skip to main content
research-article

Advanced automata minimization

Published:23 January 2013Publication History
Skip Abstract Section

Abstract

We present an efficient algorithm to reduce the size of nondeterministic Buchi word automata, while retaining their language. Additionally, we describe methods to solve PSPACE-complete automata problems like universality, equivalence and inclusion for much larger instances (1-3 orders of magnitude) than before. This can be used to scale up applications of automata in formal verification tools and decision procedures for logical theories.

The algorithm is based on new transition pruning techniques. These use criteria based on combinations of backward and forward trace inclusions. Since these relations are themselves PSPACE-complete, we describe methods to compute good approximations of them in polynomial time.

Extensive experiments show that the average-case complexity of our algorithm scales quadratically. The size reduction of the automata depends very much on the class of instances, but our algorithm consistently outperforms all previous techniques by a wide margin. We tested our algorithm on Buchi automata derived from LTL-formulae, many classes of random automata and automata derived from mutual exclusion protocols, and compared its performance to the well-known automata tool GOAL.

References

  1. RABIT tool: www.languageinclusion.org/doku.php?id=tools.Google ScholarGoogle Scholar
  2. L. Clemente and R. Mayr. Advanced Automata Minimization. Tech. Rep. EDI-INF-RR-1414, U. of Edinburgh (2012), arXiv:1210.6624 http://www.inf.ed.ac.uk/publications/report/1414.html.Google ScholarGoogle Scholar
  3. P. Abdulla, Y.-F. Chen, L. Clemente, L. Holik, C.-D. Hong, R. Mayr, and T. Vojnar. Simulation Subsumption in Ramsey-Based Buchi Automata Universality and Inclusion Testing. In T. Touili, B. Cook, and P. Jackson, editors, Computer Aided Verification, volume 6174 of LNCS, pages 132--147, 2010. ISBN 978-3-642-14294-9. doi: 10.1007/978-3-642-14295-6 14. URL http://dx.doi.org/10.1007/978-3-642-14295-6 14. Google ScholarGoogle ScholarDigital LibraryDigital Library
  4. P. Abdulla, Y.-F. Chen, L. Clemente, L. Holik, C.-D. Hong, R. Mayr, and T. Vojnar. Advanced Ramsey-based Buchi Automata Inclusion Testing. In J.-P. Katoen and B. Konig, editors, International Conference on Concurrency Theory, volume 6901 of LNCS, pages 187--202, Sept. 2011. Google ScholarGoogle ScholarDigital LibraryDigital Library
  5. P. A. Abdulla, Y.-F. Chen, L. Hol1k, and T. Vojnar. Mediating for reduction (on minimizing alternating Buchi automata). In FSTTCS, volume 4 of LIPIcs, pages 1--12. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2009.Google ScholarGoogle Scholar
  6. P. A. Abdulla, Y.-F. Chen, L. Holik, R. Mayr, and T. Vojnar. When Simulation Meets Antichains. In Tools and Algorithms for the Construction and Analysis of Systems, volume 6015 of LNCS, 2010. URL http://hal.inria.fr/inria-00460294/en/. Google ScholarGoogle ScholarDigital LibraryDigital Library
  7. D. Bustan and O. Grumberg. Simulation-based minimization. ACM Trans. Comput. Logic, 4:181--206, April 2003. ISSN 1529-3785. doi: http://doi.acm.org/10.1145/635499.635502. URL http://doi.acm.org/10.1145/635499.635502. Google ScholarGoogle ScholarDigital LibraryDigital Library
  8. L. Clemente. Buchi Automata Can Have Smaller Quotients. In L. Aceto, M. Henzinger, and J. Sgall, editors, ICALP, volume 6756 of LNCS, pages 258--270. 2011. ISBN 978-3-642-22011-1. doi: 10.1007/978-3-642-22012-8 20. URL http://arxiv.org/pdf/1102.3285. Google ScholarGoogle ScholarDigital LibraryDigital Library
  9. L. Clemente. Generalized Simulation Relations with Applications in Automata Theory. PhD thesis, University of Edinburgh, 2012.Google ScholarGoogle Scholar
  10. D. L. Dill, A. J. Hu, and H. Wont-Toi. Checking for Language Inclusion Using Simulation Preorders. In Computer Aided Verification, volume 575 of LNCS. Springer-Verlag, 1991. doi: 10.1007/3-540-55179-4 25. URL http://dx.doi.org/10.1007/3-540-55179-4 25. Google ScholarGoogle ScholarDigital LibraryDigital Library
