Stochastic Games with Lexicographic Reachability-Safety Objectives

Lecture notes in computer science, pp. 398–420

Abstract

We study turn-based stochastic zero-sum games with lexicographic preferences over reachability and safety objectives. Stochastic games are standard models in control, verification, and synthesis of stochastic reactive systems that exhibit both randomness as well as angelic and demonic non-determinism. Lexicographic order allows to consider multiple objectives with a strict preference order over the satisfaction of the objectives. To the best of our knowledge, stochastic games with lexicographic objectives have not been studied before. We establish determinacy of such games and present strategy and computational complexity results. For strategy complexity, we show that lexicographically optimal strategies exist that are deterministic and memory is only required to remember the already satisfied and violated objectives. For a constant number of objectives, we show that the relevant decision problem is in $$\mathsf {NP}\cap \mathsf {coNP}$$ , matching the current known bound for single objectives; and in general the decision problem is $$\mathsf {PSPACE}$$ -hard and can be solved in $$\mathsf {NEXPTIME}\cap \mathsf {coNEXPTIME}$$ . We present an algorithm that computes the lexicographically optimal strategies via a reduction to computation of optimal strategies in a sequence of single-objectives games. We have implemented our algorithm and report experimental results on various case studies.

Authors 4

1. Krishnendu Chatterjee

   Institute of Science and Technology Austria

   Affiliation as printed

   IST Austria, Klosterneuburg, Austria

   IST Austria; Klosterneuburg Austria

2. Joost-Pieter Katoen Aachen

   RWTH Aachen University

   Affiliation as printed

   RWTH Aachen University, Aachen, Germany

   RWTH Aachen, University, Aachen, Germany

3. Maximilian Weininger

   Technical University of Munich

   Affiliation as printed

   Technical University of Munich, Munich, Germany

4. Tobias Winkler corresponding Aachen

   RWTH Aachen University

   Affiliation as printed

   RWTH Aachen University, Aachen, Germany

   RWTH Aachen University