Approximate probabilistic verification of hybrid systems

Hybrid systems whose mode dynamics are governed by non-linear ordinary differential equations (ODEs) are often a natural model for biological processes. However such models are difficult to analyze. To address this, we develop a probabilistic analysis method by approximating the mode transitions as stochastic events. We assume that the probability of making a mode transition is proportional to the measure of the set of pairs of time points and value states at which the mode transition is enabled. To ensure a sound mathematical basis, we impose a natural continuity property on the non-linear ODEs. We also assume that the states of the system are observed at discrete time points but that the mode transitions may take place at any time between two successive discrete time points. This leads to a discrete time Markov chain as a probabilistic approximation of the hybrid system. We then show that for BLTL (bounded linear time temporal logic) specifications the hybrid system meets a specification iff its Markov chain approximation meets the same specification with probability $1$. Based on this, we formulate a sequential hypothesis testing procedure for verifying -approximately- that the Markov chain meets a BLTL specification with high probability. Our case studies on cardiac cell dynamics and the circadian rhythm indicate that our scheme can be applied in a number of realistic settings

Probabilistic regular graphs

Deterministic graph grammars generate regular graphs, that form a structural extension of configuration graphs of pushdown systems. In this paper, we study a probabilistic extension of regular graphs obtained by labelling the terminal arcs of the graph grammars by probabilities. Stochastic properties of these graphs are expressed using PCTL, a probabilistic extension of computation tree logic. We present here an algorithm to perform approximate verification of PCTL formulae. Moreover, we prove that the exact model-checking problem for PCTL on probabilistic regular graphs is undecidable, unless restricting to qualitative properties. Our results generalise those of EKM06, on probabilistic pushdown automata, using similar methods combined with graph grammars techniques.Comment: In Proceedings INFINITY 2010, arXiv:1010.611

Distribution, Approximation and Probabilistic Model Checking

AbstractAPMC is a model checker dedicated to the quantitative verification of fully probabilistic systems against LTL formulas. Using a Monte-Carlo method in order to efficiently approximate the verification of probabilistic specifications, it could be used naturally in a distributed framework. We present here the tool and its distribution scheme, together with extensive performance evaluation, showing the scalability of the method, even on clusters containing 500+ heterogeneous workstations

Exact and Approximate Abstraction for Classes of Stochastic Hybrid Systems

A stochastic hybrid system contains a collection of interacting discrete and continuous components, subject to random behaviour. The formal verification of a stochastic hybrid system often comprises a method for the generation of a finite-state probabilistic system which either represents exactly the behaviour of the stochastic hybrid system, or which approximates conservatively its behaviour. We extend such abstraction-based formal verification of stochastic hybrid systems in two ways. Firstly, we generalise previous results by showing how bisimulation-based abstractions of non-probabilistic hybrid automata can be lifted to the setting of probabilistic hybrid automata, a subclass of stochastic hybrid systems in which probabilistic choices can be made with respect to finite, discrete alternatives only. Secondly, we consider the problem of obtaining approximate abstractions for discrete-time stochastic systems in which there are continuous probabilistic choices with regard to the slopes of certain system variables. We restrict our attention to the subclass of such systems in which the approximate abstraction of such a system, obtained using the previously developed techniques of Fraenzle et al., results in a probabilistic rectangular hybrid automaton, from which in turn a finite-state probabilistic system can be obtained. We illustrate this technique with an example, using the probabilistic model checking tool PRISM

PRISM: a tool for automatic verification of probabilistic systems

Probabilistic model checking is an automatic formal verification technique for analysing quantitative properties of systems which exhibit stochastic behaviour. PRISM is a probabilistic model checking tool which has already been successfully deployed in a wide range of application domains, from real-time communication protocols to biological signalling pathways. The tool has recently undergone a significant amount of development. Major additions include facilities to manually explore models, Monte-Carlo discrete-event simulation techniques for approximate model analysis (including support for distributed simulation) and the ability to compute cost- and reward-based measures, e.g. "the expected energy consumption of the system before the first failure occurs". This paper presents an overview of all the main features of PRISM. More information can be found on the website: www.cs.bham.ac.uk/~dxp/prism

Decisive Markov Chains

We consider qualitative and quantitative verification problems for infinite-state 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, this holds for probabilistic lossy channel systems (PLCS). Furthermore, all globally coarse Markov chains are decisive. This class includes probabilistic vector addition systems (PVASS) and probabilistic noisy Turing machines (PNTM). We consider both safety and liveness problems for decisive Markov chains, i.e., the probabilities that a given set of states F is eventually reached or reached infinitely often, respectively. 1. We express the qualitative problems 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 solves 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.Comment: 32 pages, 0 figure

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

Verifying Performance Properties of Probabilistic Inference

In this extended abstract, we discuss the opportunity to formally verify that inference systems for probabilistic programming guarantee good performance. In particular, we focus on hybrid inference systems that combine exact and approximate inference to try to exploit the advantages of each. Their performance depends critically on a) the division between exact and approximate inference, and b) the computational resources consumed by exact inference. We describe several projects in this direction. Semi-symbolic Inference (SSI) is a type of hybrid inference system that provides limited guarantees by construction on the exact/approximate division. In addition to these limited guarantees, we also describe ongoing work to extend guarantees to a more complex class of programs, requiring a program analysis to ensure the guarantees. Finally, we also describe work on verifying that inference systems using delayed sampling -- another type of hybrid inference -- execute in bounded memory. Together, these projects show that verification can deliver the performance guarantees that probabilistic programming languages need

Verification and control of partially observable probabilistic systems

We present automated techniques for the verification and control of partially observable, probabilistic systems for both discrete and dense models of time. For the discrete-time case, we formally model these systems using partially observable Markov decision processes; for dense time, we propose an extension of probabilistic timed automata in which local states are partially visible to an observer or controller. We give probabilistic temporal logics 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 synthesise a controller for the model which makes it true. Our approach is based on a grid-based abstraction of the uncountable belief space induced by partial observability and, for dense-time models, an integer discretisation of real-time behaviour. The former 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 task and network scheduling, computer security and planning