Linear-time logics -- a coalgebraic perspective

TL;DR

This paper introduces a coalgebraic framework for deriving linear-time logics applicable to various state-based, quantitative systems, providing both step-wise and path-based semantics and proving their equivalence.

Contribution

It presents a novel coalgebraic approach to linear-time logics, unifying semantics and enabling reasoning about likelihood and cost of properties.

Findings

Two semantics for the logics are shown to be equivalent.

The framework supports reasoning about possibility, likelihood, and cost.

A semantic characterization of logical distance is provided.

Abstract

We describe a general approach to deriving linear-time logics for a wide variety of state-based, quantitative systems, by modelling the latter as coalgebras whose type incorporates both branching and linear behaviour. Concretely, we define logics whose syntax is determined by the type of linear behaviour, and whose domain of truth values is determined by the type of branching behaviour, and we provide two semantics for them: a step-wise semantics akin to that of standard coalgebraic logics, and a path-based semantics akin to that of standard linear-time logics. The former semantics is useful for model checking, whereas the latter is the more natural semantics, as it measures the extent with which qualitative properties hold along computation paths from a given state. Our main result is the equivalence of the two semantics. We also provide a semantic characterisation of a notion of logical distance induced by these logics. Instances of our logics support reasoning about the possibility, likelihood or minimal cost of exhibiting a given linear-time property.