Algebras for classifying regular tree languages and an application to frontier testability

Abstract

Point-tree algebras, a class of equational three-sorted algebras, are introduced for the purpose of characterizing and classifying regular tree languages. Any tree over an alphabet A can be identified with some element of the free point-tree algebra T(A) generated by A. A set L of finite binary trees over A is proved to be regular if and only if L is recognized by a finite point-tree algebra B, i.e. if L (as a subset of T(A)) is an inverse image under a homomorphism from T(A) into B. For each regular tree language a smallest recognizing point-tree algebra, its syntactic point-tree algebra, is shown to exist and to be effectively computable. For the class of frontier testable tree languages a finite set of equations characterizing the corresponding syntactic point-tree algebras is presented. This is in contrast with other algebraic approaches to the classification of tree languages (the semigroup and the universal-algebraic approach) where such equations are not possible resp. not known. As a byproduct one obtains an alternative proof of the decidability of the class of frontier testable tree languages.