Probabilistic Alternating-Time µ-Calculus

Reasoning about strategic abilities is key to an AI system consisting of multiple agents with random behaviors. We propose a probabilistic extension of Alternating mu-Calculus (AMC), named PAMC, for reasoning about strategic abilities of agents in stochastic multi-agent systems. PAMC subsumes existing logics AMC and P mu TL. The usefulness of PAMC is exemplified by applications in genetic regulatory networks. We show that, for PAMC, the model checking problem is in UP boolean AND co-UP, and the satisfiability problem is EXPTIME complete, both of which are the same as those for AMC. Moreover, PAMC admits the small model property. We implement the satisfiability checking procedure in a tool PAMCSolver.

[1]  Michael Wooldridge,et al.  ATL Satisfiability is Indeed EXPTIME-complete , 2006, J. Log. Comput..

[2]  Jian Lu,et al.  Probabilistic Alternating-time Temporal Logic and Model Checking Algorithm , 2007, Fourth International Conference on Fuzzy Systems and Knowledge Discovery (FSKD 2007).

[3]  J. Moon,et al.  On cliques in graphs , 1965 .

[4]  Valentin Goranko,et al.  Optimal Decision Procedures for Satisfiability in Fragments of Alternating-time Temporal Logics , 2014, Advances in Modal Logic.

[5]  Cheng Luo,et al.  A logic of probabilistic knowledge and strategy , 2013, AAMAS.

[6]  Tomás Brázdil,et al.  Stochastic games with branching-time winning objectives , 2006, 21st Annual IEEE Symposium on Logic in Computer Science (LICS'06).

[7]  Jan Kretínský,et al.  The Satisfiability Problem for Probabilistic CTL , 2008, 2008 23rd Annual IEEE Symposium on Logic in Computer Science.

[8]  Bernd Finkbeiner,et al.  Satisfiability and Finite Model Property for the Alternating-Time mu-Calculus , 2006, CSL.

[9]  Michael Huth,et al.  Quantitative analysis and model checking , 1997, Proceedings of Twelfth Annual IEEE Symposium on Logic in Computer Science.

[10]  Rance Cleaveland,et al.  Probabilistic temporal logics via the modal mu-calculus , 1999, Theor. Comput. Sci..

[11]  Marc Pauly,et al.  Logic for social software , 2000 .

[12]  Krishnendu Chatterjee,et al.  Stochastic omega-regular games , 2007 .

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

[14]  Xiaowei Huang,et al.  Model Checking Probabilistic Knowledge: A PSPACE Case , 2016, AAAI.

[15]  Annabelle McIver,et al.  Games, Probability and the Quantitative µ-Calculus qMµ , 2002, LPAR.

[16]  Aniello Murano,et al.  Reasoning about Strategies: on the Satisfiability Problem , 2016, Log. Methods Comput. Sci..

[17]  Joost-Pieter Katoen,et al.  On the Satisfiability of Some Simple Probabilistic Logics , 2016, 2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

[18]  Moshe Y. Vardi,et al.  LTL Satisfiability Checking , 2007, SPIN.

[19]  Krishnendu Chatterjee,et al.  Strategy logic , 2007, Inf. Comput..

[20]  Yuri Gurevich,et al.  Trees, automata, and games , 1982, STOC '82.

[21]  Sven Schewe.  ATL* Satisfiability Is 2EXPTIME-Complete , 2008, ICALP.

[22]  Christel Baier,et al.  Stochastic game logic , 2007, Fourth International Conference on the Quantitative Evaluation of Systems (QEST 2007).

[23]  Harry D. Foster,et al.  Assertion-Based Design , 2010 .

[24]  Rupak Majumdar,et al.  Quantitative solution of omega-regular games , 2004, J. Comput. Syst. Sci..

[25]  Edward R. Dougherty,et al.  Probabilistic Boolean Networks - The Modeling and Control of Gene Regulatory Networks , 2010 .

[26]  M. Mio.  Game semantics for probabilistic modal μ-calculi , 2012 .

[27]  Taolue Chen,et al.  Automatic verification of competitive stochastic systems , 2012, Formal Methods in System Design.

[28]  Lijun Zhang,et al.  Model Checking Probabilistic Epistemic Logic for Probabilistic Multiagent Systems , 2018, IJCAI.

[29]  Lijun Zhang,et al.  A Simple Probabilistic Extension of Modal Mu-calculus , 2015, IJCAI.

[30]  Kaile Su,et al.  Probabilistic Alternating-Time Temporal Logic of Incomplete Information and Synchronous Perfect Recall , 2012, AAAI.

[31]  Carroll Morgan,et al.  A Probabilistic Temporal Calculus Based on Expectations , 1997 .

[32]  Aniello Murano,et al.  Reasoning About Strategies: On the Model-Checking Problem , 2011, ArXiv.

[33]  Annabelle McIver,et al.  Results on the quantitative μ-calculus qMμ , 2007, TOCL.

[34]  Krishnendu Chatterjee,et al.  A survey of stochastic ω-regular games , 2012, J. Comput. Syst. Sci..