Regular tree languages and quasi orders (original) (raw)

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.

On Well Quasi-orders on Languages

Lecture Notes in Computer Science, 2003

Let G be a context-free grammar and let L be the language of all the words derived from any variable of G. We prove the following generalization of Higman's theorem: any division order on L is a well quasi-order on L. We also give applications of this result to some quasiorders associated with unitary grammars.

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.

Ordered Algebraic Structures and Related Topics

Contemporary Mathematics, 2017

We study the model theoretic strength of various lattices that occur naturally in topology, like closed (semi-linear or semi-algebraic or convex) sets. The method is based on weak monadic second order logic and sharpens previous results by Grzegorczyk. We also answers a question of Grzegorczyk on the 'algebra of convex sets'.

Generalized Priestley Quasi-Orders

Order, 2011

We introduce generalized Priestley quasi-orders and show that subalgebras of bounded distributive meet-semilattices are dually characterized by means of generalized Priestley quasi-orders. This generalizes the well-known characterization of subalgebras of bounded distributive lattices by means of Priestley quasiorders (Adams, Algebra Univers 3:216-228, 1973; Cignoli et al., Order 8(3):299-315, 1991; Schmid, Order 19 : 2002). We also introduce Vietoris families and prove that homomorphic images of bounded distributive meet-semilattices are dually characterized by Vietoris families. We show that this generalizes the wellknown characterization (Priestley, Proc Lond Math Soc 24(3):507-530, 1972) of homomorphic images of a bounded distributive lattice by means of closed subsets of its Priestley space. We also show how to modify the notions of generalized Priestley quasi-order and Vietoris family to obtain the dual characterizations of subalgebras and homomorphic images of bounded implicative semilattices, which generalize the well-known dual characterizations of subalgebras and homomorphic images of Heyting algebras (Esakia, Sov Math Dokl 15:147-151, 1974).

From well-quasi-ordered sets to better-quasi-ordered sets

The electronic journal of combinatorics

We consider conditions which force a well-quasi-ordered poset (wqo) to be better-quasi-ordered (bqo). In particular we obtain that if a poset PPP is wqo and the set Somega(P)S_{\omega}(P)Somega(P) of strictly increasing sequences of elements of PPP is bqo under domination, then PPP is bqo. As a consequence, we get the same conclusion if Somega(P)S_{\omega} (P)Somega(P) is replaced by mathcalJ1(P)\mathcal J^1(P)mathcalJ1(P), the collection of non-principal ideals of PPP, or by AM(P)AM(P)AM(P), the collection of maximal antichains of PPP ordered by domination. It then follows that an interval order which is wqo is in fact bqo.

From well-quasi-ordered sets to

2006

We consider conditions which force a well-quasi-ordered poset (wqo) to be betterquasi-ordered (bqo). In particular we obtain that if a poset P is wqo and the set S ω (P) of strictly increasing sequences of elements of P is bqo under domination, then P is bqo. As a consequence, we get the same conclusion if S ω (P) is replaced by J ¬↓ (P), the collection of non-principal ideals of P , or by AM (P), the collection of maximal antichains of P ordered by domination. It then follows that an interval order which is wqo is in fact bqo.

Tree algebras and varieties of tree languages

Theoretical Computer Science, 2007

We consider several aspects of Wilke's [T. Wilke, An algebraic characterization of frontier testable tree languages, Theoret. Comput. Sci. 154 (1996) 85-106] tree algebra formalism for representing binary labelled trees and compare it with approaches that represent trees as terms in the traditional way. A convergent term rewriting system yields normal form representations of binary trees and contexts, as well as a new completeness proof and a computational decision method for the axiomatization of tree algebras. Varieties of binary tree languages are compared with varieties of tree languages studied earlier in the literature. We also prove a variety theorem thus solving a problem noted by several authors. Syntactic tree algebras are studied and compared with ordinary syntactic algebras. The expressive power of the language of tree algebras is demonstrated by giving equational definitions for some well-known varieties of binary tree languages.

Well-Quasi Orders and Hierarchy Theory

Well-Quasi Orders in Computation, Logic, Language and Reasoning, 2020

We discuss some applications of WQOs to several fields were hierarchies and reducibilities are the principal classification tools, notably to Descriptive Set Theory, Computability theory and Automata Theory. While the classical hierarchies of sets usually degenerate to structures very close to ordinals, the extension of them to functions requires more complicated WQOs, and the same applies to reducibilities. We survey some results obtained so far and discuss open problems and possible research directions.