Explaining Behavioural Inequivalence Generically in Quasilinear Time

TL;DR

This paper introduces a generic, efficient algorithm for constructing distinguishing formulas for behaviorally inequivalent states across various system types using coalgebraic methods, improving runtime and formula size bounds.

Contribution

It presents a universal coalgebraic algorithm that constructs formulas for behavioral inequivalence in systems of different types with improved efficiency.

Findings

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

Constructs formulas with asymptotically optimal size.

Applies to diverse systems like automata, Markov chains, and transition systems.

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 $\mathcal{O}((m+n) \log n)$ 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 $\mathcal{O}(m n)$.