Automatic Verification of Concurrent Stochastic Systems

Automated verification techniques for stochastic games allow formal reasoning about systems that feature competitive or collaborative behaviour among rational agents in uncertain or probabilistic settings. Existing tools and techniques focus on turn-based games, where each state of the game is controlled by a single player, and on zero-sum properties, where two players or coalitions have directly opposing objectives. In this paper, we present automated verification techniques for concurrent stochastic games (CSGs), which provide a more natural model of concurrent decision making and interaction. We also consider (social welfare) Nash equilibria, to formally identify scenarios where two players or coalitions with distinct goals can collaborate to optimise their joint performance. We propose an extension of the temporal logic rPATL for specifying quantitative properties in this setting and present corresponding algorithms for verification and strategy synthesis for a variant of stopping games. For finite-horizon properties the computation is exact, while for infinite-horizon it is approximate using value iteration. For zero-sum properties it requires solving matrix games via linear programming, and for equilibria-based properties we find social welfare or social cost Nash equilibria of bimatrix games via the method of labelled polytopes through an SMT encoding. We implement this approach in PRISM-games, which required extending the tool's modelling language for CSGs, and apply it to case studies from domains including robotics, computer security and computer networks, explicitly demonstrating the benefits of both CSGs and equilibria-based properties

Qualitative Analysis of Partially-observable Markov Decision Processes

We study observation-based strategies for partially-observable Markov decision processes (POMDPs) with omega-regular objectives. An observation-based strategy relies on partial information about the history of a play, namely, on the past sequence of observations. We consider the qualitative analysis problem: given a POMDP with an omega-regular objective, whether there is an observation-based strategy to achieve the objective with probability~1 (almost-sure winning), or with positive probability (positive winning). Our main results are twofold. First, we present a complete picture of the computational complexity of the qualitative analysis of POMDP s with parity objectives (a canonical form to express omega-regular objectives) and its subclasses. Our contribution consists in establishing several upper and lower bounds that were not known in literature. Second, we present optimal bounds (matching upper and lower bounds) on the memory required by pure and randomized observation-based strategies for the qualitative analysis of POMDP s with parity objectives and its subclasses

Probabilistic model checking for strategic equilibria-based decision making:advances and challenges

Game-theoretic concepts have been extensively studied in economics to provide insight into competitive behaviour and strategic decision making. As computing systems increasingly involve concurrently acting autonomous agents, game-theoretic approaches are becoming widespread in computer science as a faithful modelling abstraction. These techniques can be used to reason about the competitive or collaborative behaviour of multiple rational agents with distinct goals or objectives. This paper provides an overview of recent advances in developing a modelling, verification and strategy synthesis framework for concurrent stochastic games implemented in the probabilistic model checker PRISM-games. This is based on a temporal logic that supports finite- and infinite-horizon temporal properties in both a zero-sum and nonzero-sum setting, the latter using Nash and correlated equilibria with respect to two optimality criteria, social welfare and social fairness. We summarise the key concepts, logics and algorithms and the currently available tool support. Future challenges and recent progress in adapting the framework and algorithmic solutions to continuous environments and neural networks are also outlined

Verification and Control of Turn-Based Probabilistic Real-Time Games

Quantitative verification techniques have been developed for the formal analysis of a variety of probabilistic models, such as Markov chains, Markov decision process and their variants. They can be used to produce guarantees on quantitative aspects of system behaviour, for example safety, reliability and performance, or to help synthesise controllers that ensure such guarantees are met. We propose the model of turn-based probabilistic timed multi-player games, which incorporates probabilistic choice, real-time clocks and nondeterministic behaviour across multiple players. Building on the digital clocks approach for the simpler model of probabilistic timed automata, we show how to compute the key measures that underlie quantitative verification, namely the probability and expected cumulative price to reach a target. We illustrate this on case studies from computer security and task scheduling

Verification and Control of Partially Observable Probabilistic Real-Time Systems

We propose automated techniques for the verification and control of probabilistic real-time systems that are only partially observable. To formally model such systems, we define an extension of probabilistic timed automata in which local states are partially visible to an observer or controller. We give a probabilistic temporal logic that can express a range of quantitative properties of these models, relating to the probability of an event's occurrence or the expected value of a reward measure. We then propose techniques to either verify that such a property holds or to synthesise a controller for the model which makes it true. Our approach is based on an integer discretisation of the model's dense-time behaviour and a grid-based abstraction of the uncountable belief space induced by partial observability. The latter is necessarily approximate since the underlying problem is undecidable, however we show how both lower and upper bounds on numerical results can be generated. We illustrate the effectiveness of the approach by implementing it in the PRISM model checker and applying it to several case studies, from the domains of computer security and task scheduling

Solving Stochastic B\"uchi Games on Infinite Arenas with a Finite Attractor

We consider games played on an infinite probabilistic arena where the first player aims at satisfying generalized B\"uchi objectives almost surely, i.e., with probability one. We provide a fixpoint characterization of the winning sets and associated winning strategies in the case where the arena satisfies the finite-attractor property. From this we directly deduce the decidability of these games on probabilistic lossy channel systems.Comment: In Proceedings QAPL 2013, arXiv:1306.241

IST Austria Thesis

This dissertation concerns the automatic verification of probabilistic systems and programs with arrays by statistical and logical methods. Although statistical and logical methods are different in nature, we show that they can be successfully combined for system analysis. In the first part of the dissertation we present a new statistical algorithm for the verification of probabilistic systems with respect to unbounded properties, including linear temporal logic. Our algorithm often performs faster than the previous approaches, and at the same time requires less information about the system. In addition, our method can be generalized to unbounded quantitative properties such as mean-payoff bounds. In the second part, we introduce two techniques for comparing probabilistic systems. Probabilistic systems are typically compared using the notion of equivalence, which requires the systems to have the equal probability of all behaviors. However, this notion is often too strict, since probabilities are typically only empirically estimated, and any imprecision may break the relation between processes. On the one hand, we propose to replace the Boolean notion of equivalence by a quantitative distance of similarity. For this purpose, we introduce a statistical framework for estimating distances between Markov chains based on their simulation runs, and we investigate which distances can be approximated in our framework. On the other hand, we propose to compare systems with respect to a new qualitative logic, which expresses that behaviors occur with probability one or a positive probability. This qualitative analysis is robust with respect to modeling errors and applicable to many domains. In the last part, we present a new quantifier-free logic for integer arrays, which allows us to express counting. Counting properties are prevalent in array-manipulating programs, however they cannot be expressed in the quantified fragments of the theory of arrays. We present a decision procedure for our logic, and provide several complexity results