  11. L. Doyen and J.-F. Raskin. Antichains Algorithms for Finite Automata. In Tools and Algorithms for the Construction and Analysis of Systems, volume 6015 of LNCS, pages 2--22. Springer-Verlag, 2010. Google ScholarGoogle ScholarDigital LibraryDigital Library
  12. K. Etessami. A Hierarchy of Polynomial-Time Computable Simulations for Automata. In International Conference on Concurrency Theory, volume 2421 of LNCS, pages 131--144. Springer-Verlag, 2002. doi: 10.1007/3-540-45694-5 10. URL http://dx.doi.org/10.1007/3-540-45694-5 10. Google ScholarGoogle ScholarDigital LibraryDigital Library
  13. K. Etessami and G. Holzmann. Optimizing Buchi Automata. In International Conference on Concurrency Theory, volume 1877 of LNCS, pages 153--168. Springer-Verlag, 2000. Google ScholarGoogle ScholarDigital LibraryDigital Library
  14. K. Etessami, T. Wilke, and R. A. Schuller. Fair Simulation Relations, Parity Games, and State Space Reduction for Buchi Automata. SIAM J. Comput., 34(5):1159--1175, 2005. doi: 10.1137/S0097539703420675. URL http://epubs.siam.org/sam-bin/dbq/article/42067. Google ScholarGoogle ScholarDigital LibraryDigital Library
  15. S. Fogarty and M. Vardi. Complementation and Size-Change Termination. In S. Kowalewski and A. Philippou, editors, Tools and Algorithms for the Construction and Analysis of Systems, volume 5505 of LNCS, pages 16--30. 2009. doi: 10.1007/978-3-642-00768-2 2. URL http://dx.doi.org/10.1007/978-3-642-00768-2 2. Google ScholarGoogle ScholarDigital LibraryDigital Library
  16. S. Fogarty and M. Y. Vardi. Efficient Buchi Universality Checking. In Tools and Algorithms for the Construction and Analysis of Systems, pages 205--220, 2010. Google ScholarGoogle ScholarDigital LibraryDigital Library
  17. S. Fogarty, O. Kupferman, M. Y. Vardi, and T. Wilke. Unifying Buchi Complementation Constructions. In M. Bezem, editor, Computer Science Logic, volume 12 of LIPIcs, pages 248--263. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2011. doi: http://dx.doi.org/10.4230/LIPIcs.CSL.2011.248.Google ScholarGoogle Scholar
  18. P. Gastin and D. Oddoux. Fast LTL to Buchi automata translation. In CAV, volume 2102 of LNCS, pages 53--65. Springer, 2001. Google ScholarGoogle ScholarDigital LibraryDigital Library
  19. S. Gurumurthy, R. Bloem, , and F. Somenzi. Fair simulation minimization. In CAV, volume 2404 of LNCS, pages 610--624. Springer, 2002. Google ScholarGoogle ScholarDigital LibraryDigital Library
  20. M. R. Henzinger, T. A. Henzinger, and P. W. Kopke. Computing simulations on finite and infinite graphs. In Foundations of Computer Science, FOCS '95, Washington, DC, USA, 1995. IEEE Computer Society. ISBN 0-8186-7183-1. URL http://portal.acm.org/citation.cfm?id=796255. Google ScholarGoogle ScholarDigital LibraryDigital Library
  21. T. A. Henzinger, O. Kupferman, and S. K. Rajamani. Fair Simulation. Information and Computation, 173: 64--81, 2002. doi: 10.1006/inco.2001.3085. URL http://dx.doi.org/10.1006/inco.2001.3085. Google ScholarGoogle ScholarDigital LibraryDigital Library
  22. G. Holzmann. The SPIN Model Checker. Addison Wesley, 2004.Google ScholarGoogle Scholar
  23. T. Jiang and B. Ravikumar. Minimal NFA Problems are Hard. In J. Albert, B. Monien, and M. Artalejo, editors, ICALP, volume 510 of LNCS, pages 629--640. 1991. doi: 10.1007/3-540-54233-7 169. Google ScholarGoogle ScholarDigital LibraryDigital Library
  24. S. Juvekar and N. Piterman. Minimizing Generalized Buchi Automata. In Computer Aided Verification, volume 4414 of LNCS, pages 45--58. Springer-Verlag, 2006. doi: 10.1007/11817963 7. URL http://dx.doi.org/10.1007/11817963 7. Google ScholarGoogle ScholarDigital LibraryDigital Library
  25. O. Kupferman and M. Vardi. Verification of Fair Transition Systems. In Computer Aided Verification, volume 1102 of LNCS, pages 372--382. Springer-Verlag, 1996. URL http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.29.9654. Google ScholarGoogle ScholarDigital LibraryDigital Library
