Exact Accepting-State Spectrum for Reversal of Permutation Automata

TL;DR

This paper determines the exact spectrum of accepting states for the reversal operation on permutation automata, confirming a conjecture and revealing the simplest possible spectrum consistent with known obstructions.

Contribution

It constructs specific permutation automata to precisely characterize the accepting-state spectrum under reversal, proving the Rauch--Holzer conjecture.

Findings

Reversal spectrum for permutation automata is exactly characterized.

Constructed automata demonstrate the spectrum for all relevant cases.

Reversal has the simplest spectrum compatible with known obstructions.

Abstract

We determine the accepting-state spectrum of reversal for permutation automata exactly, thereby proving the Rauch--Holzer conjecture on this operation. For every $m \ge 2$ and every $\alpha \ge 2$, we construct a binary permutation automaton $A_{m,\alpha}$ such that $\operatorname{asc}(L(A_{m,\alpha}))=m$ and $\operatorname{asc}(L(A_{m,\alpha})^R)=\alpha$. Combined with the trivial cases $m=0$ and $m=1$, and with the previously known fact that $1$ is magic for every $m \ge 2$, this yields the exact spectrum $g^{\operatorname{asc}}_{R,\mathrm{PFA}}(0)=\{0\}$, $g^{\operatorname{asc}}_{R,\mathrm{PFA}}(1)=\{1\}$, and $g^{\operatorname{asc}}_{R,\mathrm{PFA}}(m)=\mathbb{N}_{\ge 2}$ for every $m \ge 2$. Thus reversal has, for permutation automata, the simplest possible exact accepting-state spectrum compatible with the single nontrivial obstruction at value $1$. The proof uses a uniform group-theoretic witness family: the states of the forward automaton are the $\alpha$-subsets of $[n]$, where $n=m+\alpha-1$, under the action generated by an $n$-cycle and a transposition, while the accepting states form a single star family. After reversal, the reachable subset-states are exactly the stars. This makes it possible to count the accepting reachable states precisely and to prove minimality of the reachable reverse automaton.