Syntactic Algebra Research Articles

Decorated one-dimensional cobordisms and tensor envelopes of noncommutative recognizable power series

The paper explores the relation between noncommutative power series and topological theories of one-dimensional cobordisms decorated by labelled zero-dimensional submanifolds. These topological theories give rise to several types of tensor envelopes of noncommutative recognizable power series, including the categories built from the syntactic algebra and syntactic ideals of the series and the analogue of the Deligne category.

Weak Separation Problem for Tree Languages

Forest algebras are defined for investigating languages of forests [ordered sequences] of unranked trees, where a node may have more than two [ordered] successors. They consist of two monoids, the horizontal and the vertical, with an action of the vertical monoid on the horizontal monoid, and a complementary axiom of faithfulness. A pseudovariety is a class of finite algebras of a given signature, closed under the taking of homomorphic images, subalgebras and finitary direct products. By looking at the syntactic congruence for monoids and as the natural extension in the case of forest algebras, we could define a version of syntactic congruence of a subset of the free forest algebra, not just a forest language. Let [Formula: see text] be a finite alphabet and [Formula: see text] be a pseudovariety of finite forest algebras. A language [Formula: see text] is [Formula: see text]-recognizable if its syntactic forest algebra belongs to [Formula: see text]. Separation is a classical problem in mathematics and computer science. It asks whether, given two sets belonging to some class, it is possible to separate them by another set of a smaller class. Suppose that a forest language [Formula: see text] and a forest [Formula: see text] are given. We want to find if there exists any proof for that [Formula: see text] does not belong to [Formula: see text] just by using [Formula: see text]-recognizable languages, i.e. given such [Formula: see text] and [Formula: see text], if there exists a [Formula: see text]-recognizable language [Formula: see text] which contains [Formula: see text] and does not contain [Formula: see text]. In this paper, we present how one can use profinite forest algebra to separate a forest language and a forest term and also to separate two forest languages.

Topological rewriting systems applied to standard bases and syntactic algebras

We introduce topological rewriting systems as a generalisation of abstract rewriting systems, where we replace the set of terms by a topological space. Abstract rewriting systems correspond to topological rewriting systems for the discrete topology. We introduce the topological confluence property as an approximation of the confluence property. Using a representation of linear topological rewriting systems with continuous reduction operators, we show that the topological confluence property is characterised by lattice operations. Using this characterisation, we show that standard bases induce topologically confluent rewriting systems on formal power series. Finally, we investigate duality for reduction operators that we relate to series representations and syntactic algebras. In particular, we use duality for proving that an algebra is syntactic or not.

Fuzzy tree language recognizability

A fuzzy tree language with membership grades in an arbitrary set is syntactically recognizable (s-recognizable) if its syntactic algebra is finite. The equality problem for such languages is decidable and their syntactic algebra can be effectively constructed provided that they are s-recognizable. Linear (but non arbitrary) tree homomorphisms preserve s-recognizability. Tree automata whose transitions are weighted over the unit interval and whose behavior is computed with respect to a pair made of a t-norm distributive over a t-conorm have the syntactic recognition power and thus their equivalence problem is decidable. However, s-recognizability is more powerful when dealing with non-distributive pairs of such operations.

Tree algebras and varieties of tree languages

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.

Effective construction of the syntactic algebra of a recognizable series on trees

In this paper we exhibit two different effective constructions of the syntactic algebraA S associated to a recognizable formal series on treesS. The one method consists of a direct construction of (=a copy ofA S ) which is the subspace with the natural algebra structure. We first determine a basis of the subspace and then, using the junction isomorphism we obtain a basis for . The second method consists of considering an arbitrary surjective realization (A, φ) ofS, defining an appropriate ideal ℬ ofA and then constructing the quotient algebraA/ℬ this quotient is isomorphic toA S and thus independent of the choice of (A S φ).

Effective construction of the syntactic algebra of a recognizable series on trees

In this paper we exhibit two different effective constructions of the syntactic algebra associated to a recognizable formal series on treesS.

Constructions effectives sur les séries formelles d'arbres

In this paper we exhibit two different effective constructions of the syntactic algebra U s associated with a recognizable tree series S. In order to make the first construction we only need a natural number n satisfying dim U s⩽n . The second way to obtain U s is by minimization of a given realization of S. We finally describe (effectively) a minimal matrix representation of S.

Syntactic congruences and syntactic algebras

Syntactic congruences and syntactic algebras

Séries formelles et algèbres syntactiques

The notion of the syntactic monoid is well known to be very important for formal languages, and in particular for rational languages; examples of that importance are Kleene's theorem, Schützenberger's theorem about aperiodic monoid and Eilenberg's theorem about varieties. We introduce here, for formal power series, a similar object: to each formal power series we associate its syntactic algebra. The Kleene-Schützenberger theorem can then be stated in the following way: a series is rational if and only if its syntactic algebra has finite dimension. A rational central series (this means that the coefficient of a word depends only on its conjugacy class) is a linear combination of characters if and only if its syntactic algebra is semisimple. Fatou properties of rational series in one variable are extended to series in several variables and a special case of the rationality of the Hadamard quotient of two series is positively answered. The correspondence between pseudovarieties of finite monoids and varieties of rational languages, as studied by Eilenberg, is extended between pseudovarieties of finite dimensional algebras and varieties of rational series. We study different kinds of varieties that are defined by closure properties and prove a theorem similar to Schützenberger's theorem on aperiodic monoids.