On Frequency LTL in Probabilistic Systems

TL;DR

This paper introduces a decidable variant of frequency LTL for probabilistic systems, enabling probability computation of formulas with frequency constraints, aligning with the complexity of standard LTL in certain cases.

Contribution

The paper establishes decidability and complexity results for frequency LTL in probabilistic models, especially for Markov chains and restricted Markov decision processes.

Findings

Probability computation for fLTL in Markov chains matches LTL complexity.

For certain MDP restrictions, maximum satisfaction probability of fLTL can be computed efficiently.

Frequency constraints can be incorporated into probabilistic model checking without losing decidability.

Abstract

We study frequency linear-time temporal logic (fLTL) which extends the linear-time temporal logic (LTL) with a path operator $G^p$ expressing that on a path, certain formula holds with at least a given frequency p, thus relaxing the semantics of the usual G operator of LTL. Such logic is particularly useful in probabilistic systems, where some undesirable events such as random failures may occur and are acceptable if they are rare enough. Frequency-related extensions of LTL have been previously studied by several authors, where mostly the logic is equipped with an extended "until" and "globally" operator, leading to undecidability of most interesting problems. For the variant we study, we are able to establish fundamental decidability results. We show that for Markov chains, the problem of computing the probability with which a given fLTL formula holds has the same complexity as the analogous problem for LTL. We also show that for Markov decision processes the problem becomes more delicate, but when restricting the frequency bound $p$ to be 1 and negations not to be outside any $G^p$ operator, we can compute the maximum probability of satisfying the fLTL formula. This can be again performed with the same time complexity as for the ordinary LTL formulas.