n c e l l a t i v e . Lemma

5.5. Let R be a l e n g t h - r e d u c i n g complete r e d u c t i o n system on ~.

Then the f o l l o w i n g two statements are equivalent: (i)

the m o n o i d M p r e s e n t e d by

(ii)

there exist a letter a £ Z, and words l,r 6 Z* such that (a)

(a/,r) 6 R, and

(b)

L(RI,R)

A L(R2,R)

(Z;R)

is not l e f t - c a n c e ! l a t i v e ;

# ¢, where R I and R 2 are subsets of Z*

defined as follows:

Proof.

R I = a. (Irr(R)

R /./*),

R 2 = a.(Irr(R)

N (Z*-/.Z*)).

(i) ~

and

(ii) : A s s u m e that the m o n o i d M is not l e f t - c a n c e l l a t i v e .

Then by Lemma 5.4 there are a letter a C Z and i r r e d u c i b l e words y1,y 2 6 Z* such that Yl % Y2" aYl +

aY2'

and no rule of R can be

a p p l i e d to b o t h ay I and ay 2. W i t h o u t loss of g e n e r a l i t y we may assume that

lyll >

ly21. Since R is complete,

ay 2 have a common d e s c e n d a n t modulo reducible modulo

R. Hence,

we can c o n c l u d e that ay I and

R, which implies that ay I is

there is a rule

(a/,r)

6 R such that

Yl = lu for some u 6 Z*, and so we see that ay I £ R I := a.(IRR{R) aY2,

N 1-1*).

Since no rule of R can be a p p l i e d to both ay I and

I is not a p r e f i x of Y2' and so ay 2 £ R 2 := a-(Irr(R)

Thus, we see that L(RI,R) (ii) ~

N L(R2,R)

# ¢.

(i): Let a 6 Z and /,r 6 X* such that

L(RI,R)

N L(R2,R)

a.(Irr(R)

# ¢, where R I = a. (Irr(R)

N (Z*-/.I*)).

D (Z*-/-Z*)).

(a/,r) N 1.Z*)

6 R and and R 2 =

Then there are words ay I 6 R I and ay 2 C R 2

such that ay I and ay 2 have a common descendant, Since ay I 6 RI, we see that Yl 6 Irr(R)

i.e., ay I ~

ay 2.

and Yl = lu for some u 6 Z*.

Since ay 2 6 R 2, we see that Y2 6 Irr(R), and that 1 is not a p r e f i x of Y2" Hence, Yl # Y2' and so Yl ~ left-cancellative,

Y2" Hence, the m o n o i d ~ is not

u

N a r e n d r a n and O ' D u n l a i n g

[Na-O'Dun] have shown that it is u n d e c i d -

able in general w h e t h e r or not a m o n o i d p r e s e n t e d by a finite lengthreducing complete reduction hand,

system is l e f t - c a n c e l l a t i v e .

On the other

if R is a finite m o n a d i c c o m p l e t e r e d u c t i o n system on Z, then

for e a c h rule

(a/,r)

E R, the sets R 1 and R 2 are r e g u l a r subsets of

Z*, and finite state a c c e p t o r s r e c o g n i z i n g these sets can be constructed

246

effectively

(Theorem 3.1). But then finite state acceptors for L(R1,R)

and L(R2,R)

can be o b t a i n e d

(Theorem 3.2),

(ii) of Lemma 5.5 becomes decidable

implying that c o n d i t i o n

in this situation.

Thus, we have

shown the following. T h e o r e m 5.6

([Na-O'Dun]).

The f o l l o w i n g p r o b l e m is decidable: INSTANCE:

A finite p r e s e n t a t i o n

(Z;R), where

R is a monadic complete

reduction system on Z. QUESTION:

Is the m o n o i d M R given through this p r e s e n t a t i o n cancellative

As r e m a r k e d earlier, cancellativity.

left-

? a c o r r e s p o n d i n g result also holds for right-

C o m b i n i n g these results we obtain the following.

C o r o l l a r y 5.7. The following p r o b l e m is decidable: INSTANCE:

A finite p r e s e n t a t i o n reduction

QUESTION:

(Z;R), where R is a monadic complete

system on Z.

Is the m o n o i d M R c a n c e l l a t i v e

?

