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Energy and Mean-Payoff Parity Markov Decision Processes Laurent Doyen LSV, ENS Cachan & CNRS Krishnendu Chatterjee IST Austria MFCS 2011

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Games for system analysis Verification: check if a given system is correct reduces to graph searching System input output Spec: φ( input,output ) Environment

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Games for system analysis ? input output Spec: φ( input,output ) Environment Verification: check if a given system is correct reduces to graph searching Synthesis : construct a correct system reduces to game solving – finding a winning strategy

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Games for system analysis Spec: φ( input,output ) Verification: check if a given system is correct reduces to graph searching Synthesis : construct a correct system reduces to game solving – finding a winning strategy This talk: environment is abstracted as a stochastic process ? input output Environment input output = Markov decision process (MDP) ?

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Markov decision process

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Markov decision process (MDP) Nondeterministic (player 1) Probabilistic

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Markov decision process (MDP) Nondeterministic (player 1) Probabilistic

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Markov decision process (MDP) Nondeterministic (player 1) Probabilistic

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Markov decision process (MDP) Nondeterministic (player 1) Probabilistic

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Markov decision process (MDP) Nondeterministic (player 1) Probabilistic

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Markov decision process (MDP) Nondeterministic (player 1) Probabilistic

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Markov decision process (MDP) Nondeterministic (player 1) Probabilistic

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Markov decision process (MDP) Nondeterministic (player 1) Probabilistic Strategy (policy) = recipe to extend the play prefix

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Markov decision process (MDP) Nondeterministic (player 1) Probabilistic Strategy (policy) = recipe to extend the play prefix

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Markov decision process (MDP) Nondeterministic (player 1) Probabilistic (player 2) Strategy (policy) = recipe to extend the play prefix weak

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Objective Strategy is almost-sure winning, if with probability 1: - Büchi: visit accepting states infinitely often. - Parity: least priority visited infinitely often is even. Fix a strategy (infinite) Markov chain

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Decision problem Given an MDP, decide whether there exists an almost-sure winning strategy for parity objective.

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Decision problem End-component = set of states U s.t. if then some successor of q is in U if then all successors of q are in U strongly connected U Given an MDP, decide whether there exists an almost-sure winning strategy for parity objective.

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Decision problem End-component is good if least priority is even Almost-sure reachability to good end-components in PTIME strongly connected U Given an MDP, decide whether there exists an almost-sure winning strategy for parity objective. End-component = set of states U s.t. if then some successor of q is in U if then all successors of q are in U strongly connected

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Given an MDP, decide whether there exists an almost-sure winning strategy for parity objective. Decision problem End-component is good if least priority is even Almost-sure reachability to good end-components in PTIME strongly connected End-component = set of states U s.t. if then some successor of q is in U if then all successors of q are in U strongly connected

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Parity condition Qualitative ω-regular specifications (reactivity, liveness,…) Objectives

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Parity conditionEnergy condition QualitativeQuantitative ω-regular specifications (reactivity, liveness,…) Resource-constrained specifications Objectives

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Energy objective Positive and negative weights (encoded in binary)

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Energy objective Energy level: 20, 21, 11, 1, 2,… (sum of weights) Positive and negative weights (encoded in binary) 2021 11 12

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Energy objective Positive and negative weights Energy level: 20, 21, 11, 1, 2,… (sum of weights) Initial credit 20 21 11 12

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Energy objective Positive and negative weights Energy level: 20, 21, 11, 1, 2,… (sum of weights) Initial credit 20 A play is winning if the energy level is always nonnegative. “Never exhaust the resource (memory, battery, …)”

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Decision problem Given a weighted MDP, decide whether there exist an initial credit c 0 and an almost-sure winning strategy to maintain the energy level always nonnegative.

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Decision problem Equivalent to a two-player game: If player 2 can force a negative energy level on a path, then the path is finite and has positive probability in MDP player 2 state Given a weighted MDP, decide whether there exist an initial credit c 0 and an almost-sure winning strategy to maintain the energy level always nonnegative.

