K.U. Schulz. Makanin's algorithm: Two improvements and a generalization. In K.U. Schulz, editor, Word Equations and Related Topics, volume 572 of Lecture Notes in Computer Science,Tubingen, Germany, October 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....it reduces to a nontrivial extension of word equations [24] In the case of ground rules, the decidability of monadic SREU was established in [24] by reducing it to word equations with regular constraints . The decidability of the latter problem is an extension of Makanin s [29] result by Schulz [32]. Conversely, word equations reduce in polynomial time to monadic SREU [11] In Section 5 we show that monadic SRR with ground rules is in PSPACE, improving over the EXPTIME result that we have obtained earlier [20] The PSPACE hardness of monadic SREU with ground rules was already shown by ....

K.U. Schulz. Makanin's algorithm: Two improvements and a generalization. In K.U. Schulz, editor, Proceedings of the First International Workshop on Word Equations and Related Topics, Tubingen, number 572 in Lecture Notes in Computer Science, 1990.

....word equations [Gurevich Voronkov 1997] In the case of ground rules, the decidability of monadic SREU was established in [Gurevich Voronkov 1997] by reducing it to word equations with regular constraints . The decidability of the latter problem is an extension of Makanin s [1977] result by Schulz [1990]. Conversely, word equations reduce in polynomial time to monadic SREU [Degtyarev, Matiyasevich Voronkov 1996] The rst main result of this paper (in Section 3) is that monadic SRR with ground rules is in PSPACE, improving the EXPTIME result in Ganzinger et al. 1998] Hence, it is unlikely ....

Schulz, K. (1990), Makanin's algorithm: Two improvements and a generalization, in K. Schulz, ed., `Proceedings of the First International Workshop on Word Equations and Related Topics, Tubingen', number 572 in `Lecture Notes in Computer Science'.

....solves a string equation s 1 i s 2 if oe(s 1 ) oe(s 2 ) The problem of deciding if there is a substitution solving a given string equation was open for a long time and was finally solved by Makanin [5] who was able to construct a decision algorithm. For variants of his algorithm see [4] 7] [8]. Clearly the problem of solving string equations is equivalent to the unification problem in associative theories E = ff(x; f(y; z) j f(f(x; y) z)g where terms t 1 ; t 2 are unifiable modulo E if there is a substitution oe with oe(t 1 ) jE oe(t 2 ) Since during the last years 1 Formally S = ....

.... string equations P and given constant restrictions R(x) C, x 2 var(P ) there is substitution oe which solves all equations in P and satisfies the constant restrictions R(x) Remark 1 We learned recently that a similar and even more general extension of Makanin s algorithm was given by Schulz [8] 2 Generalized equations Consider a pair (P; R) where P is a finite set of string equations and R = fR(x) x 2 var(P )g is a set of constant restrictions. In this section we transform the pair (P; R) into a generalized equation which is a special structure to decide string equations. It was ....

Klaus U. Schulz. Makanin's Algorithm --- Two Improvements and a Generalization. Technical Report 91-39, CIS---Universitat Munchen, 1991.

....solving all equations in the system. Words will be denoted by U; V; W , word Monadic Simultaneous Rigid E Unification and Related Problems 5 variables by u; v; w and word substitutions by ae; oe; Makanin [11] proved that word equations are decidable. Analyzing Makanin s algorithm, Schultz [14] proves the following result. Lemma4 (Decidability of word equations with regular constraints) The problem of solvability of word equations where every word variable u i ranges over a regular set S i , is decidable. It is known that the problem of solvability of word equations is NP hard. No ....

K.U. Schulz. Makanin's algorithm: Two improvements and a generalization. In K.U. Schulz, editor, Word Equations and Related Topics, volume 572 of Lecture Notes in Computer Science, Tubingen, Germany, October 1990.

