W. Plandowski and W. Rytter. Application of Lempel-Ziv encodings to the solution of word equations. In Kim G. Larsen et al., editors, Proc. of the 25th ICALP, 1998, number 1443 in Lect. Not. Comp. Sc., pages 731--742. Springer, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....equations is in PSPACE, 26] One ingredient of his work is to use data compression to reduce the exponential space to polynomial space. The importance of data compression was first recognized by Rytter and Plandowski when applying Lempel Ziv encodings to the minimal solution of a word equation [27]. Another important notion is the definition of an factorization of the solution being explained below. Guti errez extended Plandowski s method to the case of free groups, 10] Thus, a non primitive recursive scheme for solving equations in free groups has been replaced by a polynomial space ....

Wojciech Plandowski and Wojciech Rytter. Application of Lempel-Ziv encodings to the solution of word equations. In Kim G. Larsen et al., editors, Proc. 25th ICALP (ICALP'98), Aalborg (Denmark) 1998.

.... LZ(h(x) The usefulness of LZ encoding for word equations is demonstrated by following two results. Theorem 2.1 ( GKPR96] Let u = v be a word equation. Given a LZ encoding of a morphism h we can check whether h is a solution of the equation in time polynomial in LZ(h) Theorem 2. 2 ( PR98] Let u = v be a word equation with lengths specified by a function f . Assume that u = v has a solution respecting the lengths given by f . Then there is a solution h respecting the lengths such that LZ(h) is polynomial in the size of the binary encoding of f and the size of the equation. ....

Wojciech Plandowski and Wojciech Rytter. Application of lempel-ziv encodings to the solution of words equations. In Automata, Languages and Programming, pages 731--742, 1998.

....equations containing several unknowns. On the other hand, the solver in [BP98] is known to be terminating on the given fragment, while it is unknown if the algorithm described in Section 4 terminates for all input equations. If it is indeed terminating then it may be used to decide word equations [Mak92, PR98]. It has been shown, however, that any non xed size solver that supports a richer set of operators as required for most hardware applications is necessarily incomplete, since the halting problem can be reduced to solve non xed size equations on bit vectors built up from concatenation, ....

W. Plandowski and W. Rytter. Application of Lempel-Ziv Encodings to the Solution of Words Equations. In Proceedings of the 25th International Colloquium on Automata, Languages 17 and Programming, ICALP'98, pages 731-742, July 1998. Aalborg, Denmark.

.... problem 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 ....

....in at most jEj v ways) Then it rest to deal with the subwords of w 2 in between variables (hence consisting only of constants and of total length jEj Gamma v) Summing up, there are no more than (jEj v) 2v (jEj Gamma v) v jEj 4v consistent superpositions. Lemma 2 (Compare Lemma 6, [12]) Assume S is a minimal (w.r.t. length) solution of E. Then 1. For each subword w = S(E) i; j] with jwj 1, there is an occurrence of w or w Gamma1 which contains a cut of (E; S ) 2. For each letter c = S(E) i] of S(E) there is an occurrence of c or c Gamma1 in E. Proof. Let 1 p q ....

W. Rytter and W. Plandowski, Applications of Lempel-Ziv encodings to the solution of word equations, In Proceedings of the 25th. ICALP, 1998.

....equations containing several unknowns. On the other hand, the solver in [BP98] is known to be terminating on the given fragment, while it is unknown if the algorithm described in Section 4 terminates for all input equations. If it is indeed terminating then it may be used to decide word equations [Mak92,PR98]. It has been shown, however, that any non fixed size solver that supports a richer set of operators as required for most hardware applications is necessarily incomplete, since the halting problem can be reduced to solve non fixed size equations on bit vectors built up from concatenation, ....

W. Plandowski and W. Rytter. Application of Lempel-Ziv Encodings to the Solution of Words Equations. In Proceedings of the 25th International Colloquium on Automata, Languages and Programming, ICALP'98, pages 731--742, July 1998. Aalborg, Denmark.

.... solving 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 ....

....in h(u) resp. h(v) namely (q; q 1) where q = jh(u[1; p] j, which is called a cut of h. Because h(u) h(v) there are no more than jEj cuts. The following proposition about cuts is a straightforward generalization for free SGA of the similar result for words due to Rytter and Plandowski [16]. The proof can be found in [4] Proposition 1 (Lemma 2 in [4] Assume S is a minimal (w.r.t. length) solution of E. Then 1. For each subword w = h(E) i; j] with jwj 1, there is and occurrence of w or w Gamma1 which contains a cut of h which is neither the initial nor the final boundary ....

W. Rytter and W. Plandowski, Applications of LempelZiv encodings to the solution of word equations, In Proceedings of the 25th. ICALP, 1998.