Next we want to deal with the p r o b l e m of the e x i s t e n c e of nontrivial idempotents of a m o n o i d given through a finite monadic complete r e d u c t i o n system. D e f i n i t i o n 5.8. Let M be a m o n o i d p r e s e n t e d by

(I;R). Then a word

u 6 Z* d e s c r i b e s an idempotent of M, if u 2 +~+ u. If, in addition, U ~

e, then u describes a n o n - t r i v i a l idempotent of M. As we saw in Section 2, it is u n d e c i d a b l e

in general whether or

not a m o n o i d M given through a finite p r e s e n t a t i o n non-trivial

idempotents.

monadic and complete,

If, however,

(Z;R)

contains

the r e d u c t i o n system R is

then those words that describe n o n - t r i v i a l

idem-

potents can be c h a r a c t e r i z e d as follows. Lemma 5.9. Let R be a monadic complete r e d u c t i o n u 6 Z* be a n o n - e m p t y

system on Z, and let

irreducible word. Then the following two state-

ments are equivalent: (i)

u describes a n o n - t r i v i a l

(ii)

(a) Hx,y 6 z*: u = xy and yx ~ e,

idempotent of the monoid M~; w

or (b) Hx,y 6 ~*, a £ Z: u = xay,

ayx~e,

and ayxa ~ a.

247

Proof.

* y x ~ e,

If u = xy w i t h

s c r i b e s an i d e m p o t e n t . 2 * = x a y x a y ~ x a y = u,

thesis

On the

there

other

= y'y

(a) This

result

will

Definition

5.10.

of the

5.11.

let a 6 E. trivial Proof.

an

that

yx

of u.

= x'y' ayxa

By h y p o -

u ~

then Since

~ e,

e. Thus,

u2 +

that = xa. (ii)

~ a.

with

u

R is m o n a d i c ,

i.e.,

= x'y'

u = x(ay)

then

M R.

idempotent,

u = xay with otherwise

(a)

If a y x ~ e ,

(ay)x ~ e,

need

the

a necessary

by a m o n a d i c

idempotents.

and

complete

However,

following

R be a m o n a d i c

INTR(a) letter

Let If

still

Let

for d e r i v i n g

M presented

non-trivial

we

for a 6 E,

Lemma

implying

of the m o n o i d

to the c h o i c e

be u s e f u l

for a m o n o i d

to c o n t a i n

factors

an i d e m p o t e n t .

u = xy w i t h

is s a t i s f i e d ,

condition,

Then

a n d a y x a ~ a,

u de-

i.e.,

is s a t i s f i e d .

condition system

ayx~e

6 E* and some a 6 E U {e} such , ~ a. Hence, x' = ay a n d y'

If a 6 E, then

(b)

i.e.,

x'y'

t h e n we h a v e

is s a t i s f i e d .

(ii)

x,x',y,y'

= u,

describes

if u d e s c r i b e s

= xay with

If a = e,

(ii)

hand,

* ~ xy

= xyxy

word

idempotent

u 2 ~ u according

are w o r d s

then

u again

irreducible

a non-trivial

that

u = xx'

this

i.e.,

u is a n o n - e m p t y

implying

2

u

If u = x a y w i t h

u

u describes

then

sufficient

reduction

before

deriving

notion.

complete

reduction

:= {w 6 Irr(R) lawa ~ a}

is the

system

on E.

set of

internal

a.

R be a m o n a d i c

complete

IINTR(a) I > 2, then

reduction

the m o n o i d

system

M R does

on E, a n d

have

a non-

idempotent. Let

a £ E with

w I # w 2. T h e n

IINTR(a) I > 2, a n d

for

i = 1,2.

Then w I ~+

irreducible,

of w I a n d

w2 .

with

the

w 2 implying But

Let

the e x i s t e n c e

by monadic

complete

R be a m o n a d i c

M R contains

following

wlaw 2 +~

a w i a ~ a.

R is c o m p l e t e .

characterize

presented

5.12.

the m o n o i d

and

we h a v e

this

w I = w2,

since

contradicts

w I and

the c h o i c e

m

N o w we can in m o n o i d s Theorem

i = 1,2,

let W l , W 2 C INTR(a)

If a w i ~ e , t h e n & w i de, s c r i b e s a n o n - t r i v i a l i d e m p o t e n t of M~, if w . a ~ ' e , t h e n w . a d e s c r i b e s i -K , 1 , a non-trivial i d e m p o t e n t of M R . So a s s u m e t h a t a w i ~ e and w i a ~ e

w 2 are

for

three

complete

a non-trivial

conditions

(i)

Hu 6 I r r ( R ) - { e }

Hx,y

(ii)

Ha 6 E:

(iii)

Ha 6 E Hw 6 Irr(R)

of n o n - t r i v i a l reduction reduction

idempotent

system

if a n d o n l y

is s a t i s f i e d :

6 E*:

u = xy a ~ d yx ~ e,

IINTR(a) I > 2, or : INTR(a)

idempotents

systems

= {w} a n d a w ~ e .

or

as follows. on Z. T h e n if one

of

248

Proof.

