Abstract. We give a simple and efficient method to prove safety properties for parameterized systems with linear topologies. A process in the system is a finite-state automaton, where the transitions are guarded by both local and global conditions. Processes may communicate via broadcast, rendez-vous and shared variables. The method derives an overapproximation of the induced transition system, which allows the use of a simple class of regular expressions as a symbolic representation. Compared to traditional regular model checking methods, the analysis does not require the manipulation of transducers, and hence its simplicity and efficiency. We have implemented a prototype which works well on several mutual exclusion algorithms and cache coherence protocols.

Abstract. We consider qualitative and quantitative verification problems for infinitestate Markov chains. We call a Markov chain decisive w.r.t. a given set of target states F if it almost certainly eventually reaches either F or a state from which F can no longer be reached. While all finite Markov chains are trivially decisive (for every set F ), this also holds for many classes of infinite Markov chains.Infinite Markov chains which contain a finite attractor are decisive w.r.t. every set F . In particular, all Markov chains induced by probabilistic lossy channel systems (PLCS) contain a finite attractor and are thus decisive. Furthermore, all globally coarse Markov chains are decisive. The class of globally coarse Markov chains includes, e.g., those induced by probabilistic vector addition systems (PVASS) with upward-closed sets F , and all Markov chains induced by probabilistic noisy Turing machines (PNTM) (a generalization of the noisy Turing machines (NTM) of Asarin and Collins).We consider both safety and liveness problems for decisive Markov chains. Safety: What is the probability that a given set of states F is eventually reached. Liveness: What is the probability that a given set of states is reached infinitely often. There are three variants of these questions. (1) The qualitative problem, i.e., deciding if the probability is one (or zero); (2) the approximate quantitative problem, i.e., computing the probability up-to arbitrary precision; (3) the exact quantitative problem, i.e., computing probabilities exactly.1. We express the qualitative problem in abstract terms for decisive Markov chains, and show an almost complete picture of its decidability for PLCS, PVASS and PNTM.2. We also show that the path enumeration algorithm of Iyer and Narasimha terminates for decisive Markov chains and can thus be used to solve the approximate quantitative safety problem. A modified variant of this algorithm can be used to solve the approximate quantitative liveness problem.3. Finally, we show that the exact probability of (repeatedly) reaching F cannot be effectively expressed (in a uniform way) in Tarski-algebra for either PLCS, PVASS or (P)NTM (unlike for probabilistic pushdown automata).2000 ACM Subject Classification: G3, D2.4, F4.1.

Abstract

Abstract. We consider turn-based stochastic games on infinite graphs induced by game probabilistic lossy channel systems (GPLCS), the game version of probabilistic lossy channel systems (PLCS). We study games with Büchi (repeated reachability) objectives and almost-sure winning conditions. These games are pure memoryless determined and, under the assumption that the target set is regular, a symbolic representation of the set of winning states for each player can be effectively constructed. Thus, turn-based stochastic games on GPLCS are decidable. This generalizes the decidability result for PLCS-induced Markov decision processes in [10].

We introduce the simple and efficient method of monotonic abstraction to prove safety properties for parameterized systems with linear topologies. A process in the system is a finite-state automaton, where the transitions are guarded by both local and global conditions. Processes may communicate via broadcast, rendez-vous and shared variables over finite domains. The method of monotonic abstraction derives an over-approximation of the induced transition system that allows the use of a simple class of regular expressions as a symbolic representation. Compared to traditional regular model checking methods, the analysis does not require the manipulation of transducers, and hence its simplicity and efficiency. We have implemented a prototype that works well on several mutual exclusion algorithms and cache coherence protocols.