A note on the attractor-property of infinite-state Markov chains

In the past 5 years, a series of verification algorithms has been proposed for infinite Markov chains that have a finite attractor, i.e., a set that will be visited infinitely often almost surely starting from any state. In this paper, we establish a sufficient criterion for the existence of an attractor. We show that if the states of a Markov chain can be given levels (positive integers) such that the expected next level for states at some level n > 0 is less than n - Δ for some positive Δ, then the states at level 0 constitute an attractor for the chain. As an application, we obtain a direct proof that some probabilistic channel systems combining message losses with duplication and insertion errors have a finite attractor.

[1]  John Odentrantz,et al.  Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues , 2000, Technometrics.

[2]  Marta Kwiatkowska,et al.  PRISM 2.0: a tool for probabilistic model checking , 2004 .

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

[4]  Christel Baier,et al.  Establishing Qualitative Properties for Probabilistic Lossy Channel Systems: An Algorithmic Approach , 1999, ARTS.

[5]  Sasha Rubin,et al.  Verifying ω-regular properties of Markov chains , 2004 .

[6]  Rajeev Alur,et al.  Model-Checking for Probabilistic Real-Time Systems (Extended Abstract) , 1991, ICALP.

[7]  BrandDaniel,et al.  On Communicating Finite-State Machines , 1983 .

[8]  Parosh Aziz Abdulla,et al.  Verifying infinite Markov chains with a finite attractor or the global coarseness property , 2005, 20th Annual IEEE Symposium on Logic in Computer Science (LICS' 05).

[9]  Philippe Schnoebelen,et al.  The Verification of Probabilistic Lossy Channel Systems , 2004, Validation of Stochastic Systems.

[10]  李幼升,et al.  Ph , 1989 .

[11]  Moshe Y. Vardi.  Probabilistic Linear-Time Model Checking: An Overview of the Automata-Theoretic Approach , 1999, ARTS.

[12]  Grégoire Sutre,et al.  An Optimal Automata Approach to LTL Model Checking of Probabilistic Systems , 2003, LPAR.

[13]  Rajeev Alur,et al.  Model-Checking for Probabilistic Real Time Systems , 1991 .

[14]  Parosh Aziz Abdulla,et al.  Verification of probabilistic systems with faulty communication , 2005, Inf. Comput..

[15]  Mihalis Yannakakis,et al.  The complexity of probabilistic verification , 1995, JACM.

[16]  R. Segala,et al.  Automatic Verification of Real-Time Systems with Discrete Probability Distributions , 1999, ARTS.

[17]  Alexander Moshe Rabinovich,et al.  Quantitative Analysis of Probabilistic Lossy Channel Systems , 2003, ICALP.

[18]  Daniel Brand,et al.  On Communicating Finite-State Machines , 1983, JACM.

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

[20]  Alexander Moshe Rabinovich.  Quantitative analysis of probabilistic lossy channel systems , 2006, Inf. Comput..

[21]  J. Esparza,et al.  Model checking probabilistic pushdown automata , 2004, LICS 2004.

[22]  Kousha Etessami,et al.  Verifying Probabilistic Procedural Programs , 2004, FSTTCS.

[23]  Kousha Etessami,et al.  Recursive Markov Decision Processes and Recursive Stochastic Games , 2005, ICALP.

[24]  Kousha Etessami,et al.  Algorithmic Verification of Recursive Probabilistic State Machines , 2005, TACAS.

[25]  Prakash Panangaden,et al.  Measure and probability for concurrency theorists , 2001, Theor. Comput. Sci..

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

[27]  Nathalie Bertrand,et al.  Model Checking Lossy Channels Systems Is Probably Decidable , 2003, FoSSaCS.