Bisimulation for Probabilistic Transition Systems: A Coalgebraic Approach

The notion of bisimulation as proposed by Larsen and Skou for discrete probabilistic transition systems is shown to coincide with a coalgebraic definition in the sense of Aczel and Mendier in terms of a set functor. This coalgebraic formulation makes it possible to generalize the concepts to a continuous setting involving Borel probability measures. Under reasonable conditions, generalized probabilistic bisimilarity can be characterized categorically. Application of the final coalgebra paradigm then yields an internally fully abstract semantical domain with respect to probabilistic bisimulation.

[1]  Bart Jacobs,et al.  Objects and Classes, Co-Algebraically , 1995, Object Orientation with Parallelism and Persistence.

[2]  Christel Baier,et al.  Domain equations for probabilistic processes , 2000, Mathematical Structures in Computer Science.

[3]  Abbas Edalat.  Domain Theory and Integration , 1995, Theor. Comput. Sci..

[4]  R. Blute,et al.  Bisimulation for Labeled Markov Processes , 1997 .

[5]  Jan J. M. M. Rutten,et al.  Initial Algebra and Final Coalgebra Semantics for Concurrency , 1993, REX School/Symposium.

[6]  Martin L. Puterman,et al.  Markov Decision Processes: Discrete Stochastic Dynamic Programming , 1994 .

[7]  A. R. D. Mathias,et al.  NON‐WELL‐FOUNDED SETS (CSLI Lecture Notes 14) , 1991 .

[8]  Abbas Edalat,et al.  Dynamical Systems, Measures and Fractals via Domain Theory , 1993, Inf. Comput..

[9]  E. Tronci,et al.  1996 , 1997, Affair of the Heart.

[10]  Kim G. Larsen,et al.  Bisimulation through Probabilistic Testing , 1991, Inf. Comput..

[11]  W. Rudin.  Real and complex analysis , 1968 .

[12]  Lawrence S. Moss,et al.  Coalgebraic Logic , 1999, Ann. Pure Appl. Log..

[13]  Thomas A. Henzinger,et al.  Hybrid Automata with Finite Bisimulatioins , 1995, ICALP.

[14]  Claire Jones,et al.  Probabilistic non-determinism , 1990 .

[15]  Horst Reichel,et al.  An approach to object semantics based on terminal co-algebras , 1995, Mathematical Structures in Computer Science.

[16]  Peter Aczel,et al.  Non-well-founded sets , 1988, CSLI lecture notes series.

[17]  Kim Guldstrand Larsen,et al.  Specification and refinement of probabilistic processes , 1991, [1991] Proceedings Sixth Annual IEEE Symposium on Logic in Computer Science.

[18]  José Meseguer,et al.  Initiality, induction, and computability , 1986 .

[19]  Robert M. Keller,et al.  Formal verification of parallel programs , 1976, CACM.

[20]  Roberto Segala,et al.  Modeling and verification of randomized distributed real-time systems , 1996 .

[21]  Karen Seidel,et al.  Probabilistic Communicating Processes , 1992, Theor. Comput. Sci..

[22]  Roberto Gorrieri,et al.  Extended Markovian Process Algebra , 1996, CONCUR.

[23]  Erik P. de Vink,et al.  Bisimulation for Probabilistic Transition Systems: A Coalgebraic Approach , 1999, Theor. Comput. Sci..

[24]  Albert Benveniste,et al.  A Calculus of Stochastic Systems for the Specification, Simulation, and Hidden State Estimation of Mixed Stochastic/Nonstochastic Systems , 1994, Theor. Comput. Sci..

[25]  Pierre America,et al.  Solving Reflexive Domain Equations in a Category of Complete Metric Spaces , 1989, J. Comput. Syst. Sci..

[26]  Abbas Edalat,et al.  Bisimulation for labelled Markov processes , 1997, Proceedings of Twelfth Annual IEEE Symposium on Logic in Computer Science.

[27]  Abbas Edalat,et al.  A logical characterization of bisimulation for labeled Markov processes , 1998, Proceedings. Thirteenth Annual IEEE Symposium on Logic in Computer Science (Cat. No.98CB36226).

[28]  B. Jacobs,et al.  A tutorial on (co)algebras and (co)induction , 1997 .

[29]  Erik P. de Vink,et al.  Control flow semantics , 1996 .

[30]  Peter Aczel,et al.  A Final Coalgebra Theorem , 1989, Category Theory and Computer Science.

[31]  Marta Z. Kwiatkowska,et al.  Probabilistic Metric Semantics for a Simple Language with Recursion , 1996, MFCS.

[32]  Scott A. Smolka,et al.  Algebraic Reasoning for Probabilistic Concurrent Systems , 1990, Programming Concepts and Methods.

[33]  M. Bonsangue,et al.  Topological Dualities in Semantics , 1996 .

[34]  Frank Harary,et al.  Graph Theory , 2016 .

[35]  Michèle Giry,et al.  A categorical approach to probability theory , 1982 .

[36]  David Williams,et al.  Probability with Martingales , 1991, Cambridge mathematical textbooks.

[37]  Kenneth Kunen,et al.  Handbook of Set-Theoretic Topology , 1988 .

[38]  Glynn Winskel,et al.  Bisimulation from Open Maps , 1994, Inf. Comput..

[39]  Michael Barr,et al.  Terminal Coalgebras in Well-Founded Set Theory , 1993, Theor. Comput. Sci..

[40]  Abbas Edalat,et al.  Semi-pullbacks and bisimulation in categories of Markov processes , 1999, Mathematical Structures in Computer Science.

[41]  Radha Jagadeesan,et al.  Metrics for Labeled Markov Systems , 1999, CONCUR.

[42]  Bernhard Steffen,et al.  Reactive, Generative and Stratified Models of Probabilistic Processes , 1995, Inf. Comput..

[43]  C. Jones,et al.  A probabilistic powerdomain of evaluations , 1989, [1989] Proceedings. Fourth Annual Symposium on Logic in Computer Science.

[44]  Robin Milner,et al.  A Calculus of Communicating Systems , 1980, Lecture Notes in Computer Science.

[45]  Jan J. M. M. Rutten,et al.  On the Foundation of Final Semantics: Non-Standard Sets, Metric Spaces, Partial Orders , 1992, REX Workshop.

[46]  S. Shelah,et al.  Annals of Pure and Applied Logic , 1991 .

[47]  David Park,et al.  Concurrency and Automata on Infinite Sequences , 1981, Theoretical Computer Science.