....we conjecture that this last problem is tractable, i.e. there is a polynomial time algorithm which solves it. 6 Conclusions We proved that the complexity of deciding if a word equation with constants is solvable is EXPSPACE. On the other side, the known lower bound is NP hard. The conjecture in [16] is that the problem is NP complete. It is interesting to note that the improvement in the complexity of the decision problem presented in this paper gives also an improvement for the upper bound on the length of minimal solutions of word equations, triple exponential as shown above. Rytter and ....

....the problem is NP complete. It is interesting to note that the improvement in the complexity of the decision problem presented in this paper gives also an improvement for the upper bound on the length of minimal solutions of word equations, triple exponential as shown above. Rytter and Plandowski [16] showed recently, using compression of solutions, that if the length of a minimal solution to a word equation E is L jE j, then there is a nondeterministic algorithm running in time polynomial in lg L. Unfortunately, due to the current bounds on lengths of minimal solutions (triple exponential ....

W. Rytter and W. Plandowski, 1998. Application of Lempel-Ziv encodings to the solution of word equations, in Larsen, Proceedings of the 25th ICALP, Aarhus, 1998.

....hard to prove. For an upper bound Guti errez [5] showed that the problem is in EXPSPACE, and Plandowski stated a better NEXPTIME result in [15] Both results imply that the satis ability problem is in 2 DEXPTIME. Plandowski s method is based on another recent result due to Rytter and Plandowski [18] showing that the minimal solution of a word equation is highly compressible in terms of Lempel Ziv encodings. It is conjectured that the length of a minimal solution is at most exponential in the denotational length of the equation. This work was partially supported by the French German ....

....there is a corresponding solution. A corollary is that if the lengths of a minimal solution of solvable quadratic systems were at most exponential, then the satis ability problem would be NP complete. In fact this is strongly conjectured. The conclusion of containment in NP follows also from [18], but the direct method yields a much simpler approach to the special situation of quadratic systems. In the second part we address the problem with regular constraints. The uniform version is PSPACE complete. We also show that xing the lengths of a possible solution doesn t make the problem ....

[Article contains additional citation context not shown here]

Wojciech Rytter and Wojciech Plandowski. Application of Lempel-Ziv encodings to the solution of word equations. In K. G. Larsen et al., editors, Proc. 25th ICALP, Aalborg (Denmark), number 1443 in Lect. Notes Comp. Sci., pages 731-742. Springer-Verlag, Berlin, Heidelberg, New York, 1998.

....xed the lengths of a possible solution, then we can decide in linear time whether there is a corresponding solution. If the lengths of a minimal solution were at most exponential, then the satis ability problem of quadratic systems would be NP complete. The inclusion in NP follows also from [21]) In the second part we address the problem with regular constraints: The uniform version is PSPACE complete. Fixing the lengths of a possible solution doesn t make the problem much easier. The non uniform version remains NP hard (in contrast to the linear time result above) The uniform ....

....result [5, 9, 24] By Ko scielski and Pacholski [10, Cor. 4.6] this went down to 3 NEXPTIME. The present state of the art is due to [7] Guti errez showed that the problem is in EXPSPACE, in particular, it is in 2 DEXPTIME. Another recent and very interesting result is due to Rytter and Plandowski [21]. It shows that the minimal solution of a word equation is highly compressible in terms of Lempel Ziv encodings. It is conjectured that the length of a minimal solution is at most exponential in the denotational length of the equation. If this were true, then the Lempel Ziv encoding has polynomial ....

[Article contains additional citation context not shown here]

Wojciech Rytter and Wojciech Plandowski. Application of Lempel-Ziv encodings to the solution of words equations. In Kim G. Larsen et al., editors, Proc. of the 25th ICALP, Aarhus, 1998, number 1443 in Lect. Notes Comp. Sci., pages 731-742. Springer-Verlag, Berlin, Heidelberg, New York, 1998.

....equations containing several unknowns. On the other hand, the solver in [BP98] is known to be terminating on the given fragment, while it is unknown if the algorithm described in Section 4 terminates for all input equations. If it is indeed terminating then it may be used to decide word equations [Mak92,PR98]. It has been shown, however, that any non xed size solver that supports a richer set of operators as required for most hardware applications is necessarily incomplete, since the halting problem can be reduced to solve non xed size equations on bit vectors built up from concatenation, ....

W. Plandowski and W. Rytter. Application of Lempel-Ziv Encodings to the Solution of Words Equations. In Proceedings of the 25th International Colloquium on Automata, Languages and Programming, ICALP'98, pages 731-742, July 1998. Aalborg, Denmark.

....consider in this paper three problem areas. First in Section 3 we introduce new methods to show that some languages are not expressible. These results are based on subword complexity modified to finite words as in [6] Combining this with a method of compressing solutions of word equations, cf. [24], we can prove a gap theorem for expressible languages: the classical subword complexity of an expressible language is either bounded by 2 log or the language must contain a pattern language, cf. 1, 13] and hence the complexity is exponential at least for certain values of n. As a ....

