Undecidability Results for Bisimilarity on Prefix Rewrite Systems

  • Petr Jančar
  • Jiří Srba
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3921)

Abstract

We answer an open question related to bisimilarity checking on labelled transition systems generated by prefix rewrite rules on words. Stirling (1996, 1998) proved the decidability of bisimilarity for normed pushdown processes. This result was substantially extended by Sénizergues (1998, 2005) who showed the decidability for regular (or equational) graphs of finite out-degree (which include unnormed pushdown processes). The question of decidability of bisimilarity for a more general class of so called Type -1 systems (generated by prefix rewrite rules of the form \(R\,{\mathop{\longrightarrow}\limits^{a}}\,w\) where R is a regular language) was left open; this was repeatedly indicated by both Stirling and Sénizergues. Here we answer the question negatively, i.e., we show undecidability of bisimilarity on Type -1 systems, even in the normed case. We complete the picture by considering classes of systems that use rewrite rules of the form \(w\,{\mathop{\longrightarrow}\limits^{a}}\,R\) and \(R_{1}\,{\mathop{\longrightarrow}\limits^{a}}\,R_{2}\) and show when they yield low undecidability (Π\(^{\rm 0}_{\rm 1}\)-completeness) and when high undecidability (Σ\(^{\rm 1}_{\rm 1}\)-completeness), all with and without the assumption of normedness.

