Simplicity and irreducibility in circular automata
Riccardo Venturi
TL;DR
The paper addresses when a synchronizing circular DFA is simple (primitive) and when it is irreducible over $\\mathbb{C}$, establishing that irreducibility implies simplicity and that $\\mathbb{C}$-irreducibility can differ from $\\mathbb{Q}$-irreducibility. It provides a complete combinatorial characterization of simplicity via the weak contracting property and, in the circular contracting case, derives necessary and sufficient conditions for irreducibility using circulant matrices and the synchronized representation. The irreducibility analysis relies on the eigenstructure of the circulating-letter matrix on the invariant subspace $\\omega^\\perp$, revealing the spectrum $\\{e^{\\frac{2\\pi i k}{n}}\\}$ and corresponding eigenvectors, and it demonstrates separations between $\\mathbb{Q}$- and $\\mathbb{C}$-irreducibility through explicit examples. The work also constructs infinite families of irreducible, non-weakly defective automata and poses open questions about reducible circular automata and extremal automata, offering avenues toward advancing toward the Černý conjecture.
Abstract
This paper investigates the conditions under which a given circular (synchronizing) DFA is \emph{simple} (sometimes referred to as \emph{primitive}) and when it is \emph{irreducible}. Our notion of irreducibility slightly differs from the classical one, since we are considering our monoid representations to be over $\mathbb{C}$ instead of $\mathbb{Q}$; nevertheless, several well-known results remain valid-for instance, the fact that every irreducible automaton is necessarily simple. We provide a complete characterization of simplicity in the circular case by means of the \emph{weak contracting property}. Furthermore, we establish necessary and sufficient conditions for a circular \emph{contracting automaton} (a stronger condition than the weakly contracting one) to be irreducible, and we present examples illustrating our results.