Solvability of Word Equations Modulo Finite Special And. Conuent String-Rewriting Systems Is Undecidable In General.

Transcription

1 Solvability of Word Equations Modulo Finite Special And Conuent String-Rewriting Systems Is Undecidable In General Friedrich Otto Fachbereich Mathematik/Informatik, Universitat GH Kassel Kassel, Germany Keywords: Formal languages, word equations, string-rewriting. Abstract A nite special and conuent string-rewriting system S is constructed such that it is undecidable in general whether a word equation is solvable mod S. Thus, (word) unication mod S is undecidable. 1 Introduction The problem of deciding whether an equation has a solution is one of the most fundamental problems in mathematics. Under the name of "unication" this problem has received much attention in the computer science literature. For various theories algorithms have been developed that not only allow to decide whether a given equation has a solution modulo the theory considered, but that in the armative also compute a basis for the set of all solutions, that is, a "complete set of most general uniers" (See [14] for an overview). One of the most important results in this area is Makanin's proof that it is decidable whether a word equation u v has a solution in a free semigroup [6]. Here u and v are strings that contain letters from a given alphabet as well as variables from a set of variables V, and the equation u v is said to be solvable if there exists a morphism (a "solution") : V! such that (u) and (v) are identical strings. This means that "uniability modulo associativity" is decidable. Since Makanin's paper appeared, his algorithm has been the object of many research activities. The objectives have been to simplify the proof of the termination and correctness of his algorithm [11, 13], to develop simpler algorithms for deciding the solvability of word equations [5, 12], and to compute a description for the set of all solutions of a solvable word equation [8]. Observe that a word equation can have a minimal complete set of most general uniers that is innite, that is, the theory of associativity is of unication type innitary. Makanin also proved that the solvability of word equations in free groups is decidable [7]. Since the free group in n generators can be seen as a factor monoid of the free monoid in 2n generators, it is only natural to ask whether the solvability of word equations can be generalized to a still larger class of non-free monoids. Book [2] introduces a restricted class of logical formulae that he calls "linear sentences." The purely existential "linear sentences" are just disjunctions of word equations containing no variable more than once. Book proves that it is decidable in polynomial time whether an existential linear sentence is valid with respect to an interpretation that is induced by a nite, monadic, and conuent string-rewriting system. In fact, he considers constrained solutions in 1

2 that he allows that, for each variable occuring in the sentence under consideration, a regular set may be specied as the domain for this variable. In a recent paper Oleshchuk applies the technique of narrowing to word equations showing that it is decidable whether a word equation has a solution modulo a nite string-rewriting system that is homogeneous of degree 2, that is, each rule of this system has a left-hand side of length 2 and the empty string as right-hand side [10]. Actually, Oleshchuk works with nite homogeneous systems of degree 2 that