Computable processes and bisimulation equivalence

Abstract

Abstract

A process is calledcomputable if it can be modelled by a transition system that has a recursive structure—implying finite branching. The equivalence relation between transition systems considered is strong bisimulation equivalence. The transition systems studied in this paper can be associated to processes specified in common specification languages such as CCS, LOTOS, ACP and PSF. As a means for defining transition systems up to bisimulation equivalence, the specification languageμCRL is used. Two simple fragments of,μCRL are singled out, yielding universal expressivity with respect to recursive and primitive recursive transition systems. For both these domains the following properties are classified in the arithmetical hierarchy:bisimilarity, perpetuity (both ∏₁⁰),regularity (having a bisimilar, finite representation, Σ₂⁰),acyclic regularity (Σ₁⁰), anddeadlock freedom (distinguishing deadlock from successful termination, ∏₁⁰). Finally, it is shown that in the domain of primitive recursive transition systems over a fixed, finite label set, a genuine hierarchy in bisimilarity can be defined by the complexity of the witnessing relations, which extends r.e. bisimilarity. Hence, primitive recursive transition systems already form an interesting class.

References

1. [AuB84] Austry D.Boudol G.Algèbre de processus et synchronisationsTheoretical Computer Science19843019113110.1016/0304-3975(84)90067-70533.68026748133
2. [BaB92] Baeten, J.C.M. and Bergstra, J.A.: Process algebra with signals and conditions. In M. Broy, editor,Programming and Mathematical Methods, Proceedings Summer School Marktoberdorf 1991, pages 273–323. Springer-Verlag, 1992. NATO ASI Series F88.
3. [BBK87] Baeten J.C.M.Bergstra J.A.Klop J.W.On the consistency of Koomen's fair abstraction ruleTheoretical Computer Science1987511/212917610.1016/0304-3975(87)90052-10621.68010908483
4. [Ber91] Bergstra, J.A.: 1991. Personal Communications.
5. [BeG94] Bezem, M.A. and Groote, J.F.: Invariants in process algebra with data. In [JoP94], pages 401–416, 1994.
6. [BoG96] Bosscher, D.J.B. and Griffioen, W.O.D.: Regularity for a class of context-free processes is decidable. In proceedings of ICALP'96, to appear.
7. [BeK84] Bergstra J.A.Klop J.W.Process algebra for synchronous communicationInformation and Computation1984601/31091370597.68027764282
8. [BeK85] Bergstra J.A.Klop J.W.Algebra of communicating processes with abstractionTheoretical Computer Science19853717712110.1016/0304-3975(85)90088-X0579.68016796314
9. [BaV95] Baeten, J.C.M. and Verhoef, C.: Concrete process algebra. In S. Abramsky, D.M. Gabbay, and T.S.E. Maibaum, editors,Handbook of Logic in Computer Science, Volume IV, Syntactical Methods, pages 149–268. Oxford University Press, 1995.
10. [BaW90] Baeten, J.C.M. and Weijland, W.P.:Process Algebra. Cambridge Tracts in Theoretical Computer Science 18. Cambridge University Press, 1990.
11. [CCI87] CCITT Working Party X/1.Recommendation Z.100 (SDL), 1987.
12. [Dar90] Darondeau Ph.Guessarian I.Concurrency and computabilitySemantics of Systems of Concurrent Processes1990La Roche Posay, FranceSpringer London22323810.1007/3-540-53479-2_9
13. [Dar91] Darondeau Ph.Recursive graphs are not stable under maximal reductionBulletin of the European Association for Theoretical Computer Science1991441861890744.68106
14. [Dav82] Davis, M.:Computability and Unsolvability. Dover Publications, Inc., 1982.
15. [Gla95] van Glabbeek, R.J.: On the expressiveness of ACP (extended abstract). In [PVV95], pages 188–217, 1995.
16. [GrP90] Groote J.F.Ponse A.The syntax and semantics ofμCRLReport CS-R90761990AmsterdamCWI
