Klaus Madlener. Internet: Friedrich Otto. Fachbereich Mathematik/Informatik, Universitat Kassel.

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1 Some Undecidability Results For Finitely Generated Thue Congruences On A Two-Letter Alphabet Klaus Madlener Fachbereich Informatik, Universitat Kaiserslautern D{67653 Kaiserslautern Internet: madlener@informatik.uni-kl.de Friedrich Otto Fachbereich Mathematik/Informatik, Universitat Kassel D{34109 Kassel Internet: otto@theory.informatik.uni-kassel.de December 18, 1996 Abstract Following the course set by A. Markov (1951), S. Adjan (1958), and M. Rabin (1958), C. O'Dunlaing (1983) has shown that certain properties of nitely generated Thue congruences are undecidable in general. Here we prove that many of these undecidability results remain valid even when only nitely generated Thue congruences on a xed twoletter alphabet 2 are considered. In contrast to a construction given by P. Schupp (1976) which applies to groups only, we use a modied version of a technical lemma from A. Markov's original paper. Based on this technical result we can carry the result of A. Sattler-Klein (1996), which says that certain Markov properties remain undecidable even when they are restricted to nitely generated Thue congruences that are decidable, over to the alphabet 2.

2 1 Introduction A string-rewriting system R on some alphabet is a set of pairs of strings over. It induces a congruence $ on R, the Thue congruence generated by R, thus dening the factor monoid M R := = $. Since the monoid M R R is uniquely determined by and R (up to isomorphism), the ordered pair (; R) is called a (monoid-) presentation of M R. Given a nite presentation (; R), what can we say about the monoid M R it presents? As shown by A. Markov [Mar51] almost all properties of interest are undecidable. For example, it is undecidable whether the monoid M R is trivial, that is, whether it consists of a single element only, whether it is nite, whether it is a free monoid, whether it is (left-, right-) cancellative, whether it is commutative, whether it is a group, or whether it has a decidable word problem. For the class of nitely presented groups, corresponding undecidability results were established by S. Adjan [Adj58] and M. Rabin [Rab58]. Later C. O'Dunlaing proved that also certain properties of nitely generated Thue congruences are undecidable [ O'D83]. For example, it is undecidable in general whether there exists a nite convergent string-rewriting system that generates a given Thue congruence. Here a string-rewriting system is called convergent if it is noetherian and conuent (see Section 2 for the denitions). Observe that this property is really a property of the Thue congruence considered and not just a property of the monoid presented by that Thue congruence [BO84]. If we consider Thue congruences on a single-letter alphabet, then some of the above properties become decidable. For example, each monoid that is generated by a single letter has a decidable word problem. On the other hand, there are nitely generated Thue congruences on a two-letter alphabet that are undecidable. Thus, with respect to the cardinality of the underlying alphabet, there is a clear-cut borderline between the decidable and the undecidable nitely generated Thue congruences. What can be said about the other decision problems mentioned above? Do they all remain undecidable even if we just consider nitely generated Thue congruences on a xed two-letter alphabet? A. Markov's as well as C. O'Dunlaing's proofs of their undecidability results cited above are based on a construction of a string-rewriting system R u;v on the alphabet :=? [ fc; dg from a given string-rewriting system R on? and two strings u; v 2? such that the following holds: if u $ v, then the congruence R $ R u;v generated by R u;v is trivial, but if u $ v does not hold, then the monoid M R R presented by (?; R) is embedded in the monoid M Ru;v presented by (; R u;v ) via the identity mapping. If the system R is chosen appropriately, then it is the collection fr u;v j u; v 2 g which shows that the property considered is undecidable. Thus, these proofs require that the Thue congruences considered are dened on some alphabet containing at least ve letters (cf. Section 2). Using small cancellation theory P. Schupp presents an ingenious construction in [Sch76] based on which he obtains the undecidability of Markov properties for groups that are presented by nite monoid-presentations containing only two generators. Obviously, this result does not cover properties of monoids that are satised by all groups like being (left-, right-) cancellative or like being a group. Further, A. Sattler-Klein has shown that certain Markov properties remain undecidable even when they are restricted to nitely generated Thue congruences that are decidable [SK]. However, the Thue congruences considered by A. Sattler- Klein do not present groups unless they are trivial. Thus, P. Schupp's construction cannot be used to carry this result over to the two-letter case. Here we present a new version of A. Markov's embedding lemma, which constructs a 2

