The Behavior of Probabilistic Systems: From Equivalences to Behavioral Distances

In this paper we synthesize our recent work on behavioral distances for probabilistic systems and present an overview of the current state of the art in the field. We mainly focus on behavioral distances for Markov chains, Markov decision processes, and Segala systems. We illustrate three different methods used for the definition of such metrics: logical, order theoretic, and measure-testing; and we discuss the relationships between them and provide the main arguments in support of each of them. We also overview the problem of computing such distances, both from a theoretical and a practical view point, including the exact and the approximated methods.

[1]  Luca Cardelli,et al.  BioAmbients: an abstraction for biological compartments , 2004, Theor. Comput. Sci..

[2]  James Worrell,et al.  On the Complexity of Computing Probabilistic Bisimilarity , 2012, FoSSaCS.

[3]  Luca Cardelli,et al.  The Measurable Space of Stochastic Processes , 2014, Fundam. Informaticae.

[4]  Doina Precup,et al.  Bisimulation for Markov Decision Processes through Families of Functional Expressions , 2014, Horizons of the Mind.

[5]  James Worrell,et al.  The Complexity of Computing a Bisimilarity Pseudometric on Probabilistic Automata , 2014, Horizons of the Mind.

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

[7]  Hongfei Fu,et al.  Computing Game Metrics on Markov Decision Processes , 2012, ICALP.

[8]  Doina Precup,et al.  Metrics for Finite Markov Decision Processes , 2004, AAAI.

[9]  Kim G. Larsen,et al.  On-the-Fly Exact Computation of Bisimilarity Distances , 2013, TACAS.

[10]  Radha Jagadeesan,et al.  Metrics for labelled Markov processes , 2004, Theor. Comput. Sci..

[11]  O. H. Brownlee,et al.  ACTIVITY ANALYSIS OF PRODUCTION AND ALLOCATION , 1952 .

[12]  Thomas A. Henzinger,et al.  Real-Time Logics: Complexity and Expressiveness , 1993, Inf. Comput..

[13]  Rupak Majumdar,et al.  Game Relations and Metrics , 2007, 22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007).

[14]  Luca Cardelli.  Two-domain DNA strand displacement , 2013, Math. Struct. Comput. Sci..

[15]  D. R. Fulkerson,et al.  Solving a Transportation Problem , 1956 .

[16]  Kim G. Larsen,et al.  Continuous Markovian Logics - Axiomatization and Quantified Metatheory , 2012, Log. Methods Comput. Sci..

[17]  Kousha Etessami,et al.  On the Complexity of Nash Equilibria and Other Fixed Points , 2010, SIAM J. Comput..

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

[19]  Luca Cardelli,et al.  On process rate semantics , 2008, Theor. Comput. Sci..

[20]  Luca Cardelli,et al.  Stochastic Pi-calculus Revisited , 2013, ICTAC.

[21]  Christos H. Papadimitriou,et al.  On the Complexity of the Parity Argument and Other Inefficient Proofs of Existence , 1994, J. Comput. Syst. Sci..

[22]  Kim G. Larsen,et al.  Computing Behavioral Distances, Compositionally , 2013, MFCS.

[23]  Kim G. Larsen,et al.  Topologies of Stochastic Markov Models: Computational Aspects , 2014, ArXiv.

[24]  James Worrell,et al.  Towards Quantitative Verification of Probabilistic Transition Systems , 2001, ICALP.

[25]  Luca Cardelli,et al.  The Cell Cycle Switch Computes Approximate Majority , 2012, Scientific Reports.

[26]  Matteo Mio,et al.  Upper-Expectation Bisimilarity and Real-valued Modal Logics , 2013, ArXiv.

[27]  Christian N. S. Pedersen,et al.  The consensus string problem and the complexity of comparing hidden Markov models , 2002, J. Comput. Syst. Sci..

[28]  David K. Smith.  Theory of Linear and Integer Programming , 1987 .

[29]  Luca Cardelli,et al.  Brane Calculi , 2004, CMSB.

[30]  Joost-Pieter Katoen,et al.  Quantitative Model Checking of Continuous-Time Markov Chains Against Timed Automata Specifications , 2009, 2009 24th Annual IEEE Symposium on Logic In Computer Science.

[31]  Joost-Pieter Katoen,et al.  Quantitative Model Checking of Continuous-Time Markov Chains Against Timed Automata Specifications , 2009, 2009 24th Annual IEEE Symposium on Logic In Computer Science.

[32]  Dexter Kozen,et al.  A probabilistic PDL , 1983, J. Comput. Syst. Sci..