A specialized completion procedure for monadic string-rewriting systems presenting groups

K. Madlener F a e h b e r e i c h Inforrnat ik . Universit~it K a i s e r s l a u t e r n

Po ' . t faah 3049. 6750 Ka i sa r s l au t e rn . G e r m a n y

P. Narendran Department of Computer Science. State University of New York

Albany. MY 12222. U.S.A.

F. Otto

Fachbereich Mathema%ik-lnformatik. Gesamthochschule I<assel

Postfach 101380, 3500 Kassel. Germany

Abstract

Based on a simplified test for determining whether a finite monadic string-rewriting system

R presanting a group is confluent on [e] R, a procedure for completing a system of this

form on [el R is derived. The correctness and completeness of this procedure are shown.

I. Introduction

Algebraic structures and rewrite methods for effectively comput ing information on such structures have been studied extensively. In the present paper we are interested in groups that are presented through certain string-rewriting systems. Our interest in groups has mainly two reasons: first of all, the theory of groups is well developed, and secondly, groups are often used as invariants for more complicated structures. For example, there are m a n y results on structural properties of groups, and knowing about these properties m a y g ive a lot of additional information on the groups considered [11]. Also it was first

for groups that rewrite methods were used for solving the word problem as exemplified by Dehn's algorithm for the class of small cancellation groups [4.10.11].

Each finitely presented group G can be presented by a finite string-rewriting system R on some alphabet 7., i.e. G is isomorphic to the factor monoid ~ := 7.*/< * ~R of the free monoid Y.* generated b y 7. modulo the Thue congruence < ~>R induced by R. Although we m a y restrict the system R to only contain special rules, i.e.. one side of each rule is the empty word e. it is in general impossible to obtain m u c h information

on the Thue congruence c * )R or on the monoid ~ R from R. For example, it is even undecidable in general whether the monoid ~ presented b y a finite special string- rewriting system R is at all a group [17]. In fact, the undecidabil i ty of Markov properties can be carried over to the class of monoids that are presented by finite special string- rewriting systems [23].

The situation improves dramatically w h e n attention is restricted to those finite string- rewriting systems R that are Noetherian and confluent. Let !~. R denote the reduction relation induced b y R. wh ich is obtained b y allowing the rules of 1~ to be applied only from left to right. Then R is called Noetherian if there are no infinite sequences of reductions modulo R. and it is called confluent if. for all u.v £ Z ~, u < ~ >R v implies that u 2-> R w and v "-->R w for some w 6 ~1". Thus. if R is Noetherian and confluent, then each congruence class n o d < * >R contains a unique "minimal" word. and g iven any word

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u E Z ~, the minimal word v congruent to u can be computed through rewriting. Hence, the word problem for R is decidable, but also some other problems become decidable in this setting. By restricting the syntactic form of the rules admitted, e.g. b y allowing only length-reducing, monadic or special rules, stronger decidability results have been obtained, (c.f., e.g., [2,3]) and even some results on structural properties have been der ived (see ['13] for an overview).

The confluence property for finite Noetherian string-rewriting systems is decidable. Based on a confluence test !,(nuth and Bendix developed a completion procedure for rewrite systems [Q]. If R is a finite Noetherian string-rewriting system on Z that is non-confluent, then this completion procedure tries to construct a finite string-rewriting system S on the

same alphabet F. such that S is equivalent to R, i.e., the two congruences < ~ >s and < ~ >R coincide, and S is Noetherian and confluent. Specialized completion procedures a la !4nuth- Bendix have been developed for groups [10]. However, even if the word problem for R is decidable and if it is allowed to change the ordering used in the completion process as well as the under ly ing alphabet, they will not a lways succed [22], By restricting the ordering used or the syntactic form of the rules al lowed we are led to completion

procedures for certain restricted classes of string-rewriting systems e,g., length-reducing, monadic or special systems. In case of success these procedures m a y g ive more informa- tion on the group considered or algorithms that are more efficient, Of course, they will not a lways succeed either.

If the monoid ~ R is a group, the word problem for R is reducible to the membership problem for the congruence class [el R. The system R is called confluent on [e l R. if, for all w E ~-~, w < ~ >R e implies w "'>R e. If a Noetherian system R is confluent on [e l R. then the process of reduction mod R yields an algorithm for testing membership in [e l R and therewith for solving the word problem for R, even if R is not confluent. In fact

Dehn's algorithm for the word problem can be interpreted as comput ing minimal words modulo a finite length-reducing system R, which in general is on ly confluent on [el R. As pointed out in [4,10] this system can be computed by only using critical pairs which involve the group axioms, provided the g iven presentation satisfies certain small cancellation conditions. An example of a system R, which presents a group, and which is confluent on [ e l R. but for which no equivalent finite string-rewriting system exists, that is both Noetherian and confluent, is described in detail in [7] and [12].

The property of confluence on a g iven congruence class is undecidable in general e v e n for finite length-reducing string-rewriting systems, and for finite monadic string-

rewriting systems only algorithms of doubly exponential time complexity are k n o w n for deciding this property [19], So far it is only for finite special string-rewriting systems that this property has been shown to be tractable [20] Based on this result a completion procedure m a y now be developed which, g iven a finite special string-rewriting system R that is not confluent on [e l R, tries to compute a special system S which is equiyalent to R and confluent on [ e ~ . For the case of finite special string-rewriting systems presenting groups such a specialized completion procedure is described in [21].

