The Decidability Frontier for Probabilistic Automata on Infinite Words

We consider probabilistic automata on infinite words with acceptance defined by safety, reachability, B\"uchi, coB\"uchi, and limit-average conditions. We consider quantitative and qualitative decision problems. We present extensions and adaptations of proofs for probabilistic finite automata and present a complete characterization of the decidability and undecidability frontier of the quantitative and qualitative decision problems for probabilistic automata on infinite words

IST Austria Technical Report

We consider probabilistic automata on infinite words with acceptance defined by safety, reachability, Büchi, coBüchi and limit-average conditions. We consider quantitative and qualitative decision problems. We present extensions and adaptations of proofs of [GO09] and present a precise characterization of the decidability and undecidability frontier of the quantitative and qualitative decision problems

On the Complexity of Branching Games with Regular Conditions

Infinite duration games with regular conditions are one of the crucial tools in the areas of verification and synthesis. In this paper we consider a branching variant of such games - the game contains branching vertices that split the play into two independent sub-games. Thus, a play has the form of~an~infinite tree. The winner of the play is determined by a winning condition specified as a set of infinite trees. Games of this kind were used by Mio to provide a game semantics for the probabilistic mu-calculus. He used winning conditions defined in terms of parity games on trees. In this work we consider a more general class of winning conditions, namely those definable by finite automata on infinite trees. Our games can be seen as a branching-time variant of the stochastic games on graphs. We address the question of determinacy of a branching game and the problem of computing the optimal game value for each of the players. We consider both the stochastic and non-stochastic variants of the games. The questions under consideration are parametrised by the family of strategies we allow: either mixed, behavioural, or pure. We prove that in general, branching games are not determined under mixed strategies. This holds even for topologically simple winning conditions (differences of two open sets) and non-stochastic arenas. Nevertheless, we show that the games become determined under mixed strategies if we restrict the winning conditions to open sets of trees. We prove that the problem of comparing the game value to a rational threshold is undecidable for branching games with regular conditions in all non-trivial stochastic cases. In the non-stochastic cases we provide exact bounds on the complexity of the problem. The only case left open is the 0-player stochastic case, i.e. the problem of computing the measure of a given regular language of infinite trees

Alternating Tree Automata with Qualitative Semantics

We study alternating automata with qualitative semantics over infinite binary trees: Alternation means that two opposing players construct a decoration of the input tree called a run, and the qualitative semantics says that a run of the automaton is accepting if almost all branches of the run are accepting. In this article, we prove a positive and a negative result for the emptiness problem of alternating automata with qualitative semantics. The positive result is the decidability of the emptiness problem for the case of Büchi acceptance condition. An interesting aspect of our approach is that we do not extend the classical solution for solving the emptiness problem of alternating automata, which first constructs an equivalent non-deterministic automaton. Instead, we directly construct an emptiness game making use of imperfect information. The negative result is the undecidability of the emptiness problem for the case of co-Büchi acceptance condition. This result has two direct consequences: The undecidability of monadic second-order logic extended with the qualitative path-measure quantifier and the undecidability of the emptiness problem for alternating tree automata with non-zero semantics, a recently introduced probabilistic model of alternating tree automata

A Survey on Continuous Time Computations

We provide an overview of theories of continuous time computation. These theories allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuous time analog models. We survey the existing models, summarizing results, and point to relevant references in the literature

Decisiveness of Stochastic Systems and its Application to Hybrid Models (Full Version)

In [ABM07], Abdulla et al. introduced the concept of decisiveness, an interesting tool for lifting good properties of finite Markov chains to denumerable ones. Later, this concept was extended to more general stochastic transition systems (STSs), allowing the design of various verification algorithms for large classes of (infinite) STSs. We further improve the understanding and utility of decisiveness in two ways. First, we provide a general criterion for proving decisiveness of general STSs. This criterion, which is very natural but whose proof is rather technical, (strictly) generalizes all known criteria from the literature. Second, we focus on stochastic hybrid systems (SHSs), a stochastic extension of hybrid systems. We establish the decisiveness of a large class of SHSs and, under a few classical hypotheses from mathematical logic, we show how to decide reachability problems in this class, even though they are undecidable for general SHSs. This provides a decidable stochastic extension of o-minimal hybrid systems. [ABM07] Parosh A. Abdulla, Noomene Ben Henda, and Richard Mayr. 2007. Decisive Markov Chains. Log. Methods Comput. Sci. 3, 4 (2007).Comment: Full version of GandALF 2020 paper (arXiv:2001.04347v2), updated version of arXiv:2001.04347v1. 30 pages, 6 figure

Approximating values of generalized-reachability stochastic games

Simple stochastic games are turn-based 2½-player games with a reachability objective. The basic question asks whether one player can ensure reaching a given target with at least a given probability. A natural extension is games with a conjunction of such conditions as objective. Despite a plethora of recent results on the analysis of systems with multiple objectives, the decidability of this basic problem remains open. In this paper, we present an algorithm approximating the Pareto frontier of the achievable values to a given precision. Moreover, it is an anytime algorithm, meaning it can be stopped at any time returning the current approximation and its error bound

A unified view of parameterized verification of abstract models of broadcast communication

We give a unified view of different parameterized models of concurrent and distributed systems with broadcast communication based on transition systems. Based on the resulting formal models, we discuss related verification methods and tools based on abstractions and symbolic state exploration