If there exists a w o r d u 6 Irr(R)-{e}

word u d e s c r i b e s 5.11

a non-trivial

(ii) implies

if INTR(a)

idempotent

So a s s u m e i.e.,

that M R , c o n t a i n s

= {w} with a w ~ e

non-trivial

ayx~e,

Thus,

either

Thus,

are w o r d s x,y

is s u f f i c i e n t Define taking

operation (i')

idempotent.

a non-trivial

describing

to Lemma

this

By Lemma Finally, a

idempotent,

a non-trivial

5.9 u satisfies

condition

a 6 Z such that u = xay,

and awa ~ a i m p l y i n g

(ii)

is s a t i s f i e d

system

with a, or c o n d i t i o n

whether

a monoid

R on Z c o n t a in s

to check c o n d i t i o n s

(i) to

CYCLE on the

= {yxlxy

descendant

that w 6 INTR(a).

6 L},

M given by a monadic

a non-trivial (iii)

idempotent,

of T h e o r e m

set P(I*)

on Z by

i.e.,

CYCLE(L)

is the language

of w o r d s

of L. By u s i n g

condition

(i) of T h e o r e m

it

5.12.

of languages

all the cyclic p e r m u t a t i o n s we can express

(iii)

u

to d e t e r m i n e

an o p e r a t i o n

CYCLE(L)

5.9.

a. Take w 6 Z* to be the i r r e d u c i b l e

condition

reduction

contains

M R contains

6 Z* and a letter

w i t h a and w.

in o r d e r

complete

that

R. Then a w ~ e ,

is satisfied

a non-trivial

a word u 6 Irr(R)-{e}

and ayxa ~

of yx m o d u l o

(i), then

for some a 6 Z, then u = aw d e s c r i b e s

of M R . Then a c c o r d i n g

(i), or there

satisfying

of M R by Lemma

of M R.

conversely,

there exists

idempotent

idempotent

that this

5.12 as follows:

e 6 L(CYCLE(Irr(R)-{e}),R). Given

a finite

effectively

set Irr(R)-{e}

theorey

acce p t o r

A 2 for the

a finite

(Theorem 3.2). Lemma

5.13.

complete

a finite

(Theorem 3.1).

auto m a t a

derive

monadic

construct

(c.f.,

[Ho-UI]),

we have

The f o l l o w i n g

INSTANCE:

A finite Does

monadic

there

the

from

a finite

state

from w h i c h we finally

A 3 for the

set L ( C Y C L E ( I r r ( R ) - { e } ) , R )

the f o l l o w i n g

problem

techniques

we then o b t a i n

set C Y C L E ( I r r ( R ) - { e } ) ,

Thus,

system R on £, we can A 1 for r e c o g n i z i n g

By u s i n g w e l l - k n o w n

e.g.,

state a c c e p t o r

QUESTION:

reduction

state a cc e p t o r

result.

is decidable:

complete

reduction

system

exist a word u 6 Irr(R)-{e}

R on Z.

that can be f a c t o r e d

as u = xy w i t h yx ~ e ? Lemma Lemma Then

5.13

5.14.

Let

shows that c o n d i t i o n R be a finite

(i) of T h e o r e m

monadic

complete

for each a 6 Z, one can e f f e c t i v e l y

pushdown INTR(a)

automaton

M a that r e c o g n i z e s

is a d e t e r m i n i s t i c

reduction

construct

the

context-free

5.12

is decidable. system on I.

a deterministic

set INTR(a) ° In particular,

language

for each a 6 Z.

249 Proof.

Given

letter

a 6 Z, one can e f f e c t i v e l y

automaton ceptor

a finite m o n a d i c

(dpda)

complete

M I recognizing

M 2 recognizing

the

set

with awa + ~

6 Irr(R)

niques

for d e a l i n g

nizing

the

Given

compute

with awa ~ a} with

of the

dpda's

complete

reduction

Theorem

5.15

INSTANCE:

of T h e o r e m

([Ot 85a]).

the

set

3.!

pushdown

state ac-

and 3.3).

From

[a] R N a - I r r ( R ) . a

In the p r o o f s solving

comp l e t e

algorithm

able any more. above p r o b l e m involving

we have

from M 3.

M we can d e t e r m i n e

of L. Hence,

the

we can

conditions

for finite monadic

the f o l l o w i n g

problem

(Z;R),

result.

is decidable:

where

R is a m o n a d i c

complete

by

used

problem.

Since

any non-

set Irr(R),

solving

systems

whether

there

remains

open of w h e t h e r

complete

exists

a

exponena more

length-re-

that are not monadic,

the above a l g o r i t h m

can be solved when b e i n g c o n s i d e r e d length-reducing

in general

If finite

are c o n s i d e r e d

to develop

monadic

at first derives

it n e e d s

this problem.

is d e v e l o p e d

given a finite

R on Z, this a l g o r i t h m

So the q u e s t i o n

contain

5.14 an a l g o r i t h m

It is not yet k n o w n for

(Z;R)

?

