Approximative Symbolic Model Checking of Continuous-Time Markov Chains

Abstract

This paper presents a symbolic model checking algorithm for continuous-time Markov chains for an extension of the continuous stochastic logic CSL of Aziz et al [1]. The considered logic contains a time-bounded until-operator and a novel operator to express steadystate probabilities. We show that the model checking problem for this logic reduces to a system of linear equations (for unbounded until and the steady state-operator) and a Volterra integral equation system for timebounded until. We propose a symbolic approximate method for solving the integrals using MTDDs (multi-terminal decision diagrams), a generalisation of MTBDDs. These new structures are suitable for numerical integration using quadrature formulas based on equally-spaced abscissas, like trapezoidal, Simpson and Romberg integration schemes.

Faster and Symbolic CTMC Model Checking

“…A (labelled) CTMC C is a tuple (S, R, L) where S and L are as for DTMCs, and R : S × S → IR 0 is the rate matrix. (We adopt the same conventions as in [6,7], i.e. we do allow self-loops.)…”

“…The temporal logic CSL (Continuous Stochastic Logic) developed originally by Aziz et al [2,3] and extended by Baier et al [7] provides a powerful means to specify path-based as well as traditional state-based measures on CTMCs in a concise, flexible and unambiguous way. CSL is based on the well-known branching-time temporal logic CTL (Computation Tree Logic [11]) and PCTL (Probabilistic CTL [17]); a steady-state operator, a time-bounded until, and a probabilistic (path) operator constitute its main ingredients.…”

“…The model checking problem for CSL is decidable for rational time bounds [2,3]. Approximate CSL model-checking algorithms have been studied in [7] where the satisfaction of time-bounded until formulas is shown to be based on solving a (recursive) Volterra equation system. More recently, Baier et al [6] reduced verifying time-bounded until formulas to the problem of computing transient-state probabilities for CTMCs.…”

“…This reduces to graph analysis and iterative matrix-vector multiplication which can be implemented with standard MTBDD operations. Variants of MTBDDs tailored to numerical integration [7] are not needed. This paper reports on the implementation of symbolic CSL model checking as part of PRISM 1 (PRobabilistIc Symbolic Model checker), a prototype tool for the symbolic verification of Markov decision processes (MDPs) with DTMCs as a subset thereof.…”

“…This paper reports on the implementation and the experiments with symbolic model checking of continuous-time Markov chains using multi-terminal binary decision diagrams (MTBDDs). Properties are expressed in Continuous Stochastic Logic (CSL) [7] which includes the means to express both transient and steady-state performance measures. We show that all CSL operators can be treated using standard operations on MTBDDs, thus allowing a rather straightforward implementation of symbolic CSL model checking on existing MTBDD-based platforms such as the verifier PRISM.…”