Satisfiability of word equations with constants is in PSPACE

Plandowski, Wojciech

doi:10.1145/990308.990312

Cited by 124 publications

(82 citation statements)

References 14 publications

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“…Over the years the complexity and the decidability of combinatorial problems over finite-length strings has been deeply studied [31,48,52,78,91,99,108,110,139]. Much progress has been made, but many questions remain open -especially when the language is enriched with new predicates [58].…”

Section: Theoretical Aspectsmentioning

confidence: 99%

A Novel Approach to String Constraint Solving

Amadini

Gange

Stuckey

et al. 2017

Lecture Notes in Computer Science

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String constraint solving refers to solving combinatorial problems involving constraints over string variables. String solving approaches have become popular over the last years given the massive use of strings in different application domains like formal analysis, automated testing, database query processing, and cybersecurity. This paper reports a comprehensive survey on string constraint solving by exploring the large number of approaches that have been proposed over the last decades to solve string constraints.

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Section: Theoretical Aspectsmentioning

confidence: 99%

A Novel Approach to String Constraint Solving

Amadini

Gange

Stuckey

et al. 2017

Lecture Notes in Computer Science

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“…The procedure to decide equivalence of MTT yp is essentially the same as we discussed in this paper but instead of conjunctions of equations of trees over ∆ in ∪Y we obtain conjunctions equations of words. Equations of words is a well studied problem [25,27,24]. In particular, the confirmed Ehrenfeucht conjecture states that each conjunction of a set of word equations over a finite alphabet and using a finite number of variables, is equivalent to the conjunction of a finite subset of word equations [20].…”

Section: Corollary 3 the Equivalence Of Total Separated Basic Mtts Wmentioning

confidence: 99%

Deciding Equivalence of Separated Non-nested Attribute Systems in Polynomial Time

Seidl

Palenta

Maneth

2019

Lecture Notes in Computer Science

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In 1982, Courcelle and Franchi-Zannettacci showed that the equivalence problem of separated non-nested attribute systems can be reduced to the equivalence problem of total deterministic separated basic macro tree transducers. They also gave a procedure for deciding equivalence of transducer in the latter class. Here, we reconsider this equivalence problem. We present a new alternative decision procedure and prove that it runs in polynomial time. We also consider extensions of this result to partial transducers and to the case where parameters of transducers accumulate strings instead of trees.

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“…Equations with rational constraints in a free monoid were first considered by Schulz [44]. Following an approach of Plandowski for free monoids, Diekert, Gutierrez, and Hagenah proved that systems of equations, inequations, and rational constraints in free groups are algorithmically solvable [16,36].…”

Section: Rational Constraintsmentioning

confidence: 99%

“…Makanin's algorithm (in its corrected version [32]) is known not to be primitive recursive [28], and we do not see why ours should be better. Using some data compression on words, Plandowski proposed an algorithm of much better complexity (polynomial in space) [16,36].…”

Section: Overview Of the Main Algorithmmentioning

confidence: 99%

Foliations for solving equations in groups: free, virtually free, and hyperbolic groups

Dahmani

Guirardel

2010

Journal of Topology

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We give an algorithm for solving equations and inequations with rational constraints in virtually free groups. Our algorithm is based on Rips' classification of measured band complexes. Using canonical representatives, we deduce an algorithm for solving equations and inequations in all hyperbolic groups (possibly with torsion). Additionally, we can deal with quasi-isometrically embeddable rational constraints. Introduction The equations problemGiven a group G, the equations problem in G consists in deciding algorithmically whether a system of equations with constants has a solution in G or not. An equation is an equality w = 1 where w is a word on a set of variables an their inverses together with constants taken from G. When inequations (i. e. negation of equations) are allowed, we call this problem the problem of equations and inequations. In other words, the problem of equations and inequations is equivalent to the decidability of the existential (or universal) theory of G with constants in G.A solution to the equations problem is quite powerful, as it vastly generalises the word problem, the conjugacy problem, the simultaneous conjugacy problem, etc. When G is abelian, the problem of equations and inequations is easily solved using linear algebra. But already, if G a free 3-step nilpotent group of rank 2, the equations problem is undecidable ([Tru95], see also [Rom79b]). This is based on Matiyasevich's Theorem saying that one cannot decide the solubility of polynomial equations in Z [Mat70]. For a free 2-step nilpotent group, the equations problem is solvable if and only if one can decide the solubility of polynomial equations over Q [Rom79b]. Additionally, if G is a non-commutative free metabelian group, the equations problem in G is unsolvable [Rom79a], but the problem of equations and inequations without constants is solvable [Cha95].The problem of equations and inequations in free groups is natural, and has attracted attention of many people (Lyndon, Appel, Lorents...). The solution of this problem by Makanin in 1982 [Mak82] (with the appropriate correction in [Mak84]) certainly constitutes a milestone in the theory. It has been a source of inspiration for Rips for his study of group actions on R-trees, and his solution to Morgan and Shalen's conjecture [BF95,GLP94], which found applications in many branches of geometry. It has been a decisive step towards algorithmic, and theoretical description of the set of homomorphisms of a group into a free group (Razborov [Raz84]). 1It has also been, together with these developments, a prelude to the far reaching recent solutions to Tarski's problems on the elementary theory of free groups, by Sela [Sel06], and by Kharlampovich and Miasnikov [KM06]. Makanin's algorithm is also the basis of Rips and Sela's solution to the equations problem in torsion free hyperbolic groups that they manage to reduce to the equations problem in a free group [RS95]. Finally, it is crucial in Sela's solution of the isomorphism problem for torsion free hyperbolic groups with f...

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Satisfiability of word equations with constants is in PSPACE

Abstract: We prove that satisfiability problem for word equations is in PSPACE.

Cited by 124 publications

References 14 publications

A Novel Approach to String Constraint Solving

A Novel Approach to String Constraint Solving

Deciding Equivalence of Separated Non-nested Attribute Systems in Polynomial Time

Foliations for solving equations in groups: free, virtually free, and hyperbolic groups

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