5.13 and

for the

reduction

finite

a dpda M 4 recog-

by M, and if L is finite,

MR presented

decision

system

time and space.

complete

Thus,

idempotents

the above

the t e c h n i q u e

automaton

tech-

system on ~.

state a c c e p t o r

efficient

[Ha 78])

be c o n s t r u c t e d

The f o l l o w i n g

of Lemmas

reduction

e.g.,

=

By u s i n g w e l l - k n o w n

5.12 are also d e c i d a b l e

the m o n o i d

trivial

is not applicor not the

for p r e s e n t a t i o n s

reduction

systems

that are

not monadic. N o w we turn to the p r o b l e m given

=

we have

[a]R N a. Irr(R).a

all the e l e m e n t s

presentation

reduction Does

pushdown

systems.

A finite

QUESTION:

(cf.,

set L r e c o g n i z e d

a list c o n t a i n i n g (iii)

then

(Theorems

= a'INTR(a).a.

can e f f e c t i v e l y

a deterministic

(ii) and

ducing

a deterministic

R is complete,

if awa ~ a. Thus,

set INTR(a)

cardinality

tial

R on Z and a

[a] R and a finite

recognizes Since

system

w

{awalw

finite

set

a. Irr(R)'a

a}.

w

awa +~+ a if and only

for

construct

the

M I and M 2 we get a dpda M 3 , t h a t {awalw £ Irr(R)

reduction

through

non-trivial Definition

a finite

elements 5.16.

w 6 z* d e s c r i b e s

monadic

of finite

of d e c i d i n g complete

or not a m o n o i d

system c o n t a i n s

order.

Let M be a m o n o i d an e l e m e n t

of w h e t h e r

reduction

presented

of finite

by

(X;R) . Then a w o r d

order of M,

if there exist

any

250 integers

n > O and k >

I such that w n+k +

w n.

If,

in addition,

W

w ~

e,

then

w is

said

to

describe

a non-trivial

element

of

finite

orde r of M. As we saw in Section not a f i n i t e l y finite

order.

acterization for w h i c h

the m o n o i d

this

decidable

whether

given

through

monadic

presented

If M R c o n t a i n s

a non-trivial

elements

finite words

group.

Therefore,

5.77.

before

(X;R)

derive

elements

us with the n e c e s s a r y

Let R be a finite

the

by

(X;R)

following

two

(i)

The m o n o i d

(ii)

There

non-trivial Proof.

statements

Obviously

it suffices

w 6 X* be a shortest order

of M R . If

of finite

Here

system on X. Acwhether

or

idempotent. is a

M R m a y have n o n -

the m o n o i d

idem-

M R is a

information

order

on

of the m o n o i d

idempotents.

The

MR

following

information.

6 left(R)}.

complete

reduction

If the m o n o i d

any n o n - t r i v i a l

MR

idempotents,

then

are equivalent: a non-trivial

element

of finite order.

lwl < ~ such that w d e s c r i b e s

a

order of M R . to prove

word describing

lwl < ~

is

that are

idempotent

additional

of finite

is a word w 6 X* of length element

when

length-reducing

does not c o n t a i n

M R contains

systems.

it has no n o n - t r i v i a l

any n o n - t r i v i a l

s y s t e m on X, and let ~ = m a x { l / I l l presented

then this

occurs

it

problem

a non-trivial

order of M R . However,

we must

Since

R = {(/,r)}

it is d e c i d a b l e

although

char-

on X,

systems.

reduction

contains

idempotent,

non-trivial

reduction

reduction

situation

M R does not c o n t a i n

lemma p r o v i d e s Lemma

this

one-rule

complete

of f i n i t e order,

describing

in case

by

of finite

For example,

[La 74].

to the class of all m o n o i d s

complete

M R presented

trivial

such e l e m e n t s

system

to the results

potents.

R = {(/,r)}

decision

monadic

element

systems

the above

this result

or

of

has given a syntactic

reduction

reduction

n o t the m o n o i d

non-trivial

Lallement

whether

elements

or not a o n e - r u l e

for m o n o i d s

by finite

in general

non-trivial

characterization,

So let R be a finite cording

hand,

one-rule

M~ does contain

syntactic

we want to e x t e n d presented

2, it is u n d e c i d a b l e monoid contains

On the other for those

is d e c i d a b l e meets

presented

that

(i) implies

a non-trivial

then we are done.

Hence,

(ii).

element assume

So let

of finite

that

lwl > ~.

