Some Remarks on First-Order Definable Tree Languages

TL;DR

This paper investigates the definability of regular finite tree languages in first-order logic, introducing a decidable algebraic framework to establish necessary and sufficient conditions for such definability.

Contribution

It presents a novel algebraic approach that yields decidable conditions for first-order definability of regular tree languages, advancing the theoretical understanding.

Findings

Developed algebraic conditions for definability

Derived necessary and sufficient conditions

Conditions are decidable

Abstract

We study the question of whether a given regular language of finite trees can be defined in first-order logic. We develop an algebraic approach to address this question and we use it to derive several necessary and sufficient conditions for definability (but unfortunately no condition that is both). The main difference of our results to those from the literature is that our conditions are decidable.