# Synchronization Problems in Automata without Non-trivial Cycles

## TL;DR

This paper explores the computational complexity of synchronization problems in weakly acyclic automata, providing bounds, hardness results, and NP-completeness proofs for various synchronization-related tasks.

## Contribution

It offers new bounds and complexity results for synchronization problems specifically in weakly acyclic automata, a subclass of aperiodic automata.

## Key findings

- Bounds on shortest synchronizing words established
- Approximation of synchronization length shown to be hard
- NP-completeness results for recognizing synchronizing subsets

## Abstract

We study the computational complexity of various problems related to synchronization of weakly acyclic automata, a subclass of widely studied aperiodic automata. We provide upper and lower bounds on the length of a shortest word synchronizing a weakly acyclic automaton or, more generally, a subset of its states, and show that the problem of approximating this length is hard. We investigate the complexity of finding a synchronizing set of states of maximum size. We also show inapproximability of the problem of computing the rank of a subset of states in a binary weakly acyclic automaton and prove that several problems related to recognizing a synchronizing subset of states in such automata are NP-complete.