Lattice-Theoretic Progress Measures and Coalgebraic Model Checking (with Appendices)

TL;DR

This paper introduces a general lattice-theoretic framework for progress measures in model checking, unifying various fixed point notions and extending to coalgebraic systems, potentially enabling new decision procedures and proof methods.

Contribution

It formalizes progress measures in a general, possibly infinitary, lattice-theoretic setting and applies this to coalgebraic model checking, bridging branching and linear-time frameworks.

Findings

Unified lattice-theoretic notion of progress measures.

Smooth transfer from branching-time to linear-time systems.

Potential for sound proof methods in undecidable/infinitary problems.

Abstract

In the context of formal verification in general and model checking in particular, parity games serve as a mighty vehicle: many problems are encoded as parity games, which are then solved by the seminal algorithm by Jurdzinski. In this paper we identify the essence of this workflow to be the notion of progress measure, and formalize it in general, possibly infinitary, lattice-theoretic terms. Our view on progress measures is that they are to nested/alternating fixed points what invariants are to safety/greatest fixed points, and what ranking functions are to liveness/least fixed points. That is, progress measures are combination of the latter two notions (invariant and ranking function) that have been extensively studied in the context of (program) verification. We then apply our theory of progress measures to a general model-checking framework, where systems are categorically presented as coalgebras. The framework's theoretical robustness is witnessed by a smooth transfer from the branching-time setting to the linear-time one. Although the framework can be used to derive some decision procedures for finite settings, we also expect the proposed framework to form a basis for sound proof methods for some undecidable/infinitary problems.