References

  1. 1.
    Balcazar, J., Gabarro, J., Santha, M.: Deciding bisimilarity is P-complete. Formal Aspects of Computing 4(6A), 638–648 (1992)CrossRefMATHGoogle Scholar
  2. 2.
    Bouajjani, A., Esparza, J., Maler, O.: Reachability analysis of pushdown automata: Application to model-checking. In: Mazurkiewicz, A., Winkowski, J. (eds.) CONCUR 1997. LNCS, vol. 1243, pp. 135–150. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  3. 3.
    Burkart, O., Caucal, D., Moller, F., Steffen, B.: Verification on infinite structures. In: Bergstra, J., Ponse, A., Smolka, S. (eds.) Handbook of Process Algebra, ch.9, pp. 545–623. Elsevier Science, Amsterdam (2001)CrossRefGoogle Scholar
  4. 4.
    Burkart, O., Caucal, D., Steffen, B.: An elementary decision procedure for arbitrary context-free processes. In: Hájek, P., Wiedermann, J. (eds.) MFCS 1995. LNCS, vol. 969, pp. 423–433. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  5. 5.
    Burkart, O., Caucal, D., Steffen, B.: Bisimulation collapse and the process taxonomy. In: Sassone, V., Montanari, U. (eds.) CONCUR 1996. LNCS, vol. 1119, pp. 247–262. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  6. 6.
    Cachat, T.: Uniform solution of parity games on prefix-recognizable graphs. Electronic Notes in Theoretical Computer Science 68(6) (2002)Google Scholar
  7. 7.
    Caucal, D.: On the regular structure of prefix rewriting. Theoretical Computer Science 106(1), 61–86 (1992)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Caucal, D.: Bisimulation of context-free grammars and of pushdown automata. In: Modal Logic and Process Algebra. CSLI Lectures Notes, vol. 53, pp. 85–106. University of Chicago Press (1995)Google Scholar
  9. 9.
    Caucal, D.: On infinite transition graphs having a decidable monadic theory. In: Meyer auf der Heide, F., Monien, B. (eds.) ICALP 1996. LNCS, vol. 1099, pp. 194–205. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  10. 10.
    Esparza, J., Hansel, D., Rossmanith, P., Schwoon, S.: Efficient algorithms for model checking pushdown systems. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 232–247. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  11. 11.
    Harel, D.: Effective transformations on infinite trees, with applications to high undecidability, dominoes, and fairness. Journal of the ACM (JACM) 33(1), 224–248 (1986)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Hennessy, M., Milner, R.: Algebraic laws for nondeterminism and concurrency. Journal of the Association for Computing Machinery 32(1), 137–161 (1985)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Hirshfeld, Y., Jerrum, M., Moller, F.: A polynomial algorithm for deciding bisimilarity of normed context-free processes. Theoretical Computer Science 158(1–2), 143–159 (1996)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Jančar, P., Srba, J.: Highly undecidable questions for process algebras. In: Proc. of the 3rd IFIP International Conference on Theoretical Computer Science (TCS 2004). Exploring New Frontiers of Theoretical Informatics, pp. 507–520. Kluwer Academic Publishers, Dordrecht (2004)Google Scholar
  15. 15.
    Kanellakis, P.C., Smolka, S.A.: CCS expressions, finite state processes, and three problems of equivalence. Information and Computation 86(1), 43–68 (1990)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Kučera, A., Mayr, R.: On the complexity of semantic equivalences for pushdown automata and BPA. In: Diks, K., Rytter, W. (eds.) MFCS 2002. LNCS, vol. 2420, pp. 433–445. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  17. 17.
    Milner, R.: Communication and Concurrency. Prentice-Hall, Englewood Cliffs (1989)MATHGoogle Scholar
  18. 18.
    Ruohonen, K.: Reversible machines and post’s correspondence problem for biprefix morphisms. Elektronische Informationsverarbeitung und Kybernetik 21(12), 579–595 (1985)MathSciNetMATHGoogle Scholar
  19. 19.
    Sénizergues, G.: Decidability of bisimulation equivalence for equational graphs of finite out-degree. In: Proc. of the 39th Annual Symposium on Foundations of Computer Science(FOCS 1998), pp. 120–129. IEEE Computer Society, Los Alamitos (1998)Google Scholar
  20. 20.
    Sénizergues, G.: L(A)=L(B)? Decidability results from complete formal systems. Theoretical Computer Science 251(1–2), 1–166 (2001)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Senizergues, G.: The bisimulation problem for equational graphs of finite out-degree. SIAM Journal on Computing 34(5), 1025–1106 (2005)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Srba, J.: Strong bisimilarity and regularity of basic process algebra is PSPACE-hard. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 716–727. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  23. 23.
    Srba, J.: Undecidability of weak bisimilarity for pushdown processes. In: Brim, L., Jančar, P., Křetínský, M., Kucera, A. (eds.) CONCUR 2002. LNCS, vol. 2421, pp. 579–593. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  24. 24.
    Srba, J.: Completeness results for undecidable bisimilarity problems. In: Proc. of the 5th International Workshop on Verification of Infinite-State Systems (INFINITY 2003). ENTCS, vol. 98, pp. 5–19. Elsevier Science Publishers, Amsterdam (2004)Google Scholar
  25. 25.
    Srba, J.: Roadmap of Infinite results, vol. 2: Formal Models and Semantics. World Scientific Publishing Co, Updated version can be downloaded from the author’s home-page (2004)Google Scholar
  26. 26.
    Stirling, C.: Local model checking games. In: Lee, I., Smolka, S.A. (eds.) CONCUR 1995. LNCS, vol. 962, pp. 1–11. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  27. 27.
    Stirling, C.: Decidability of bisimulation equivalence for normed pushdown processes. In: Sassone, V., Montanari, U. (eds.) CONCUR 1996. LNCS, vol. 1119, pp. 217–232. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  28. 28.
    Stirling, C.: Decidability of bisimulation equivalence for normed pushdown processes. Theoretical Computer Science 195(2), 113–131 (1998)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Stirling, C.: Decidability of bisimulation equivalence for pushdown processes. Research Report EDI-INF-RR-0005, School of Informatics, Edinburgh University, The latest version is downloadable from the author’s home-page (January 2000)Google Scholar
  30. 30.
    Stirling, C.: Bisimulation and language equivalence. In: Logic for Concurrency and Synchronisation. Trends in Logic, vol. 18, pp. 269–284. Kluwer Academic Publishers, Dordrecht (2003)CrossRefGoogle Scholar
  31. 31.
    Thomas, W.: On the Ehrenfeucht-Fraïssé game in theoretical computer science (extended abstract). In: Gaudel, M.-C., Jouannaud, J.-P. (eds.) CAAP 1993, FASE 1993, and TAPSOFT 1993. LNCS, vol. 668, pp. 559–568. Springer, Heidelberg (1993)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Petr Jančar
    • 1
  • Jiří Srba
    • 2
  1. 1.Center of Applied Cybernetics, Department of Computer ScienceTechnical University of OstravaCzech Republic
  2. 2.BRICS Department of Computer ScienceAalborg UniversityDenmark

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