Parametric Markov Chains: PCTL Complexity and Fractionfree Gaussian Elimination
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 closedform 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 fractionfree Gaussian elimination, a longknown technique for linear equation systems with parametric coefficients. Our contribution to (2) and (3) is a complexitytheoretic discussion of the model checking problem for parametric Markov chains and probabilistic computation tree logic (PCTL) formulas. We present an exponentialtime 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 NPhardness for other PCTL fragments.
 Publication:

arXiv eprints
 Pub Date:
 September 2017
 DOI:
 10.48550/arXiv.1709.02093
 arXiv:
 arXiv:1709.02093
 Bibcode:
 2017arXiv170902093H
 Keywords:

 Computer Science  Logic in Computer Science
 EPrint:
 In Proceedings GandALF 2017, arXiv:1709.01761