35,898 research outputs found
Algebraic recognizability of regular tree languages
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
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
Proving that a Tree Language is not First-Order Definable
We explore from an algebraic viewpoint the properties of the tree languages
definable with a first-order formula involving the ancestor predicate, using
the description of these languages as those recognized by iterated block
products of forest algebras defined from finite counter monoids. Proofs of
nondefinability are infinite sequences of sets of forests, one for each level
of the hierarchy of quantification levels that defines the corresponding
variety of languages. The forests at a given level are built recursively by
inserting forests from previous level at the ports of a suitable set of
multicontexts. We show that a recursive proof exists for the syntactic algebra
of every non-definable language. We also investigate certain types of uniform
recursive proofs. For this purpose, we define from a forest algebra an algebra
of mappings and an extended algebra, which we also use to redefine the notion
of aperiodicity in a way that generalizes the existing ones
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