Parametric Markov Chains: PCTL Complexity and Fraction-free Gaussian Elimination

TL;DR

This paper improves the computation of reachability probabilities in parametric Markov chains using fraction-free Gaussian elimination and analyzes the complexity of PCTL model checking, identifying tractable cases and hardness results.

Contribution

It introduces an improved method for computing rational functions in parametric Markov chains and provides a complexity analysis of PCTL model checking with new algorithmic results.

Findings

Fraction-free Gaussian elimination enhances rational function computation.

Exponential-time algorithm for model checking PCTL formulas.

NP-hardness results for certain PCTL fragments.

Abstract

Parametric Markov chains have been introduced as a model for families of stochastic systems that rely on the same graph structure, but differ in the concrete transition probabilities. The latter are specified by polynomial constraints for the parameters. Among the tasks typically addressed in the analysis of parametric Markov chains are (1) the computation of closed-form solutions for reachabilty probabilities and other quantitative measures and (2) finding symbolic representations of the set of parameter valuations for which a given temporal logical formula holds as well as (3) the decision variant of (2) that asks whether there exists a parameter valuation where a temporal logical formula holds. Our contribution to (1) is to show that existing implementations for computing rational functions for reachability probabilities or expected costs in parametric Markov chains can be improved by using fraction-free Gaussian elimination, a long-known technique for linear equation systems with parametric coefficients. Our contribution to (2) and (3) is a complexity-theoretic discussion of the model checking problem for parametric Markov chains and probabilistic computation tree logic (PCTL) formulas. We present an exponential-time algorithm for (2) and a PSPACE upper bound for (3). Moreover, we identify fragments of PCTL and subclasses of parametric Markov chains where (1) and (3) are solvable in polynomial time and establish NP-hardness for other PCTL fragments.