Confidence Bounds for Statistical Model Checking of Probabilistic Hybrid Systems

Abstract

Model checking of technical systems is a common and demanding task. The behavior of such systems can often be characterized using hybrid automata, leading to verification problems within the first-order logic over the reals. The applicability of logic-based formalisms to a wider range of systems has recently been increased by introducing quantifiers into satisfiability modulo theory (SMT) approaches to solve such problems, especially randomized quantifiers, resulting in stochastic satisfiability modulo theory (SSMT). These quantifiers combine non-determinism and stochasticity, thereby allowing to represent models such as Markov decision processes. While algorithms for exact model checking in this setting exist, their scalability is limited due to the computational complexity which increases with the number of quantified variables. Additionally, these methods become infeasible if the domain of the quantified variables, randomized variables in particular, becomes too large or even infinite. In this paper, we present an approximation algorithm based on confidence intervals obtained from sampling which allow for an explicit trade-off between accuracy and computational effort. Although the algorithm gives only approximate results in terms of confidence intervals, it is still guaranteed to converge to the exact solution. To further increase the performance of the algorithm, we adopt search strategies based on the upper bound confidence algorithm UCB, originally used to solve a similar problem, the multi-armed bandit. Preliminary results show that the proposed algorithm can improve the performance in comparison to existing SSMT solvers, especially in the presence of many randomized quantified variables.