3 nite string-rewriting system R u;v on the two-letter alphabet 2 = fa; bg and a mapping :?! from a nite string-rewriting system R on? and two strings u; v 2 2? such that the following holds: if u $ v, then the Thue congruence R $ R u;v on 2 is trivial, but if u $ v does not hold, then, for all x; y 2 R?, (x) $ R u;v (y) if and only if x $ R y. Even though the mapping does not induce an embedding of the monoid M R presented by (?; R) in the monoid M Ru;v presented by ( 2 ; R u;v ) (this is the price we have to pay for getting down into a two-letter alphabet), this construction suces to prove that certain Markov properties are undecidable even when they are considered only for nitely generated Thue congruences on a two-letter alphabet. The examples of such properties include the properties of being (left-, right-) cancellative and of being a group. With this construction we can also carry A. Sattler-Klein's result mentioned above over to the case of a xed two-letter alphabet. We feel that these results, which cannot be shown by P. Schupp's method, clearly demonstrate the value of our construction. This paper is organized as follows. In Section 2 we dene the notion of Markov properties, describe P. Schupp's construction of an embedding, and based on that prove the undecidability of certain Markov properties for nitely generated Thue congruences on a two-letter alphabet. As an additional application we show that the divergence problem of the Knuth- Bendix completion procedure [KB70] is undecidable for the case of two-letter alphabets, even when a xed partial ordering is considered. For a discussion of the Knuth-Bendix completion procedure and the fundamental role that the given ordering plays during completion see, e.g., the paper by J. Avenhaus [Ave86]. In Section 3 the revised version of A. Markov's embedding lemma is presented. Its proof makes heavy use of rewriting techniques. Then, in Section 4 we give some applications of this result. We close this introduction by restating a few basic denitions in order to establish notation. For a nite alphabet, denotes the set of strings over including the empty string, and + := r fg denotes the set of non-empty strings over. A stringrewriting system R on is a set of pairs of strings over. By dom(r) we denote the set dom(r) := f` 2 j 9r 2 : (`; r) 2 Rg of left-hand sides of rules of R. The reduction relation! on R that is generated by R is the reexive, transitive closure of the single-step reduction relation! R := f(x`y; xry) j x; y 2 ; (`; r) 2 Rg. The reexive, symmetric, and transitive closure $ of! R R is a congruence relation on, the Thue congruence generated by R. Finally, by IRR(R) we denote the set of strings that are irreducible mod R, that is, x 2 IRR(R) if there is no y 2 satisfying x! R y. For more information regarding the denitions and notation concerning string-rewriting systems the interested reader is asked to consult the literature, e.g., the monograph by R.V. Book and F. Otto [BO93], which is our main reference in these matters. 2 Markov properties for monoids and Thue congruences A Thue congruence on some nite alphabet is an equivalence relation on the set of strings over that is compatible with the operation of concatenation of strings. It is called nitely generated if there exists a nite string-rewriting system R on such that coincides with the Thue congruence $ generated by R. By M R R we denote the factor monoid = $ R. This monoid is uniquely determined by and R (up to isomorphism), 3

4 and accordingly, the ordered pair (; R) is called a (monoid-) presentation of M R with generators and dening relations R. If R is nite, we say that the monoid M R is nitely presented. If only contains a single letter, then each nitely generated Thue congruence on is decidable, that is, there exists an algorithm that decides whether or not two strings u; v 2 are congruent mod. Hence, each monoid with a single generator has a decidable word problem. On the other hand, if contains two or more letters, then there are nitely generated Thue congruences on that are undecidable, and so there are nitely presented monoids with generating set that have an undecidable word problem. In fact, in this situation (almost) all interesting properties of monoids are undecidable. By `interesting properties' we mean properties such as { having a decidable word problem, { being nite, { being commutative, or { being a group. These are all examples of Markov properties. A property P of nitely presented monoids is called a Markov property if it satises the following conditions: (0) P is invariant, that is, every monoid that is isomorphic to a monoid possessing property P itself possesses this property; (1) there exists a nitely presented monoid M 1 which does not have property P, and which is not isomorphic to a submonoid of any nitely presented monoid having property P ; (2) there exists a nitely presented monoid M 2 which possesses property P. A. Markov has established the following fundamental undecidability result [Mar51], which has been extended by S. Adjan [Adj58] and M. Rabin [Rab58] to the special case of nitely presented groups. Proposition 1 [Mar51]. Let P be a Markov property of nitely presented monoids. Then the following problem is undecidable: INSTANCE: A nite monoid-presentation (; R). QUESTION: Does the monoid M R possess property P? A. Markov's proof of Proposition 1 is based on the following technical lemma (see, for example, [BO93] Section 7.3 for a detailed presentation of this proof). Lemma 2 [Mar51]. Let? be a nite alphabet, let c; d 62? be two additional letters, and let :=? [ fc; dg. Given a nite string-rewriting system R on? and two strings u; v 2?, a nite string-rewriting system R u;v on and a morphism ' :?! can be constructed eectively such that either (i) u $ R v, and the congruence $ R u;v on is trivial, or 4

