Parametric Markov chains: PCTL complexity and fraction-free Gaussian elimination

Christel BaierChristel Baier, Christian HenselChristian Hensel, Lisa HutschenreiterLisa Hutschenreiter, Sebastian JungesSebastian Junges, Joost-Pieter KatoenJoost-Pieter Katoen, Joachim KleinJoachim Klein

$Parametric Markov chains: PCTL complexity and fraction-free Gaussian elimination$

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Christel Baier, Christian Hensel, Lisa Hutschenreiter, Sebastian Junges, Joost-Pieter Katoen, Joachim Klein
Parametric Markov chains: PCTL complexity and fraction-free Gaussian elimination
Information and Computation, 272:104504, 2020

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KurzfassungAbstract
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 over a finite set of parameters. Important tasks in the analysis of parametric Markov chains are (1) computing closed-form solutions for reachability 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.
Weitere Informationen unter:Further Information: Link
Forschungsgruppe:Research Group: Algebraische und logische Grundlagen der InformatikAlgebraic and Logical Foundations of Computer Science

@article{BHHJKK2020,
  author  = {Christel Baier and Christian Hensel and Lisa Hutschenreiter and
             Sebastian Junges and Joost-Pieter Katoen and Joachim Klein},
  title   = {Parametric Markov chains: {PCTL} complexity and fraction-free
             Gaussian elimination},
  journal = {Information and Computation},
  volume  = {272},
  year    = {2020},
  pages   = {104504},
  doi     = {https://doi.org/10.1016/j.ic.2019.104504}
}