Finite-Horizon First-Order Rank Profiles of Regular Languages

Abstract

We introduce the finite-horizon first-order rank profile of a language $L \subseteq \Sigma^*$: the least quantifier rank needed by an $\mathrm{FO}[<]$ sentence to classify membership in $L$ correctly on all words of length at most $n$. The invariant measures quantifier depth only; formula size is deliberately not bounded. First, we prove a rank calculus that is independent of regularity. Every language satisfies $\rho_L(n) \le \lceil \log_2 n \rceil + 4$, via balanced first-order distance formulas and exact-word definitions. Moreover, $\sup_n \rho_L(n) < \infty$ holds exactly when $L$ is globally $\mathrm{FO}[<]$-definable, and the supremum equals the minimum quantifier rank of such a definition. Second, for regular languages we prove a sharp aperiodicity gap: if the syntactic monoid of $L$ is aperiodic, then $\rho_L(n) = O(1)$; otherwise $\rho_L(n) = \log_2 n + O_L(1)$. The lower bound extracts a nontrivial cyclic component from the syntactic monoid and combines it with an Ehrenfeucht-Fraisse power lemma for long repetitions of a fixed word. Thus, for full $\mathrm{FO}[<]$ quantifier rank, regular languages admit no intermediate finite-horizon growth between bounded and logarithmic rank.