5 (ii) u = v, and ' induces an embedding from R into R R u;v, that is, for all x; y 2?, x $ y if and only if '(x) R $ R u;v '(y). In fact, ' is just the identity on?. Thus, in (ii) the monoid M R is embedded in the monoid M Ru;v by '. Let P be a Markov property of nitely presented monoids, let M 1 and M 2 be the two nitely presented monoids for P according to the denition of Markov property, and let M 3 = ( 3 ; R 3 ) be a nitely presented monoid with an undecidable word problem. For u; v 2 3, let M u;v = (?; R u;v ) be the nitely presented monoid that is obtained from u, v, and the free product M 1 M 3 of the monoids M 1 and M 3 by the construction of Lemma 2. Then the free product ^M u;v := M u;v M 2 has property P if and only if u $ R 3 v. Since M 3 has an undecidable word problem, this proves the undecidability of the property P. Observe that the presentation of the monoid ^Mu;v we obtain this way contains at least 5 generators. So the question arises whether Markov properties of nitely presented monoids remain undecidable when they are restricted to nitely presented monoids that are generated by just two generators. P. Schupp has answered this question in the armative, even for the restricted case of groups, by proving the following strong undecidability result. Proposition 3 [Sch76] Let P be a Markov property of groups that is possessed by the trivial monoid (group). Then the following problem is undecidable: INSTANCE: A nite set of strings L fa; bg +. QUESTION: Does the group G L that is presented by (a; b; f(w; ) j w 2 L [ fa 2 ; b 3 gg) possess property P? Recall that denotes the empty string. Observe that the monoid presented by (a; b; f(w; ) j w 2 L [ fa 2 ; b 3 gg) is actually a factor group of the free product C 2 C 3 of the cyclic group C 2 of order 2 and the cyclic group C 3 of order 3. P. Schupp's proof of Proposition 3 is based on a construction that does the following: INPUT: A nite presentation (; R) of a group G, and a string u 2. OUTPUT: A nite presentation (a; b; f(w; ) j w 2 L u [ fa 2 ; b 3 gg) of a group G u satisfying the following conditions: (i) if u = in G, then G u is the trivial group, and (ii) if u 6= in G, then G is embedded in G u. If G = (; R) is the free product of a nitely presented group G 1 which cannot be embedded in any nitely presented group possessing property P and a nitely presented group G 3 which has a undecidable word problem, then, for all u 2, u = holds in G if and only if G u is the trivial group if and only if the group G u possesses property P. Actually, using the above construction we can even show that certain properties of Thue congruences are undecidable that are not invariant under monoid-isomorphisms in general. A property P of Thue congruences is called a Markov property of Thue congruences if it satises the following conditions: 5

6 (1) there exists a nitely generated Thue congruence 1 on some nite alphabet 1 such that the monoid M 1 := = 1 1 is a group which cannot be embedded in any monoid that has a nite presentation (?; R), for which the Thue congruence $ R possesses property P ; (2) the trivial Thue congruence on any nite alphabet possesses property P. Observe that the properties of string-rewriting systems considered by C. O'Dunlaing in [ O'D83] can be interpreted as a special class of Markov properties of Thue congruences. Using the above construction we now derive the following undecidability result. Proposition 4 Let P be a Markov property of Thue congruences. Then the following problem is undecidable: INSTANCE: A nite string-rewriting system S on 2 = fa; bg. QUESTION: Does the Thue congruence $ S possess property P? Proof. Let 1 be a nitely generated Thue congruence on? 1 such that the monoid M 1 :=? 1 = 1 is a group which cannot be embedded in any nitely presented monoid (?; R), where the Thue congruence $ R has property P, and let (? 2; R 2 ) be a nite monoid-presentation of a group G 2 with an undecidable word problem. Without loss of generality we may assume that? 1 and? 2 are disjoint. Hence, (? 1 [? 2 ; R 1 [ R 2 ) is a nite presentation of the free product G of the groups M 1 and G 2, where R 1 is a nite string-rewriting system on 1 generating the Thue congruence 1. For u 2? 2, we apply the construction above to the presentation (? 1 [? 2 ; R 1 [ R 2 ) and the string u. In this way we obtain a nite presentation ( 2 ; S u ) := ( 2 ; f(w; ) j w 2 L u [fa 2 ; b 3 gg) of a group G u satisfying the following conditions: (i) If u = in G 2, then u = in G, and hence, G u is the trivial group, that is, $ S u is the trivial Thue congruence on. Hence, in this case the Thue congruence 2 $ S u possesses property P. (ii) If u 6= in G 2, then u 6= in G, and hence, G is embedded in G u. Therewith M 1 is embedded in G u, and thus, we see from the choice of M 1 that the Thue congruence $ S u does not possess property P. Hence, for all u 2?, u = in G 2 2 if and only if the Thue congruence $ S u possesses property P. This proves the undecidability of P for nitely generated Thue congruences on 2. 2 Before discussing the shortcomings of P. Schupp's construction let us consider some interesting consequences of Proposition 4. Corollary 5 The following decision problems are undecidable: INSTANCE: A nite string-rewriting system R on 2. QUESTION 1: QUESTION 2: QUESTION 3: Is the Thue congruence $ R decidable? Is the monoid 2= $ R nite? Does there exist a nite convergent string-rewriting system R 0 on 2 that also generates the Thue congruence $ R? 6