are in addition conuent, and then he uses a result of Book [3] which states that for each nite and homogeneous string-rewriting system of degree 2 there exists a conuent system of the same type that is isomorphic to the original system under a morphism that identies some of the letters. To which classes of nite string-rewriting systems can Oleshchuk's decidability result be generalized? On the one hand, it is not hard to see that the Post Correspondence Problem can be reduced to the problem of deciding whether a word equation is solvable modulo a nite monadic and conuent string-rewriting system (see, e.g., [4] Section 4.5). On the other hand, Adian has shown that there exists a nite homogeneous string-rewriting system S 3 of degree 3 such that the word problem for S 3 is undecidable [1], and hence, since the word problem of S 3 is obviously a special case of the problem of deciding whether a word equation has a solution modulo S 3, the latter problem is also undecidable. Here we show that Oleshchuk's result cannot be carried over to the class of all nite and special string-rewriting systems that are conuent (see the next section for the denitions). We construct a xed nite string-rewriting system S that is special and conuent such that it is undecidable in general whether a given word equation has a solution mod S. In particular, this shows that for nite special and conuent string-rewriting systems the validity of nonlinear sentences in the sense of Book is undecidable. 2 Denitions and Results Let be a nite alphabet, and let X := fx i j i 1g be a countable set of (existential) variables such that \ X = ;. A word equation (over ) is an expression of the form u v, where u and v are strings from ( [ X). For w 2 ( [ X), X(w) denotes the set of variables that occur in the string w, that is, X(w) := fx i 2 X j jwj xi > 0g. A mapping : X(u) [ X(v)! is called a solution of the word equation u v if the strings (u) and (v) coincide. Here is extended to a mapping from (X(u) [ X(v) [ ) into in the obvious way. The solvability problem for word equations is the following decision problem: INSTANCE: A word equation u v over. QUESTION: Does this word equation have a solution, that is, is there a mapping : X(u) [ X(v)! such that the strings (u) and (v) are identical? As mentioned in the introduction Makanin has shown that this problem is decidable [6]. Here we are interested in a generalization of this problem. Let S be a string-rewriting system on. Then! S denotes the Thue congruence on that is generated by S. The solvability problem for word equations modulo S is the following decision problem: INSTANCE: A word equation u v over. QUESTION: Does this equation have a solution modulo S, that is, is there a mapping : X(u) [ X(v)! such that (u)! S (v) holds? Obviously the word problem for S is reducible to this problem. Thus, the solvability problem for word equations modulo S is undecidable for each nite string-rewriting system 2

3 S that has an undecidable word problem. Therefore, we are interested in certain restricted classes of nite string-rewriting systems. A nite string-rewriting system S is called length-reducing if j`j > jrj holds for each rule (`! r) 2 S. Here jwj denotes the length of the string w. The system S is called monadic if it is length-reducing and r 2 [ fg holds for each rule (`! r) 2 S, where denotes the empty string. Further, S is a special system if it is length-reducing, and r = for each rule (`! r) 2 S, and it is homogeneous of degree k, if it is