Verification and Refutation of Probabilistic Specifications via Games

We develop an abstraction-based framework to check probabilistic specifications of Markov Decision Processes (MDPs) using the stochastic two-player game abstractions (\ie ``games'') developed by Kwiatkowska et al.\ as a foundation. We define an abstraction preorder for these game abstractions which enables us to identify many new game abstractions for each MDP --- ranging from compact and imprecise to complex and precise. This added ability to trade precision for efficiency is crucial for scalable software model checking, as precise abstractions are expensive to construct in practice. Furthermore, we develop a four-valued probabilistic computation tree logic (PCTL) semantics for game abstractions. Together, the preorder and PCTL semantics comprise a powerful verification and refutation framework for arbitrary PCTL properties of MDPs.

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

[2]  Annabelle McIver,et al.  Proofs and Refutations for Probabilistic Refinement , 2008, FM.

[3]  Marta Z. Kwiatkowska,et al.  A game-based abstraction-refinement framework for Markov decision processes , 2010, Formal Methods Syst. Des..

[4]  Nuel D. Belnap,et al.  A Useful Four-Valued Logic , 1977 .

[5]  Mark Kattenbelt,et al.  Abstraction Framework for Markov Decision Processes and PCTL via Games , 2009 .

[6]  Joseph Sifakis,et al.  Specification and verification of concurrent systems in CESAR , 1982, Symposium on Programming.

[7]  Lijun Zhang,et al.  Probabilistic CEGAR , 2008, CAV.

[8]  Moshe Y. Vardi.  Automatic verification of probabilistic concurrent finite state programs , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[9]  J. Kemeny,et al.  Denumerable Markov chains , 1969 .

[10]  Edmund M. Clarke,et al.  Design and Synthesis of Synchronization Skeletons Using Branching-Time Temporal Logic , 1981, Logic of Programs.

[11]  Daniel Kroening,et al.  SATABS: SAT-Based Predicate Abstraction for ANSI-C , 2005, TACAS.

[12]  Orna Grumberg,et al.  Abstract interpretation of reactive systems , 1997, TOPL.

[13]  Mahesh Viswanathan,et al.  Least Upper Bounds for Probability Measures and Their Applications to Abstractions , 2008, CONCUR.

[14]  Michael Huth,et al.  On finite-state approximants for probabilistic computation tree logic , 2005, Theor. Comput. Sci..

[15]  Joost-Pieter Katoen,et al.  Counterexamples in Probabilistic Model Checking , 2007, TACAS.

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

[17]  Anne Condon,et al.  The Complexity of Stochastic Games , 1992, Inf. Comput..

[18]  Bengt Jonsson,et al.  A logic for reasoning about time and reliability , 1990, Formal Aspects of Computing.

[19]  Kim G. Larsen,et al.  Reduction and Refinement Strategies for Probabilistic Analysis , 2002, PAPM-PROBMIV.

[20]  Marta Z. Kwiatkowska,et al.  Stochastic Games for Verification of Probabilistic Timed Automata , 2009, FORMATS.

[21]  Kim G. Larsen,et al.  Modal Specifications , 1989, Automatic Verification Methods for Finite State Systems.

[22]  Martin Leucker,et al.  Don't Know in Probabilistic Systems , 2006, SPIN.

[23]  P. S. Thiagarajan.  Proceedings of the 15th Conference on Foundations of Software Technology and Theoretical Computer Science , 1995 .

[24]  Kim G. Larsen,et al.  A modal process logic , 1988, [1988] Proceedings. Third Annual Information Symposium on Logic in Computer Science.

[25]  Marta Z. Kwiatkowska,et al.  Game-based Abstraction for Markov Decision Processes , 2006, Third International Conference on the Quantitative Evaluation of Systems - (QEST'06).

[26]  Andrea Bianco,et al.  Model Checking of Probabalistic and Nondeterministic Systems , 1995, FSTTCS.

[27]  Radha Jagadeesan,et al.  Three-valued abstractions of games: uncertainty, but with precision , 2004, Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004..

[28]  Marta Z. Kwiatkowska,et al.  Game-Based Probabilistic Predicate Abstraction in PRISM , 2008, QAPL.

[29]  Edmund M. Clarke,et al.  Counterexample-guided abstraction refinement , 2003, 10th International Symposium on Temporal Representation and Reasoning, 2003 and Fourth International Conference on Temporal Logic. Proceedings..