7 Here a string-rewriting system R is called convergent if it is noetherian and conuent, that is, there is no innite sequence of reduction steps of the form w 0! R w 1! R : : :, and for all u; v; w 2, if u! v and u R! w, then there exists some z 2 R such that v! z and R w! R z. See e.g. the monograph by R.V. Book and F. Otto [BO93] for more information on these notions. Since we have a xed alphabet 2 we can in fact consider decision problems that are more specialized than the three above. In the Knuth-Bendix completion procedure compatible well-founded partial orderings are used to orient pairs (u; v) of strings into rules. In fact, together with the nite string-rewriting system R that is to be completed into an equivalent convergent system R 0, the user is asked to supply the compatible well-founded ordering > that is to be used. The successful termination of the completion procedure crucially depends on this ordering. Unfortunately, we have the following undecidability result. Corollary 6 Let > be a compatible well-founded partial ordering on 2 such that a > and b > both hold. Then the following problem is undecidable: INSTANCE: A nite string-rewriting system R on 2. QUESTION: Does there exist a nite convergent system S on 2 such that S is equivalent to R, and S is based on >, that is, ` > r holds for each rule (`! r) 2 S? Proof. The trivial system R 2 = fa! ; b! g is certainly convergent, and it is based on the given ordering > by the hypothesis. On the other hand, if a system S of the form required exists for R, then the Thue congruence $ R is decidable. Thus, we see from the proof of Proposition 4 that the problem above is undecidable. 2 If the given compatible well-founded partial ordering > is actually a linear (or total) ordering, then the Knuth-Bendix completion procedure terminates successfully on input R and > if and only if there exists a nite convergent system S that is equivalent to R and that is based on >. Thus, we obtain the following undecidability result for the divergence problem. Corollary 7 Let > be a compatible well-founded linear ordering on 2. Then the following problem is undecidable: INSTANCE: A nite string-rewriting system R on 2. QUESTION: Will the Knuth-Bendix completion procedure terminate on input R and >? Observe that a > and b > are necessarily satised if > is a compatible well-founded linear ordering on 2. Examples of such orderings are the length-lexicographical orderings and the syllable orderings [SK91]. 3 An improved version of Markov's lemma As seen above P. Schupp's construction yields very strong undecidability results. An advantage is certainly the fact that these undecidability results do not only hold for monoids and Thue congruences in general, but that they remain valid even if we require in addition that the monoids considered are actually groups. However, this advantage turns into a problem when we are dealing with a situation in which groups are explicitly excluded. In the next 7

8 section we will present examples of such situations. In this case we would rather like to have a modied version of Markov's lemma (Lemma 2). Here we present such a version of Markov's lemma which is based on a carefully chosen encoding of a nite alphabet? n into. 2 Let be a nitely generated Thue congruence on? n = fs 1 ; : : :; s n g, and let R be a nite string-rewriting system on? n generating. We dene an encoding ' :?! n through 2 '(s i ) := ab i ab 2n+1?i, 1 i n. Then j'(s i )j = 2n + 3 for all i, and for i; j 2 f1; : : :; ng, if '(s i ) = uv and '(s j ) = vw for some v 6=, then u = = w, and hence, i = j. This means that there are no non-trivial overlaps between the strings '(s 1 ); : : :; '(s n ). Let R ' denote the following nite string-rewriting system on 2 : R ' := f'(`)! '(r) j (`! r) 2 Rg: Then it is straightforward to check the following. Lemma 8 For all strings u; v 2? n, u $ R v if and only if '(u) $ R ' '(v). Thus, if the Thue congruence = $ R is undecidable, then so is the Thue congruence $ R ' on 2. In fact, it is easily veried that the Thue congruence $ R is undecidable if and only if $ R ' is undecidable. Our main technical result is the following version of Markov's lemma. Theorem 9 Let? be a nite alphabet. Given a nite string-rewriting system R on? and two strings u; v 2?, a nite string-rewriting system R u;v on 2 and a mapping :?! 2 can be constructed in linear time such that either (i) u $ v, and the congruence R $ R u;v on 2 is trivial, that is, z $ R u;v holds for all z 2 2, or (ii) u = R v, and induces an embedding from R into R u;v, that is, for all x; y 2?, x $ R y if and only if (x) $ R u;v (y). Proof. Let? = fs 1 ; : : :; s n g, and let :=? [ fs n+1 g, where s n+1 is an additional letter. We dene a morphism ' :! 2 through '(s i ) := ab i ab 2n+3?i ; i = 1; : : :; n + 1: Then j'(s i )j = 2n + 5 for all i 2 f1; : : :; n + 1g, and there are no non-trivial overlaps between the strings '(s 1 ); : : :; '(s n+1 ). Since?, ' yields a morphism from? into 2, which for simplicity will also be denoted by '. On we dene an ordering as follows: x y i jxj > jyj or (jxj = jyj and x lex y), where lex denotes the lexicographical ordering that is induced by s n+1 > s n > : : : > s 1. Thus, is a compatible well-ordering on. Let R = f`i! r i j i = 1; : : :; mg be a nite string-rewriting system on?, and let M R be the monoid that is presented by (; R), that is, M R is the factor monoid of the free monoid? modulo the Thue congruence $ R. We dene a nite string-rewriting system S on as follows: S := f`is n+1! r i s n+1 j i = 1; : : :; mg [ fs n+1 s i! s i s n+1 j i = 1; : : :; ng: Then S has the following properties as can be veried easily: 8