special, and j`j = k holds for each rule (`! r) 2 S. Finally, S is conuent if u! S v implies that there exists a string w such that u?! S w and v?! S w. Here?! S denotes the reduction relation that is induced by S, which is the reexive and transitive closure of the following single-step reduction relation: u?! S v i 9g; h 2 9(`! r) 2 S : u = g`h and v = grh: A length-reducing and conuent system S denes a unique normal form for each congruence class [w] S := fu 2 j u! S wg, since each such class contains one and only one string that is irreducible modulo S. By IRR(S) we denote the set of strings that are irreducible modulo S. Thus, for each system of this form the word problem is decidable in linear time [4]. For more information on string-rewriting systems the reader is asked to consult the literature, e.g., [4]. Here we want to establish the following undecidability result. Theorem 2.1 There exists a nite, special, and conuent string-rewriting system S such that the solvability problem for word equations modulo S is undecidable. Since the solvability problem is a special case of the problem of determining whether a nonlinear sentence in the sense of Book ([2], see also [4]) is valid, we obtain the following consequence. Corollary 2.2 There exists a nite special and conuent string-rewriting system S such that it is undecidable in general whether or not a nonlinear sentence is valid under the interpretation induced by S. In the rest of this note we present a proof of Theorem The Proof Theorem 2.1 will be proved by a reduction from the Post Correspondence Problem (PCP). In fact, it is known (see, e.g., [9]) that there exists a set of pairs of nonempty strings P = f(u i ; v i ) j i = 2; : : :; kg fa; bg + fa; bg + such that the following problem is undecidable: INSTANCE: Two strings u 1 ; v 1 2 fa; bg +. QUESTION: Does the modied PCP f(u 1 ; v 1 )g [ P have a solution, that is, does there exist a sequence of integers i 1 ; : : :; i n 2 f2; : : :; kg such that u 1 u i1 : : :u in = v 1 v i1 : : : v in? We dene a special string-rewriting system Sp(P ) on the alphabet as follows:? := fa; b; a 0 ; b 0 ; z 2 ; : : :; z k ; z 0 2 ; : : :; z0 k g 3

4 u i z i! ; i = 2; : : :; k zi 0(v0 i )! ; i = 2; : : :; k zi 0 z i! ; i = 2; : : :; k aa 0! ; bb 0! ; where :?!? denotes the function reversal, and 0 : fa; bg! fa 0 ; b 0 g denotes the obvious isomorphism. There are no overlaps between left-hand sides of rules of Sp(P ), and hence, Sp(P ) is a nite special and conuent system. Let :=? [ f$g, where $ is an additional symbol, and, for u 1 ; v 1 2 fa; bg +, let (u 1 ; v 1 ) denote the following word equation over : (u 1 ; v 1 ) := $x 1 x 3 $x 4 x 2 $x 4 x 3 $u 1 x 1 x 2 (v 0 1)$ $ 5, where x 1 ; : : :; x 4 are variables. Finally, we dene four regular subsets of : R 1 := fa; bg ; R 2 := fa 0 ; b 0 g ; R 3 := fz 2 ; : : :; z k g ; R 4 := fz 0 2 ; : : :; z0 k g. Lemma 3.1 If the modied PCP f(u 1 ; v 1 )g [ P has a solution, then there exist strings s i 2 R i, i = 1; : : :; 4, such that $s 1 s 3 $s 4 s 2 $s 4 s 3 $u 1 s 1 s 2 (v 0 1)$! Sp(P ) $5. Proof. Let i 1 ; : : :; i n 2 f2; : : :; kg be such that u 1 u i1 : : : u in = v 1 v i1 : : : v in. Take s 1 := u i1 : : : u in 2 R 1, s 2 := (v 0 i 1 : : : v 0 i n ) 2 R 2, s 3 := z in : : :z i1 2 R 3, and s 4 := z 0 i 1 : : : z 0 i n 2 R 4. Then $s 1 s 3 $s 4 s 2 $s 4 s 3 $u 1 s 1 s 2 (v 0 1 )$ = $u i1 : : :u in z in : : :z i1 $z 0 i 1 : : :z 0 i n (v 0 i n ) : : :(v 0 i 1 )$z 0 i 1 : : : z 0 i n z in : : :z i1 $u 1 u i1 : : : u in (v 0 i