Here we consider the case of finite monadic string-rewriting systems that present groups, i.e., each rule is of the form (£,b) with ~ E 7-* and b E 7. u {eL For example, let Z - {a,b,c} and R = {(ab,e),(ba,e),(c2,e),(acb,c)}. Obviously, this system presents a group. In fact, this group is isomorphic to the direct product of the free group F of rank 1 and the cycl ic group Z 2 of order 2, and hence, it cannot be presented b y a finite monadic

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and confluent s t r ing-rewri t ing system on a n y set of generators [13]. However , let

R o := R u {(bca,c)}. Then R o is a finite monadic sys tem that is equ iva len t to R. Obvious ly ,

R o is not confluent either, but for all w C Z*. if w < ~ : ~ e. then w "'>Re e. i.e. R e is

confluent on [e]Ra. Hence, the process of reduct ion mod R o g ives a l inear- t ime a lgor i thm

for solving the word p rob lem for R. In fact m a n y decision problems can be solved, w h e n

they are restricted to the class of finite monadic s t r ing-rewri t ing systems R that present

groups, and that are confluent on [ e l R. For example , all problems wh ich can be expressed

through l inear sentences in the sense of Book [2] can be so lved in a uniform w a y in this sett ing [14]. In addition, the class of groups that can be presented b y these systems

is strictly larger than the class of groups that can be presented b y finite l eng th - reduc ing and confluent s t r ing-rewri t ing systems. In fact. the structural and l anguage theoretical

propert ies of this class of groups are also we l l -known , since it is exac t ly the class of

context-free groups [1]. w h i c h has also been character ized as the class of finite extensions

of f ini tely genera ted free groups [15].

Here w e present a p rocedure which, g iven a finite monadic s t r ing-rewri t ing sys tem

R on Z such that the monoid 9 ~ is a group as input, tries to construct a finite monadic

sys tem S on Z such that S is equ iva len t to R and confluent on [b ] R for all b 6 Z t, {e}.

This p rocedure consists of two subroutines cal led CONTEXT_RESOLVING and NORMALIZA-

TION. w h e r e the former introduces n e w monadic and special rules to make the sys tem

confluent on the re levan t equ iva l ence classes, whi le the latter deletes superfluous rules in

order to keep the sys tem as small as possible. It is shown that this p rocedure either

terminates wi th a finite monadic sys tem S, or it enumerates an infinite monadic sys tem S.

In either case, S is equ iva len t to R and confluent on [b] R for all b 6 Z u {e}. In addition, our p rocedure terminates w h e n e v e r there exists a finite monadic sys tem that is equ iva len t

to R and that is confluent on [ e l R. Thus, w e have a complet ion procedure that is correct

and complete.

This paper is organized as follows. After establishing the necessary notation in Section 2,

w e de r ive some conditions in Section 3 that are necessary and sufficient to guarantee

confluence on [ e l R for finite monadic s t r ing-rewri t ing systems R present ing groups. Since

these condit ions can be ver i f ied in po lynomia l time. w e thus have a po lynomia l - t ime

algor i thm for dec id ing confluence on [e l R for this class of s t r ing-rewri t ing systems. In

Section 4 the announced complet ion procedure is presented together wi th some examples.

Final ly , in Section 5 w e point out the relation to the notion of symmet r i zed group-

presentat ion as it is considered in small cancel la t ion theory ([lO,11]). Also some possible improvements of our comple t ion procedure and some problems for future research are

ment ioned. Due to space restrictions no proofs have been included, t hey can be found

in a more de ta i led version [15].

2 . P r e l i m i n a r y results

After establ ishing notation w e present some basic results about reductions in finite monad ic

s t r ing-rewri t ing systems. We assume the reader to be familiar wi th the foundations of

automata theory as presented for example b y Hopcrofl and Ullman [5], and the theory of

s t r ing-rewri t ing systems as in Book's seminal paper [3].

Let ~.. be a finite a lphabet . Then Z denotes the set of words over Z inc lud ing the

e m p t y word e. A monaclio . t r ing- rewr i t ing sys tem R on Z is a subset of Z+×(Z u {eD, where ~--* - Z*\{e} denotes the set of non -empty words over ~--. The e lements (E,b) of R

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are ca l led (rewrite) rules. If b - e , the rule is ca l led special. For all u ,v 6 7~* and (e,b) 6 R. u e v - ->Rubv , i.e., -->1~ is the s ingle-step reduct ion relat ion i nduc e d b y R. Its re~ex ive

and t ransi t ive closure -->R is the reduct ion relat ion induced b y R. For u ,v 6 7-~, if u "->R v,

then u is an ancestor of v, and v is a descendan t of u. By <v> R w e denote the set of all ancestors of v, and A~(u) denotes the set of all descendants of u. For a subset L c 7.*

<L> R = [J <u> R, and AI~(L) =u~6LA~(u). u6L

Since we allow monadic rules of the form (a,b), where a.b 6 7. are both letters, we

will always assume a fixed ordering :. on Y~, and for each rule of this form, we will require

that a ~ b. This slightly extends the usual definition of monadic systems, but 2-> R is

terminating and the usual properties still hold . For example, if R is finite and monadic.

and if L c 7.* is a regular set that is given through a non-deterministic finite state acceptor

fanfsa) ~, then the set A~f.L) is regular as well, and an nfsa ~ for this set can be construc-

ted in polynomial time [2]. If there is no word y 6 Y.* such that x -->q y, then x is

cal led i r reducible , otherwise, it is reducible . If R is finite, then the set IRR{R) of i r reducible

words is regular, and from R a determinist ic finite state acceptor (dfsa) for this set can be

ob ta ined in po lynomia l t ime [5]. By < ~ >R we denote the equ iva l ence relation induced

b y "-->R' w h i c h is ac tua l ly a congruence on 7- '~. It is cal led the Thue congruence genera ted

by R. For w 6 Z*, [w] R = {u 6 7.* I u < * >R w} is the congruence class of w nod R.

