Nondeterministic automata: equivalence, bisimulations, and uniform relations

TL;DR

This paper explores the equivalence of nondeterministic automata using bisimulations and uniform relations, providing decision procedures and characterizations for various types of bisimulations, including forward and weak forward bisimulations.

Contribution

It introduces a unified framework combining bisimulations with uniform relations for automata equivalence, along with decision procedures and isomorphism characterizations.

Findings

Decidable procedures for three types of bisimulations

Characterization of automata equivalence via isomorphisms of factor automata

Examples illustrating relationships between bisimulation types

Abstract

In this paper we study the equivalence of nondeterministic automata pairing the concept of a bisimulation with the recently introduced concept of a uniform relation. In this symbiosis, uniform relations serve as equivalence relations which relate states of two possibly different nondeterministic automata, and bisimulations ensure compatibility with the transitions, initial and terminal states of these automata. We define six types of bisimulations, but due to the duality we discuss three of them: forward, backward-forward, and weak forward bisimulations. For each od these three types of bisimulations we provide a procedure which decides whether there is a bisimulation of this type between two automata, and when it exists, the same procedure computes the greatest one. We also show that there is a uniform forward bisimulation between two automata if and only if the factor automata with respect to the greatest forward bisimulation equivalences on these automata are isomorphic. We prove a similar theorem for weak forward bisimulations, using the concept of a weak forward isomorphism instead of an isomorphism. We also give examples that explain the relationships between the considered types of bisimulations.