Energy and Mean-Payoff Games with Imperfect Information

We consider two-player games with imperfect information and quantitative objective. The game is played on a weighted graph with a state space partitioned into classes of indistinguishable states, giving players partial knowledge of the state. In an energy game, the weights represent resource consumption and the objective of the game is to maintain the sum of weights always nonnegative. In a mean-payoff game, the objective is to optimize the limit-average usage of the resource. We show that the problem of determining if an energy game with imperfect information with fixed initial credit has a winning strategy is decidable, while the question of the existence of some initial credit such that the game has a winning strategy is undecidable. This undecidability result carries over to meanpayoff games with imperfect information. On the positive side, using a simple restriction on the game graph (namely, that the weights are visible), we show that these problems become EXPTIME-complete.

[1]  John H. Reif,et al.  The Complexity of Two-Player Games of Incomplete Information , 1984, J. Comput. Syst. Sci..

[2]  Wolfgang Reisig,et al.  Lectures on Petri Nets I: Basic Models , 1996, Lecture Notes in Computer Science.

[3]  Philippe Schnoebelen,et al.  Well-structured transition systems everywhere! , 2001, Theor. Comput. Sci..

[4]  PatersonMike,et al.  The complexity of mean payoff games on graphs , 1996 .

[5]  Thomas Wilke,et al.  Automata logics, and infinite games: a guide to current research , 2002 .

[6]  Alex K. Simpson,et al.  Computational Adequacy in an Elementary Topos , 1998, CSL.

[7]  Krishnendu Chatterjee,et al.  Algorithms for Omega-Regular Games with Incomplete Information ∗ , 2006 .

[8]  E. Allen Emerson,et al.  Tree automata, mu-calculus and determinacy , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[9]  Kousha Etessami,et al.  Analysis of Recursive Game Graphs Using Data Flow Equations , 2004, VMCAI.

[10]  L. Dickson.  Finiteness of the Odd Perfect and Primitive Abundant Numbers with n Distinct Prime Factors , 1913 .

[11]  Rajeev Alur,et al.  On Omega-Languages Defined by Mean-Payoff Conditions , 2009, FoSSaCS.

[12]  A. Prasad Sistla,et al.  On Model-Checking for Fragments of µ-Calculus , 1993, CAV.

[13]  Charles Rackoff,et al.  The Covering and Boundedness Problems for Vector Addition Systems , 1978, Theor. Comput. Sci..

[14]  Javier Esparza,et al.  Decidability and Complexity of Petri Net Problems - An Introduction , 1996, Petri Nets.

[15]  Jakub Chaloupka,et al.  Faster Algorithm for Mean-Payoff Games , 2009, MEMICS.

[16]  Krishnendu Chatterjee,et al.  Algorithms for Omega-Regular Games with Imperfect Information , 2006, Log. Methods Comput. Sci..

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

[18]  Uri Zwick,et al.  The Complexity of Mean Payoff Games on Graphs , 1996, Theor. Comput. Sci..

[19]  Dietmar Berwanger,et al.  On the Power of Imperfect Information , 2008, FSTTCS.

[20]  Jan Maluszy¿ski.  Verification, Model Checking, and Abstract Interpretation , 2009, Lecture Notes in Computer Science.

[21]  Robin Milner,et al.  On Observing Nondeterminism and Concurrency , 1980, ICALP.

[22]  Orna Kupferman,et al.  Lattice Automata , 2007, VMCAI.

[23]  L. Brim,et al.  Faster algorithms for mean-payoff games , 2011, Formal Methods Syst. Des..

[24]  Krishnendu Chatterjee,et al.  Quantitative languages , 2008, TOCL.

[25]  A. Ehrenfeucht,et al.  Positional strategies for mean payoff games , 1979 .

[26]  Krishnendu Chatterjee,et al.  Better Quality in Synthesis through Quantitative Objectives , 2009, CAV.

[27]  Luca de Alfaro,et al.  How to Specify and Verify the Long-Run Average Behavior of Probabilistic Systems , 1998, LICS.

[28]  Daniel Krob,et al.  The Equality Problem for Rational Series with Multiplicities in the tropical Semiring is Undecidable , 1992, Int. J. Algebra Comput..

[29]  Paul Gastin,et al.  Weighted automata and weighted logics , 2005, Theor. Comput. Sci..

[30]  Krishnendu Chatterjee,et al.  Compositional Quantitative Reasoning , 2006, Third International Conference on the Quantitative Evaluation of Systems - (QEST'06).

[31]  Ian Stark,et al.  Free-Algebra Models for the pi-Calculus , 2005, FoSSaCS.