The set {[w] R I w 6 F.*} of congruence classes forms a monoid 9)7 R under the operation

[U]R* [v] R - [uv] R with identity [el R. This monoid is uniquely determined (up to iso-

morphism) by 7. and R. and hence, whenever M is a monoid that is isomorphic to ~t R,

we call the ordered pair (Z~R) a (monoid-) presentation of M with generators 7. and

defining relations R. Two systems R and S on the same alphabet Z are called equivalent

if they generate the same Thue congruence, i.e. <'*>R = <-/-'>S and ~R " 9J~S"

The monoid ~ is a group if and only if, for each letter a £ 7., there exists a word

u a 6 Z* such that au a <'-~R e. In this case there exists a function -l: Z* --> Z* such that

for all w 6 Y--*, w "1 is a formal inverse of w, i.e., w w -1 < * >R w - l w ~ ~ >R e. In fact ,

for e v e r y letter a 6 7. a candida te w a for u a of length less than (max{l£1: (£,b) 6 RD Ixl

can be computed from R [18]. A subset L c Z ~ is closed under cyc l i c permutat ions if

u v 6 L implies v u 6 L for all u,v E 7-*. The set [ e l R is closed under cyc l i c permutat ions

if ~ R is a group. A s t r ing-rewri t ing sys tem R on 7. is confluent on [ w ] R for some word w E 7- *, if

there exists a word w o 6 IRR(R) such that A~( [w] R) a IRR(R) = {Wo}. Thus, R is confluent on [ w ] R if all words in that class reduce to the same i r reducible word. w h i c h then can

serve as a normal form for this class. R will be cal led w e a k l y confluent if it is confluent

on [b] R for all b E Y.. u {e}, and it wil l be cal led e-confluent if it is confluent on [ e l R.

From now on w e will assume that the monoid 9)~ R is a group. This group is ca l led

context-free if the set [ e l R c Z ~ is a context-free language. This p roper ty is i ndependen t of

the ac tua l ly chosen finite presentation. The impor tance of monadic s t r ing-rewri t ing systems that are w e a k l y confluent or e-confluent is due to their relation to context-free groups and

to the dec idab i l i ty of the l inear sentences of Book for this class [14]. Autebert , Boasson, and Senizergues [1] establ ished the fol lowing fundamental result on context-free groups.

Theorem 2.1 [Auteber t et al. 87]. A finitely genera ted group (~ is context-free if and on ly if it has a presentat ion of the form (Z~R), whe re R is a finite, monad ic and w e a k l y

confluent s t r ing-rewri t ing system on Z.

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An a lgebra ic character izat ion of this class of groups has been g i v e n b y Muller and

Schupp [1 6 ] using the concept of v i r tua l ly free groups (i.e. groups w h i c h contain a f ini tely

genera ted free subgroup of finite index). So it is fairly easy to construct examples of such

groups. They all h a v e presentations b y finite and confluent s t r ing-rewri t ing systems.

p r o v i d e d appropr ia te order ings are used, w h i c h in genera l are not l eng th -compa t ib le . Note

that the class of groups that can be presented b y finite, monadic and confluent systems is a proper subclass of the class of context-free groups (See the example of Section 1 and [13]).

Example 2.2. a) Let 7 . . {a.b,c} and R = {(ab.e),(ba,e),(c2,e),(aca.c)}. Then (E~R) presents a

group w h i c h is an extension of the free group of rank 1 and the cyc l i c group Z 2. R is

not confluent , s ince ac < '~ >Rcb and ca < ~ >R be. It is nei ther confluent on [ e l R. since

cbcb < '~ >R e~ nor on the congruence class of any letter, because bcb < ~>R c. cac < ~ >R b

and cbc < ~ >R a. By add ing the rule (bcb.c) w e get a sys tem which is confluent on [ e l

and [b ] . and if (cac.b) and (cbc,a) are also taken, w e get a w e a k l y confluent system. (See

also Example 4.57.

b) Let E = {a,b.c} and R = {(ab,e).(ba,e).(c3.e).(c2ac,a).(c2bc.b)}.Then (7.oR) presents a g roup

isomorphic to ZxZ 3. the direct product of 7. wi th the cyc l i c group of order 3. For all n ~ 1, canc2bn < ~ >R e <'-~-->R cbncZan" Since bc2a < ~ >R c2 <-2-'>R ac2b' no factor u of canc2b n

or cbnc2a n satisfying I < ]ul ~ n is congruent to any letter. Thus. there is no finite monadic

system S that is both equivalent to R and confluent on [el R. Since ZxZ 3 is a context-free

group, there must be a monadic presentation of this group which is confluent on [el. In

fact. b y in t roducing a n e w letter d and the rules (c2.d).(cd.e).(dc.e).(d2,c) together wi th

{(axb,x).(bxa,x) I x E {c.d}} w e get a different presentat ion of Zx~'. 3 for w h i c h conf luence

on [ e l and e v e n w e a k confluence can be shown.