Since w d e s c r i b e s exist This

integers implies

a n o n - t r i v i a l e l e m e n t of finite o r d e r of M R, there n > 0 and k > I such that wn+k ~ w n, and w ~ e.

in p a r t i c u l a r

that n+k > 2. Since

R is complete,

the

words w n+k and w n have a common d e s c e n d a n t m o d u l o R, w h i c h means that n+k 2 w is r e d u c i b l e m o d u l o R. But }wl > ~, and so w is r e d u c i b l e modulo

251

R. Thus, w = WlW 2 = w 3 w 4 with r 6 Z* w i t h Now

Irl <

(w2w3,r)

6 R for some words Wl,W2,W3,W4,

lw2w31.

lw2w31 < Z ~

Jwl = iWlW21

implying that

tw31 <

lWlf. Hence,

there is a w o r d x 6 E* such that w I = w3x, i.e., w = w 3 x w 2. This gives: (xr)n+k+1 ~-~ (xw2w3)n+k+1 = x w 2 ( w 3 x w 2 ) n + k w 3 : xw2wn+kw3 R x w 2 w n w 3 = (xw2w3)n+1

+

If xr ~ *

(xr) n+1.

e, this shows that xr

does d e s c r i b e a n o n - t r i v i a l e l e m e n t of finite order of M R, a l t h o u g h Ixrl <

Ixw2w31

other hand,

•

if xr +~+ e, then we have the following:

w3(xw2w3)xw 2 ~ trivial

= lwl, thus c o n t r a d i c t i n g the choice of w. On the

w3(xr)xw 2 ~

w

2

= (w3xw 2)

2

=

w 3 x w 2 = w, i.e., w d e s c r i b e s a non-

idempotent of M R, thus c o n t r a d i c t i n g the a s s u m p t i o n s about

~R" So in order to decide whether or not a m o n o i d M given through a finite m o n a d i c c o m p l e t e r e d u c t i o n system R on Z c o n t a i n s any nontrivial e l e m e n t s of finite order, we can p r o c e e d as follows. we d e t e r m i n e w h e t h e r or not M c o n t a i n s any n o n - t r i v i a l If it does,

At first

idempotents.

then the answer to our q u e s t i o n is "yes". If it does not,

then we e n u m e r a t e all the words from Z* of length up to ~ = max{Ill I 1 6 left(R)} and test whether any of these words describe n o n - t r i v i a l e l e m e n t s of finite order of M. The f o l l o w i n g lemma shows that this test can be p e r f o r m e d effectively. Lemma 5.18. The f o l l o w i n g p r o b l e m is decidable: INSTANCE:

A finite m o n a d i c c o m p l e t e r e d u c t i o n system R on Z, and a w o r d w 6 Z*.

QUESTION:

Does w describe a n o n - t r i v i a l element of finite order of the m o n o i d M R ?

Proof. Given R and w, we can easily decide w h e t h e r or not w ~ holds.

If w ~+

e

e, then o b v i o u s l y w does not describe a n o n - t r i v i a l

element of finite order of M R . So assume that w ~

e. F r o m w we can

c o n s t r u c t a n o n d e t e r m i n i s t i c finite state a c c e p t o r A w for the language {w} +. F r o m A w and R we get a n o n d e t e r m i n i s t i c A~w for the language L({w}+,R)

finite state a c c e p t o r

and a finite state acceDtor_ A R for the

set Irr(R). By c o m b i n i n g A ~ . a n d A R we finally get a n o n d e t e r m i n i s t i c finite state a c c e p t o r A(R,w) Irr(R)

that r e c o g n i z e s the language L({w}+,R)

= {u 6 Irr(R) I There exists an integer k ~ I such that k

N

~ u}.

It is o b v i o u s that w d e s c r i b e s an element of finite order of M R if and only if this language

is finite.

But the c a r d i n a l i t y of this language

can be d e t e r m i n e d e f f e c t i v e l y from A(R,w).

D

252 N o w T h e o r e m 5.15, Lemma 5.17, and Lemma 5.18 t o g e t h e r y i e l d the f o l l o w i n g result. T h e o r e m 5.19 INSTANCE:

([Ot 85a]). The following p r o b l e m is decidable: A finite p r e s e n t a t i o n reduction

(E;R), where R is a monadic complete

system on E.

Does the monoid M R p r e s e n t e d by

QUESTION:

(E;R)

contain any non-

trivial e l e m e n t s of finite order ? Since a m o n o i d that does not contain an}f n o n - t r i v i a l e l e m e n t s of finite order is c a l l e d torsion-free,

the above result can also be

stated as follows. C o r o l l a r y 5.20. The following p r o b l e m is decidable: INSTANCE:

A finite p r e s e n t a t i o n

(Z;R), where R is a monadic complete

reduction system on Z. Is the monoid M R p r e s e n t e d by

QUESTION:

(Z;R)

torsion-free

?

