Perturbation Analysis in Verification of Discrete-Time Markov Chains

Abstract

Abstract. Perturbation analysis in probabilistic verification addresses the robustness and sensitivity problem for verification of stochastic models against qualitative and quantitative properties. We identify two types of perturbation bounds, namely non-asymptotic bounds and asymptotic bounds. Non-asymptotic bounds are exact, pointwise bounds that quantify the upper and lower bounds of the verification result subject to a given perturbation of the model, whereas asymptotic bounds are closed-form bounds that approximate non-asymptotic bounds by assuming that the given perturbation is sufficiently small. We perform perturbation analysis in the setting of Discrete-time Markov Chains. We consider three basic matrix norms to capture the perturbation distance, and focus on the computational aspect. Our main contributions include algorithms and tight complexity bounds for calculating both non-asymptotic bounds and asymptotic bounds with respect to the three perturbation distances.

Asymptotic Perturbation Bounds for Probabilistic Model Checking with Empirically Determined Probability Parameters

“…Section 8 concludes the paper. Preliminary results in the paper have been reported in three previous conference papers [17], [18], [19].…”

“…First, they enjoy simple closed forms that uniformly characterize the sensitivity and robustness of a verification result, regardless of the actual model perturbation. Second, their computation has relatively low complexity upper-bound (compared with the point-wise exact bounds [19]) and can employ efficient numerical iteration methods in practice.…”

“…Indeed, this relationship has been exploited in our previous work [19] to compute pointwise exact bounds for the probability function. However, point-wise bounds are not in closed form, and thus are less informative and useful for characterizing the consequences of model perturbations on its verification if the value ranges of the parameters are unknown.…”

Quantitative Verification for Monitoring Event-Streaming Systems

“…Asymptotic perturbation analysis [6], [29] extends the standard matrix-iteration methods (e.g., the power method, Jacobi method and Gauss-Seidel method) in probabilistic model checking to compute the partial derivatives of a verification problem. This approach aims to estimate an accurate worst-case bound for the verification output when the model parameters are subject to small perturbations.…”

“…This approach relies on a symbolic computation technique called parametric model checking, [9] which computes a closed-form rational function for all possible output values. Different from the symbolic computation, another approach called asymptotic perturbation analysis [6], [29] extends the standard matrix-iteration method of probabilistic model checking to compute the partial derivatives of an output value against the perturbed parameters. However, these approaches do not address the aforementioned statistical significance problem.…”