Termination Criteria for Solving Concurrent Safety and Reachability Games

We consider concurrent games played on graphs. At every round of a game, each player simultaneously and independently selects a move; the moves jointly determine the transition to a successor state. Two basic objectives are the safety objective to stay forever in a given set of states, and its dual, the reachability objective to reach a given set of states. We present in this paper a strategy improvement algorithm for computing the value of a concurrent safety game, that is, the maximal probability with which player~1 can enforce the safety objective. The algorithm yields a sequence of player-1 strategies which ensure probabilities of winning that converge monotonically to the value of the safety game. Our result is significant because the strategy improvement algorithm provides, for the first time, a way to approximate the value of a concurrent safety game from below. Since a value iteration algorithm, or a strategy improvement algorithm for reachability games, can be used to approximate the same value from above, the combination of both algorithms yields a method for computing a converging sequence of upper and lower bounds for the values of concurrent reachability and safety games. Previous methods could approximate the values of these games only from one direction, and as no rates of convergence are known, they did not provide a practical way to solve these games

Strategy improvement for concurrent reachability and turn based stochastic safety games

We consider concurrent games played on graphs. At every round of a game, each player simultaneously and independently selects a move; the moves jointly determine the transition to a successor state. Two basic objectives are the safety objective to stay forever in a given set of states, and its dual, the reachability objective to reach a given set of states. First, we present a simple proof of the fact that in concurrent reachability games, for all ε>0, memoryless ε-optimal strategies exist. A memoryless strategy is independent of the history of plays, and an ε-optimal strategy achieves the objective with probability within ε of the value of the game. In contrast to previous proofs of this fact, our proof is more elementary and more combinatorial. Second, we present a strategy-improvement (a.k.a. policy-iteration) algorithm for concurrent games with reachability objectives. Finally, we present a strategy-improvement algorithm for turn-based stochastic games (where each player selects moves in turns) with safety objectives. Our algorithms yield sequences of player-1 strategies which ensure probabilities of winning that converge monotonically (from below) to the value of the game. © 2012 Elsevier Inc

Strategy Improvement for Concurrent Safety Games

We consider concurrent games played on graphs. At every round of the game, each player simultaneously and independently selects a move; the moves jointly determine the transition to a successor state. Two basic objectives are the safety objective: ``stay forever in a set F of states'', and its dual, the reachability objective, ``reach a set R of states''. We present in this paper a strategy improvement algorithm for computing the value of a concurrent safety game, that is, the maximal probability with which player 1 can enforce the safety objective. The algorithm yields a sequence of player-1 strategies which ensure probabilities of winning that converge monotonically to the value of the safety game. The significance of the result is twofold. First, while strategy improvement algorithms were known for Markov decision processes and turn-based games, as well as for concurrent reachability games, this is the first strategy improvement algorithm for concurrent safety games. Second, and most importantly, the improvement algorithm provides a way to approximate the value of a concurrent safety game from below (the known value-iteration algorithms approximate the value from above). Thus, when used together with value-iteration algorithms, or with strategy improvement algorithms for reachability games, our algorithm leads to the first practical algorithm for computing converging upper and lower bounds for the value of reachability and safety games.Comment: 19 pages, 1 figur

Magnifying Lens Abstraction for Stochastic Games with Discounted and Long-run Average Objectives

Turn-based stochastic games and its important subclass Markov decision processes (MDPs) provide models for systems with both probabilistic and nondeterministic behaviors. We consider turn-based stochastic games with two classical quantitative objectives: discounted-sum and long-run average objectives. The game models and the quantitative objectives are widely used in probabilistic verification, planning, optimal inventory control, network protocol and performance analysis. Games and MDPs that model realistic systems often have very large state spaces, and probabilistic abstraction techniques are necessary to handle the state-space explosion. The commonly used full-abstraction techniques do not yield space-savings for systems that have many states with similar value, but does not necessarily have similar transition structure. A semi-abstraction technique, namely Magnifying-lens abstractions (MLA), that clusters states based on value only, disregarding differences in their transition relation was proposed for qualitative objectives (reachability and safety objectives). In this paper we extend the MLA technique to solve stochastic games with discounted-sum and long-run average objectives. We present the MLA technique based abstraction-refinement algorithm for stochastic games and MDPs with discounted-sum objectives. For long-run average objectives, our solution works for all MDPs and a sub-class of stochastic games where every state has the same value

Taming denumerable Markov decision processes with decisiveness

Decisiveness has proven to be an elegant concept for denumerable Markov chains: it is general enough to encompass several natural classes of denumerable Markov chains, and is a sufficient condition for simple qualitative and approximate quantitative model checking algorithms to exist. In this paper, we explore how to extend the notion of decisiveness to Markov decision processes. Compared to Markov chains, the extra non-determinism can be resolved in an adversarial or cooperative way, yielding two natural notions of decisiveness. We then explore whether these notions yield model checking procedures concerning the infimum and supremum probabilities of reachability properties

Computer Aided Verification

This open access two-volume set LNCS 10980 and 10981 constitutes the refereed proceedings of the 30th International Conference on Computer Aided Verification, CAV 2018, held in Oxford, UK, in July 2018. The 52 full and 13 tool papers presented together with 3 invited papers and 2 tutorials were carefully reviewed and selected from 215 submissions. The papers cover a wide range of topics and techniques, from algorithmic and logical foundations of verification to practical applications in distributed, networked, cyber-physical, and autonomous systems. They are organized in topical sections on model checking, program analysis using polyhedra, synthesis, learning, runtime verification, hybrid and timed systems, tools, probabilistic systems, static analysis, theory and security, SAT, SMT and decisions procedures, concurrency, and CPS, hardware, industrial applications

Computer Aided Verification

This open access two-volume set LNCS 10980 and 10981 constitutes the refereed proceedings of the 30th International Conference on Computer Aided Verification, CAV 2018, held in Oxford, UK, in July 2018. The 52 full and 13 tool papers presented together with 3 invited papers and 2 tutorials were carefully reviewed and selected from 215 submissions. The papers cover a wide range of topics and techniques, from algorithmic and logical foundations of verification to practical applications in distributed, networked, cyber-physical, and autonomous systems. They are organized in topical sections on model checking, program analysis using polyhedra, synthesis, learning, runtime verification, hybrid and timed systems, tools, probabilistic systems, static analysis, theory and security, SAT, SMT and decisions procedures, concurrency, and CPS, hardware, industrial applications