The Existential Theory of Equations with Rational Constraints in Free Groups is PSPACE—Complete
This paper extends extends known results on the complexity of word equations and equations in free groups in order to include the presence of rational constraints, i.e., such that a possible solution has to respect a specification given by a rational language. Our main result states that the existential theory of equations with rational constraints in free groups is PSPACE-complete.
KeywordsFormal languages equations regular language free group
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- 2.J. Berstel. Transductions and context-free languages. Teubner Studienbücher, Stuttgart, 1979.Google Scholar
- 3.V. Diekert. Makanin’s Algorithm. In M. Lothaire, Algebraic Combinatorics on Words. Cambridge University Press, 2001. To appear. A preliminary version is on the web: http://www-igm.univ-mlv.fr/~berstel/Lothaire/index.html.
- 4.V. Diekert and M. Lohrey. A note on the existential theory of plain groups. Submitted for publication, 2000.Google Scholar
- 6.Yu. Gurevich and A. Voronkov. Monadic simultaneous rigid E-unification and related problems. In P. Degano et al., editor, Proc. 24th ICALP, Bologna,Italy)1997, number 1256 in Lect. Not. Comp. Sc., pages 154–165. Springer, 1997.Google Scholar
- 7.C. Gutiérrez. Satisfiability of word equations with constants is in exponential space. In Proc. of the 39th Ann. Symp. on Foundations of Computer Science, FOCS’98, pages 112–119, Los Alamitos, California, 1998. IEEE Computer Society Press.Google Scholar
- 8.C. Gutiérrez. Equations in free semigroups with anti-involution and their relation to equations in free groups. In G. H. Gonnet et al., editor, Proc. Lat. Am. Theor. Inf., LATIN’2000, number 1776 in LNCS, pages 387–396. Springer, 2000.Google Scholar
- 9.C. Gutiérrez. Satisfiability of equations in free groups is in PSPACE. In 32nd ACM Symp. on Theory of Computing (STOC’2000), pages 21–27. ACM Press, 2000.Google Scholar
- 10.Ch. Hagenah. Gleichungen mit regulären Randbedingungen über freien Gruppen. PhD-thesis, Institut für Informatik, Universität Stuttgart, 2000.Google Scholar
- 11.A. Kościelski and L. Pacholski. Complexity of Makanin’s algorithm. Journal of the Association for Computing Machinery, 43(4):670–684, 1996. Preliminary version in Proc. of the 31st Ann. Symp. on Foundations of Computer Science, FOCS 90, pages 824-829, Los Alamitos, 1990. IEEE Computer Society Press.MATHGoogle Scholar
- 12.D. Kozen. Lower bounds for natural proof systems. In Proc. of the 18th Ann. Symp. on Foundations of Computer Science, FOCS 77, pages 254–266, Providence, Rhode Island, 1977. IEEE Computer Society Press.Google Scholar
- 16.P. Narendran and F. Otto. The word matching problem is undecidable for finite special string-rewriting systems that are confluent. In P. Degano et al., editor, Proc. 24th ICALP, Bologna (Italy)1997, number 1256 in Lect. Not. Comp. Sc., pages 638–648. Springer, 1997.Google Scholar
- 17.W. Plandowski. Satisfiability of word equations with constants is in NEXPTIME. In Proc. 31st Ann. Symp. on Theory of Computing, STOC’99, pages 721–725. ACM Press, 1999.Google Scholar
- 18.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.Google Scholar
- 19.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.Google Scholar
- 20.W. Rytter. On the complexity of solving word equations. Lecture given at the 16th British Colloquium on Theoretical Computer Science, Liverpool (http://www.csc.liv.ac.uk/~bctcs16/abstracts.html), 2000.
- 21.K.U. Schulz. Makanin’s algorithm for word equations-Two improvements and a generalization. In Klaus U. Schulz, editor, Word Equations and Related Topics, number 572 in Lect. Not. Comp. Sc., pages 85–150. Springer, 1991.Google Scholar