17. [GrP91a] Groote, J.F. and Ponse, A.:μCRL: A base for analysing processes with data. In E. Best and G. Rozenberg, editors,Proceedings 3^rdWorkshop on Concurrency and Compositionality, Goslar, GMD-Studien Nr. 191, pages 125–130. Universität Hildesheim, 1991.
18. [GrP91b] Groote, J.F. and Ponse, A.: Proof theory forμCRL. Report CS-R9138, CWI, 1991.
19. [GrP93] Groote, J.F. and Ponse, A.: Proof theory forμCRL: a language for processes with data. In D.J. Andrews, J.F. Groote, and C.A. Middelburg, editors,Proceedings of the International Workshop on Semantics of Specification Languages, pages 232–251. Workshops in Computing, Springer-Verlag, 1994.
20. [GrP95] Groote, J.F. and Ponse, A.: The syntax and semantics ofμCRL. In [PVV95], pages 26–62, 1995. (Appeared earlier as [GrP90].)
21. [GrV92] Groote J.F.Vaandrager F.W.Structured operational semantics and bisimulation as a congruenceInformation and Computation1992100220226010.1016/0890-5401(92)90013-60752.680531181993
22. [HHJ87] Hoare C.A.R.Hayes I.J.Jifeng H.E.Morgan C.C.Roscoe A.W.Sanders J.W.Sorensen I.H.Spivey J.M.Sufrin B.A.Laws of programmingCommunications of the ACM198730867268610.1145/27651.276530629.68006
23. [HU79] Hopcroft, J.E. and Ullman, J.D.:Introduction to Automata Theory, Languages and Computation. Addison-Wesley, 1979.
24. [ISO87] ISO.Information processing systems — open systems interconnection — LOTOS — a formal description technique based on the temporal ordering of observational behaviour ISO/TC97/SC21/N DIS8807, 1987.
25. [JoP94] Jonsson B.Parrow J.Proceedings CONCUR 941994Uppsala, SwedenSpringer London0825.68132
26. [Kle52] Kleene, S.C.:Introduction to Meta Mathematics. North-Holland, 1952.
27. [Mil83] Milner R.Calculi for synchrony and asynchronyTheoretical Computer Science19832526731010.1016/0304-3975(83)90114-70512.68026716132
28. [Mil89] Milner R.Communication and Concurrency1989Englewood CliffsPrentice-Hall International0683.68008
29. [MaM94] Mauw, S. and Mulder, H.: Regularity of BPA-systems is decidable. In [JoP94], pages 34–47, 1994.
30. [MaV90] Mauw S.Veltink G.J.A process specification formalismFundamenta Informaticae1990XIII85139
31. [MaV93] Mauw, S. and Veltink, G.J.: editors,Algebraic Specification of Communication Protocols. Cambridge Tracts in Theoretical Computer Science 36. Cambridge University Press, 1993.
32. [Par81] Park, D.M.R.: Concurrency and automata on infinite sequences. In P. Deussen, editor, 5^thGI Conference, LNCS 104, pages 167–183. Springer-Verlag, 1981.
33. [Plo81] Plotkin, G.D.: A structural approach to operational semantics. Report DAIMI FN-19, Computer Science Department, Aarhus University, 1981.
34. [PVV95] Ponse, A., Verhoef, C. and van Vlijmen, S.F.M.: editors,Algebra of Communicating Processes, Utrecht 1994. Workshops in Computing, Springer-Verlag, 1995.
35. [Rog67] Rogers, H.:Theory of Recursive Functions and Effective Computability. McGraw-Hill Book Co., 1967.
36. [Sim85] de Simone R.Higher-level synchronising devices inMeije-SCCSTheoretical Computer Science19853724526710.1016/0304-3975(85)90093-30598.68027824475
37. [Vaa93] Vaandrager F.W.de Bakker J.W.de Roever W.P.Rozenberg G.Expressiveness results for process algebras1993Beekbergen, The NetherlandsSpringer London609638

Index Terms

• Mathematics of computing

  • Probability and statistics

    • Probabilistic representations

• Software and its engineering

• Theory of computation

  • Formal languages and automata theory

    • Formalisms

      • Algebraic language theory

  • Models of computation

  • Semantics and reasoning

    • Program semantics

      • Algebraic semantics