Abstract
We give an axiomatic category theoretic account of bisimulation in process algebras based on the idea of functional bisimulations as open maps. We work with 2-monads, T, on Cat. Operations on processes, such as nondeterministic sum, prefixing and parallel composition are modelled using functors in the Kleisli category for the 2-monad T. We may define the notion of open map for any such 2-monad; in examples of interest, that agrees exactly with the usual notion of functional bisimulation. Under a condition on T, namely that it be a dense KZ-monad, which we define, it follows that functors in Kl(T) preserve open maps, i.e., they respect functional bisimulation. We further investigate structures on Kl(T) that exist for axiomatic reasons, primarily because T is a dense KZ-monad, and we study how those structures help to model operations on processes. We outline how this analysis gives ideas for modelling higher order processes. We conclude by making comparison with the use of presheaves and profunctors to model process calculi.
This work is supported by EPSRC grant GR/J84205: Frameworks for programming language semantics and logic.
Basic Research in Computer Science, Centre of the Danish National Research Foundation.
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References
R. Blackwell, G.M. Kelly, and A.J. Power. Two-dimensional monad theory. Journal of Pure and Applied Algebra, 59:1–41, 1989.
M. Barr, C. Wells. Toposes, Triples and Theories. Springer-Verlag, 1985.
F. Borceux. Handbook of Categorical Algebra, vol. 1. CUP, 1994.
G. L. Cattani, M. Fiore, and G. Winskel. A Theory of Recursive Domains with Applications to Concurrency. To appear in Proceedings of LICS '98.
G. L. Cattani, I. Stark, and G. Winskel. Presheaf Models for the π-Calculus. In Proceedings of CTCS '97, LNCS 1290, pages 106–126, 1997.
G. L. Cattani and G. Winskel. Presheaf Models for Concurrency. In Proceedings of CSL' 96, LNCS 1258, pages 58–75, 1997.
G. L. Cattani and G. Winskel. On bisimulation for higher order processes. Manuscript, 1998.
B. Jacobs and J. Rutten. A tutorial on (Co)algebras and (Co)induction. EACTS Bulletin 62 (1997) 222–259.
A. Joyal and I. Moerdijk. A completeness theorem for open maps. Annals of Pure and Applied Logic, 70:51–86, 1994.
A. Joyal, M. Nielsen, and G. Winskel. Bisimulation from open maps. Information and Computation, 127:164–185, 1996.
G.M. Kelly. Basic concepts of enriched category theory. London Math. Soc. Lecture Note Series 64, CUP, 1982.
G.M. Kelly and R. Street. Review of the elements of 2-categories. In Proceedings of Sydney Category Theory Seminar 1972/73, LNM 420, pages. 75–103, Springer-Verlag, 1974.
A. Kock. Monads for which structures are adjoint to units. Journal of Pure and Applied Algebra, 104:41–59, 1995
A. Kock. Closed categories generated by commutative monads. Journal of the Australian Mathematical Society, 12:405–424, 1971.
S. Mac Lane. Categories for the working mathematician. Springer-Verlag, 1971.
R. Milner. Communication and concurrency. Prentice Hall, 1989.
A. J. Power, G. L. Cattani and G. Winskel. A representation result for free cocompletions. Submitted for publication.
D. Sangiorgi. Bisimulation for higher-order process calculi. Information and Computation, 131(2):141–178, 1996.
D. S. Scott. Continuous lattices. In F.W. Lawvere, editor, Toposes, Algebraic Geometry and Logic, LNM 274, pages 97–136. Springer-Verlag, 1972.
M.B. Smyth and G. D. Plotkin. The category-theoretic solution of recursive domain equations. SIAM Journal of Computing, 11(4):761–783, 1982.
R. Street. Fibrations in Bicategories. Cahiers de Topologie et Géométrie Différentielle, XXI(2):111–160, 1980.
G. Winskel. A presheaf semantics of value-passing processes. In Proceedings of CONCUR'96, LNCS 1119, pages 98–114, 1996.
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Cattani, G.L., Power, J., Winskel, G. (1998). A categorical axiomatics for bisimulation. In: Sangiorgi, D., de Simone, R. (eds) CONCUR'98 Concurrency Theory. CONCUR 1998. Lecture Notes in Computer Science, vol 1466. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0055649
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DOI: https://doi.org/10.1007/BFb0055649
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