The a l g o r i t h m d e v e l o p e d above for d e c i d i n g t o r s i o n - f r e e n e s s of monoids given through finite monadic complete reduction e x p o n e n t i a l time and space in general.

systems needs

So far it is not known whether

there exists a more e f f i c i e n t a l g o r i t h m for solving this problem.

How-

ever, when we restrict our a t t e n t i o n to c a n c e l l a t i v e monoids that are p r e s e n t e d by finite monadic c o m p l e t e reduction

systems,

then those

monoids that are not t o r s i o n - f r e e can be r e c o g n i z e d n o n - d e t e r m i n i stica!ly in p o l y n o m i a l time,

since a c a n c e l l a t i v e m o n o i d

contain any n o n - t r i v i a l idempotents other hand,

(for details cf.

does not

[Ot 85b]). On the

it is an open p r o b l e m of w h e t h e r or not the p r o p e r t y of

t o r s i o n - f r e e n e s s is decidable for monoids p r e s e n t e d by finite lengthr e d u c i n g complete r e d u c t i o n systems that are not monadic, Lemma 5.17 still holds

although

in this situation.

T h e o r e m 4.12 and C o r o l l a r y 5.20 can now be c o m b i n e d to show that it is decidable w h e t h e r or not a m o n o i d M given t h r o u g h a finite monadic complete reduction a monoid form

system R on X is a free group. Recall that

M is a free ~ r 0 u p if and only if it has a p r e s e n t a t i o n of the

(Z t' ~; {(aa,e),(aa,e) la E ~}), where ~ is an a l p h a b e t in l-to-]

c o r r e s p o n d e n c e with I, Z n ~ = 9, and -: Z ~ ~ is a b i j e c t i o n realizing this correspondence. Let R be a finite monadic complete r e d u c t i o n system on Z. Then each c o n q r u e n c e class In particular,

[u] R is a d e t e r m i n i s t i c context-free

[el R is a c o n t e x t - f r e e

language.

language.

So if the m o n o i d M

253

p r e s e n t e d by group. Hence, T h e o r e m 5.21

(Z;R)

h a p p e n s to be a group,

then it is a c o n t e x t - f r e e

the following result of Muller and Schupp is applicable. ([Mu-Schu]).

A f i n i t e l y g e n e r a t e d t o r s i o n - f r e e group G is free if and only if it is c o n t e x t - f r e e . Since each free group is torsion-free,

this implies the following.

C o r o l l a r y 5.22. A finitely g e n e r a t e d c o n t e x t - f r e e group is free if and only if it is torsion-free. Thus,

the m o n o i d M p r e s e n t e d by

(Z;R)

if it is a group and it is torsion-free.

is a free group if and only Thus, by c o m b i n i n g the cor-

r e s p o n d i n g d e c i d a b i l i t y results we obtain the following. T h e o r e m 5.23. The f o l l o w i n g p r o b l e m is decidable: INSTANCE:

A finite p r e s e n t a t i o n

(E;R), where

R is a monadic c o m n l e t e

r e d u c t i o n system on E. QUESTION:

Is the m o n o i d M R given through this p r e s e n t a t i o n a free group

?

Finally we want to look at the existence of elements of infinite order. Definition

5.24. Let M be a monoid p r e s e n t e d by

(E;R) . Then a word

w 6 E* d e s c r i b e s an element of infinite order of M if wi ~ all i,j 6 ~,

w j for

i % j.

If R is a finite c o m p l e t e r e d u c t i o n system on E, then we can decide e f f e c t i v e l y w h e t h e r or not the m o n o i d M p r e s e n t e d by finite. Obviously,

if M is finite,

(E;R)

is

then all its e l e m e n t s have finite

order. But does also the reverse i m p l i c a t i o n h o l d ? In general this p r o b l e m is known as the B u r n s i d e problem.

It has

been shown that there exist infinite monoids all the elements of which have finite order,

i.e., the reverse

not hold in general.

In fact, Adjan

implication m e n t i o n e d above does [Ad 79] has proven that for each

integer m > I and each odd integer n > 665, the group B(m,n) by

(E U ~;

{(aa,e), (aa,e)la 6 E} U { (xn,e)Ix 6

presented

(E U ~)*}), where

I = { a l , a 2 , . . . , a m} and ~ = {al,a2,...,am} , is infinite. All these groups are finitely g e n e r a t e d only, they are not f i n i t e l y p r e s e n t e d lAd 79]. This leaves the p r o b l e m of w h e t h e r or not there exists a f i n i t e l y p r e s e n t e d infinite group all of which e l e m e n t s have finite

254 order. Here, we will show that such a group,

if it should exist,

not be p r e s e n t e d by a finite complete reduction

can-

system.