Confluence on one equ iva l ence class is much harder to dec ide than conf luence e v e r y -

where . In fact, in [19] it is shown that this p roper ty is undec idab l e e v e n for l eng th - reduc ing

systems, whi l e for monadic systems it can be de c ide d using the dec idab i l i t y of the

e q u i v a l e n c e p rob lem for f in i te- turn determinist ic pushdown automata. For stating this

result in detai l w e need some more notation. Let R be a finite monad ic s t r ing-rewri t ing

sys tem on Z . and let (~,.b t) and (e2.b 2) be two rules of R. If ~1 = x82Y for some x.y E F-~.

or if {~lx = Y~2 for some x,y 6 Z ~* satisfying 0 < lyl < Jell, then the pair (bpxb2Y),

r e spec t ive ly (blX,Yb2), is ca l led a crit ical pair of R. By UCP(R) w e denote the set {(x,y) ]

(x,y) is a crit ical pair of R such that AR(X) n A~(y) = O} of unreso lvab le crit ical pairs of

R. Obvious ly , this set can be computed in po lynomia l time. For w 6 IRR(PJ w e define a

l anguage Lu(W) as follows: Lu(W) = {x~y I x.y E IRR(R), xuy -->R w}. Here ~ is an

addi t ional letter not in ~-. Then x~y 6 Lu(W) if (x,y) is an i r reducib le context of u in

<w> R. Using sets of this form the conf luence on [ w ] R can be charac ter ized as follows:

Proposi t ion 2.3 [Otto 87]. Let R be a monadic s t r ing-rewri t ing sys tem on 7.. and let

w 6 IRR(R). Then the fol lowing two statements are equivalent :

(i) The sys tem R is confluent on [ w ] R. (it) V(u.v) 6 UCP(R): Lu(W) = Lv(W).

Since the sets Lu(W) are in genera l context-free languages , this character izat ion will

not be useful in a comple t ion procedure . In the next section w e wil l de r ive a s impler test

b y us ing the fact that ~ is a group.

As a first simplification w e wou ld l ike to keep the sys tem R as small as possible. A

s t r ing-rewri t ing sys tem R is ca l led r educed if. for each rule (£,r) 6 R, r 6 IRR(R) and

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e 6 IRR(R\{(g,r)}). In genera l there does not exist a r educed sys tem w h i c h is equ iva len t

to a g i v e n one. However , such a r educed sys tem exists w h e n the sys tem R is confluent

on [r] R for e v e r y r ight -hand side r. In fact, for monadic systems the fol lowing holds.

Theorem 2.4. There is a po lynomia l - t ime a lgor i thm that solves the fol lowing task:

INPUT: A finite monad ic s t r ing-rewri t ing sys tem R on Z such th~t ~ R is a group.

TASK: Either recognize correct ly that R is not w e a k l y confluent ( respect ive ly

e-confluent) or de te rmine a r educed finite monadic sys tem 1Z 2 on Y- such that

R and R 2 are equivalent , and R 2 is w e a k l y confluent ( respec t ive ly e-confluent)

iff R is.

If (£1.bl) 6 R and (E2.b 2) 6 R such thai g I = xg2Y for some x.y 6 E ~. then we obtain

the critical pair (xb2Y.bl). which can always be oriented into a monadic rule. To obtain

a r educed sys tem the rule (gl ,bl) is deleted, and if 81 7~R2 bl (where R 2 := R\{gi.bl)}), then the rule result ing from this crit ical pair must be a d d e d to p rese rve the congruence .

If no letter is congruent to e or to a different letter, then e v e n reduct ions are preserved,

wh ich is not the case if a letter becomes reducib le to e or to a different letter.

3 . A polynomial fee% for e-confluence

Let R be a finite reduced monadic string-rewriting system on Y. such %hat ~ is a group.

If R is confluent on (el, then for each a 6 Y.., there exists a word u a 6 Y--~ such that

SUe ~'>R e. In fact, au a i--> R 8 for some i ~ IY-I, ~d hence, lUal ~ ]7.1 .(~-l), where It =

max{]gl I (g,b) 6 R}. Further, if au 2-> R e, then us -->R e, since (el is closed under cyclic

permutations, and therewith <e> also has this property. If one of these conditions is not

satisfied, then R is not e-confluent. It is easy to see that R is e-confluent iff for all a 6 Y-

and w 6 [a -l ] n IRRCR): aw -L> R e. This set might not be easy to construct, so we will

use an approximation of it, namely the set of right-inverses which will play a central

role in a test for e-confluence.

Definition 3.1. For u 6 Z ~, let RIR(U), the set of r ight inverses of u, consist of all words v

such that u v --->R e, and no step of the reduct ion sequence is performed ent i re ly wi th in

u or wi th in v. To be precise

RIR(U) = {v 6 Y-~ I 3k > 1 3ul.....u k. VI,...,V k 6 y-~ with u = Uk...U 1, V = Vl...V k. (UlVl.a 1) 6 R.

(u2alv2,a 2) 6 R...,(Ukak_IVk,e) 6 R and u i • e * v i if at_ I = e}.

