Constrained Synchronization for Commutative Automata and Automata with Simple Idempotents

TL;DR

This paper studies the constrained synchronization problem in automata, showing polynomial-time solvability for commutative automata and certain automata with simple idempotents, contrasting with the general PSPACE-complete complexity.

Contribution

It identifies classes of automata where the constrained synchronization problem is efficiently solvable, expanding understanding of automata synchronization complexity.

Findings

Constrained synchronization is polynomial-time solvable for commutative automata.

For automata with simple idempotents over a binary alphabet, the problem is polynomial-time solvable with small constraint automata.

General case remains PSPACE-complete or NP-complete, highlighting complexity boundaries.

Abstract

For general input automata, there exist regular constraint languages such that asking if a given input automaton admits a synchronizing word in the constraint language is PSPACE-complete or NP-complete. Here, we investigate this problem for commutative automata over an arbitrary alphabet and automata with simple idempotents over a binary alphabet as input automata. The latter class contains, for example, the \v{C}ern\'y family of automata. We find that for commutative input automata, the problem is always solvable in polynomial time, for every constraint language. For input automata with simple idempotents over a binary alphabet and with a constraint language given by a partial automaton with up to three states, the constrained synchronization problem is also solvable in polynomial time.