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Topologies, Continuity and Bisimulations

Published online by Cambridge University Press:  15 August 2002

J. M. Davoren
J. M. Davoren*
Affiliation:
Center for Foundations of Intelligent Systems, 626 Rhodes Hall, Cornell University, Ithaca, NY 14853, U.S.A.; e-mail: davoren@hybrid.cornell.eduJune – Dec. 1999: Computer Sciences Laboratory, Research School of Information Sciences and Engineering, Australian National University, Canberra ACT 0200, Australia; e-mail: davoren@arp.anu.edu.au
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Abstract

The notion of a bisimulation relation is ofbasic importance in many areas of computation theory and logic. Of late, it has come to take a particular significance in work on the formal analysis and verification of hybrid control systems, where system properties are expressible by formulas of the modal μ-calculus or weaker temporal logics. Our purpose here is to give an analysis of the concept of bisimulation, starting with the observation that the zig-zag conditions are suggestive of some form of continuity. We give a topological characterization of bisimularity for preorders, and then use the topology as a route to examining the algebraic semantics for the µ-calculus, developed in recent work of Kwiatkowska et al., and its relation to the standard set-theoretic semantics. In our setting, μ-calculus sentencesevaluate as clopen sets of an Alexandroff topology, rather than as clopensof a (compact, Hausdorff) Stone topology, as arises in the Stone space representationof Boolean algebras (with operators). The paper concludes by applying the topological characterization to obtain the decidability of μ-calculus properties for a class of first-order definable hybrid dynamicalsystems, slightly extending and considerably simplifying the proof of a recent result of Lafferriere et al.

Keywords

Modal logicμ-calculusbisimulationset-valued mapssemi-continuityAlexandroff topologyhybrid systemsdecidability.

Information

Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 33 , Issue 4-5 , July 1999, pp. 357 - 381
Copyright
© EDP Sciences, 1999

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References

Alur, R., Courcoubetis, C., Halbwachs, N., Henzinger, T., Ho, P.-H., Nicollin, X., Olivero, A., Sifakis, J. and Yovine, S., The algorithmic analysis of hybrid systems. Theoret. Comput. Sci. 138 (1995) 3-34. CrossRef
Ambler, S., Kwiatkowska, M.Z. and Measor, N., Duality and the completeness of the modal µ-calculus. Theoret. Comput. Sci. 151 (1995) 3-27. CrossRef
J.-P. Aubin and H. Frankowska, Set-Valued Analysis. Birkhäuser, Boston (1990).
M. Bonsangue and M. Kwiatkowska, Reinterpreting the modal µ-calculus, A. Ponse, M. de Rijke and Y. Venema, Eds., Modal Logic and Process Algebra. CLSI Publications, Stanford (1995) 65-83.
J. Davoren, Modal Logics for Continuous Dynamics. Ph.D. Thesis, Department of Mathematics Cornell University (1998).
Davoren, J.M., On hybrid systems and the modal µ-calculus, P. Antsaklis, W. Kohn, M. Lemmon, A. Nerode and S. Sastry, Eds., Hybrid Systems V. Springer-Verlag, Berlin, Lecture Notes in Comput. Sci. 1567 (1999) 38-69. CrossRef
Daws, C., Olivero, A., Tripakis, S. and Yovine, S., The tool KRONOS, R. Alur, T. Henzinger and E.D. Sontag, Eds., Hybrid Systems III. Springer-Verlag, Berlin, Lecture Notes in Comput. Sci. 1066 (1996) 208-219. CrossRef
T. Henzinger, The theory of hybrid automata, in Proc. of 11 th Annual IEEE Symposium on Logic in Computer Science (LICS'96). IEEE Computer Society Press (1996) 278-292.
T. Henzinger, P. Kopke, A. Puri and P. Varaiya, What's decidable about hybrid automata? J. Comput. System Sci. 57 (1998) 94-124.
M. Hollenberg, Logic and Bisimulation. Ph.D. Thesis, Department of Philosophy, Utrecht University (1998).
Jónsson, B. and Tarski, A., Boolean algebras with operators, part i. Amer. J. Math. 73 (1951) 891-939. CrossRef
Kozen, D., Results on the propositional µ-calculus. Theoret. Comput. Sci. 27 (1983) 333-354. CrossRef
G. Lafferriere, G. Pappas and S. Sastry, O-minimal hybrid systems. Technical Report UCB/ERL M98/29, Dept. EECS, UC Berkeley (1998).
G. Lafferriere, G. Pappas and S. Yovine, Decidable hybrid systems. Technical Report UCB/ERL M98/39, Dept. EECS, UC Berkeley (1998).
Nerode, A. and Kohn, W., Models for hybrid systems: Automata, topologies, controllability, observability, R. Grossman, A. Nerode, A. Ravn and H. Rischel, Eds., Hybrid Systems. Springer-Verlag, Berlin, Lecture Notes in Comput. Sci. 736 (1993) 297-316. CrossRef
M.B. Smyth, Topology, S. Abramsky, D. Gabbay and T. Maibaum, Eds. Oxford University Press, Clarendon Press, Oxford, Handb. Log. Comput. Sci. 1 (1992) 641-761.
C. Stirling, Modal and temporal logics, S. Abramsky, D. Gabbay and T. Maibaum, Eds. Oxford University Press, Clarendon Press, Oxford, Handb. Log. Comput. Sci. 2 (1992) 477-563.
L. van den Dries, Tame Topology and O-minimal Structures. Cambridge Univ. Press, Cambridge, London Math. Soc. Lecture Note Ser. 248 (1998).
Walukiewicz, I., A note on the completeness of Kozen's axiomatization of the propositonal µ-calculus. Bull. Symbolic Logic 2 (1996) 349-366. CrossRef