Continuous Capacities on Continuous State Spaces

We propose axiomatizing some stochastic games, in a continuous state space setting, using continuous belief functions, resp. plausibilities, instead of measures. Then, stochastic games are just variations on continuous Markov chains. We argue that drawing at random along a belief function is the same as letting the probabilistic player P play first, then letting the non-deterministic player C play demonically. The same holds for an angelic C, using plausibilities instead. We then define a simple modal logic, and characterize simulation in terms of formulae of this logic. Finally, we show that (discounted) payoffs are defined and unique, where in the demonic case, P maximizes payoff, while C minimizes it.

[1]  K. Hofmann,et al.  A Compendium of Continuous Lattices , 1980 .

[2]  Samson Abramsky,et al.  Domain theory , 1995, LICS 1995.

[3]  Joël Ouaknine,et al.  Domain theory, testing and simulation for labelled Markov processes , 2005, Theor. Comput. Sci..

[4]  Joël Ouaknine,et al.  Axioms for Probability and Nondeterminism , 2004, EXPRESS.

[5]  Kousha Etessami,et al.  Optimizing Büchi Automata , 2000, CONCUR.

[6]  Nancy A. Lynch,et al.  Probabilistic Simulations for Probabilistic Processes , 1994, Nord. J. Comput..

[7]  Glenn Shafer,et al.  A Mathematical Theory of Evidence , 2020, A Mathematical Theory of Evidence.

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

[9]  Philipp Sünderhauf.  Spaces of valuations as quasimetric domains , 1998, Electron. Notes Theor. Comput. Sci..

[10]  Itzhak Gilboa,et al.  Additive representations of non-additive measures and the choquet integral , 1994, Ann. Oper. Res..

[11]  Eugene A. Feinberg,et al.  Handbook of Markov Decision Processes , 2002 .

[12]  Alan Bundy,et al.  Constructing Induction Rules for Deductive Synthesis Proofs , 2006, CLASE.

[13]  Radha Jagadeesan,et al.  Approximating labelled Markov processes , 2003, Inf. Comput..

[14]  Abbas Edalat,et al.  Bisimulation for Labelled Markov Processes , 2002, Inf. Comput..

[15]  Ariel Rubinstein,et al.  A Course in Game Theory , 1995 .

[16]  Bengt Jonsson,et al.  A calculus for communicating systems with time and probabilities , 1990, [1990] Proceedings 11th Real-Time Systems Symposium.

[17]  Arthur P. Dempster,et al.  A Generalization of Bayesian Inference , 1968, Classic Works of the Dempster-Shafer Theory of Belief Functions.

[18]  David Schmeidleis.  SUBJECTIVE PROBABILITY AND EXPECTED UTILITY WITHOUT ADDITIVITY , 1989 .

[19]  Achim Jung,et al.  Stably Compact Spaces and the Probabilistic Powerspace construction , 2004, DTMPP.

[20]  Arthur P. Dempster,et al.  Upper and Lower Probabilities Induced by a Multivalued Mapping , 1967, Classic Works of the Dempster-Shafer Theory of Belief Functions.

[21]  Marta Z. Kwiatkowska,et al.  Stochastic Transition Systems for Continuous State Spaces and Non-determinism , 2005, FoSSaCS.

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

[23]  G. Choquet.  Theory of capacities , 1954 .

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

[25]  Ian Stark,et al.  Free-Algebra Models for the pi-Calculus , 2005, FoSSaCS.

[26]  Radha Jagadeesan,et al.  Approximating labeled Markov processes , 2000, Proceedings Fifteenth Annual IEEE Symposium on Logic in Computer Science (Cat. No.99CB36332).

[27]  A. Shwartz,et al.  Handbook of Markov decision processes : methods and applications , 2002 .

[28]  Michael W. Mislove,et al.  Topology, domain theory and theoretical computer science , 1998 .

[29]  Abbas Edalat,et al.  Domain theory and integration , 1994, Proceedings Ninth Annual IEEE Symposium on Logic in Computer Science.

[30]  Vincent Danos,et al.  Bisimulation and cocongruence for probabilistic systems , 2006, Inf. Comput..

[31]  Roberto Segala,et al.  Comparative analysis of bisimulation relations on alternating and non-alternating probabilistic models , 2005, Second International Conference on the Quantitative Evaluation of Systems (QEST'05).

[32]  R. Koenker,et al.  Pessimistic Portfolio Allocation and Choquet Expected Utility , 2004 .

[33]  Ronald Fagin,et al.  Two Views of Belief: Belief as Generalized Probability and Belief as Evidence , 1992, Artif. Intell..

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

[35]  Insup Lee,et al.  Weak Bisimulation for Probabilistic Systems , 2000, CONCUR.