W. Plandowski. Satisfiability of word equations with constants is in PSPACE. In Proc. of the 40th Ann. Symp. on Foundations of Computer Science, FOCS 99, pages 495--500. IEEE Computer Society Press, 1999.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to perform the search in a graph of at most exponential size, we get automatically a doubly exponential upper bound for the length of a minimal solution by backwards computation on such a path. This is still the best known upper bound (although an singly exponential bound is conjectured) see [25]. 12 Free Intervals In this section we introduce the notion of free interval in order to cope with long factors in the solution which are not related to any cut. If there were no constraints, then these factors would not appear in a minimal solution. In our setting we cannot avoid these ....

Wojciech Plandowski. Satisfiability of word equations with constants is in NEXPTIME. In Proceedings 31st Annual ACM Symposium on Theory of Computing, STOC'99, pages 721--725. ACM Press, 1999.

.... It is well known that higher order unification and second order unification are undecidable [Gol81,Far91,LV00] String unification was shown to be decidable by Makanin [Mak77] Recent upper complexity estimations are that it is in EXPSPACE [Gut98] in NEXPTIME [Pla99a] and even in PSPACE [Pla99b]. Context unification problems are restricted second order unification problems. Context variables represent terms with exactly one hole in contrast to a term with an arbitrary number of (equally named) holes in the general secondorder case. The name contexts was coined in [Com93] Currently, it ....

W. Plandowski. Satisfiability of word equations with constants is in PSPACE. In FOCS 99, pages 495--500, 1999.

.... (see e.g. Pie73,Hue75,SG89,Pre95] It is well known that higher order unification and second order unification are undecidable [Gol81,Far91,LV00] String unification was shown to be decidable by Makanin [Mak77] Recent upper complexity estimations are that it is in EXPSPACE [Gut98] in NEXPTIME [Pla99a] and even in PSPACE [Pla99b] Context unification problems are restricted second order unification problems. Context variables represent terms with exactly one hole in contrast to a term with an arbitrary number of (equally named) holes in the general secondorder case. The name contexts was ....

W. Plandowski. Satisfiability of word equations with constants is in NEXPTIME. In T. Leighton, editor, Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing (STOC'99), pages 721--725, Atlanta, Georgia, 1999. ACM Press.

.... for equations over free groups and again Makanin proved decidability [12, 13] The scheme of Makanin in the case of free groups is however known to be non primitiverecursive, see [9] Only when Plandowski invented a new method for solving word equations by some polynomial space bounded algorithm [19], the corresponding problem for free groups was reconsidered and Guti errez [8] succeeded in extending Plandowski s polynomial space algorithm to free groups. In fact, it has been possible to prove decidability (and PSPACE completeness) of equations with rational constraints in free groups, see ....

W. Plandowski. Satisfiability of word equations with constants is in PSPACE. In Proc. 40th Ann. Symp. on Found. of Comp. Sci., FOCS'99, pages 495--500. IEEE Computer Society Press 1999.

....have carried out both steps successfully but report only on the second step. A similar automata theoretic approach is already known for satisfiability but not for entailment [7,8] Our extension to entailment yields a new characterization of NSSE that uses regular expressions and word equations [17,18]. Word equations raise the real di#culty behind NSSE since they spoil the usual pumping arguments from automata theory. They also clarify why NSSE di#ers so significantly from seemingly similar entailment problems [19,20] A tree automata based approach to non structural subtype entailment was ....

W. Plandowski, Satisfiability of word equations with constants is in PSPACE, in: IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, 1999, pp. 495--500.

....have carried out both steps successfully but report only on the second step. A similar automata theoretic approach is already known for satisfiablity but not for entailment [9, 16] Our extension to entailment yields a new characterization of NSSE that uses regular expressions and word equations [11, 17]. Word equations raise the real difficulty behind NSSE since they spoil the usual pumping arguments from automata theory. They also clarify why NSSE differs so significantly from seemingly similar entailment problems [13, 14] 2 Characterization We now formulate the main result of this paper and ....

W. Plandowski. Satisfiability of word equations with constants is in PSPACE. In Proc. of the 40 IEEE Symp. on Found. of Comp. Science, pages 495--500, 1999.

