Sound approximate and asymptotic probabilistic bisimulations for PCTL

TL;DR

This paper introduces a soundness framework for approximate probabilistic bisimulations in PCTL, providing linear error bounds and extending to asymptotic equivalences for families of states.

Contribution

It proposes a new parametric bisimilarity notion for PCTL, establishes its soundness, and extends it to asymptotic equivalences for state families.

Findings

Soundness of approximate bisimilarity with respect to PCTL established.

Error bounds depend linearly on the observation depth n.

Asymptotic equivalence preserves satisfaction of PCTL formulas.

Abstract

We tackle the problem of establishing the soundness of approximate bisimilarity with respect to PCTL and its relaxed semantics. To this purpose, we consider a notion of bisimilarity inspired by the one introduced by Desharnais, Laviolette, and Tracol, and parametric with respect to an approximation error $\delta$, and to the depth $n$ of the observation along traces. Essentially, our soundness theorem establishes that, when a state $q$ satisfies a given formula up-to error $\delta$ and steps $n$, and $q$ is bisimilar to $q'$ up-to error $\delta'$ and enough steps, we prove that $q'$ also satisfies the formula up-to a suitable error $\delta"$ and steps $n$. The new error $\delta"$ is computed from $\delta$, $\delta'$ and the formula, and only depends linearly on $n$. We provide a detailed overview of our soundness proof. We extend our bisimilarity notion to families of states, thus obtaining an asymptotic equivalence on such families. We then consider an asymptotic satisfaction relation for PCTL formulae, and prove that asymptotically equivalent families of states asymptotically satisfy the same formulae.