Some Remarks on First-Order Definable Tree Languages
Achim Blumensath
TL;DR
This work tackles the problem of deciding whether a regular language of finite trees is definable in $FO$ by building an algebraic framework of tree algebras and applying $\text{Tame Congruence Theory}$. It develops a translation between clone- and preclone-based algebras, introduces syntactic algebras, and derives decidable necessary and sufficient conditions for FO-definability, as well as non-definability criteria. The results include a decidable necessary condition and two decidable sufficient conditions, plus partial sufficiency results under additional assumptions, illustrating progress toward a complete characterization. The findings connect FO-definability with algebraic properties such as aperiodicity and semilattice-type minimal divisors, offering tools and directions for further work on chain logic and fuller algebraic characterizations with practical decidability implications.
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.