Graded Monads and Graded Logics for the Linear Time -- Branching Time Spectrum

TL;DR

This paper introduces a unified framework using graded monads for modeling various process equivalences in concurrent systems, extending modal logics to characterize these equivalences and generalizing classical theorems.

Contribution

It demonstrates that graded semantics subsume key equivalences in the linear time -- branching time spectrum and extends modal logic expressiveness criteria.

Findings

Framework unifies various process equivalences

Extends modal logic to graded semantics

Provides generic expressiveness criterion

Abstract

State-based models of concurrent systems are traditionally considered under a variety of notions of process equivalence. In the particular case of labelled transition systems, these equivalences range from trace equivalence to (strong) bisimilarity, and are organized in what is known as the linear time -- branching time spectrum. A combination of universal coalgebra and graded monads provides a generic framework in which the semantics of concurrency can be parametrized both over the branching type of the underlying transition systems and over the granularity of process equivalence. We show in the present paper that this framework of graded semantics does subsume the most important equivalences from the linear time -- branching time spectrum. An important feature of graded semantics is that it allows for the principled extraction of characteristic modal logics. We have established invariance of these graded logics under the given graded semantics in earlier work; in the present paper, we extend the logical framework with an explicit propositional layer and provide a generic expressiveness criterion that generalizes the classical Hennessy-Milner theorem to coarser notions of process equivalence. We extract graded logics for a range of graded semantics on labelled transition systems and probabilistic systems, and give exemplaric proofs of their expressiveness based on our generic criterion.