.... e.g. Pie73,Hue75,SG89,Wol93,Pre95] It is well known that general higher order unification and second order unification are undecidable [Gol81,Far91,LV99] and that string unification is decidable [Mak77] Recent upper complexity estimations for string unification are NEXPTIME [Pla99a] and PSPACE [Pla99b]. Context unification problems are restricted second order unification problems: context variables represent terms with exactly one hole in contrast to a term with an arbitrary number of (equally named) holes in the general case. The name contexts was coined in [Com93] Currently, it is not known ....

W. Plandowski. Satisfiability of word equations with constants is in PSPACE, 1999. To appear.

.... unification (see e.g. Pie73,Hue75,SG89,Wol93,Pre95] It is well known that general higher order unification and second order unification are undecidable [Gol81,Far91,LV99] and that string unification is decidable [Mak77] Recent upper complexity estimations for string unification are NEXPTIME [Pla99a] and PSPACE [Pla99b] Context unification problems are restricted second order unification problems: context variables represent terms with exactly one hole in contrast to a term with an arbitrary number of (equally named) holes in the general case. The name contexts was coined in [Com93] ....

W. Plandowski. Satisfiability of word equations with constants is in NEXPTIME. In T. Leighton, editor, Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing (STOC'99), Atlanta, Georgia, 1999. ACM Press. To appear.

.... SG89, Pre95] It is well known that general higher order unification and second order unification are undecidable [Gol81, Far91, LV99] String unification was shown to be decidable by Makanin [Mak77] Recent upper complexity estimations are that it is in NEXPTIME [Pla99a] and even in PSPACE [Pla99b]. Context unification problems are restricted second order unification problems: context variables represent terms with exactly one hole in contrast to a term with an arbitrary number of (equally named) holes in the general case. The name contexts was coined in [Com93] Currently, it is not known ....

W. Plandowski. Satisfiability of word equations with constants is in PSPACE, 1999. To appear.

.... (see e.g. Pie73, Hue75, SG89, Pre95] It is well known that general higher order unification and second order unification are undecidable [Gol81, Far91, LV99] String unification was shown to be decidable by Makanin [Mak77] Recent upper complexity estimations are that it is in NEXPTIME [Pla99a] and even in PSPACE [Pla99b] Context unification problems are restricted second order unification problems: context variables represent terms with exactly one hole in contrast to a term with an arbitrary number of (equally named) holes in the general case. The name contexts was coined in ....

W. Plandowski. Satisfiability of word equations with constants is in NEXPTIME. In T. Leighton, editor, Proceedings of the ThirtyFirst Annual ACM Symposium on Theory of Computing (STOC'99), Atlanta, Georgia, 1999. ACM Press. To appear. 17

.... solving equations in free SGA (unification in free SGA) and its computational complexity is a problem closely related to the problem of solving equations in free semigroups and in free groups, which lately have attracted much attention of the theoretical computer science community [3] 12] 13] [14]. Free semigroups with anti involution is a structure which lies in between that of free semigroups and free groups. Besides the relationship with semigroups and groups, the axioms defining SGA show up in several important theories, like algebras of binary relations, transpose in matrices, ....

.... have analyzed the complexity of Makanin s algorithm for semigroups [6] 16] 1] being EXPSPACE the best upper bound so far [3] Very recently Plandowski, without using Makanin s algorithm, presented an upper bound of PSPACE for the problem of satisfiability of equations in free semigroups [14]. On the other hand, the analysis of the complexity of Makanin s algorithm for groups was done by Koscielski and Pacholski [8] who showed that it is not primitive recursive. With respect to lower bounds, the only known lower bound for both problems is NP hard, which seems to be weak for the case ....

Plandowski, W., Satisfiability of word equations with constants is in PSPACE, in Proc. FOCS'99.

.... of solving equations in free SGA (unification in free SGA) and its computational complexity is a problem closely related to the problem of solving equations in free semigroups and in free groups, which lately have attracted much attention of the theoretical computer science community [3] 12] [13], 14] Free semigroups with anti involution is a structure which lies in between that of free semigroups and free groups. Besides the relationship with semigroups and groups, the axioms defining SGA show up in several important theories, like algebras of binary relations, transpose in matrices, ....

Plandowski, W., Satisfiability of word equations with constants is in NEXPTIME, in Proc. STOC'99.

