Statistical model checking for stochastic hybrid systems involving nondeterminism over continuous domains

Abstract

Behavioral verification of technical systems invol ving both discrete and continuous components is a common and demanding task. The behavior of such systems can often be characterized using stochastic hybrid automata, leading to verification problems which can be formalized and solved using stochastic logic calculi such as stochastic satisfiability modulo theory (SSMT). While algorithms for discharging proof obligations in SSMT form exist, their applicability is limited due to the computational complexity, which often increases exponentially with the number of quantified variables. Recently, statistical model checking has been successfully applied to stochastic hybrid systems, thereby increasing the size of the system for which verification problems is tractable. However, being based on randomized simulation, these methods usually cannot handle non-determinism. In previous work, we have deviated from the usual approach of simulating the model and rather proposed a statistical method for SSMT solving which, being based on statistical AI planning algorithms, can also treat non-determinism over a finite domain. Here, we extend this previous work to the case of continuous domains. In particular, using ideas from noisy optimization, we adaptively build up a decision tree recording the findings and guiding further exploration, thereby favoring the currently most promising sub-domain. The non-determinism is resolved by translating the satisfac-C. Ellen (B) · S. Gerwinn Transportation, tion problem into an optimization problem, thereby computing both optimistic and pessimistic bounds on the probability of satisfaction. At each stage of the evaluation process, we show how to obtain confidence statements about the probability of satisfaction for the overall SSMT formula, including reliable estimates on the optimal resolution of any nondeterministic choice involved.

ProbReach

“…However, it supports only discrete random variables, while ProbReach accepts continuous and discrete random initial parameters. A recent work [2] proposes a statistical model checking technique for verifying hybrid systems with continuous nondeterminism, thereby significantly expanding the class of systems analysable. However, the approach is based on statistical planning algorithms from AI, and therefore it cannot offer the absolute guarantees provided by ProbReach.…”

SMT-based Synthesis of Safe and Robust PID Controllers for Stochastic Hybrid Systems

“…There has been a sizable amount of work on tools for formal analysis of probabilistic reachability, although they all have limitations that make them unsuitable for our approach. SiSAT [15] uses an SMT approach for probabilistic hybrid systems with discrete nondeterminism, while continuous nondeterminism is handled via Monte Carlo techniques only [11]; UPPAAL [7] uses statistical model checking to analyze nonlinear stochastic hybrid automata; ProHVer [36] computes upper bounds for maximal reachability probabilities, but continuous random parameters are analyzed via discrete overapproximations [14]; U-Check [5] enables parameter synthesis and statistical model checking of stochastic hybrid systems [4]). However, this approach is based on Gaussian process emulation and optimisation, and provides only statistical guarantees and requires certain smoothness conditions on the satisfaction probability function.…”

Statistically Sound Verification and Optimization for Complex Systems

“…The observation supports H 1 if (7) holds. In this case, the variable count is incremented, terminating when it reaches K. Otherwise, count is reset to 0, and and u are updated to satisfy (7).…”

“…We refer the reader to recent papers by Zuliani et al [8] and Ellen et al [7] that use reinforcement learning techniques to verify the correctness properties under the worst case values of non-deterministic parameters.…”