If u.v 6 IRR(R) and uv -->R e, then v 6 RIR(U) since R is monadic. RIR(U) may be infinite.

but it is easy to compute.

Lemma 3.2. For e v e r y u £ Z ~, RIR(u) is a regular set. F rom R and u a nfsa for this set

can be constructed in po lynomia l time.

Now w e are r e a d y to formulate a test for e-confluence.

Theorem 3.3. R is confluent on ( e l iff the fol lowing conditions are satisfied:

(i) ¥a 6 Y-: A~(RIn(a) 'a) n IRR(R) = {e} if RIR(a) :~ ~ and

(it) V(p.q) 6 UCP(R) Vpl 6 AR(p) Vq I 6 ZkR(q): A~(q-RIR(Pl)) n IRR(R) = {e} = A~(p.RIR(ql)) n IRR(R) if RIR(P l) $ 0 t RIR(ql).

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Accord ing to the discussion at the end of Section 2, w e m a y assume that R is reduced.

Thus, if (p,q) E UCP(R), then there exist words x,y E Z ~ and rules (el,bl),(4~2,b 2) E R such

that Rlx - y e 2. O < lyl < I~iI. p - b lx and q = y b 2. In part icular , y is a proper prefix of

e 1. and x is a proper suffix of e 2. i.e. x and y are both i r reducible . If b 1 = e, then p is

i r reducible , and if b 2 = e, then q is i rreducible. Otherwise, the sets A~(p) and A~(q) are

of size b o u n d e d from a b o v e b y ~, whe re ~ = max{{4~l I (e,b) E R}. Hence, to ve r i fy the

condit ions (i) and (it) of Theorem 3.3 on ly a po lynomia l ly bounde d number of tests must be performed. Since w e can construct nfsa~s recogniz ing the i n v o l v e d testsets in po lynomia l t ime. w e obta in the fol lowing result.

Corol lary 3.4: The fol lowing p rob lem is dec idab le in po lynomia l time.

INSTANCE: A finite monadic s t r ing-rewri t ing system R on Z such that the monoid 9}~ R

is a group.

QUESTION: Is R confluent on [e]i~ ?

Let u s now consider the p rob lem of dec id ing w e a k conf luence of such a system. We

m a y assume that R is confluent on [e] and reduced, and so w e on ly h a v e to check the

conf luence of R on the congruence classes of i r reducible letters. Let b be such a letter

and let b "1 be an i r reducib le inverse of b. Then RIp(b-l) n IRR(R) = [b ] n IRR(R). Thus. w e

h a v e the fol lowing characterization.

Theorem 3.5. R is w e a k l y confluent iff condit ions (i) and (it) of Theorem 3.3 and (iii)

Va 6 Z n IRR(R): RIR(a't) n IRR(R) = {a} for some i r reducible inverse a -1 of a are satisfied.

Since e-conf luence is dec idab le in po lynomia l time, and since condit ion (iii) of

Theorem 3.5 is also dec idab le in po lynomia l t ime w e get:

Coro l la ry 3.6: The fol lowing p rob lem is dec idab le in po lynomia l time.

INSTANCE: A finite monad ic s t r ing-rewri t ing sys tem R on ~ such that the monoid 9 ~

is a group,

QUESTION: Is R w e a k l y confluent ?

One might ask whe the r e-conf luence implies w e a k confluence. This is not the case as

shown b y Example 2.2a). But the existence of an e-confluent monad ic sys tem R implies the

existence of a w e a k l y confluent monadic sys tem R ~ w h i c h is equ iva len t to R. In fact. R' m a y

be constructed in po lynomia l t ime from R. W.l.o.g. w e m a y assume that R is reduced.

Thus, for each reduc ib le letter b, R contains exact ly one rule wi th lef t-hand side b. and

this letter does not occur in a n y other rule. Let a be the smallest i r reducible letter (~ is ordered) such that R is not confluent on [a], and let a -1 be an i r reducible inverse of a.

Then RIR(a -1) n IRR(R) = [a] n IRR(R) p roper ly contains {a}. This set is finite in this case,

since otherwise w e wou ld h a v e i r reducible words uxnv < '~ >R u v +-~-->R a. Since 9Yt R is a

group, and since R is confluent on [el , this would i m p l y that x n < ~ >R e contradic t ing the

fact that u x n v is i r reducible . So let w 1.....w k be the i r reducible words in [a] different from

a, w h i c h can be c o m p u t e d in po lynomia l time. Then R v {(wi,a) I i = 1,...,k} is monadic , e-confluent and also confluent on [a]. This process m a y be i terated wi th the next i r reducib le

letter on w h i c h the result ing r educed sys tem is not confluent. For the result ing sys tem R'

w e obtain "->R c "~R" is equ iva len t to R. and R I is w e a k l y confluent.

286

4. The completion procedure

Based on our confluence test. we now present a procedure which on input a finite monadic string-rewriting system R o presenting a group, tries to construct a w e a k l y confluent monadic system R that is equivalent to R o. This procedure contains two main subroutines: NORMALIZATION and CONTEXT.RESOLVING. The first one realizes the reduction

process explained at the end of Section 2. The second one introduces new rules if necessary based on the test of Theorem 3.5. There are three types of regular sets which m a y contribute new rules depending on the condition actually checked:

E a := A~(RIR(a). a) n IRR(R)\{e} for a 6 Z

Lar 1 := RIR(a'I) n IRR(R)\{a} for irreducible a 6 7-

SPi := A~(q.RIR(Pi)) n IRR(R)\{e} and AR( p .RIR(qi)) n IRR(R)\{e} Sqi :=

for (p,q) 6 UCP(R) and Pi 6 A~(p) , qi 6 A~(q).

