Deciding FO-definability of regular languages

TL;DR

This paper establishes that deciding FO(<,C)- and FO(<,MOD)-definability of regular languages is PSpace-complete, extending known results and providing algebraic characterisations via localisable properties of transition monoids.

Contribution

It introduces a new algebraic criterion for FO-definability based on localisable properties of transition monoids, generalising previous hardness proofs to two-way NFAs.

Findings

Deciding FO(<,C)- and FO(<,MOD)-definability is PSpace-complete.

Algebraic characterisations of FO-definability can be captured by localisable properties.

Results apply to both DFAs and two-way NFAs.

Abstract

We prove that, similarly to known PSpace-completeness of recognising FO(<)-definability of the language L(A) of a DFA A, deciding both FO(<,C)- and FO(<,MOD)-definability are PSpace-complete. (Here, FO(<,C) extends the first-order logic FO(<) with the standard congruence modulo n relation, and FO(<,MOD) with the quantifiers checking whether the number of positions satisfying a given formula is divisible by a given n>1. These FO-languages are known to define regular languages that are decidable in AC0 and ACC0, respectively.) We obtain these results by first showing that known algebraic characterisations of FO-definability of L(A) can be captured by `localisable' properties of the transition monoid of A. Using our criterion, we then generalise the known proof of PSpace-hardness of FO(<)-definability, and establish the upper bounds not only for arbitrary DFAs but also for two-way NFAs.