An evaluation of approximate probabilistic reachability techniques for stochastic parametric hybrid systems

Abstract

Stochastic parametric hybrid systems allow formalising automata with discrete interruptions, continuous nonlinear dynamics and parametric uncertainty (e.g. randomness and/or nondeterminism), and are a useful framework for cyber-physical systems modelling. The problem of designing safe cyber-physical systems is very timely, given that such systems are ubiquitous in modern society, often in safety-critical contexts (e.g., aircraft and cars) with possibly some level of decisional autonomy. Therefore, the verification of cyber-physical systems (and consequently of hybrid systems) is a problem urgently demanding innovative solutions. Unfortunately, this problem is also extremely challenging. Reachability checking is a crucial element of designing safe systems. Given a system model, we specify a set of "goal" states (indicating (un)wanted behaviour) and ask whether the system evolution can reach these states or not. Probabilistic reachability is the corresponding problem for stochastic systems, and it amounts to computing the probability that the system reaches a goal state. The main problem researched in this thesis is probabilistic reachability analysis of hybrid systems with random and/or nondeterministic parameters. For nondeterministic systems, this problem amounts to computing a range of reachability probabilities depending on how nondeterminism is resolved. In this thesis I have investigated and developed three distinct techniques: Statistical methods, involving Monte Carlo, Quasi-Monte Carlo and Randomised Quasi-Monte Carlo sampling with interval estimation techniques which give statistical guarantees; An analytical approximation method, utilising Gaussian Processes that offer a statistical approximation for an (unknown) smooth function over its entire domain; A promising combination of a formal approach, based on formal reasoning which provides absolute numerical guarantees, and the Gaussian Regression method. This research offers contributions on two different levels to the verification of stochastic parametric hybrid systems. From a theoretical point of view, it offers a proof that the reachability probability function is a smooth function of the uncertain parameters of the model, and hence Gaussian Processes techniques can be used to obtain an efficient analytical approximation of the function. From a practical point of view, I have implemented all the above described statistical and approximation techniques as part of the publicly available ProbReach tool, including a Gaussian Process Expectation Propagation algorithm that performs Gaussian Process classification and regression for uni-variate and multiple class labels. My empirical evaluation of the presented techniques to a number of case studies has shown a great Gaussian Process approach advantage with respect to standard statistical model checking techniques.