Since these sets can be infinite, we have to determine a finite number of special and monadic rules which can reduce all the computed divergences. For doing this, notice

that E a. SPi and Sqi are subsets of [el, and that La_ 1 c [a]. From the nfsa's for these sets a finite number of simple accepting paths and of simple loops which generate all accept ing paths can be extracted in polynomial time, Since we have a group, the irreducible

words corresponding to simple loops are equivalent to e. (The argument for this being similar to the one at the end of Section 3). In the nfsa for La_ 1 the irreducible words corresponding to simple accepting paths are equivalent to a, so they lead to proper monadic rules.

Let GENSPATH and GENSLOOP be procedures which compute the irreducible words corresponding to the simple paths, respectively simple loops, w h e n applied to a nfsa accept ing one of the above sets. Since the subroutine CONTEXT_RESOLVING m a y introduce new rules which destroy the property of being reduced and also add new unresolved critical pairs, w e have to keep app ly ing both subroutines until a stable system is obtained.

Procedure 41:

INPUT: A finite monadic string-rewriting system R on an ordered alphabet Y- such that

the monoid 9)I R is a group.

begin i <-- 0 ~ Ri +-- R; NORMALIZATION: Reduce right-hand sides using first applicable rules~

while 3el,e2,x,y C Y..*: e 2 - xe ly ^ (81,b 1) C R i ^ (E2.b 2) C R i do

begin R i + - Ri\{(£2,b2)}, if b 2 ¢ A~i(XblY) then R i ~ - R i u {<xbly,b2>}~ Reduce right-hand sides

end

comment: At this point the system R i is reduced. Here, <xbly,b 2> is the monadic rule resulting from this pair using the ordering on "~ if both sides are letters.

287

CONTEXT_RESOLVING: For each a 6 F. n IRR(R i) compute an irreducible inverse a'l~

Compute UCP(R i) = R!I 4 - O

For all a C ~- do {E a <-.- AR(RIR(a)-a) n IRR(R)\{e}~ ~'a <'- GENSPATH(Ea) u GENSLOOP(Ea)~ RI 4 - o ¢Ce.e I e C

For all a E E n IRR(R i) do {La_ I 4 -

ARMa-I 4 - 4.--

RIR(a-1) n IRR(R)\{a}~

GENSPATH(La_ I)~ R I u {<8.a> I g Cl0Ia_l},

For all (p,q) 6 UCP(R i) do

La_ l 4 - GENSLOOP(La_I)~ R i 4 - R~ u {(8.e) 1 8 E La_ 1 }

{For all Pi C A~(p) . qi C ~ ( q ) do

{SPi <-" A~(q'RIR(Pi)) n IRR(R)\{e}~ Sqi 4 - A~(p'RIR(qi)) n IRR(R)\{e}, SPi <-- GENSPATH(SPi] u GENSLOOP(SPi), ~qi 4 - GENSPATH(Sqi) u GENSLOOP(Sqi), A R~ <--- R i u {(g,e) I e E SPi u ~qi }}}

comment: The new rules are now collected in R: all left- and right-hand sides of the rules 1' in R i' are Ri-irreducible.

if RI ~: O then { Ri. 1 4 - R i u RI, i 4 - i*I, goto NORMALIZATION }

comment : At this point R i is weak ly confluent and reduced

OUTPUT: R i end

We claim that the above procedure determines a finite monadic system R i that is weak ly confluent and that is equivalent to R wheneve r an equivalent e-confluent monadic system exists. Otherwise it enumerates an infinite monadic system Rco hav ing both these properties. The subroutine NORMALIZATION applied to some system R, a lways terminates with a reduced system R' which is equivalent to R and IRR(R') c IRR(R). So a rule wh ich was once deleted will never be introduced again, neither b y NORMALIZATION nor b y CONTEXT_RESOLVING. If no letter is equivalent to e or to a different letter then "-->R c "-->R"

Lamina 4.2. Let R be a finite monadic string-rewriting system on Z such that the monoid is a group. If Procedure 4.1 terminates on input R. then it yields a finite monadic

system R i on ~ that is equivalent to R. weak ly confluent and reduced.

Thus, w h e n e v e r Procedure 4.1 terminates, the system R i constructed has indeed all the properties we want. It remains to show that this algorithm does terminate w h e n e v e r a monadic system S exists that is finite, equivalent to R, and confluent on [ e l R. Because of the discussion at the end of Section 3 we m a y assume that S is in fact weak ly confluent. Notice also that the existence of such a system does not depend on the fixed ordering on ~-, since a different ordering induces just a renaming. As a first step towards proving this fact, we analyse the situation w h e n Procedure 4.1 does not terminate.

288

Lemma 4,3. Let R be a finite monadic string-rewriting system on Z such that the monoid

9J~ R is a group. If Procedure 4.1 does not terminate on input R, then it enumerates an

infinite monadic system Rco that is reduced, equivalent to R and weak ly confluent.

