Computational Complexity of Synchronization under Sparse Regular Constraints

TL;DR

This paper analyzes the computational complexity of the constrained synchronization problem under sparse regular constraints, providing classifications for when the problem is NP-complete or solvable in polynomial time.

Contribution

It offers a new characterization of sparse regular sets, classifies CSP complexity for letter-bounded and strongly self-synchronizing codes, and extends understanding of synchronization under these constraints.

Findings

CSP is NP-complete for some constraint languages.

CSP is polynomial-time solvable for others.

A full classification of CSP complexity for sparse regular constraints is provided.

Abstract

The constrained synchronization problem (CSP) asks for a synchronizing word of a given input automaton contained in a regular set of constraints. It could be viewed as a special case of synchronization of a discrete event system under supervisory control. Here, we study the computational complexity of this problem for the class of sparse regular constraint languages. We give a new characterization of sparse regular sets, which equal the bounded regular sets, and derive a full classification of the computational complexity of CSP for letter-bounded regular constraint languages, which properly contain the strictly bounded regular languages. Then, we introduce strongly self-synchronizing codes and investigate CSP for bounded languages induced by these codes. With our previous result, we deduce a full classification for these languages as well. In both cases, depending on the constraint language, our problem becomes NP-complete or polynomial time solvable.