Language Theoretic Properties Research Articles

Node Replacement Graph Languages Squeezed with Chains, Trees, and Forests

We present results on language-theoretic properties (such as closure, membership, and other decision properties) of node replacement graph languages (such as NLC, B-NLC, and Lin-NLC languages) squeezed with chains, trees, and forests.

RECOGNIZABLE PICTURE LANGUAGES

The purpose of this paper is to propose a new notion of recognizability for picture (two-dimensional) languages extending the characterization of one-dimensional recognizable languages in terms of local languages and alphabetic mappings. We first introduce the family of local picture languages (denoted by LOC) and, in particular, prove the undecidability of the emptiness problem. Then we define the new family of recognizable picture languages (denoted by REC). We study some combinatorial and language theoretic properties of REC such as ambiguity, closure properties or undecidability results. Finally we compare the family REC with the classical families of languages recognized by four-way automata.

Parallel generation of infinite images

We introduce a syntactic model for generating sets of finite and infinite images where a finite image can be viewed as an array over finite alphabet and an infinite image is an array with finite number of columns and infinite number of rows. This model, called image grammar, can be considered as a generalization of classical Chromsky grammar. We study various types of infinite image grammars using Nivat's derivation which is an infinite unrestrictive derivation, Bvichi's (respectively Muller's) which is defined as infinite derivations selected by some repetitive set (respectively sets of rewriting rules). Then we study the combinatorial and language theoretical properties such as complexity measure, closure properties and decidability results. In terms of complexity measure we give a strict infinite hierarchy. We have extended Nivat's and Eilenberg's theorems to infinite images. Unfortunately we prove that Biichi's and McNaughton's theorems cannot be extended to infinite images. We also characterize these families in terms of deterministic image co-substitution and ω-languages

PARALLEL GENERATION OF FINITE IMAGES

We define a syntactic model for generating sets of images, where an image can be viewed as an array over finite alphabet. This model is called image grammar. Image grammar can be considered as a generalization of classical Chomsky grammar. Then we study some combinatorial and language theoretical properties such as reduction, pumping lemmas, complexity measure, we give a strict infinite hierarchy. We also characterize these families in terms of deterministic substitutions and Chomsky languages.

Representations of language families by homomorphic equality operations and generalized equality sets

In this paper the operations of homomorphic equality and inverse homomorphic equality are introduced and are studied in all detail as part of formal language theory. These operations are defined from n-tuples of homomorphisms by a matching of the homomorphic images. They model a simple and powerful equality mechanism and incorporate the notion of generalized equality sets. Single homomorphisms on free monoids are natural and simple operations. However, combining n-tuples of homomorphisms into a homomorphic equality or inverse homomorphic equality may result in very complex morphic mappings, which map, e.g., σ ∗ to the set of solutions of an instance of the Post Correspondence Problem. In this paper particular emphasis is laid upon strong representation theorems. In the general case every recursively enumerable set can be obtained from the trivial set {$} by applying an inverse homomorphism, a (inverse) homomorphic equality of two nonerasing homomorphisms and an erasing homomorphism. This is a uniform morphic representation of the recursively enumerable sets with special regular sets as a basis, and it is minimal in its constituents. Hence, the recursively enumerable sets are the smallest class containing all regular sets of the form σ ∗$σ ∗ and closed under homomorphic equality or under inverse homomorphic equality and homomorphism. Every nonempty language is a generator of the recursively enumerable sets, when the full trio operations are extended to include homomorphic equality or inverse homomorphic equality. In a similar manner, the regular sets can be characterized in terms of homomorphic equality and inverse homomorphic equality operations with bounded balance. Finally, in the nonerasing case, the target are classes of the form H( L ⋀ BNP ). For a class of languages L , H( L ⋀ BNP ) is the smallest class containing L and closed under nonerasing homomorphic equality or under inverse homomorphic equality and nonerasing homomorphism. It includes the intersection closure of L . Every language in H( L ⋀ BNP ) can be obtained from a language in L by a nonerasing homomorphic equality or an inverse homomorphic equality of three homomorphisms and followed by a nonerasing homomorphism, provided that L is closed under inverse homomorphism and endmarking. This representation goes through to trio generators of L , which are the generators of H( L ⋀ BNP ), if the trio operations are extended to include nonerasing homomorphic equality or inverse homomorphic equality. The operations of homomorphic equality and inverse homomorphic equality are based on generalized equality sets. These are equality sets of finite sets of homomorphisms, which are used forwards or backwards. We investigate basic language-theoretic properties of these sets including the star event property, commutativity and hardest sets, and provide a complete classification of the generalized equality sets according to the numbers of homomorphisms that are used forwards or backwards.

Subset languages of Petri nets part I: The relationship to string languages and normal forms

In this paper we are interested in language theoretical properties of Petri nets. However, in this research we consider a step to consist of concurrent firing of a set of transitions. In this way a firing sequence will be a sequence of subsets of the set of all transitions. The set of all such subset firing sequences (or the set of all those subset firing sequences that lead to one of the finite number of final markings) forms the subset language of a given net (to consider such a set to be a language one has to consider subsets to be letters of the alphabet used to record behaviour of nets). The aim of the paper is to present a systematic research exploring the above point of view. First of all we point out several differences between the standard ‘string language’ approach and the ‘subset language’ approach. Then we proceed to the investigation of properties of Petri nets as generators of subset languages. We provide here some normal form results (and notice that those results and their proofs are quite different from the case when one considers Petri nets as generators of string languages).

Closure properties of selective substitution grammars part I

Let be the class of languages generated by selective substitution grammars, where: 1) arbitrary productions of the form are allowed, where b is a letter and w is a word and 2) the selectors used are from . This is the first part of a paper consisting of two parts that investigates how various language-theoretical properties of influence the closure properties of In this part basic techniques for manipulating selectors of selective substitution grammars are established. Then we investigate how properties of influence the closure of under union and concatenation.

Unary multiple equality sets: The languages of rational matrices

Unary multiple equality sets are equality sets of finite sets of homomorphisms, which map into the free monoids over single letter alphabets. These languages can equally well be defined by rational matrices. Unary multiple equality sets have extraordinary language theoretic properties. They coincide with the commutative multiple equality sets, each unary multiple equality set is precisely classified by the rank of its defining matrices, and all fundamental decision questions are polynomially decidable. The least trio containing all unary multiple equality sets equals the class of languages accepted by blind multicounter machines and by simple multihead finite automata. Finally crossing-free unary multiple equality sets are introduced, which vary from unary multiple equality sets in a similar way as the one-sided Dyck set varies from the two-sided Dyck set.