....Theorem 5.6 Monadic SREU with ground left sides is decidable. 2 The idea of the proof is the following. It is shown that solutions to rigid equations with ground left hand sides can be described by word equations plus restrictions that word variables belong to regular sets. Then the result of [32] on the decidability of word equations with regular constraints is used. As for the general case, the decidability remains an open problem. 20] proves polynomial time equivalence of monadic SREU to a word problem described below. Denote by W a set of pairs of words on F . Introduce on W a binary ....

K.U. Schulz. Makanin's algorithm: Two improvements and a generalization. In K.U. Schulz, editor, Proceedings of the First International Workshop on Word Equations and Related Topics, Tubingen, number 572 in Lecture Notes in Computer Science, 1990.

.... membership problem is decidable, then it is also decidable with regular constraints (every word variable ranges over a regular set) Gurevich Voronkov 1997a) However, this does not give the undecidability result for FI( 2; 0) since word equations with regular constraints are decidable (Schulz 1990). Thus, we have Open problem 3 Is FI( 2; 0) decidable This problem is also related to the classification of decidable prenex fragments of intuitionistic logic depending on the signature and the quantifier prefix (Degtyarev Voronkov 1996a, Degtyarev, Gurevich Voronkov 1996) As for ....

Schulz, K. (1990), Makanin's algorithm: Two improvements and a generalization, in K. Schulz, ed., `Proceedings of the First International Workshop on Word Equations and Related Topics, Tubingen', number 572 in `Lecture Notes in Computer Science'.

.... for the exponent of periodicity for a minimal uni er for context uni cation from [SSS98] which is a generalization of a lemma that appeared in the decidability proof of word uni cation by Makanin [Mak77] An improvement of the latter result was given in [KP96] This link to word uni cation ([Mak77,Sch90,Sch93,Gut98,Pla99]) is not accidental. The relationship between word uni cation and bounded higherorder uni cation is indicated in Figure 1 where we also mention some other problems in order to position the results of this paper. Higher order unification Monadic 2nd order unification word unification ....

Klaus U. Schulz. Makanin's algorithm - two improvements and a generalization. In Proc. of IWWERT 1990, volume 572 of Lecture Notes in Computer Science, pages 85-150. Springer-Verlag, 1990.

.... bound for the exponent of periodicity for a minimal uni er for context uni cation from [SSS98] which is a generalization of a lemma that appeared in the decidability proof of word uni cation by Makanin [Mak77] An improvement of the latter result was given in [KP96] This link to word uni cation ([Mak77,Sch90,Sch93,Gut98,Pla99]) is not accidental. The relationship between word uni cation and bounded higherorder uni cation is indicated in Figure 1 where we also mention some other problems in order to position the results of this paper. Higher order unification eliminate types restrict signature exclude function ....

Klaus U. Schulz. Makanin's algorithm - two improvements and a generalization. In Proc. of IWWERT

....of second order variables may have an arbitrary number of bound variables, each having exactly one occurrence. 2 at least some significant difference between context unification and second order unification. Context unification can also be considered as a generalization of word unification [15, 1, 11, 25, 26, 12, 5]. Decidability of word unification had been an open problem for many years. The problem was raised by A. A. Markov in the late 1950 s who hoped to prove the undecidability of Hilbert s tenth problem by showing undecidability of the word unification problem. In this context, Y. Matiyasevich [17] ....

....Later, J.I. Hmelevskii (see [10] proved decidability of word unification for problems with two and three variables with an arbitrary number of occurrences. In his famous paper [15] G. S. Makanin then fully solved the problem, showing that solvability of arbitrary word equations is decidable. In [25] Makanin s result was generalized in the following way: given a word equation W 1 = W 2 with variables in fX 1 ; X n g and constants in the finite alphabet C, and given regular languages L 1 ; L n over C, it is decidable if there exists a solution S of W 1 = W 2 where S(X i ) 2 L i ....

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K. U. Schulz. Makanin's algorithm - two improvements and a generalization. In Proc. of IWWERT 1990, volume 572 of LNCS, pages 85--150. Springer-Verlag, 1990.

