Synthesising Strategy Improvement and Recursive Algorithms for Solving 2.5 Player Parity Games

2.5 player parity games combine the challenges posed by 2.5 player reachability games and the qualitative analysis of parity games. These two types of problems are best approached with different types of algorithms: strategy improvement algorithms for 2.5 player reachability games and recursive algorithms for the qualitative analysis of parity games. We present a method that - in contrast to existing techniques - tackles both aspects with the best suited approach and works exclusively on the 2.5 player game itself. The resulting technique is powerful enough to handle games with several million states

Mixing Probabilistic and non-Probabilistic Objectives in Markov Decision Processes

In this paper, we consider algorithms to decide the existence of strategies in MDPs for Boolean combinations of objectives. These objectives are omega-regular properties that need to be enforced either surely, almost surely, existentially, or with non-zero probability. In this setting, relevant strategies are randomized infinite memory strategies: both infinite memory and randomization may be needed to play optimally. We provide algorithms to solve the general case of Boolean combinations and we also investigate relevant subcases. We further report on complexity bounds for these problems.Comment: Paper accepted to LICS 2020 - Full versio

Qualitative Analysis of Partially-observable Markov Decision Processes

We study observation-based strategies for partially-observable Markov decision processes (POMDPs) with omega-regular objectives. An observation-based strategy relies on partial information about the history of a play, namely, on the past sequence of observations. We consider the qualitative analysis problem: given a POMDP with an omega-regular objective, whether there is an observation-based strategy to achieve the objective with probability~1 (almost-sure winning), or with positive probability (positive winning). Our main results are twofold. First, we present a complete picture of the computational complexity of the qualitative analysis of POMDP s with parity objectives (a canonical form to express omega-regular objectives) and its subclasses. Our contribution consists in establishing several upper and lower bounds that were not known in literature. Second, we present optimal bounds (matching upper and lower bounds) on the memory required by pure and randomized observation-based strategies for the qualitative analysis of POMDP s with parity objectives and its subclasses

Obligation Blackwell Games and p-Automata

We recently introduced p-automata, automata that read discrete-time Markov chains. We used turn-based stochastic parity games to define acceptance of Markov chains by a subclass of p-automata. Definition of acceptance required a cumbersome and complicated reduction to a series of turn-based stochastic parity games. The reduction 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 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 definition, we show that 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. Based on this simpler definition we give an exponential time algorithm to analyze finite turn-based stochastic parity games with obligations. Finally, we show that obligation games provide the necessary framework for reasoning about p-automata and that they generalize the previous definition

Qualitative Analysis of Concurrent Mean-payoff Games

We consider concurrent games played by two-players on a finite-state graph, where in every round the players simultaneously choose a move, and the current state along with the joint moves determine the successor state. We study a fundamental objective, namely, mean-payoff objective, where a reward is associated to each transition, and the goal of player 1 is to maximize the long-run average of the rewards, and the objective of player 2 is strictly the opposite. The path constraint for player 1 could be qualitative, i.e., the mean-payoff is the maximal reward, or arbitrarily close to it; or quantitative, i.e., a given threshold between the minimal and maximal reward. We consider the computation of the almost-sure (resp. positive) winning sets, where player 1 can ensure that the path constraint is satisfied with probability 1 (resp. positive probability). Our main results for qualitative path constraints are as follows: (1) we establish qualitative determinacy results that show that for every state either player 1 has a strategy to ensure almost-sure (resp. positive) winning against all player-2 strategies, or player 2 has a spoiling strategy to falsify almost-sure (resp. positive) winning against all player-1 strategies; (2) we present optimal strategy complexity results that precisely characterize the classes of strategies required for almost-sure and positive winning for both players; and (3) we present quadratic time algorithms to compute the almost-sure and the positive winning sets, matching the best known bound of algorithms for much simpler problems (such as reachability objectives). For quantitative constraints we show that a polynomial time solution for the almost-sure or the positive winning set would imply a solution to a long-standing open problem (the value problem for turn-based deterministic mean-payoff games) that is not known to be solvable in polynomial time

A survey of stochastic ω regular games

We summarize classical and recent results about two-player games played on graphs with ω-regular objectives. These games have applications in the verification and synthesis of reactive systems. Important distinctions are whether a graph game is turn-based or concurrent; deterministic or stochastic; zero-sum or not. We cluster known results and open problems according to these classifications