This paper presents a simple algorithm to check whether reachability probabilities in parametric Markov chains are monotonic in (some of) the parameters. The idea is to construct-only using the graph structure of the Markov chain and local transition probabilities-a pre-order on the states. Our algorithm cheaply checks a sufficient condition for monotonicity. Experiments show that monotonicity in several benchmarks is automatically detected, and monotonicity can speed up parameter synthesis up to orders of magnitude faster than a symbolic baseline.
Parametric Markov chains (pMCs) are Markov chains with symbolic (aka: parametric) transition probabilities. They are a convenient operational model to treat robustness against uncertainties. A typical objective is to find the parameter values that maximize the reachability of some target states. In this paper, we consider automatically proving robustness, that is, an $$\varepsilon $$
ε
-close upper bound on the maximal reachability probability. The result of our procedure actually provides an almost-optimal parameter valuation along with this upper bound.We propose to tackle these ETR-hard problems by a tight combination of two significantly different techniques: monotonicity checking and parameter lifting. The former builds a partial order on states to check whether a pMC is (local or global) monotonic in a certain parameter, whereas parameter lifting is an abstraction technique based on the iterative evaluation of pMCs without parameter dependencies. We explain our novel algorithmic approach and experimentally show that we significantly improve the time to determine almost-optimal synthesis.
We consider probabilistic timed automata (PTA) in which probabilities can be parameters, i.e. symbolic constants. They are useful to model randomised real-time systems where exact probabilities are unknown, or where the probability values should be optimised. We prove that existing techniques to transform probabilistic timed automata into equivalent finite-state Markov decision processes (MDPs) remain correct in the parametric setting, using a systematic proof pattern. We implemented two of these parameter-preserving transformations-using digital clocks and backwards reachability-in the Modest Toolset. Using Storm's parameter space partitioning approach, parameter values can be efficiently synthesized in the resulting parametric MDPs. We use several case studies from the literature of varying state and parameter space sizes to experimentally evaluate the performance and scalability of this novel analysis trajectory for parametric PTA.
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