Tree algebras and varieties of tree languages (original) (raw)
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VARIETIES OF REGULAR ALGEBRAS AND UNRANKED TREE LANGUAGES
In this paper we develop a variety theory for unranked tree languages and unranked algebras. In an unranked tree any symbol may label a node with any number of successors. Such trees appear in markup languages such as XML and as syntactic descriptions of natural languages. In the corresponding algebras each operation is defined for any number of arguments, but in the regular algebras used as tree recognizers the operations are finite-state computable. We develop the basic theory of regular algebras for a setting in which algebras over different operator alphabets are considered together. Using syntactic algebras of unranked tree languages we establish a bijection between varieties of unranked tree languages and varieties of regular algebras. As varieties of unranked tree languages are usually defined by means of congruences of term algebras, we introduce also varieties of congruences and a general device for defining such varieties. Finally, we show that the natural unranked counterparts of several varieties of ranked tree languages form varieties in our sense.
On the complexity of the syntax of tree languages
Lecture Notes in Computer Science, 2009
The syntactic complexity of a tree language is defined according to the number of the distinct syntactic classes of all trees with a fixed yield length. This leads to a syntactic classification of tree languages and it turns out that the class of recognizable tree languages is properly contained in that of languages with bounded complexity. A refined syntactic complexity notion is also presented, appropriate exclusively for the class of recognizable tree languages. A tree language is recognizable if and only if it has finitely many refined syntactic classes. The constructive complexity of a tree automaton is also investigated and we prove that for any reachable tree automaton it is equal with the refined syntactic complexity of its behavior. © 2009 Springer.
A Completeness Property of Wilke's Tree Algebras
Mathematical Foundations of Computer Science 2003, 2003
Abstract. Wilke's tree algebra formalism for characterizing families of tree languages is based on six operations involving letters, binary trees and binary contexts. In this paper a completeness property of these oper-ations is studied. It is claimed that all functions involving letters, ...
2011 IEEE 26th Annual Symposium on Logic in Computer Science, 2011
We study finite automata running over infinite binary trees and we relax the notion of accepting run by allowing a negligible set (in the sense of measure theory) of non-accepting branches. In this qualitative setting, a tree is accepted by the automaton if there exists a run over this tree in which almost every branch is accepting. This leads to a new class of tree languages, called the qualitative tree languages that enjoys many properties.
A First-Order Axiomatization of the Theory of Finite Trees
Journal of Logic, Language and Information, 1995
We provide first-order axioms for the theories of finite trees with bounded branching and finite trees with arbitrary (finite) branching. The signature is chosen to express, in a natural way, those properties of trees most relevant to linguistic theories. These axioms provide a foundation for results in linguistics that are based on reasoning formally about such properties. We include some observations on the expressive power of these theories relative to traditional language complexity classes.
Demonstratio Mathematica
A tree language of a fixed type τ is any set of terms of type τ. We consider here a binary operation + n on the set W τ (X n) of all n-ary terms of type τ , which results in semigroup (W τ (X n), + n). We characterize languages which are idempotent with respect to this binary operation, and look at varieties of tree languages containing idempotent languages. We also compare properties of semigroup homomorphisms from (P(W τ (X n)); + n) to (P(W τ (X m)); + m) with properties of homomorphisms between the corresponding absolutely free algebras F τ (X n) and F τ (X m).
Systolic tree -languages: the operational and the logical view
Theoretical Computer Science, 2000
The class of !-languages recognized by systolic (binary) tree automata is introduced. This class extends the class of B uchi !-languages though maintaining the closure under union, intersection and complement and the decidability of emptiness. The class of systolic tree !-languages is characterized in terms of a (suitable) concatenation of (ÿnitary) systolic tree languages. A generalization of B uchi Theorem is provided which establishes a correspondence between systolic tree !-languages and a suitable extension of the sequential calculus S1S.
Positive varieties of tree languages
Theoretical Computer Science, 2005
Pin's variety theorem for positive varieties of string languages and varieties of finite ordered semigroups is proved for trees, i.e., a bijective correspondence between positive varieties of tree languages and varieties of finite ordered algebras is established. This, in turn, is extended to generalized varieties of finite ordered algebras, which corresponds to Steinby's generalized variety theorem. Also, families of tree languages and classes of ordered algebras that are definable by ordered (syntactic or translation) monoids are characterized.