Lemma 5.25. Let R be a finite c o m p l e t e r e d u c t i o n system on Z such that the m o n o i d M R p r e s e n t e d by

(I;R)

is infinite. Then this m o n o i d con-

tains an e l e m e n t of infinite order. Proof.

Since R is a complete r e d u c t i o n system on I, the set Irr(R) of

irreducible words modulo R is a set of r e p r e s e n t a t i v e s M R . Thus,

Irr(R)

with M R also Irr(R)

is infinite.

for the m o n o i d

Since R is finite,

the set

is a regular subset of ~*. Hence, by the pumping lemma of reg-

ular sets, there exists an integer n > I such that each word u 6 Irr(R) of length

lul > n can be factored as u = xyz, where y # e and {xykzl

k > O} c Irr(R). u 6 Irr(R)

Since Irr(R)

of length

is infinite,

there exists a word

lul > n. Then u = xyz s a t i s f y i n g y % e and

{xykzlk > O} ~ Irr(R).

But the set Irr(R)

is s u b w o r d - c ! o s e d implying

that {yklk > O} E Irr(R). Since R is c o m p l e t e we can c o n c l u d e that i~yj y for all i,j 6 N with i % j. Thus, y describes an element of infinite order of M R • Hence,

s

when a m o n o i d M is p r e s e n t e d by a finite complete r e d u c t i o n

system, then this m o n o i d contains an e l e m e n t of infinite order if and onlv if it is an infinite monoid. Thus,

T h e o r e m 4.1 immediately gives

the following corollary. Corollary 5.26. The following p r o b l e m is decidable: INSTANCE:

A finite p r e s e n t a t i o n

(~;R), where R is a complete reduc-

tion system on ~. QUESTION:

Does the m o n o i d M R p r e s e n t e d by

(Z;R)

contain an element

of infinite order ? We have seen that all of the ten d e c i s i o n p r o b l e m s listed at the b e g i n n i n g of Section 2 become decidable,

when they are r e s t r i c t e d to

monoids p r e s e n t e d by finite complete r e d u c t i o n systems or certain s p e c i a l i z a t i o n s thereof.

Clearly,

this o b s e r v a t i o n raises the q u e s t i o n

of which monoids and groups can at all be p r e s e n t e d in this way. More formally, and groups

is there an algebraic c h a r a c t e r i z a t i o n ? This fundamental question

for these monoids

is still unanswered,

only a few partial results could be obtained.

Cochet

that a group G can be d e s c r i b e d by a p r e s e n t a t i o n

so far

[Co 76~ p r o v e d

(Z;R), where R is a

finite special complete reduction system on Z, if and only if G is the free p r o d u c t of finitely many cyclic groups. Avenhaus, Otto

M a d l e n e r and

[Av-Ma-Ot] p r o v e d that a group G can be d e s c r i b e d by a p r e s e n t a -

255

tion

(Z;R), where

R is a finite t w o - m o n a d i c c o m p l e t e r e d u c t i o n system

on Z, if and only if G is the free p r o d u c t of a f i n i t e l y g e n e r a t e d free group and a finite number of finite groups. Here, duction system R is called two-monadic, hand side of length two.

a monadic re-

if each rule of R has a left-

It has been c o n j e c t u r e d

(cf., e.g.,

[Gil 84])

that the same a l g e b r a i c c h a r a c t e r i z a t i o n h o l d s for the class of groups p r e s e n t e d by finite m o n a d i c c o m p l e t e r e d u c t i o n systems, but so far this has not been shown. We close this c h a p t e r with a short list c o n t a i n i n g some of the most i n t e r e s t i n g open p r o b l e m s

in c o n n e c t i o n w i t h the m a t e r i a l pre-

sented. (i)

Is it p o s s i b l e to e x t e n d Theorems

5.15,

5.19, and 5.23 to the

class of all monoids p r e s e n t e d by finite l e n g t h - r e d u c i n g complete r e d u c t i o n systems further (ii)

? Is it p o s s i b l e to e x t e n d these results even

?

Does G i l m a n ' s c o n j e c t u r e hold

?

(iii) W h i c h groups can be p r e s e n t e d by finite l e n g t h - r e d u c i n g complete r e d u c t i o n systems

?

(iv)

Which groups can be p r e s e n t e d by finite c o m p l e t e r e d u c t i o n

(v)

W h i c h m o n o i d s can be p r e s e n t e d by finite

systems

?

length-reducing)

(special, monadic,

c o m p l e t e r e d u c t i o n systems ? Note that it has

been shown only r e c e n t l y by Squier

[Sq-1] that there exists a

finitely p r e s e n t e d m o n o i d w i t h a decidable w o r d p r o b l e m such that this m o n o i d cannot be p r e s e n t e d by a finite c o m p l e t e reduction system.