9 (a) [] S = fw 2 j w $ S g = fg. (b) For all x; y 2, x $ y if and only if jxj S s n+1 = jyj sn+1 and ((jxj sn+1 > 0 and? (x) $ R?(y)) or x = y). Here jwj sn+1 denotes the number of occurrences of the letter s n+1 in w, and? :!? denotes the projection dened by? (s i ) := ( si if 1 i n; if i = n + 1: (c) For all x; y 2?, x $ y if and only if xs R n+1 $ ys S n+1. Let S ' denote the following nite string-rewriting system on 2 : S ' := f'(`)! '(r) j (`! r) 2 Sg: From Lemma 8 we obtain the following statement: (d) For all x; y 2, x $ S y if and only if '(x) $ S ' '(y). Finally, let u; v 2? be two strings, and let S u;v := fa'(us n+1 )b! ; aa'(vs n+1 )b! a'(vs n+1 )b; ba'(vs n+1 )b! a'(vs n+1 )bg: We dene the nite string-rewriting system R u;v on 2 as follows: R u;v := S ' [ S u;v : The mapping :?! 2 is dened through (w) := '(ws n+1 ) for all w 2?. Obviously, the string-rewriting system R u;v and the mapping can be obtained from R and u; v in linear time. Observe that the alphabet? is xed. This is necessary to guarantee the linear time-bound, since this bound obviously depends on the size of this alphabet. In the following we abbreviate the string (w) = '(ws n+1 ) by writing w (w 2? ). It remains to verify conditions (i) and (ii) in the statement of the theorem. We proceed as follows. For each w 2, let w 2 denote the minimal string with respect to the well-ordering that satises w $ w. We dene a string-rewriting system S S on through S := fw! w j w 2 such that w 6= wg: Then S will in general be an innite system, but obviously w > (w! w) 2 S. Thus, the system S is noetherian. Claim 1. The system S is equivalent to the system S. w holds for each rule Proof. If (w! w) 2 S, then we see from the denition of the system S that w $ S w holds. Conversely, if (`! r) 2 S, then ` = r. If ` = r, then there is nothing to prove. If ` 6= r, then either ` = ` or S contains the rule (`! `), and analogously, either r = r or S contains the rule (r! r). Thus, in any case ` $ S r holds. Hence, the two systems are indeed equivalent. 2 Claim 2. The system S is locally conuent. 9

10 Proof. Assume that x! S y and x! S z. Then x = y = z. If y 6= y, then (y! y) 2 S, and if z 6= y, then (z! y) 2 S. In any case y is a common descendant of both, y and z. 2 Thus, the system S is convergent, equivalent to S, and compatible with the well-ordering. Now we dene a string-rewriting system S ' on 2 as follows: S ' := f'(w)! '( w) j (w! w) 2 Sg: Claim 3. The system S ' is equivalent to the system S '. Proof. Let x; y 2. Then '(x) $ S ' '(y) if and only if x $ S y (by (d)) if and only if x $ S y (by Claim 1) if and only if '(x) $ S' '(y). Since S ' and S' are contained in '( ) '( ), this proves that these systems are indeed equivalent. 2 Let S u;v denote the system S u;v := fa'(us n+1 )b! ; aa'(vs n+1 )b! a'(vs n+1 )b; ba'(vs n+1 )b! a'(vs n+1 )bg; and let T denote the system T := S' [ S u;v : On 2 let denote the length-lexicographical ordering that is induced by b > a. Then '(s n+1 ) = ab n+1 ab n+2 > ab n ab n+3 = '(s n ) > : : : > '(s 1 ). Thus, for all x; y 2, x y if and only if '(x) '(y). Hence, since S >, we see that T > as well, that is, the system T is noetherian. Claim 4. The system T is equivalent to the system R u;v. Proof. From the denition of the system S we see that us n+1! S us n+1 and vs n+1! S vs n+1. Hence, '(us n+1 )! S' '(us n+1 ) and '(vs n+1 )! S' '(vs n+1 ). Thus, Claim 3 implies that the systems T and R u;v are equivalent. 2 Claim 5. The system T is locally conuent, if u = R v. Proof. We have to verify that, for each critical pair (p; q) of T, p and q have a common descendant modulo T. Since there are no nontrivial overlaps between the code words '(s 1 ); : : :; '(s n+1 ), the critical pairs of the subsystem S ' are just the images of the critical pairs of the system S. Since S is convergent, we see that all the critical pairs of S' resolve, that is, the subsystem S' of T is convergent. Thus, it remains to deal with those critical pairs that result from overlaps involving at least one rule from the subsystem S u;v. Since us n+1 ; vs n+1 are normal forms with respect to S, their images '(usn+1 ) and '(vs n+1 ) are normal forms with respect to S '. Thus, none of the rules of S ' applies to any of the strings occurring in S u;v. On the other hand, from the form of the code words '(s 1 ); : : :; '(s n+1 ) we see that the only proper overlaps between the left-hand sides of the rules of S ' and those of S u;v are of the following form: '(`)a'(vs n+1 )b = '(`1)ab i ab 2n+3?i a'(vs n+1 )b # # '(r)a'(vs n+1 )b '(`1)ab i ab 2n+2?i a'(vs n+1 )b &. a'(vs n+1 )b 10