Proof: Assume that Procedure 4.1 does not terminate on input R. Then if enumerates an

infinite sequence Ro,R1,R 2 .... of finite monadic string-rewriting systems on E satisfying the

following conditions for all j ~ O.

Rj is equiva lent to R and reduced

IRR(Rj.I) C IRR{Rj)

<e>R j C <e>Rj÷l

<a>Rj ~ <a>Rj÷l for a E Y- ¢~ IRR(Rj.1).

Let Roo "= {(g,b) [ 3j > 0 Vi ~ j: (g,b) C Ri}, i.e. Roo is the set of persistent rules.

Procedure 4.1 can be interpreted as enumera t ing this system. Rco is an infinite monadic

system, since deleted rules are never introduced again. Now it can be shown that Rco is

equivalent to R, that it is reduced, and that it is weak ly confluent. •

Thus, on input a finite monadic string-rewriting system R presenting a group,

Procedure 4.1 a lways "computes" a monadic system Rco that is reduced, equiva lent to R

and weak ly confluent. Procedure 4.1 terminates iff this system Rco is finite. Hence, it

remains to characterize the condition under which this system Rco is indeed finite.

Theorem 4.4. Let R be a finite monadic string-rewriting system on Z such that the

monoid 9 ~ is a group. On input R, Procedure 4.1 terminates if and only if there exists

a finite monadic system S on Y- that is equivalent to R and e-confluent.

It can easily be verified, that the system Rco is un ique ly determined b y R and the

ordering on ~... i.e. if S and T are reduced monadic systems on Z that are both equivalent

to R and that are both weak ly confluent, then S and T are in fact identical. This coincides

with more general situations for systems that are confluent eve rywhere (see e.g. [8]).

We close this section by presenting an example to illustrate the w a y Procedure 4.1 works.

Example 4.5: Let Z = {a,b.c} and R - {(ab,e),(ba.e).(c2,e),(cac,b)}. ~ is a group. R is reduced.

a -1 = b, b -1 = a. c -1 = c and UCP(R) = {(ac.cb),(bc.ca)}.

The procedure first computes the sets RI(u) for u C Z u {ac,cb,bc,ca}:

RI(a) = {b} , RI(b) - {a} . RI(c) = {c.aca}

RI(ac) - {cb,ac,acab} , RI(cb) - {ac,aaca}

RI(bc) - {ca.acaa} , RKca) - {ca,bc,baca}.

Now the check AR(RI(u).u) n IRR(R) = {e} for u 6 ~-- is done. In the present case this

is true, so no rules are introduced b y this test. Since aca is irreducible in RI(c), we get

the monadic rule (aca,c) as a candidate. Finally, from the test A~(p.RI(q)). respect ively

A~(q. RI(p)), for the two unresolvable critical pairs in R we get as rules (acaaca.e),(cbcb.e).

(caacaa.e) and (bcbc.e). From these last four rules, two are deleted by NORMALIZATION

and so R 1 = R u {(aca,c),(cbcb,e),(bcbc,e)}.

In the next call of CONTEXT_RESOLVING the inverses stay as they were. but new

critical pairs are added, e.g. (bcb,c) and (cbc,a). In fact these rules will be added, since

bcb and cbc are irreducible right-inverses of c, respectively b. After the second step and

289

NORMALIZATION w e get the sys tem

R 2 - {(ab.e),(ba,e),(c2,e),(cac,b),(aca.c),(cbc,a),(bcb.c)}

wi th unreso lvab le crit ical pairs {(ac,cb).(bc,ca).(cbb,aac),(caa.bbc)}.

For this monad ic sys tem RI(a) n IRRCP~ 29 ~ {b). RI(h) n IRR(R 2) - {a) and RI(c) n IRR(R 2) -

{c}, so no proper monadic rules are added.

F ina l ly because of

RI(ac) n IRRCR 2) - RI(cb) n IRR(R 2) - {ac,cb), RI(bc) n IRR(R 2) - RI(ca) n IRRCR 2) - [bc,ca}, RI(cbb) n IRR(R 2) = RICaac) n IRR(R 2) - {cbb,aac,acb} and

RI(caa) n IRR(R 2) - RI(bbc) n IRR(R 2) - {bbc.caa.bca}

no further rule is added. The procedure terminates with the weakly confluent system R 2.

5. C o n c l u d i n g R e m a r k s

We h a v e d e v e l o p e d a special ized complet ion procedure for monadic s t r ing-rewri t ing

systems present ing groups, based on a po lynomia l test for conf luence on the congruence

class of the ident i ty for such systems. The main purpose for such a p rocedure is to find

equ iva len t presentations w h i c h are syn tac t ica l ly restricted and hence p rov ide much more

structural and algori thmical information than genera l presentations. The comple t ion

p rocedure itself can be seen as a k ind of unfai l ing complet ion, w h e r e the role of non-

or ientable equations is taken b y the unresotvable crit ical pairs, and ground conf luence is

r ep laced b y conf luence on [e l . Thu~ the d i v e r g e n c y of usual complet ion procedures m a y be a v o i d e d in some cases. A general izat ion to other classes of systems, e.g. l eng th - reduc ing

ones, seems to be quite hard, since no dec idab le criteria for conf luence on a single congruence class are k n o w n for other classes.

