State Complexity of the Set of Synchronizing Words for Circular Automata and Automata over Binary Alphabets

TL;DR

This paper investigates the structure and complexity of synchronizing words in circular and generalized automata over binary alphabets, establishing conditions for maximal state complexity and analyzing specific automata families.

Contribution

It proves that completely reachable automata over binary alphabets are necessarily circular and provides criteria for maximal complexity of synchronizing words.

Findings

Over binary alphabets, completely reachable automata are circular.

Sufficient conditions for maximal state complexity are identified.

The automata family $\\mathscr K_n$ is shown to have maximal complexity under certain conditions.

Abstract

Most slowly synchronizing automata over binary alphabets are circular, i.e., containing a letter permuting the states in a single cycle, and their set of synchronizing words has maximal state complexity, which also implies complete reachability.Here, we take a closer look at generalized circular and completely reachable automata. We derive that over a binary alphabet every completely reachable automaton must be circular, a consequence of a structural result stating that completely reachable automata over strictly less letters than states always contain permutational letters. We state sufficient conditions for the state complexity of the set of synchronizing words of a generalized circular automaton to be maximal. We apply our main criteria to the family $\mathscr K_n$ of automata that was previously only conjectured to have this property.