11 where (`! r) = (`1s i! r) 2 S. Hence, the critical pairs obtained in this way all resolve modulo T. Finally, we have to deal with those critical pairs that result from overlaps between the left-hand sides of the rules of the subsystem S u;v. If u = R v, then the only critical pairs of the subsystem S u;v are obtained as follows: and a'(us n+1 )ba'(vs n+1 )b?! a'(us n+1 )a'(vs n+1 )b {z } z } { #. a'(vs n+1 )b ca'(vs n+1 )ba'(vs n+1 )b?! ca'(vs n+1 )a'(vs n+1 )b {z } z } { # # (c 2 2 ) a'(vs n+1 )ba'(vs n+1 )b?! a'(vs n+1 )b Thus, if u = R v, then the system T is locally conuent. 2 Now we are prepared to prove conditions (i) and (ii). If u $ R v, then us n+1 $ S vs n+1, and hence, us n+1 = vs n+1. Hence, we obtain the following congruences: aa'(us n+1 )b = aa'(vs n+1 )b! T a'(vs n+1 )b # T q a a'(us n+1 )b # T and ba'(us n+1 )b = ba'(vs n+1 )b! T a'(vs n+1 )b # T q b a'(us n+1 )b Hence, if u $ R v, then $ T = $ R u;v is the trivial Thue congruence on 2. If u = v, then the system T is equivalent to R R u;v, it is compatible with the well-ordering on 2, and it is convergent. Let x; y 2?. If x $ y, then xs R n+1 $ ys S n+1 by (c), and (x) = '(xs n+1 ) $ S ' '(ys n+1 ) = (y) by (d), that is, (x) $ R u;v (y). Conversely, if (x) $ R u;v (y), then there exists a string z 2 2 such that (x) = '(xs n+1 )! z and T (y) = '(ys n+1 )! z. Since (x) 2 T '( ), we can conclude from the form of the rules of T that z 2 '( ) as well, and that (x)! S' z. Analogously, we obtain (y)! S' z. From Claim 3 we obtain (x) $ S ' (y), which yields xs n+1 $ ys S n+1 because of (d). Hence, (c) yields that x $ y. Thus, x R $ y holds if and only if (x) R $ R u;v (y). This completes the proof of Theorem 9. 2 # T : 11

12 Observe that, for u; v 2?, if u = v, then the mapping : R?! 2 constructed in the proof of Theorem 9 is not a morphism. Indeed () = '(s n+1 ) = ab n+1 ab n+2 6=. Further, s n+1 is minimal in the congruence class [s n+1 ] S with respect to the well-ordering, and so '(s n+1 ) is irreducible modulo S'. In fact, '(s n+1 ) is even irreducible modulo T, and hence, since T is equivalent to R u;v and convergent, we see that () = '(s n+1 ) = R u;v. Thus, does not give an embedding of the monoid M R in the monoid M Ru;v. Hence, Theorem 9 is not suciently strong to prove the undecidability of Markov properties in general. However, we will discuss some specic applications of this result in the next section. 4 Some applications Let R be a nite string-rewriting system on a nite alphabet? such that the Thue congruence $ R is undecidable, and let u; v 2?. Assume that u = R v. Then the Thue congruence $ R u;v on 2 is undecidable as well. In addition, the monoid M u;v that is presented by ( 2 ; R u;v ) has the following properties: (1.) M u;v is neither left- nor right-cancellative. (2.) M u;v is not a group. (3.) M u;v is not commutative. (4.) M u;v is innite. (5.) M u;v is not a free monoid. The system T on 2 (see the proof of Theorem 9) is convergent and equivalent to R u;v. Since a [ b IRR(T ), we see that M u;v is innite. In fact, it is generated by the two generators a and b, which both have innite order in M u;v. Since ab; ba 2 IRR(T ), we have ab = R u;v ba, and hence, M u;v is not commutative. Further, aa'(vs n+1 )b $ R u;v a'(vs n+1 )b, but a'(vs n+1 )b = R u;v '(vs n+1 )b and aa'(vs n+1 ) = R u;v a'(vs n+1 ), since these four strings are irreducible modulo T. Thus, M u;v is neither left- nor right-cancellative, and so it is certainly not a group. Finally, it is obvious that M u;v is not a free monoid. On the other hand, if u $ R v holds, then $ R u;v is the trivial Thue congruence on 2, and hence, M u;v is the trivial monoid, which is a group, which is commutative, and which is nite. Thus, we have the following undecidability results as a consequence of Theorem 9. Corollary 10 The following problems are undecidable: INSTANCE: A nite string-rewriting system S on 2. QUESTION 1: Is the monoid M S presented by ( 2 ; S) nite? QUESTION 2: Is the monoid M S commutative? QUESTION 3: Is the monoid M S (left-, right-) cancellative? QUESTION 4: Is the monoid M S a group? QUESTION 5: Is the monoid M S a free monoid? QUESTION 6: Does the monoid M S have a decidable word problem? 12

