Counterexample Generation in Probabilistic Model Checking

Abstract

Abstract-Providing evidence for the refutation of a property is an essential, if not the most important, feature of model checking. This paper considers algorithms for counterexample generation for probabilistic CTL formulas in discrete-time Markov chains. Finding the strongest evidence (i.e., the most probable path) violating a (bounded) until-formula is shown to be reducible to a single-source (hopconstrained) shortest path problem. Counterexamples of smallest size that deviate most from the required probability bound can be obtained by applying (small amendments to) k-shortest (hop-constrained) paths algorithms. These results can be extended to Markov chains with rewards, to LTL model checking, and are useful for Markov decision processes. Experimental results show that, typically, the size of a counterexample is excessive. To obtain much more compact representations, we present a simple algorithm to generate (minimal) regular expressions that can act as counterexamples. The feasibility of our approach is illustrated by means of two communication protocols: leader election in an anonymous ring network and the Crowds protocol.

Farkas Certificates and Minimal Witnesses for Probabilistic Reachability Constraints

“…Minimal probabilistic counterexamples given as sets of paths can be computed by reframing the problem as a k-shortest-path problem [42,43]. Regular expressions have been considered to succinctly represent the set of paths in [32], and extensions were proposed in [18,75].…”

Compositional Stochastic Model Checking Probabilistic Automata via Assume-guarantee Reasoning

“…(2) It checks the assume-guarantee triple through multi-objective model checking [18]. (3) In the whole learning process, Feng et al [22] adopts the method proposed in Han et al [24] to generate probabilistic counterexamples for refining the current assumption, i.e., the PRISM [25] is used to obtain the error state nodes in the model, and then the probabilistic counterexamples are constructed by using Eppstein's [26] algorithm. (4) The verification problem of a stochastic system composed of n ≥ 2 components is not solved.…”

“…(2) Through the weighted extension of the classical simulation relation, He et al [20] presents a verification method of the assume-guarantee triple containing the weighted assumption. (3) Similarly to Feng et al [22], He et al [20] also constructs the necessary probabilistic counterexamples in the learning process through Han et al [24]. 4The verification problem of a stochastic system composed of n ≥ 2 components is not solved.…”

Debugging of Markov Decision Processes (MDPs) Models

“…Based on these works, they built an open source tool, DiPro [4], for generating indicative counterexamples for DTMCs, CTMCs and MDPs. Similar to the previous works, [20] has proposed the notion of smallest most indicative counterexample that reduces to the problem of finding K shortest paths. Instead of generating path-based counterexamples, the authors in [28] have proposed a novel approach based on critical subsystems.…”

“…In this paper we will focus on probabilistic safety properties with upper threshold. The properties with lower threshold can be easily transformed to properties with upper threshold [6], [20].…”