Algebraic characterization of logically defined tree languages

TL;DR

This paper provides an algebraic framework using preclones and block products to characterize tree languages definable by certain logical formulas, extending known results from word languages to trees.

Contribution

It introduces a novel algebraic characterization of logically defined tree languages using preclones and block products, generalizing prior word language results.

Findings

Characterization applies to first-order definable tree languages

Utilizes preclones and block product operations

Does not provide an decision algorithm for definability

Abstract

We give an algebraic characterization of the tree languages that are defined by logical formulas using certain Lindstr\"om quantifiers. An important instance of our result concerns first-order definable tree languages. Our characterization relies on the usage of preclones, an algebraic structure introduced by the authors in a previous paper, and of the block product operation on preclones. Our results generalize analogous results on finite word languages, but it must be noted that, as they stand, they do not yield an algorithm to decide whether a given regular tree language is first-order definable.