13 Observe that the undecidability of Questions 3 and 4 does not follow from Proposition 4. In the proof of Proposition 4 as well as in the considerations above a Markov property is shown to be undecidable by reducing the undecidable word problem of a nite presentation to it. In both cases this means that the monoid constructed has an undecidable word problem, if it does not have the Markov property considered. This observation raises the question of which Markov properties remain undecidable even when they are restricted to nite systems (nitely presented monoids or groups) with decidable word problems. Based on a careful analysis of the Knuth-Bendix completion procedure A. Sattler-Klein has shown the following. Proposition 11 [SK] There exist a nite alphabet containing the letters a; b; c; d; e and o, a length-lexicographical ordering > len on, and a nite, length-reducing, and conuent string-rewriting system R on such that the class C := f R [ fcbdan bbe! og j n 2 Ng of nite length-reducing string-rewriting systems has the following properties: (1.) For each R 2 C, the word problem for R is decidable. (2.) For each R 2 C, the Knuth-Bendix completion procedure with interreduction generates a possibly innite, length-reducing, conuent, and interreduced string-rewriting system on input (R; > len ). (3.) The following restricted version of the divergence problem is undecidable: INSTANCE: A string-rewriting system R 2 C. QUESTION: Will the Knuth-Bendix completion procedure terminate on input (R; > len )? (4.) For the class C, the uniform word problem, the niteness problem, the trivial monoid problem, the free monoid problem, the group problem, and the problem of commutativity are all undecidable. Let R n := R [ fcbda n bbe! og. Then it turns out that the Knuth-Bendix completion procedure terminates on input (R n ; > len ) if and only if o $ R n if and only if the congruence $ R n is trivial if and only if the monoid M Rn is trivial (nite, free, commutative, a group). Thus, the undecidability results in (4.) are an immediate consequence of the undecidability of the divergence problem for C [SK]. In fact, the problem of being a (left-, right-) cancellative monoid can be added to the list in (4.). The alphabet constructed in the proof of Proposition 11 in [SK] is fairly large. Using Theorem 9 we will now establish a corresponding result for the two-letter alphabet 2. For that we need the following observation. Lemma 12 Let? be a nite alphabet, let R be a nite string-rewriting system on?, let u; v 2?, and let R u;v be the nite string-rewriting system on 2 that is constructed from R and u; v according to Theorem 9. If the Thue congruence $ R on? is decidable, then so is the Thue congruence $ R u;v on 2. Proof. If u $ v, then R $ R u;v is the trivial Thue congruence on 2, which is decidable. So assume that u = v. Then the innite string-rewriting system T on R 2 is convergent and equivalent to R u;v (recall the proof of Theorem 9). 13

14 Claim. dom(t ) and T are recursive sets. Proof. Since T = S' [S u;v, where S u;v is a nite system, it suces to prove that dom( S' ) and S ' are recursive sets. Now S ' = f'(w)! '( w) j (w! w) 2 Sg and S = fw! w j w 2 such that w 6= wg, where w denotes the minimal string in [w] S with respect to the lengthlexicographical ordering on. With R also S has a decidable word problem. This implies that dom( S) as well as S are recursive. Hence, dom( S ' ) and S ' are recursive sets. 2 From the claim it follows immediately that the reduction! T can be performed eectively. Hence, the `normal form algorithm' induced by T is indeed an eective process that solves the word problem for R u;v. 2 By applying the construction of Theorem 9 to the systems R n := R [ fcbda n bbe! og (n 2 N) and the strings u := o and v :=, we obtain the class C 2 := fr n;o; j n 2 Ng of nite string-rewriting systems on 2. It has the following properties. Corollary 13 (a) Each S 2 C 2 has a decidable word problem. (b) The Questions 1 to 5 of Corollary 10 remain undecidable even when they are restricted to string-rewriting systems S 2 C 2. Proof. (a) This follows from Proposition 11 (1.) and Lemma 12. (b) Let n 2 N. Then the monoid M Rn presented by (; R n ) is trivial (nite, free, (left-, right-) cancellative, commutative, a group) if and only if o $ R n. If o $ R n, then the monoid presented by ( 2 ; R n;o; ) is trivial, and hence, it also has all the other properties considered. If, however, o = R n, then this latter monoid has none of these properties as observed at the beginning of this section. Hence, the undecidability result in (b) follows from Proposition 11 (4.). 2 The important thing to notice here is the fact that a result corresponding to Proposition 11 is not known for groups. Accordingly, the same applies to Corollary Concluding remark For n 2 N, if o $ R n, then $ R n;o; is the trivial congruence, and hence, the Knuth-Bendix completion procedure will terminate on input (R n;o; ; >) for any compatible well-founded partial ordering on 2 satisfying a > and b >. However, it is not clear whether or not the Knuth-Bendix completion procedure will terminate on input (R n;o; ; >), if o = R n. But there is a much simpler way for proving that the divergence problem is undecidable in general even for nite string-rewriting systems on 2 that have decidable word problems. Consider the class C of Proposition 11. Dene an encoding ' :! through '(s 2 i) := ab i ab 2n+1?i, i = 1; : : :; n, where the letters of are renamed s 1 ; : : :; s n. Let C 3 denote the class C 3 := fr ' j R 2 Cg of nite string-rewriting systems on 2. From the discussion at the beginning of Section 3 it follows that each system R ' has a decidable word problem, since R 14