....all indecomposable arguments l i are variables. A solution of such a system is a mapping oe that assigns to each variable v 2 W 0 a word over its licensed alphabet F v and solves all equations of ffi 1 . Solvability of these kind of constrained systems of word equations is known to be decidable ([Sc90]) Thus, in order to prove Lemma 9.3 it suffices to show that Algorithm 3 is sound and complete. Lemma 9.4 (Completeness of Algorithm 3) If the input system ffi( x 1 ; v 1 ; x k ; v k ; v) of Algorithm 3, with given sets of li 60 censed stabilizers D v , has a solution in L hfnwl ....

K.U. Schulz, "Makanin's Algorithm - Two Improvements and a Generalization, " (Habilitationsschrift), CIS-Report 91-39, University of Munich, also in Proc. IWWERT '90, Tubingen 1990, Springer LNCS 572.

....satisfies the linear constant restriction induced by : to obtain 0 from we just replace all occurrences of variables by the constant c. In the converse direction, all occurrences of c are replaced by the same variable. We are now in a position to use the following general result from [Sh90]: Theorem: If WE is a word equation with variables x 1 ; xn and constants in the alphabet C 0 , and if L 1 ; Ln are regular languages over C 0 , then it is decidable whether WE has a solution such that (x i ) 2 L i for i = 1; n. Let L i : C i , where C i : ....

....induced by iff (x i ) 2 L i for i = 1; n. Thus the theorem implies that unifiability of word equations with linear constant restrictions is decidable, and thus by disjunctive treatment also with partially specified linear constant restrictions. The algorithm which was used in [Sh90] in order to establish the theorem mentioned above is, however, more complicated than it would be necessary for the special purpose of regular languages of the form C i . In connection with the initial translation of systems of word equations into a single word equation and with the ....

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K.U. Schulz, "Makanin's Algorithm -- Two Improvements and a Generalization, " Proceedings of the First International Workshop on Word Equations and Related Topics IWWERT '90, Tubingen 1990, Springer LNCS 572.

....in order to get decidability of general unification modulo E 1 [ E n , apply the theorem to E 1 ; E n ; F Omega . 3. General A unifiability is decidable. For A, decidability of unification problems with constant restriction is an easy consequence (see [BS91] of a result by Schulz [Sh91] on a generalization of Makanin s procedure. This result shows that it is still decidable whether a given A unification problem with constants has a solution for which the words substituted for the variables in the problem are elements of given regular languages over the constants. It is easy to ....

K. Schulz, "Makanin's Algorithm -- Two Improvements and a Generalization, " CIS-Report 91-39, CIS, University of Munich, 1991.

....a conceptual point of view, the effect of a transformation step may be described very easily: ffl At a transformation step, a non empty left part of the generalized equation is simultaneously carried towards the right side of the generalized equation. 3 A similar procedure was introduced in [Sc90], based on P ecuchet s notion of a position equation. But it turns out that the procedure is much simpler if based on Jaffar s representation. With this property the new transformation is very similar to the transformation steps which are used in Plotkin s (Lentin s) procedure. Our main aim, ....

....the new transformation makes it very natural and easy to implement. The Problem of a Complete and Correct Proof The proof that Makanin s algorithm is correct and complete has its own history. It has turned out that the definition of a generalized equation has to contain two subtle conditions (see [Sc90], pg. 126) whose sense becomes only clear when technical details of the transformation algorithm are considered. Unfortunately, this point was not treated correctly in the classical papers on Makanin s algorithm. Already Makanin s description [Ma77] at least in its English version) contained a ....

[Article contains additional citation context not shown here]

K.U. Schulz, "Makanin's Algorithm - Two Improvements and a Generalization ", (Habilitationsschrift), CIS-Report 91-39, University of Munich, also in Proceedings of the First International Workshop on Word Equations and Related Topics IWWERT '90, Tubingen 1990, Springer LNCS 572.

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K.U. Schulz. Makanin's algorithm: Two improvements and a generalization. In K.U. Schulz, editor, Word Equations and Related Topics, volume 572 of Lecture Notes in Computer Science,Tubingen, Germany, October 1990.

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