Regular Expressions for PCTL Counterexamples

Abstract

Counterexamples for probabilistic reachability in Markov chains are sets of paths that all reach a goal state and whose cumulative likelihood exceeds a threshold. This paper is concerned with the issue of how to conveniently represent these sets. Experiments, partially substantiated with combinatorial arguments, show that the cardinality of such sets may be excessive. To obtain compact representations of counterexamples we suggest to use regular expressions. We present a simple algorithm to generate minimal regular expressions and adopt a recursive scheme to determine their likelihood. Markov chain reduction prior to counterexample generation may yield even shorter regular expressions. The feasibility of the approach is illustrated by means two protocols: leader election and the Crowds protocol.

Computing the Leakage of Information-Hiding Systems

“…In [9], regular expressions containing probability values are used to reason about traces in Markov Chains. This idea is used in [8] in the context of counterexample generation to group together paths with the same observable behaviour. The regular expression there are over pairs p, q with p a probability value and q a state, to be able to track both probabilities and observables.…”

“…Regular expressions represent sets of paths as in [8]. We also take the probability value of such a regular expression from that paper.…”

“…Indeed, this is the aim of the counterexamples generation techniques previously presented. For further details on how to debug sets of paths and why these techniques meet that purpose we refer to [1,8,2]. Example 5.8.…”

Probabilistic Reachability for Parametric Markov Models

“…For parametric continuous-time Markov chains, Han et al [12] have provided approximation algorithms to find valid valuations with respect to time bounded reachability properties. Recently, Damman et al [7] have extended the approach of [8] to generate counterexamples for MC. The regular expressions generated can be seen as a more compact and structured representation of counterexamples than providing a set of paths.…”

Diagnosis, synthesis and analysis of probabilistic models

“…. , s·(u·s) k−1 ·u·t, where (u·s) i is a short form of traversing the loop s·u·s for i times and k is the smallest integer such To illustrate that such phenomena also occur in real-life cases, we made a prototypical implementation (in Python [Dam08]) to enable generating counterexamples for more practical case studies. Our implementation uses the same input format as the probabilistic model checker MRMC [KKZ05].…”