Permissive strategies: from parity games to safety games

Abstract

It is proposed to compare strategies in a parity game by comparing the sets of behaviours they allow. For such a game, there may be no winning strategy that encompasses all the behaviours of all winning strategies. It is shown, however, that there always exists a permissive strategy that encompasses all the behaviours of all memoryless strategies. An algorithm for finding such a permissive strategy is presented. Its complexity matches currently known upper bounds for the simpler problem of finding the set of winning positions in a parity game. The algorithm can be seen as a reduction of a parity to a safety game and computation of the set of winning positions in the resulting game.

Succinct progress measures for solving parity games

“…The high-level idea of the algorithm of Calude et al [4] bears similarity to the approach of Bernet et al [2]: first devise a finite safety automaton that recognizes infinite sequences of priorities that result in a win for Even (in the case of Bernet et al, given an explicit upper bound on the number of occurrences of each odd priority before an occurrence of a higher priority), and then solve the safety game obtained from the product automaton that simulates the safety automaton on the game graph.…”

Measuring Permissivity in Finite Games

“…This is not the case here, where an early cut of a branch/edge of the game may avoid a large penalty later on for blocking many edges. However our notion of multi-strategy coincides with the non-deterministic strategies of [BJW02] and [Lut08].…”

“…This quantitative approach to permissivity is rather original, and does not compare to either of the approaches found in the literature [BJW02,PR05]. Indeed classical notions of permissivity imply the largest sets of generated plays.…”

Robust optimal policies for Markov decision processes with safety-threshold constraints

“…Under Assumption II.2, we can compute a memoryless policy from the graph G = ( S, E) of the given aug-UMDP which ensures satisfaction of (4). To this end, we compute a function f safe : S → 2 A that maps each state to a set of "safe actions" from that state.…”

“…It is noted that the penalty method in [13] does not exclude the chance of constraint violation: One has to assign reward −∞ to do so. We propose a twostep method: In the first step, the feasible set of memoryless policies is computed, by computing a strategy in a safety game on the graph of the uncertain MDP [4]. Within the set of feasible policies, we compute the optimal policy with respect to the given reward criterion.…”