....let u = u 1 u 2 . A position in h(u) between h(u 1 ) and h(u 2 ) is called a left cut. Similarly, if v = v 1 v 2 a position in h(u) h(v) between h(v 1 ) and h(v 2 ) is called a right cut. A cut in h(u) is left cut or right cut. The proof of the following lemma is anologuous to that of Lemma 6 in [24]. Lemma 3.1. Let L be a pattern free language expressible by a variable X in an equation e : u = v. Then for each word w 2 L, there is a solution h of e such that each subword of h(u) of length at least two has an occurrence that overlaps a cut. Theorem 3.2. Let L be an expressible pattern free ....

[Article contains additional citation context not shown here]

Plandowski, W., and Rytter W., Application of Lempel-Ziv encodings to the solution of word equations, in: Proc. ICALP'98, LNCS 1443, 731--742.

....four exponential functions) 7, 17] 3NEXPTIME [9] 2 EXPSPACE [4] EXPSPACE [8] The exact complexity of the algorithm is still not known. Current version of the algorithm, the full version of which can be found in [5] is still very complicated. Recently, another algorithm has been proposed in [15]. It works nondeterministically in time polynomial in n log N where n is the size of an input equation and N is the size of a minimal solution of the input equation. It is much more simple than Makanin s algorithm but its full version requires an estimation for N and the algorithm in [13] Till ....

.... of the algorithm is triple exponential (Corrollary 1 in [8] Very recently a double exponential estimation for N has been proved [14] The proof does not use Makanin s approach but uses the optimal bound for the index of periodicity of a minimal solution [9] With this estimation the algorithm in [15] places the problem in NEXPTIME. We propose the third algorithm. The full version of the algorithm requires only a proof of the upper bound for index of periodicity of a minimal solution [9] Our algorithm is the first one which is proved to work in polynomial space. A lower bound for the ....

[Article contains additional citation context not shown here]

Plandowski W., Rytter W., Application of Lempel-Ziv encodings to the solution of word equations, in: Proc. ICALP'98, LNCS 1443, 731-742, 1998.

.... string matching (FCSM) when both pattern and text strings are compressed, see e.g. 14, 15] Although it may seem that CSM stream has more practical motivation however fully compressed pattern matching appears to be much more interesting from the point of view of computation theory, e.g. see [14, 15, 4, 22]. Moreover, techniques developed in FCSM (including this paper) give firm foundations to algorithms dealing with other string problems including: computing periodicities, symmetries, repetitions, counting subwords and multi pattern matching (for respective definitions check [8] where all ....

W. Plandowski and W. Rytter, Application of Lempel-Ziv encodings to the solution of words equations, Proc. of 25th International Colloquium on Automata, Languages, and Programming, to appear.

No context found.

W. Plandowski and W. Rytter, Application of Lempel-Ziv encodings to the solution of words equations, Proc. of 25th International Colloquium on Automata, Languages, and Programming, to appear.

.... string matching (FCSM) when both pattern and text strings are compressed, see e.g. 14, 15] Although it may seem that CSM stream has more practical motivation however fully compressed pattern matching appears to be much more interesting from the point of view of computation theory, e.g. see [14, 15, 4, 22]. Moreover, techniques developed in FCSM (including this paper) give rm foundations to algorithms dealing with other string problems including: computing periodicities, symmetries, repetitions, counting subwords and multi pattern matching (for respective de nitions check [8] where all processed ....

W. Plandowski and W. Rytter, Application of Lempel-Ziv encodings to the solution of words equations, Proc. of 25th International Colloquium on Automata, Languages, and Programming, to appear.

No context found.

W. Plandowski and W. Rytter. Application of Lempel-Ziv encodings to the solution of word equations. In Kim G. Larsen et al., editors, Proc. of the 25th ICALP, 1998, number 1443 in Lect. Not. Comp. Sc., pages 731--742. Springer, 1998.

No context found.

W. Rytter and W. Plandowski, Applications of LempelZiv encodings to the solution of word equations, In Proceedings of the 25th. ICALP, 1998.

No context found.

Wojciech Plandowski and Wojciech Rytter. Application of Lempel-Ziv encodings to the solution of word equations. In Kim G. Larsen et al., editors, Proc. 25th ICALP (ICALP'98), Aalborg (Denmark) 1998.

No context found.

W. Plandowski and W. Rytter. Application of Lempel-Ziv encodings to the solution of word equations. In Kim G. Larsen et al., editors, Proc. of the 25th ICALP, 1998, number 1443 in Lect. Not. Comp. Sc., pages 731-742. Springer, 1998.

No context found.

Wojciech Plandowski and Wojciech Rytter. Application of Lempel-Ziv encodings to the solution of words equations. In Automata, Languages and Programming, pages 731--742, 1998.

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