n ) : : :(v 0 i 1 )(v 0 1 )$?! Sp(P ) $5. Lemma 3.2 If there exist strings s i 2 R i, i = 1; : : :; 4, such that $s 1 s 3 $s 4 s 2 $s 4 s 3 $u 1 s 1 s 2 (v 0 1)$! Sp(P ) $5, then the modied PCP f(u 1 ; v 1 )g [ P has a solution. Proof. Let s i 2 R i (i = 1; : : :; 4) be chosen such that the above congruence holds. Since $ 5 2 IRR(Sp(P )), and since no left-hand side of a rule of Sp(P ) contains an occurrence of the symbol $, we see that s 1 s 3?! Sp(P, s ) 4 s 2?! Sp(P, s ) 4 s 3?! Sp(P, and ) u 1 s 1 s 2 (v 0 ) 1?! Sp(P. Since s ) 1 2 fa; bg and s 2 2 fa 0 ; b 0 g, u 1 s 1 s 2 (v 0 ) 1?! Sp(P by only ) using the rules aa 0! and bb 0!, that is, u 0 1 s0 1 = (s 2 (v1)) 0 = v 0 1 (s 2). Since s 1 2 fa; bg and s 3 2 fz 2 ; : : :; z k g, s 1 s 3?! Sp(P implies that s ) 3 = z in : : :z i1 and s 1 = u i1 : : :u in for some indices i 1 ; : : :; i n 2 f2; : : :; kg. Analogously, s 4 s 2?! Sp(P implies that s ) 4 = zj 0 1 : : : zj 0 m and s 2 = (vj 0 m ) : : :(vj 0 1 ) for some indices j 1 ; : : :; j m 2 f2; : : :; kg. Finally, s 4 s 3?! Sp(P ) implies that n = m and i h = j h for all h = 1; : : :; n, that is, s 1 = u i1 : : : u in and s 2 = (vi 0 1 : : : vi 0 n ). Hence, we conclude that u 1 s 1 = u 1 u i1 : : : u in = v 1 v i1 : : : v in, which means that i 1 ; : : :; i n is a solution for the modied PCP f(u 1 ; v 1 )g [ P. 2 Together these two lemmata give the following undecidability result. 4 2

5 Corollary 3.3 Let Sp(P ) be the above nite special and conuent string-rewriting system on, and let R 1 ; : : :; R 4 be the above regular subsets of. Then the following problem is undecidable: INSTANCE: Two strings u 1 ; v 1 2 fa; bg +. QUESTION: Is the existential sentence 9x 1 ; x 2 ; x 3 ; x 4 : $x 1 x 3 $x 4 x 2 $x 4 x 3 $u 1 x 1 x 2 (v 0 1 )$ $5 true under the interpretation induced by the system Sp(P ) and the regular sets R 1 ; : : :; R 4? Thus, the problem of validity for nonlinear existential sentences is undecidable for the nite special and conuent string-rewriting system Sp(P ). Thus, the decidability result of [2] does not carry over to nonlinear sentences, even if attention is restricted to a xed nite, special, and conuent string-rewriting system. The validity problem for nonlinear existential sentences diers from the solvability problem for word equations modulo Sp(P ) in that, for each variable, a regular subset R is chosen as the domain for this variable. However, the subsets R 1 ; : : :; R 4 chosen above are of a very restricted form only. We now introduce another four new symbols, and we add some more rules to the system Sp(P ). Let := [ f; ; ; g, and let S denote the following string-rewriting system on : S := Sp(P ) [ fa! ; b! ; a 0! ; b 0! ; z i! ; zi 0! j i = 2; : : :; kg. Then S is a nite special system, and it is easily veried that S is conuent, since it does not have any nontrivial critical pairs. For u 1 ; v 1 2 fa; bg +, let w 1 (u 1 ; v 1 ) 2 ( [ fx 1 ; : : :; x 4 ; y 1 ; : : :; y 4 g) and w 2 2 ( [ fy 1 ; : : :; y 4 g) denote the following strings, where x 1 ; : : :; x 4 ; y 1 ; : : :; y 4 are variables: w 1 (u 1 ; v 1 ) := y 1 $y 2 $y 3 $y 4 $x 1 y 1 $y 2 x 2 $y 3 x 3 $x 4 y 4 $x 1 x 3 $x 4 x 2 $x 4 x 3 $u 1 x 1 x 2 (v 0 1 )$; w 2 := y 1 $y 2 $y 3 $y 4 $ 9 : Lemma 3.4 If there are strings s i 2 R i, i = 1; : : :; 4, such that $s 1 s 3 $s 4 s 2 $s 4 s 3 $u 1 s 1 s 2 (v 0 1)$! Sp(P ) $5, then the equation w 1 (u 1 ; v 1 ) w 2 has a solution modulo S. Proof. Let s i 2 R i, i = 1; : : :; 4, be such that the above congruence holds. We dene a morphism : fx 1 ; : : :; x 4 ; y 1 ; : : :; y 4 g! as follows: (x i ) := s i ; i = 1; : : :; 4; (y 1 ) := js 1 j ; (y 2 ) := js 2 j ; (y 3 ) := js 3 j, and (y 4 ) := js 4 j : Then Further, (y 1 ) = js 1 j+1 = (y 1 ); (y 2 ) = js 2 j+1 = (y 2 ); (y 3 ) = js 3 j+1 = (y 3 ); (y 4 ) = js 4 j+1 = (y 4 ): (x 1 y 1 ) = s 1 js 1 j?! S ; (y 2 x 2 ) = js 2 j s 2?! S ; (y 3 x 3 ) = js 3 j s 3?! S ; (x 4 y 4 ) = s 4 js 4 j?! S ; 5