In the subroutine CONTEXT_RESOLVING Procedure 4.t adds special rules w h e n ax --->r~ e

but xa ~[>R e. In this w a y one tries to m a k e <e> closed under cyc l i c permutat ions. Here

is a possible i m p r o v e m e n t of the procedure: Wheneve r an i r reducible word w 6 [ e l is

found, add special or monad ic rules w h i c h guarantee that w and all cyc l i c permutat ions

of it reduce to e. In fact this idea is similar to the one used in the comple t ion procedures of ([4,10]) and is based on the notion of symmet r i zed group presentations [11].

If w e start wi th a special s t r ing-rewri t ing sys tem R such that the lef t -hand sides form

a symmet r i zed set ( eve ry e lement is cyc l i c a l l y r educed and the set is closed ~ under cyc l i c

permutat ions and tak ing inverses), then <e> R is closed under cyc l i c permutat ions.

LeChenadec [tO] presents a process he calls the group symmetr iza t ion a lgor i thm that on

input a finite symmet r i zed group presentat ion <E,L> satisfying certain small cancel la t ion

conditions generates the finite l eng th- reduc ing sys tem S used in Dehn's a lgori thm to solve

the word p rob lem for such groups: Rules of the form w --> e are split as w - u v ---> e, whe re u is min imal wi th u ~ v "1, and the rule u "--> v "1 is generated. We are doing in

fact the same if v -1 is a letter.

There are examples where the sets RI(a) or RI(Pi) are indeed infinite. One interest ing question is whe the r the conf luence cri terion can be special ized to h a v e finite test sets and

not just regula r ones. This is i ndeed the case for special systems [20]. The same conf luence

criterion holds, if one restricts the elements of RI to be i rreducible , but these sets still m a y

be infinite in the monadic case.

290

The examples presented here are fairly simple ones. The reason for this is due to the number and size of the sets RIRi(U) involved. An implementation of the procedure is currently under way and we hope to gain further insights into how the procedure behaves in practice.

References

[1] ].M. Autebert, L. Boasson. G. Senizergues, Groups and NTS languages; J. Comput. System Sci. 35 (19879. 243.267.

[2] R.V. Book; Decidable sentences of Church-Rosser congruences, Theoretical Computer Science 23 (1983). 301-312.

[3] R.V. Book; Thue systems as rewriting systems; ]. Symbolic Computation 3 (1989, 39-68.

[4] H. B~cken; Reduction systems and small cancellation theory; in: Proceedings 4th Workshop on Automated DeducHon (1979), 53-59.

[5] R.H. Gilman; Presentations of groups and monoids: J. of Algebra 57 (1W9). 544-554. [6] J.E. Hopcrofi, J.D. Ullman, Introduction to Automata Theory. Languages and

Computation (Addison-Wesley. Reading, MA, 1979). [7] M. Jantzen. Confluent String-Rewriting (Springer, Berlin, 1988). [8] D. Kapur, P. Narendran; The Knuth-Bendix completion procedure and Thue systems,

SIAM J. on Computing 14 (1985), 1052-1072. [9] D. Knuth, P. Bendix; Simple word problems in universal algebras; in: J. Leech (ed.),

Computational Problems in Abstract Algebra (Pergamon, New York, IWO), 263-297. [10] Ph. LeChenadec; Canonical Forms in Finitely Presented Algebras (Pitman: London.

Wiley: New York, Toronto, 1986). [11] R.C. Lyndon. P.E. Schupp; Combinatorial Group Theory (Springer. Berlin. 1977). [12] K. Madlener. F. Otto, Using string-rewriting for solving the word problem for finitely

presented groups, Information Processing Letters 24 (1987), 281-284. [13] K. Madlener, F. Otto, About the descriptive power of certain classes of finite

string-rewriting systems, Theoretical Computer Science 67 (1989), 143-172. [14] K. Madlener, F. Otto, Decidable sentences for context-free groups, in: C. O~offrut,

M. ]antzen (eds.) Proceedings of STACS '91, Lecture Notes in Computer Science 480 (1991], 160-171.

[15] K. Madlener, P. Narendran, F. Otto: A specialized completion procedure for monadic string-rewriting systems presenting groups, SEKI-Report SR40--24, University of I(aiserslautern (1990).

[16] D.E. Muller, P.E. Schupp; Groups the theory of ends, and context-free languages, J. Comput. Systems Ci. 26 (1983), 295-310.

[17] P. Narendran. C. OIDunlalng, F. Otto; It is undecidable whether a finite special string-rewriting system presents a group: Discrete Math., to appear.

[18] F. Otto~ On deciding whether a monoid is a free monoid or is a group; Acta Informatica 23 (1986). 99-110.

[19] F. Otto; On deciding the confluence of a finite string-rewriting system on a given congruence class, J. Comp. Sci. Sciences 35 (1987). 285-310.

[20] F. Otto, The problem of deciding confluence on a given congruence class is tractable fox finite special string-rewriting systems, Preprint No. 4/90, FB Math., GhI<, Kassel. West Germany, 1990.

[21] F. Otto: Completing a finite special string-rewriting system presenting a group on the congruence class of the empty word; Preprint No. 8/90, FB Math.. GhK. IKasset, West Germany, 1990.

[22] C.C. Squier, Word problems and a homological finiteness condition for monoids, J. Pure Appl. Algebra 49 (1987). 201-217.

[23] L. Zhang; The word problem and undecidability results for finitely presented special monoids~ submitted for publication.