Subset Synchronization in Monotonic Automata

TL;DR

This paper investigates subset and careful synchronization problems in monotonic automata, providing polynomial algorithms, tight bounds, and complexity results, including NP-hardness, for various classes of these automata.

Contribution

It introduces new polynomial-time solutions for certain synchronization problems in monotonic automata and establishes complexity bounds and NP-hardness results for related problems.

Findings

Polynomial-time algorithms for subset and careful synchronization in monotonic automata.

Asymptotically tight bounds on shortest synchronizing words.

NP-completeness and inapproximability results for related problems.

Abstract

We study extremal and algorithmic questions of subset and careful synchronization in monotonic automata. We show that several synchronization problems that are hard in general automata can be solved in polynomial time in monotonic automata, even without knowing a linear order of the states preserved by the transitions. We provide asymptotically tight bounds on the maximum length of a shortest word synchronizing a subset of states in a monotonic automaton and a shortest word carefully synchronizing a partial monotonic automaton. We provide a complexity framework for dealing with problems for monotonic weakly acyclic automata over a three-letter alphabet, and use it to prove NP-completeness and inapproximability of problems such as {\sc Finite Automata Intersection} and the problem of computing the rank of a subset of states in this class. We also show that checking whether a monotonic partial automaton over a four-letter alphabet is carefully synchronizing is NP-hard. Finally, we give a simple necessary and sufficient condition when a strongly connected digraph with a selected subset of vertices can be transformed into a deterministic automaton where the corresponding subset of states is synchronizing.