6 since s 1 2 fa; bg, s 2 2 fa 0 ; b 0 g, s 3 2 fz 2 ; : : :; z k g, and s 4 2 fz 0 2 ; : : :; z0 k g. Thus, we see that (w 1 (u 1 ; v 1 ))! S (w 2), that is, w 1 (u 1 ; v 1 ) w 2 has a solution modulo S. 2 Lemma 3.5 If : fx 1 ; : : :; x 4 ; y 1 ; : : :; y 4 g! is a solution of the equation w 1 (u 1 ; v 1 ) w 2 modulo S, then s i := (x i ) 2 R i, i = 1; : : :; 4, and $s 1 s 3 $s 4 s 2 $s 4 s 3 $u 1 s 1 s 2 (v 0 1)$! Sp(P ) $5. Proof. Assume that : fx 1 ; : : :; x 4 ; y 1 ; : : :; y 4 g! is a solution of the equation w 1 (u 1 ; v 1 ) w 2 modulo S. Let s i := (x i ) and t i := (y i ), i = 1; : : :; 4. Since S is conuent, we can assume without loss of generality that these strings are irreducible mod S. Hence, (w 2 ) = t 1 $t 2 $t 3 $t 4 $ 9 is irreducible, implying that (w 1 (u 1 ; v 1 ))?! S (w 2). None of the rules of S contains an occurrence of the symbol $, and hence, j (w 2 ) j $ = 12 + P 4 i=1 j t i j $ = P 4 i=1 j t i j $ + 3 P 4 i=1 j s i j $ = j (w 1 (u 1 ; v 1 )) j $ implying that j s i j $ = j t i j $ = 0, i = 1; : : :; 4. Thus, (w 1 (u 1 ; v 1 ))?! S (w 2) implies the following: (1) t 1?! S t 1; t 2?! S t 2; t 3?! S t 3; t 4?! S t 4, (2) s 1 t 1?! S ; t 2s 2?! S ; t 3s 3?! S ; s 4t 4?! S ; and (3) $s 1 s 3 $s 4 s 2 $s 4 s 3 $u 1 s 1 s 2 (v 0 1)$?! S $5. Since S is length-reducing, and j t 1 j = j t 1 j, we conclude that t 1 = t 1, which in turn yields that t 1 = r 1 for some r 1 2 IN. Analogously, we obtain that t 2 = r 2, t 3 = r 3, and t 4 = r 4 for some r 2 ; r 3 ; r 4 2 IN. Since s 1 t 1 = s 1 r 1?! S, and since a! and b! are the only rules of S containing occurrences of the symbol, this means that s 1 2 fa; bg and r 1 = j s 1 j. Analogously, we see that s 2 2 fa 0 ; b 0 g and r 2 = j s 2 j, s 3 2 fz 2 ; : : :; z k g and r 3 = j s 3 j, and s 4 2 fz 0 2 ; : : :; z0 k and r g 4 = j s 4 j. Thus, s i 2 R i, i = 1; : : :; 4, and in the reduction (3) above only rules from the subsystem Sp(P ) of S are used. 2 The above lemmata imply that the modied PCP f(u 1 ; v 1 )g [ P has a solution if and only if the equation w 1 (u 1 ; v 1 ) w 2 has a solution modulo S. This completes the proof of Theorem Concluding Remark As dened above the solvabitily problem for word equations modulo a string-rewriting system R is in fact the word unication problem modulo the system R, that is, it corresponds to equational unication modulo the set of equations R plus associativity. If we interpret each symbol a 2 as a unary function symbol a(:), then the only terms containing variables are those of the form a i1 (a i2 ( (a im (x)) )), where a ij 2, j = 1; : : :; m. Thus, in this case the only equations that we obtain are of the form u(x) v(y), where u; v 2 and x and y are variables that are not necessarily distinct. However, if S is a nite special and conuent string-rewriting system on, then, given u; v 2, it is decidable whether or not there exists a morphism : fx; yg! such that (ux) = u(x)! S v(y) = (vy) holds. Thus, unication modulo the term-rewriting system S 0 := f`(x)! r(x) j (`! r) 2 Sg is decidable. 6

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