.... equations in free groups (or in unification language: is the unification problem for groups decidable ) This problem and the related one for free semigroups has lately attracted much attention from the theoretical computer science community, see for example [2] 8] 9] 3] 16] 17] [18]. Special particular cases were answered positively by Lyndon [12] Lorents [10] Kmelevskii [6] 7] In 1982 Makanin [14] corrections in [15] presented an algorithm that solves the general case, still the only one known. Koscielski and Pacholski [9] by showing that contrary to the common ....

....in free SGA (i.e. no restriction on the solutions) Claim 2: For each equation E 0 in free SGA there is 1 a set E 00 1 ; E 00 k of equations in free SGA such that E 0 has a non contractible solution iff one of E 00 j has a (ordinary) solution. 3. Generalize the method used in [18] for deciding satisfiability of word equations to a method for deciding satisfiability of equations in free SGA. Claim: Satisfiability of equations in free SGA is in PSPACE. The size of the set of equations in Step 1 is exponentially bigger than the size of E. Same for Step 2. The good news is ....

[Article contains additional citation context not shown here]

Plandowski, W., Satisfiability of word equations with constants is in PSPACE, in Proc. FOCS'99.

.... arbitrary equations in free groups (or in unification language: is the unification problem for groups decidable ) This problem and the related one for free semigroups has lately attracted much attention from the theoretical computer science community, see for example [2] 8] 9] 3] 16] [17], 18] Special particular cases were answered positively by Lyndon [12] Lorents [10] Kmelevskii [6] 7] In 1982 Makanin [14] corrections in [15] presented an algorithm that solves the general case, still the only one known. Koscielski and Pacholski [9] by showing that contrary to the ....

Plandowski, W., Satisfiability of word equations with constants is in NEXPTIME, in Proc. STOC'99.

.... 1985] The known upper bound is still considerably higher, even though there has recently been considerable progress in lowering the bound: Unification theory 39 the 3 NEXPTIME result by Koscielski and Pacholski [1990] was first improved to EXPSPACE by Guti errez [1998] and then to NEXPTIME by Plandowski [1999]. Interestingly, the last result no longer needs Makanin s algorithm, i.e. it yields a new decision procedure that is independent of Makanin s result. Unification type: infinitary for all three kinds of unification problems [Plotkin 1972] see also example 3.7) Unification procedure: Plotkin ....

Plandowski W. [1999], Satisfiability of word equations with constants is in NEXPTIME, in T. Leighton, ed., `Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing (STOC'99)', ACM Press, Atlanta, Georgia.

....have been studied extensively during the past ten years or so. The seminal Makanin s result, cf. 20] has inspired a lot of research on word equations, see [7] for a survey, and very recently the known complexity of the satisfiability problem was drastically lowered, the problem is in PSPACE, see [22, 23]. Resaearch on different aspects of word equations proposed in [14] where the question what kinds of languages or relations can be expressed as values of unknowns in solutions of a word equation, i.e. what kinds of properties are expressible by word equations. It is well known that many simple ....

Plandowski, W., Satisfiability of word equations with constants is in NEXPTIME, in: Proc. STOC'99, 1999.

No context found.

W. Plandowski. Satisfiability of word equations with constants is in PSPACE. In Proc. of the 40th Ann. Symp. on Foundations of Computer Science, FOCS 99, pages 495--500. IEEE Computer Society Press, 1999.

No context found.

W. Plandowski. Satisfiability of word equations with constants is in NEXPTIME. In Proceedings 31st Annual ACM Symposium on Theory of Computing, STOC'99, pages 721--725. ACM Press, 1999.

No context found.

W. Plandowski, Satisfiability of Word equations with constants is in PSPACE, In Proceedings of the FOCS'99, pp. 495-500, IEEE Comp. Soc. Press, 1999.

No context found.

Plandowski, W., Satisfiability of word equations with constants is in PSPACE, in Proc. FOCS'99.

No context found.

Plandowski, W., Satisfiability of word equations with constants is in NEXPTIME, in Proc. STOC'99.

No context found.

W. Plandowski. Satisfiability of word equations with constants is in PSPACE. In Proc. 40th Ann. Symp. on Found. of Comp. Sci., FOCS'99, pages 495--500. IEEE Computer Society Press 1999.

No context found.

W. Plandowski. Satisfiability of word equations with constants is in PSPACE, 1999. To appear.

No context found.

W. Plandowski. Satisfiability of word equations with constants is in NEXPTIME. In T. Leighton, editor, Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing (STOC'99), Atlanta, Georgia, 1999. ACM Press. To appear. 19

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