Minimization and Canonization of GFG Transition-Based Automata

TL;DR

This paper presents a polynomial-time minimization algorithm for GFG co-B"uchi automata with transition-based acceptance, establishing conditions under which minimal automata are unique up to isomorphism.

Contribution

It introduces a novel polynomial minimization method for GFG co-B"uchi automata with transition-based acceptance and proves their canonicity.

Findings

Polynomial minimization algorithm for GFG co-B"uchi automata

Minimal automata have isomorphic safe components

Saturation with $ ext{α}$-transitions yields automata isomorphism

Abstract

While many applications of automata in formal methods can use nondeterministic automata, some applications, most notably synthesis, need deterministic or good-for-games (GFG) automata. The latter are nondeterministic automata that can resolve their nondeterministic choices in a way that only depends on the past. The minimization problem for deterministic B\"uchi and co-B\"uchi word automata is NP-complete. In particular, no canonical minimal deterministic automaton exists, and a language may have different minimal deterministic automata. We describe a polynomial minimization algorithm for GFG co-B\"uchi word automata with transition-based acceptance. Thus, a run is accepting if it traverses a set $\alpha$ of designated transitions only finitely often. Our algorithm is based on a sequence of transformations we apply to the automaton, on top of which a minimal quotient automaton is defined. We use our minimization algorithm to show canonicity for transition-based GFG co-B\"uchi word automata: all minimal automata have isomorphic safe components (namely components obtained by restricting the transitions to these not in $\alpha$) and once we saturate the automata with $\alpha$-transitions, we get full isomorphism.