Algebraic recognizability of regular tree languages

TL;DR

This paper introduces a new algebraic framework called preclones for classifying recognizable tree languages, extending classical concepts and enabling algebraic characterizations of logical classes.

Contribution

It develops the theory of preclones, proves a variety theorem analogous to Eilenberg's, and connects logical definability with algebraic pseudovarieties of preclones.

Findings

Established a variety theorem for preclones.

Characterized classes of recognizable tree languages algebraically.

Linked logical definability to pseudovarieties of preclones.

Abstract

We propose a new algebraic framework to discuss and classify recognizable tree languages, and to characterize interesting classes of such languages. Our algebraic tool, called preclones, encompasses the classical notion of syntactic Sigma-algebra or minimal tree automaton, but adds new expressivity to it. The main result in this paper is a variety theorem \`{a} la Eilenberg, but we also discuss important examples of logically defined classes of recognizable tree languages, whose characterization and decidability was established in recent papers (by Benedikt and S\'{e}goufin, and by Bojanczyk and Walukiewicz) and can be naturally formulated in terms of pseudovarieties of preclones. Finally, this paper constitutes the foundation for another paper by the same authors, where first-order definable tree languages receive an algebraic characterization.