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.
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References
Balcazar, J., Gabarro, J., Santha, M.: Deciding bisimilarity is P-complete. Formal Aspects of Computing 4(6A), 638–648 (1992)
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)
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)
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)
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)
Cachat, T.: Uniform solution of parity games on prefix-recognizable graphs. Electronic Notes in Theoretical Computer Science 68(6) (2002)
Caucal, D.: On the regular structure of prefix rewriting. Theoretical Computer Science 106(1), 61–86 (1992)
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)
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)
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)
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)
Hennessy, M., Milner, R.: Algebraic laws for nondeterminism and concurrency. Journal of the Association for Computing Machinery 32(1), 137–161 (1985)
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)
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)
Kanellakis, P.C., Smolka, S.A.: CCS expressions, finite state processes, and three problems of equivalence. Information and Computation 86(1), 43–68 (1990)
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)
Milner, R.: Communication and Concurrency. Prentice-Hall, Englewood Cliffs (1989)
Ruohonen, K.: Reversible machines and post’s correspondence problem for biprefix morphisms. Elektronische Informationsverarbeitung und Kybernetik 21(12), 579–595 (1985)
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)
Sénizergues, G.: L(A)=L(B)? Decidability results from complete formal systems. Theoretical Computer Science 251(1–2), 1–166 (2001)
Senizergues, G.: The bisimulation problem for equational graphs of finite out-degree. SIAM Journal on Computing 34(5), 1025–1106 (2005)
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)
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)
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)
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)
Stirling, C.: Local model checking games. In: Lee, I., Smolka, S.A. (eds.) CONCUR 1995. LNCS, vol. 962, pp. 1–11. Springer, Heidelberg (1995)
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)
Stirling, C.: Decidability of bisimulation equivalence for normed pushdown processes. Theoretical Computer Science 195(2), 113–131 (1998)
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)
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)
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)
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Jančar, P., Srba, J. (2006). Undecidability Results for Bisimilarity on Prefix Rewrite Systems. In: Aceto, L., Ingólfsdóttir, A. (eds) Foundations of Software Science and Computation Structures. FoSSaCS 2006. Lecture Notes in Computer Science, vol 3921. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11690634_19
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