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262 [or 85b]

F. Otto:

Decision Problems and their Complexity for Monadic Church-Rosser Thue Systems Habilitationsschrift, Univ. Kaiserslautern, 1985 [or 86]

F. Otto:

On Deciding whether a Monoid is a Free Monoid or is a Group Acta Inform. 23 (1986), 99-110 [O'Dun 83] C. O ' D u n l a i n g : Infinite Regular Thue-systems Theor. Comp. Science 25 (1983), [Pe-Sti 77] G. Peterson, Complete sets Theories with Tech. Report, [Rab 58]

171-192

M. Stickel:

of Reductions for Equational Complete Unification Algorithms Univ. of Arizona 1977

M. Rabin:

Recursive Unsolvability of Group Theoretic Problems Ann. of Math.67 (1958), 172-194 [Ri 78]

M.M.

Richter:

Logikkalk~le, [Ri 82]

M.M.

Teubner Verlag 1978

Richter:

Complete and Incomplete Syystem of Reductions In: Informatikfachberichte 57 (1982), ed. J. Nehmer, Springer Verlag [Ro 65]

J.A.

Robinson:

A Machine-Oriented Logic Based on the Resolution Principle JACM 12 (1965), 23-41 [Rot 73]

J. Rotman:

The Theory of Groups Boston 1973 [Sh 77]

R.

Shostak:

On the SUP-INF Method for Proving Presburger Formulas J. ACM 24 (1977), 529-543 [Sie-Sz 83]J. Siekmann, P. Szabo: A Noetherian and Confluent Rewrite System for Idempotent Semigroups Semigroup Forum 25 (1983), 83-110

263

[Sq]

C. Squier:

Word Problems and a Homological Condition for Monoids submitted for publication [Sti 81]

M.E.

Finiteness

Stickel:

A Unification Algorithm for AssociativeCommutative Functions. J. ACM 28 (1981), 423-434 [Ti 08]

H. Tietze:

0ber die topologischen Invarianten mehrdimensionaler Mannigfaltigkeiten Monatshefte f~r Mathematik und Physik 19 (1908), 1-118 [Wo 68]

J.A. Wolf:

Growth of Finitely Generated Soluble Groups and Curvature of Riemannian Manifolds J. Differential Geometry 2 (1968), 421-446

Subject Index Absolutely free algebra

3

Reducible

28,50,121

Block

92

Reduction

27,49

Cancellative

243

Reduction graph

172

Canonical mapping

2

Redundant reduction

41

Church Rosser Property

(CR) 29,50

Regular reduction scheme 117

Collaps

108

Regular reduction system 119

Compatible

37

Repetition-free deri-

Complete System

30,50 vation

147

Replacement

Congruence relation

1

Critical pair

38,141 Small constellation con-

Cycle number

198

9

ditions

65,82

Cyclically reduced

66

Special reduction

229

Dehn's algorithm

82

Structural induction

3

Derivation equivalent

28

Substitution

8

Equationally equivalent

28

Superpos

38,140

Superpos-Deduction

Finite Termination Pro29,50

Chains

86

Forward-backward system

147

Symmetrized

64

Ground equation

5

Term algebra

4

Growth function

190,191 Term ordering

perty

(FTP)

i0

197

Tietze transformation

219

Initial algebra

4

Theory

5

Internal factor

247

Thue system

49

Irreducible

28,50

Todd Coxeter algorithm

106

121

Unifier

9

KB-ordering

12,51 Unique termination pro-

Knuth-Bendix completion

41,42 perty

algorithm

52

Weak Church-Rosser pro-

Malcev-coordinates

206

perty

Markov property

222

Weak FTP

Minimal path

198

Weight function

ii

Monadic system

186

Word graph

170

Most general unifier

9

Word tree

171

Narrow derivation

155

(UTP)

(WCR)

30,50

30 147

List

of S y m b o l s

and Abbreviations

KBCA r

41

K1

KB CA

42

K2,

51

RTC

ii0

51

T

(h)

170

SGCA r

53

F

(A)

170

SGCA

53

F (R)

171

54

T

171

54

7T

54

h

(C)

242

CYCLE

248

SG w Mw

(E,R) (E,R)

TSG

Tm

(7-)

(E)

SG T

(T.)

mT

(~-)

54

Gw

(7.,R)

63

PA r Pi

(Q, R)

(Q, R)

64 64

Sym

65

c' (I)

65

GCA r

67

T(p)

82

89 K

(i)

92

190