Obligation Blackwell Games and p-Automata

We recently introduced p-automata, automata that read discrete-time Markov chains and showed they provide an automata-theoretic framework for reasoning about pCTL model checking and abstraction of discrete time Markov chains. We used turn-based stochastic parity games to define acceptance of Markov chains by a special subclass of p-automata. Definition of acceptance required a reduction to a series of turn-based stochastic parity games. The reduction was cumbersome and complicated and could not support acceptance by general p-automata, which was left undefined as there was no notion of games that supported it. Here we generalize two-player games by adding a structural acceptance condition called obligations. Obligations are orthogonal to the linear winning conditions that define whether a play is winning. Obligations are a declaration that player 0 can achieve a certain value from a configuration. If the obligation is met, the value of that configuration for player 0 is 1. One cannot define value in obligation games by the standard mechanism of considering the measure of winning paths on a Markov chain and taking the supremum of the infimum of all strategies. Mainly because obligations need definition even for Markov chains and the nature of obligations has the flavor of an infinite nesting of supremum and infimum operators. We define value via a reduction to turn-based games similar to Martin’s proof of determinacy of Blackwell games with Borel objectives. Based on this value definition we show that obligation games are determined. We show that for Markov chains with Borel objectives and obligations, and finite turn-based stochastic parity games with obligations there exists an alternative and simpler characterization of the value function without going through a Martin-like reduction. Based on this simpler definition we give an exponential time algorithm to analyze finite turn-based stochastic parity games with obligations and show that the decision problem of winning parity games with obligations is in NP∩co-NP. Finally, we show that obligation games provide the necessary framework for reasoning about p-automata and that they generalize the previous definition. 1 ar X iv :1 20 6. 51 74 v3 [ cs .L O ] 3 N ov 2 01 3