  26. C. S. Lee, N. D. Jones, and A. M. Ben-Amram. The size-change principle for program termination. POPL '01, pages 81--92, 2001. doi: http://doi.acm.org/10.1145/360204.360210. Google ScholarGoogle ScholarDigital LibraryDigital Library
  27. J. Leroux and G. Point. TaPAS: The Talence Presburger Arithmetic Suite. In Proceedings of the 15th International Conference on Tools and Algorithms for the Construction and Analysis of Systems (TACAS), volume 5505 of LNCS. Springer, 2009. Google ScholarGoogle ScholarDigital LibraryDigital Library
  28. I. Niven. Mathematics of Choice. The Mathematical Association of America, 1965.Google ScholarGoogle Scholar
  29. N. Piterman. From nondeterministic Buchi and Streett automata to deterministic parity automata. In LICS, pages 255--264. IEEE, 2006. Google ScholarGoogle ScholarDigital LibraryDigital Library
  30. R. Sebastiani and S. Tonetta. More deterministic vs. smaller Buchi automata for efficient LTL model checking. In Correct Hardware Design and Verification Methods, volume 2860 of LNCS, 2003.Google ScholarGoogle Scholar
  31. A. P. Sistla, M. Y. Vardi, and P. Wolper. The complementation problem for Buchi automata with applications to temporal logic. Theor. Comput. Sci., 49:217--237, Jan. 1987. ISSN 0304-3975. doi: 10.1016/0304 3975(87)90008-9. URL http://dx.doi.org/10.1016/0304-3975(87)90008-9. Google ScholarGoogle ScholarDigital LibraryDigital Library
  32. F. Somenzi and R. Bloem. Efficient Buchi Automata from LTL Formulae. In Computer Aided Verification, volume 1855 of LNCS, pages 248--263. Springer-Verlag, 2000. doi: 10.1007/10722167 21. URL http://dx.doi.org/10.1007/10722167_21. Google ScholarGoogle ScholarDigital LibraryDigital Library
  33. D. Tabakov and M. Vardi. Model Checking Buchi Specifications. In LATA, volume Report 35/07. Research Group on Mathematical Linguistics, Universitat Rovira i Virgili, Tarragona, 2007.Google ScholarGoogle Scholar
  34. Y.-K. Tsay, Y.-F. Chen, M.-H. Tsai,W.-C. Chan, and C.-J. Luo. GOAL extended: Towards a research tool for omega automata and temporal logic. In C. Ramakrishnan and J. Rehof, editors, Tools and Algo- rithms for the Construction and Analysis of Systems, volume 4963 of LNCS, pages 346--350. 2008. ISBN 978-3-540-78799-0. URL http://dx.doi.org/10.1007/978-3-540-78800-3 26. Google ScholarGoogle ScholarDigital LibraryDigital Library
  35. Y.-K. Tsay, M.-H. Tsai, J.-S. Chang, and Y.-W. Chang. Buchi store: An open repository of Buchi automata. In P. Abdulla and K. Leino, editors, Tools and Algorithms for the Construction and Analysis of Systems, volume 6605 of LNCS, pages 262--266. 2011. ISBN 978-3-642-19834-2. URL http://dx.doi.org/10.1007/978-3-642-19835-9 23.10.1007/978-3-642-19835-9 23. Google ScholarGoogle ScholarDigital LibraryDigital Library

Index Terms

  1. Advanced automata minimization

        Recommendations

        Comments

        Login options

        Check if you have access through your login credentials or your institution to get full access on this article.

        Sign in

        Full Access

        • Published in

          cover image ACM SIGPLAN Notices
          ACM SIGPLAN Notices  Volume 48, Issue 1
          POPL '13
          January 2013
          561 pages
          ISSN:0362-1340
          EISSN:1558-1160
          DOI:10.1145/2480359
          Issue’s Table of Contents
          • cover image ACM Conferences
            POPL '13: Proceedings of the 40th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages
            January 2013
            586 pages
            ISBN:9781450318327
            DOI:10.1145/2429069

          Copyright © 2013 ACM

          Publisher

          Association for Computing Machinery

          New York, NY, United States

          Publication History

          • Published: 23 January 2013

          Check for updates

          Qualifiers

          • research-article

        PDF Format

        View or Download as a PDF file.

        PDF

        eReader

        View online with eReader.

        eReader
        About Cookies On This Site

        We use cookies to ensure that we give you the best experience on our website.

        Learn more

        Got it!