15 does. Further, if the Knuth-Bendix completion procedure terminates on input (R; > len ) giving a nite convergent system S, then S ' is a nite convergent system on 2 that is equivalent to R '. Conversely, if the Knuth-Bendix completion procedure terminates on input (R ' ; >), where > is the length-lexicographical ordering on 2 induced by b > a, then it is easily veried that all the rules of the resulting system S 0 are of the form '(u)! '(v) for some u; v 2. Hence, there exists a nite system S on such that S 0 = S '. Further, with S 0 also S is convergent. Thus, we have the following undecidability result. Corollary 14 There exists a nite, length-reducing, and conuent string-rewriting system R 3 on 2 such that the class C 3 := f R3 [ f'(cbda n bbe)! '(o) j n 2 Ng of nite, lengthreducing string-rewriting systems has the following properties: (1.) For each R 3 2 C 3, the word problem for R 3 is decidable. (2.) For each R 3 2 C 3, the Knuth-Bendix completion procedure with interreduction generates a possibly innite, length-reducing, conuent, and interreduced string-rewriting system on input (R 3 ; >). (3.) The following restricted version of the divergence problem is undecidable: INSTANCE: A string-rewriting system R 3 2 C 3. QUESTION: Will the Knuth-Bendix completion procedure terminate on input (R 3 ; >)? In addition to restricting the size of the alphabet considered, one can also restrict the form of the rules of the string-rewriting systems that are used to generate Thue congruences. A string-rewriting system is called special, if each of its rules is of the form (`! ) for some non-empty string `. Observe that in Lemma 2 the resulting string-rewriting system R u;v is not special, even if the original system R is. P. Narendran et al have shown that the group property is undecidable for nite special string-rewriting systems [N OO91], and L. Zhang has then generalized A. Markov's undecidability result to nite special string-rewriting systems [Zha92]. Acknowledgement. The authors are indebted to an anonymous referee for bringing the paper by P.E. Schupp [Sch76] to their attention. In addition, the proof of Proposition 4 follows essentially his (or her) outline. References [Adj58] S.I. Adjan. On algorithmic problems in eectively complete classes of groups. Doklady Akademii Nauk SSSR, 123:13{16, [Ave86] [BO84] J. Avenhaus. On the descriptive power of term rewriting systems. J. Symbolic Computation, 2:109{122, G. Bauer and F. Otto. Finite complete rewriting systems and the complexity of the word problem. Acta Informatica, 21:521{540,

16 [BO93] [KB70] [Mar51] R.V. Book and F. Otto. String-Rewriting Systems. Springer-Verlag, New York, D. Knuth and P. Bendix. Simple word problems in universal algebras. In J. Leech, editor, Computational Problems in Abstract Algebra, pages 263{297. Pergamon Press, New York, A. Markov. Impossibility of algorithms for recognizing some properties of associative systems. Doklady Adakemii Nauk SSSR, 77:953{956, [N OO91] P. Narendran, C. O'Dunlaing, and F. Otto. It is undecidable whether a nite special string-rewriting system presents a group. Discrete Mathematics, 98:153{159, [ O'D83] C. O'Dunlaing. Undecidable questions related to Church-Rosser Thue systems. Theoretical Computer Science, 23:339{345, [Rab58] M.O. Rabin. Recursive unsolvability of group theoretic problems. Annals of Math., 67:172{174, [Sch76] P.E. Schupp. Embeddings into simple groups. J. London Math. Soc., 13:90{94, [SK] [SK91] [Zha92] A. Sattler-Klein. A systematic study of innite canonical systems generated by Knuth-Bendix completion and related problems. Doctoral Dissertation, Fachbereich Informatik, Universitat Kaiserslautern, February A. Sattler-Klein. Divergence phenomena during completion. In R.V. Book, editor, Rewriting Techniques and Applications, pages 374{385. Springer-Verlag, Berlin, Lecture Notes in Computer Science 488. L. Zhang. Some properties of nite special string-rewriting systems. J. Symbolic Computation, 14:359{369,