Finite-Horizon First-Order Rank Profiles of Regular Languages

TL;DR

This paper introduces the finite-horizon first-order rank profile for languages, providing bounds and characterizations based on syntactic monoids and logical definability, with implications for regular languages.

Contribution

It establishes a rank calculus independent of regularity, characterizes when the rank is bounded, and reveals a sharp aperiodicity gap in regular languages.

Findings

Every language satisfies _L(n)   \, ext{log}_2 n + 4

Regular languages with an aperiodic syntactic monoid have constant rank

Regular languages with non-aperiodic monoids have rank  ext{log}_2 n + O(1)

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.