Behavioural equivalence for infinite systems — Partially decidable!
For finite-state systems non-interleaving equivalences are computationally at least as hard as interleaving equivalences. In this paper we show that when moving to infinite-state systems, this situation may change dramatically.
We compare standard language equivalence for process description languages with two generalizations based on traditional approaches capturing non-interleaving behaviour, pomsets representing global causal dependency, and locality representing spatial distribution of events.
We first study equivalences on Basic Parallel Processes, BPP, a process calculus equivalent to communication free Petri nets. For this simple process language our two notions of non-interleaving equivalences agree. More interestingly, we show that they are decidable, contrasting a result of Hirshfeld that standard interleaving language equivalence is undecidable. Our result is inspired by a recent result of Esparza and Kiehn, showing the same phenomenon in the setting of model checking.
We follow up investigating to which extent the result extends to larger subsets of CCS and TCSP. We discover a significant difference between our non-interleaving equivalences. We show that for a certain non-trivial subclass of processes between BPP and TCSP, not only are the two equivalences different, but one (locality) is decidable whereas the other (pomsets) is not. The decidability result for locality is proved by a reduction to the reachability problem for Petri nets.
KeywordsProcess Calculi Petri Nets Behavioural Equivalence Partial Order Methods Decidability
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- 1.S. Abramsky, Eliminating Local Non-determinism: a New Semantics for CCS, Computer Systems Laboratory, Queen Mary College, Report no. 290 (1981).Google Scholar
- 2.L. Aceto, A Static View of Localities, Formal Aspects of Computing, 6 (2), 202–222 (1994).Google Scholar
- 3.G. Boudol, I. Castellani, M. Hennessy and A. Kiehn, Observing Localities, Theoretical Computer Science, 114, 31–61, 1993.Google Scholar
- 4.S. Christensen. Decidability and Decomposition in Process Algebras Ph.D. Thesis, University of Edinburgh, CST-105-93, 1993.Google Scholar
- 5.S. Christensen, H. Hüttel, Decidability Issues for Infinite-State Processes — a Survey, EATCS Bulletin 51, 156–166 (1993).Google Scholar
- 6.S. Christensen, Y. Hirshfeld and F. Moller, Bisimulation equivalence is decidable for basic parallel processes, CONCUR '93, Springer LNCS 715, 143–157 (1993).Google Scholar
- 7.J. Engelfriet, Tree automata and tree grammars, University of Aarhus, DAIMI FN-10 (1975).Google Scholar
- 8.J. Esparza, Petri nets, commutative context-free grammars, and Basic Parallel Processes, in Proceedings of Fundamentals of Computation Theory, (FCT'95), LNCS 965, Springer Verlag 1995.Google Scholar
- 9.J. Esparza and A. Kiehn, On the Model Checking Problem for Branching Logics and Basic Parallel Processes, CAV '95, Springer LNCS 939, 353–366 (1995).Google Scholar
- 10.Y. Hirshfeld, Petri Nets and the Equivalence Problem, CSL '93, Springer LNCS 882, 165–174 (1994).Google Scholar
- 11.C. A. R. Hoare, Communicating Sequential Processes, Prentice Hall, International Series in Computer Science (1985).Google Scholar
- 12.H. Hütte, Undecidable Equivalences for Basic Parallel Processes, TACS '94, Springer LNCS 789, 454–464 (1994).Google Scholar
- 13.P. Jancar and F. Moller, Checking Regular Properties of Petri Nets, CONCUR '95, Springer LNCS 962, 348–362 (1995).Google Scholar
- 14.L. Jategaonkar and A. Meyer, Deciding true concurrency equivalences on finite safe nets. ICALP '93, Springer LNCS 700, 519–531 (1993).Google Scholar
- 15.A. Kiehn and M. Hennessy, On the Decidability on Non-interleaving Equivalences, CONCUR '94, Springer LNCS 836, 18–33 (1994)Google Scholar
- 16.S.R. Kosaraju. Decidability of Reachability in Vector Addition Systems. 14th Annual ACM Symposium on Theory of Computing, San Francisco, 267–281 (1982).Google Scholar
- 17.E.W. Mayr, Persistence of Vector Replacement Systems is Decidable. Acta Informatica 15, 309–318 (1981).Google Scholar
- 18.E.W. Mayr and A.R. Meyer, The Complexity of the Finite Containment Problem for Petri Nets, Journal of the ACM, 28(3), 561–576 (1981).Google Scholar
- 19.A. Mazurkiewicz, Basic notions of trace theory, in de Bakker, de Roever and Rozenberg (eds.), Linear Time, Branching Time and Partial Orders in Logics and Models for Concurrency, Springer LNCS 354, 285–363 (1988).Google Scholar
- 20.A.R.G. Milner, Communication and concurrency, Prentice Hall (1989).Google Scholar
- 21.M.L. Minsky, Computation — Finite and Infinite Machines, Prentice Hall (1967).Google Scholar
- 22.M. Mukund and M. Nielsen, CCS, Locations and Asynchronous Transition Systems, FST & TCS '92, Springer LNCS 652, 328–341 (1992).Google Scholar
- 23.E.R. Olderog and C.A.R. Hoare, Specification-Oriented Semantics for Communicating Processes, Acta Informatica, 23, 9–66 (1986).Google Scholar
- 24.V.R. Pratt, Modeling concurrency with partial orders, International Journal of Parallel Programming, 15(1), 33–71 (1986).Google Scholar
- 25.W. Reisig, A note on the representation of finite tree automata. Information Processing Letters, 8(5):239–240, June 1979Google Scholar
- 26.W. Reisig, Petri Nets — an Introduction, EATCS Monograph in Computer Science, Springer (1985).Google Scholar
- 27.K. Sunesen and M. Nielsen, Behavioural equivalence for infinite systems — partially decidable!, Technical Report RS-95-55, BRICS, Aarhus University (1995).Google Scholar
- 28.D. Taubner, Finite Representations of CCS and TCSP Programs by Automata and Petri Nets, Springer LNCS 369 (1989).Google Scholar
- 29.W. Thomas, Automata on Infinite Objects, in Handbook of Theoretical Computer Science, vol B, ed. J. van Leeuwen, Elsevier, 133–192 (1990).Google Scholar
- 30.R.J. van Glabbeek, Comparative concurrency semantics and refinement of actions, PhD thesis, CWI Amsterdam (1990).Google Scholar
- 31.R.J. van Glabbeek and U. Goltz, Equivalence Notions for Concurrent Systems and Refinement of Actions, MFCS '89, Springer LNCS 379, 237–248 (1989).Google Scholar
- 32.P. Wolper and P. Godefroid, Partial Order Methods for Temporal Verification, Concur '93, Springer LNCS 715, 233–246 (1993).Google Scholar