Constrained Synchronization and Subset Synchronization Problems for Weakly Acyclic Automata

TL;DR

This paper studies the complexity of constrained and subset synchronization problems in weakly acyclic automata, revealing that many problems become easier (NP or P) in this setting compared to general automata.

Contribution

It provides a complete complexity classification for constrained synchronization problems in weakly acyclic automata, including cases with small automata and alphabet sizes.

Findings

Most constrained problems are NP-complete instead of PSPACE-complete.

Some problems become polynomial time solvable in weakly acyclic automata.

Subset synchronization problems vary in complexity, with some becoming easier.

Abstract

We investigate the constrained synchronization problem for weakly acyclic, or partially ordered, input automata. We show that, for input automata of this type, the problem is always in NP. Furthermore, we give a full classification of the realizable complexities for constraint automata with at most two states and over a ternary alphabet. We find that most constrained problems that are PSPACE-complete in general become NP-complete. However, there also exist constrained problems that are PSPACE-complete in the general setting but become polynomial time solvable when considered for weakly acyclic input automata. We also investigate two problems related to subset synchronization, namely if there exists a word mapping all states into a given target subset of states, and if there exists a word mapping one subset into another. Both problems are PSPACE-complete in general, but in our setting the former is polynomial time solvable and the latter is NP-complete.