Algebras for Classifying Regular Tree Languages and an Application to Frontier Testability
Point-tree algebras, a class of equational three-sorted algebras are defined. The elements of sort t of the free point-tree algebra T generated by a set A are identified with finite binary trees with labels in A. A set L of finite binary trees over A is recognized by a point-tree algebr B if there exists a homomorphism h from T in B such that L is an inverse image of h. A tree language is regular if and only if it is recognized by a finite point-tree algebra. There exists a smallest recognizing point-tree algebra for every tree language, the so-called syntactic point-tree algebra. For regular tree languages, this point-tree algebra is computable from a (minimal) recognizing tree automaton. The class of finite point-tree algebras recognizing frontier testable (also known as reverse definite) tree languages is described by means of equations. This gives a cubic algorithm deciding whether a given regular tree language (over a fixed alphabet) is frontier testable. The characterization of the class of frontier testable languages in terms of equations 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 or not known.