Quasilinear-time Computation of Generic Modal Witnesses for Behavioural Inequivalence

TL;DR

This paper introduces a generic, efficient algorithm for constructing modal formulas that distinguish behaviorally inequivalent states across various transition systems using coalgebraic methods, improving on previous bounds.

Contribution

It presents a universal coalgebraic algorithm for generating distinguishing formulas with optimal complexity, applicable to multiple transition system types.

Findings

Runs in O((m+n) log n) time for systems with n states and m transitions

Constructs formulas with size proportional to the system's complexity

Improves bounds over previous algorithms for transition systems and Markov chains

Abstract

We provide a generic algorithm for constructing formulae that distinguish behaviourally inequivalent states in systems of various transition types such as nondeterministic, probabilistic or weighted; genericity over the transition type is achieved by working with coalgebras for a set functor in the paradigm of universal coalgebra. For every behavioural equivalence class in a given system, we construct a formula which holds precisely at the states in that class. The algorithm instantiates to deterministic finite automata, transition systems, labelled Markov chains, and systems of many other types. The ambient logic is a modal logic featuring modalities that are generically extracted from the functor; these modalities can be systematically translated into custom sets of modalities in a postprocessing step. The new algorithm builds on an existing coalgebraic partition refinement algorithm. It runs in time O((m+n) log n) on systems with n states and m transitions, and the same asymptotic bound applies to the dag size of the formulae it constructs. This improves the bounds on run time and formula size compared to previous algorithms even for previously known specific instances, viz. transition systems and Markov chains; in particular, the best previous bound for transition systems was O(mn).