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Decision problem Equivalent to a two-player game: If player 2 can force a negative energy level on a path, then the path is finite and has positive probability in MDP player 2 state

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Energy Parity MDP

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Objectives Energy parity MDP QualitativeQuantitative Mixed qualitative-quantitative ω-regular specifications (reactivity, liveness,…) Resource-constrained specifications Parity condition Energy condition

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Strategy is almost-sure winning with initial credit c 0, if with probability 1: energy condition and parity condition hold Energy parity MDP “never exhaust the resource” and “always eventually do something useful”

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Algorithm for Energy Büchi MDP For parity, probabilistic player is my friend For energy, probabilistic player is my opponent

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Algorithm for Energy Büchi MDP Replace each probabilistic state by the gadget: For parity, probabilistic player is my friend For energy, probabilistic player is my opponent

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Algorithm for Energy Büchi MDP Reduction of energy Büchi MDP to energy Büchi game

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Algorithm for Energy Büchi MDP Reduction of energy Büchi MDP to energy Büchi game Player 1 can guess an even priority 2i, and win in the energy Büchi MDP where: Büchi states are 2i-states, and transitions to states with priority <2i are disallowed Reduction of energy parity MDP to energy Büchi MDP

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Mean-payoff Parity MDP

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Mean-payoff Mean-payoff value of a play = limit- average of the visited weights Optimal mean-payoff value can be achieved with a memoryless strategy. Decision problem: Given a rational threshold, decide if there exists a strategy for player 1 to ensure mean-payoff value at least with probability 1.

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Mean-payoff Mean-payoff value of a play = limit- average of the visited weights Optimal mean-payoff value can be achieved with a memoryless strategy. Decision problem: Given a rational threshold, decide if there exists a strategy for player 1 to ensure mean-payoff value at least with probability 1.

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Mean-payoff games Memoryless strategy σ ensures nonnegative mean-payoff value iff all cycles are nonnegative in G σ iff memoryless strategy σ is winning in energy game Mean-payoff games with threshold 0 are equivalent to energy games.

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Mean-payoff vs. Energy EnergyMean-Payoff Games NP coNP MDP NP coNP PTIME

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Mean-payoff parity MDPs Find a strategy which ensures with probability 1: - parity condition, and - mean-payoff value ≥ Gadget reduction does not work: Player 1 almost- surely wins Player 1 loses

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Algorithm for mean-payoff parity End-component analysis almost-surely all states of end-component can be visited infinitely often expected mean-payoff value of all states in end-component is same strongly connected End-component is good if - least priority is even - expected mean-payoff value ≥ Almost-sure reachability to good end-component in PTIME

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Algorithm for mean-payoff parity End-component analysis almost-surely all states of end-component can be visited infinitely often expected mean-payoff value of all states in end-component is same strongly connected End-component is good if - least priority is even - expected mean-payoff value ≥ Almost-sure reachability to even end-component in PTIME

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MDP Parity MDPsEnergy MDPs Mean-payoff MDPs Energy parity MDPs Mean-payoff parity MDPs QualitativeQuantitative Mixed qualitative-quantitative

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MDP Parity MDPsEnergy MDPs Mean-payoff MDPs Energy parity MDPs Mean-payoff parity MDPs QualitativeQuantitative Mixed qualitative-quantitative

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Summary Energy parityMean-Payoff parity Games NP coNP MDP NP coNP PTIME Algorithmic complexity

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Summary Energy parityMean-Payoff parity Games NP coNP MDP NP coNP PTIME Strategy complexity Energy parityMean-Payoff parity Gamesn·d·Winfinite MDP2·n·Winfinite Algorithmic complexity

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Thank you ! Questions ? The end

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References [CdAHS03]A.Chakrabarti, L. de Alfaro, T.A. Henzinger, and M. Stoelinga. Resource interfaces, Proc. of EMSOFT: Embedded Software, LNCS 2855, Springer, pp.117-133, 2003 [EM79]A. Ehrenfeucht, and J. Mycielski, Positional Strategies for Mean-Payoff Games, International Journal of Game Theory, vol. 8, pp. 109-113, 1979 [BFL+08]P. Bouyer, U. Fahrenberg, K.G. Larsen, N. Markey, and J. Srba, Infinite Runs in Weighted Timed Automata with Energy Constraints, Proc. of FORMATS: Formal Modeling and Analysis of Timed Systems, LNCS 5215, Springer, pp. 33- 47, 2008 [CHJ05]K. Chatterjee, T.A. Henzinger, M. Jurdzinski. Mean-payoff Parity Games, Proc. of LICS: Logic in Computer Science, IEEE, pp. 178-187, 2005.

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Energy parity

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Mean-payoff parity

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Complexity StrategyAlgorithmic Player 1Player 2complexity Energymemoryless NP coNP Paritymemoryless NP coNP Energy parityexponentialmemoryless NP coNP Mean-payoffmemoryless NP coNP Mean-payoff parity infinitememoryless NP coNP

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