Tightening the Complexity of Equivalence Problems for Commutative Grammars

Authors Christoph Haase, Piotr Hofman



PDF
Thumbnail PDF

File

LIPIcs.STACS.2016.41.pdf
  • Filesize: 0.71 MB
  • 14 pages

Document Identifiers

Author Details

Christoph Haase
Piotr Hofman

Cite AsGet BibTex

Christoph Haase and Piotr Hofman. Tightening the Complexity of Equivalence Problems for Commutative Grammars. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 41:1-41:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.STACS.2016.41

Abstract

Given two finite-state automata, are the Parikh images of the languages they generate equivalent? This problem was shown decidable in coNEXP by Huynh in 1985 within the more general setting of context-free commutative grammars. Huynh conjectured that a Pi_2^P upper bound might be possible, and Kopczynski and To established in 2010 such an upper bound when the size of the alphabet is fixed. The contribution of this paper is to show that the language equivalence problem for regular and context-free commutative grammars is actually coNEXP-complete. In addition, our lower bound immediately yields further coNEXP-completeness results for equivalence problems for regular commutative expressions, reversal-bounded counter automata and communication-free Petri nets. Finally, we improve both lower and upper bounds for language equivalence for exponent-sensitive commutative grammars.
Keywords
  • language equivalence
  • commutative grammars
  • presburger arithmetic
  • semi-linear sets
  • petri nets

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Eric Domenjoud. Solving systems of linear Diophantine equations: An algebraic approach. In Mathematical Foundations of Computer Science (MFCS), pages 141-150, 1991. Google Scholar
  2. Samuel Eilenberg and M.P. Schützenberger. Rational sets in commutative monoids. Journal of Algebra, 13(2):173-191, 1969. Google Scholar
  3. Javier Esparza. Petri nets, commutative context-free grammars, and basic parallel processes. Fundamenta Informaticae, 31(1):13-25, 1997. URL: http://dx.doi.org/10.3233/FI-1997-3112.
  4. M. R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, 1979. Google Scholar
  5. Seymour Ginsburg. The mathematical theory of context free languages. McGraw-Hill, 1966. Google Scholar
  6. Seymour Ginsburg and Edwin H. Spanier. Bounded ALGOL-like languages. Transactions of the American Mathematical Society, pages 333-368, 1964. Google Scholar
  7. Erich Grädel. Dominoes and the complexity of subclasses of logical theories. Annals of Pure Applied Logic, 43(1):1-30, 1989. URL: http://dx.doi.org/10.1016/0168-0072(89)90023-7.
  8. Christoph Haase. Subclasses of Presburger arithmetic and the weak EXP hierarchy. In Proceedings of the Joint Meeting of the 23rd EACSL Annual Conference on Computer Science Logic (CSL) and the 29th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) (CSL-LICS), pages 47:1-47:10. ACM, 2014. URL: http://dx.doi.org/10.1145/2603088.2603092.
  9. Christoph Haase and Simon Halfon. Integer vector addition systems with states. In Proceedings of the 8th International Workshop on Reachability Problems (RP), volume 8762 of Lecture Notes in Computer Science, pages 112-124. Springer, 2014. URL: http://dx.doi.org/10.1007/978-3-319-11439-2_9.
  10. Michel Hack. The equality problem for vector addition systems is undecidable. Theoretical Computer Science, 2(1):77-95, 1976. URL: http://dx.doi.org/10.1016/0304-3975(76)90008-6.
  11. Matthew Hague and Anthony Widjaja Lin. Model checking recursive programs with numeric data types. In Proccedings of the 23rd International Conference on Computer Aided Verification (CAV), volume 6806 of Lecture Notes in Computer Science, pages 743-759. Springer, 2011. URL: http://dx.doi.org/10.1007/978-3-642-22110-1_60.
  12. John E. Hopcroft, Rajeev Motwani, and Jeffrey D. Ullman. Introduction to automata theory, languages, and computation - international edition (2. ed). Addison-Wesley, 2003. Google Scholar
  13. Dung T. Huynh. Commutative grammars: The complexity of uniform word problems. Information and Control, 57(1):21-39, 1983. URL: http://dx.doi.org/10.1016/S0019-9958(83)80022-9.
  14. Dung T. Huynh. The complexity of equivalence problems for commutative grammars. Information and Control, 66(1-2):103-121, 1985. URL: http://dx.doi.org/10.1016/S0019-9958(85)80015-2.
  15. Dung T. Huynh. A simple proof for the Σ^p₂ upper bound of the inequivalence problem for semilinear sets. Elektronische Informationsverarbeitung und Kybernetik, 22(4):147-156, 1986. Google Scholar
  16. Oscar H. Ibarra. Reversal-bounded multicounter machines and their decision problems. Journal of the ACM, 25(1):116-133, 1978. URL: http://dx.doi.org/10.1145/322047.322058.
  17. Eryk Kopczyński. Complexity of problems of commutative grammars. Logical Methods in Computer Science, 11(1), 2015. URL: http://dx.doi.org/10.2168/lmcs-11(1:9)2015.
  18. Eryk Kopczyński and Anthony Widjaja To. Parikh images of grammars: Complexity and applications. In Proceedings of the 25th Annual IEEE Symposium on Logic in Computer Science, (LICS), pages 80-89. IEEE Computer Society, 2010. URL: http://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=5570020, URL: http://dx.doi.org/10.1109/LICS.2010.21.
  19. Jérôme Leroux and Sylvain Schmitz. Demystifying reachability in vector addition systems. In 30th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), pages 56-67. IEEE, 2015. URL: http://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=7174833.
  20. Richard Lipton. The reachability problem is exponential-space-hard. Technical report, Yale University, New Haven, CT, 1976. Google Scholar
  21. Ernst W. Mayr and Jeremias Weihmann. Completeness results for generalized communication-free Petri nets with arbitrary edge multiplicities. In Proceedings of the 7th International Workshop on Reachability Problems (RP), volume 8169 of Lecture Notes in Computer Science, pages 209-221. Springer, 2013. URL: http://dx.doi.org/10.1007/978-3-642-41036-9_19.
  22. Ernst W. Mayr and Jeremias Weihmann. Completeness results for generalized communication-free Petri nets with arbitrary edge multiplicities. Technical Report TUM-I1335, Technische Universität München, 2013. URL: http://mediatum.ub.tum.de/node?id=1169599.
  23. Ernst W. Mayr and Jeremias Weihmann. Complexity results for problems of communication-free Petri nets and related formalisms. Fundamenta Informaticae, 137(1):61-86, 2015. URL: http://dx.doi.org/10.3233/FI-2015-1170.
  24. Loïc Pottier. Minimal solutions of linear Diophantine systems: Bounds and algorithms. In Proceedings of the 4th International Conference on Rewriting Techniques and Applications (RTA), volume 488 of Lecture Notes in Computer Science, pages 162-173. Springer, 1991. URL: http://dx.doi.org/10.1007/3-540-53904-2_94.
  25. Jirí Srba. Complexity of weak bisimilarity and regularity for BPA and BPP. Mathematical Structures in Computer Science, 13(4):567-587, 2003. URL: http://dx.doi.org/10.1017/S0960129503003992.
  26. Larry J. Stockmeyer and Albert R. Meyer. Word problems requiring exponential time: Preliminary report. In 5th Annual ACM Symposium on Theory of Computing, STOC, pages 1-9. ACM, 1973. Google Scholar
  27. Hsu-Chun Yen. On reachability equivalence for BPP-nets. Theoretical Computer Science, 179(1-2):301-317, 1997. URL: http://dx.doi.org/10.1016/S0304-3975(96)00147-8.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail