Complete Axiomatization for the Total Variation Distance of Markov Chains

Abstract We propose a complete axiomatization for the total variation distance of finite labelled Markov chains. Our axiomatization is given in the form of a quantitative deduction system, a framework recently proposed by Mardare, Panangaden, and Plotkin (LICS 2016) to extend classical equational deduction systems by means of inferences of equality relations t ≡ e s indexed by rationals, expressing that “t is approximately equal to s up to an error e”. Notably, the quantitative equational system is obtained by extending our previous axiomatization (CONCUR 2016) for the probabilistic bisimilarity distance with a distributivity axiom for the prefix operator over the probabilistic choice inspired by Rabinovich's (MFPS 1983). Finally, we propose a metric extension to the Kleene-style representation theorem for finite labelled Markov chains w.r.t. trace equivalence due to Silva and Sokolova (MFPS 2011).

[1]  S C Kleene,et al.  Representation of Events in Nerve Nets and Finite Automata , 1951 .

[2]  Prakash Panangaden,et al.  Quantitative Algebraic Reasoning , 2016, 2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

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

[4]  Yuxin Deng,et al.  Axiomatizations for probabilistic finite-state behaviors , 2007, Theor. Comput. Sci..

[5]  Robin Milner,et al.  A Complete Inference System for a Class of Regular Behaviours , 1984, J. Comput. Syst. Sci..

[6]  Alexandra Silva,et al.  Sound and Complete Axiomatizations of Coalgebraic Language Equivalence , 2011, TOCL.

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

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

[9]  James Worrell,et al.  Approximating and computing behavioural distances in probabilistic transition systems , 2006, Theor. Comput. Sci..

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

[11]  Luca Aceto,et al.  Equational Axioms for Probabilistic Bisimilarity , 2002, AMAST.

[12]  Alexandra Silva,et al.  Deriving Syntax and Axioms for Quantitative Regular Behaviours , 2009, CONCUR.

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

[14]  Arto Salomaa,et al.  Two Complete Axiom Systems for the Algebra of Regular Events , 1966, JACM.

[15]  Gordon D. Plotkin,et al.  Semantics for Algebraic Operations , 2001, MFPS.

[16]  Kim G. Larsen,et al.  Complete Axiomatization for the Bisimilarity Distance on Markov Chains , 2016, CONCUR.

[17]  Kim G. Larsen,et al.  Converging from branching to linear metrics on Markov chains , 2015, Mathematical Structures in Computer Science.

[18]  Scott A. Smolka,et al.  A complete axiom system for finite-state probabilistic processes , 2000, Proof, Language, and Interaction.

[19]  Kim G. Larsen,et al.  On the Total Variation Distance of Semi-Markov Chains , 2015, FoSSaCS.

[20]  Scott A. Smolka,et al.  Equivalences, Congruences, and Complete Axiomatizations for Probabilistic Processes , 1990, CONCUR.

[21]  Roberto Segala,et al.  Axiomatizations for Probabilistic Bisimulation , 2001, ICALP.

[22]  M. H. Stone.  Postulates for the barycentric calculus , 1949 .

[23]  Mads Tofte,et al.  A Complete Axiom System for Finite-State Probabilistic Processes , 2000 .

[24]  Dexter Kozen.  A Completeness Theorem for Kleene Algebras and the Algebra of Regular Events , 1994, Inf. Comput..

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

[26]  Klaus Keimel,et al.  Mixed powerdomains for probability and nondeterminism , 2016, Log. Methods Comput. Sci..

[27]  A. Sokolova,et al.  Sound and Complete Axiomatization of Trace Semantics for Probabilistic Systems , 2011, MFPS.

[28]  Alexander Moshe Rabinovich.  A Complete Axiomatisation for Trace Congruence of Finite State Behaviors , 1993, MFPS.

[29]  Alexandra Silva,et al.  Quantitative Kleene coalgebras , 2011, Inf. Comput..