Alexandru Mateescu

Shuffle on trajectories: syntactic constraints

Abstract We introduce and investigate new methods to define parallel composition of words and languages as well as of ω-words and ω-languages. The operation of parallel composition leads to new shuffle-like operations defined by syntactic constraints on the usual shuffle operation. The approach is applicable to concurrency, providing a method to define parallel composition of processes. It is also applicable to parallel computation. The operations are introduced using a uniform method based on the notion of trajectory . As a consequence, we obtain a very intuitive geometrical interpretation of the parallel composition operation. These operations lead in a natural way to a large class of semirings. The approach is amazingly flexible, diverse concepts from the theory of concurrency can be introduced and studied in this framework. For instance, we provide examples of applications to fairness property and to parallelization of non-context-free languages in terms of context-free and even regular languages. This paper concentrates on syntactic constraints. Semantic constraints will be dealt with in a forthcoming contribution.

A SHARPENING OF THE PARIKH MAPPING

In this paper we introduce a sharpening of the Parikh map- ping and investigate its basic properties. The new mapping is based on square matrices of a certain form. The classical Parikh vector appears in such a matrix as the second diagonal. However, the matrix prod- uct gives more information about a word than the Parikh vector. We characterize the matrix products and establish also an interesting in- terconnection between mirror images of words and inverses of matrices. Mathematics Subject Classification. 68Q45, 68Q70.

Aspects of classical language theory

The purpose of this chapter is to give an overview on some types of grammars and families of languages arising in classical language theory and not covered elsewhere in this Handbook. Since we will discuss in this chapter a large number of topics, we cannot penetrate very deeply in any one of them. Topics very related to the ones discussed in this chapter, such as regular languages and context-free languages, have their own chapters in this Handbook, where the presentation is more detailed than in the present chapter. Among the topics covered in this chapter (Section 3 below) will also be the general theory of language families, AFL-theory. In view of the whole language theory, there is a huge number of topics possible for this chapter. It is clear that our choice of topics and the amount of detail in which each of them is presented reflect, at least to some extent, our personal tastes.

Simple splicing systems

Abstract We consider one of the most restrictive classes of splicing ( H ) systems, namely based on splicing rules of the form ( a , λ ; a , λ ), where a is a symbol in a given set and λ is the empty string. They correspond to a special class of “null context splicing systems”, as introduced in Head (1987). A series of language-theoretic properties of languages generated by such systems with finite sets of axioms are investigated.

MATRIX INDICATORS FOR SUBWORD OCCURRENCES AND AMBIGUITY

The paper investigates inequalities between the numbers of different (scattered) subword occurrences. The Parikh matrix recently introduced is an efficient tool. We give various characterizations for Parikh matrices. Of special interest is the case where the matrix determines the word uniquely. We investigate such matrix unambiguous words. The considerations are extended to concern languages. Several open problems and problem areas are indicated.

Formal languages: an introduction and a synopsis

What is a language? By consulting a dictionary one finds, among others, the following explanations: 1 The body of words and systems for their use common to people who are of the same community or nation, the same geographical area, or the same cultural tradition. 2 Any set or system of signs or symbols used in a more or less uniform fashion by a number of people who are thus enabled to communicate intelligibly with one other. 3 Any system of formalized symbols, signs, gestures, or the like, used or conceived as a means of communicating thought, emotion, etc.

PARALLEL FINITE AUTOMATA SYSTEMS COMMUNICATING BY STATES

An accepting device based on the communication between finite automata working in parallel is introduced. It consists of several finite automata working independently but communicating states to each other by request. Several variants of parallel communicating finite automata systems are investigated from their computational power point of view. We prove that all of them are at most as powerful as multi-head finite automata. Homomorphical characterizations of recursively enumerable languages are obtained starting from languages recognized by all variants of parallel communicating finite automata systems having at most three components. We present a brief comparison with the parallel communicating grammar systems. Some remarks suggesting that these devices might be mildly context-sensitive ones as well as a few open problems and directions for further research are also discussed.

Lexical Analysis with a Simple Finite-Fuzzy-Automaton Model

Many fuzzy automaton models have been introduced in the past. Here, we discuss two basic finite fuzzy automaton models, the Mealy and Moore types, for lexical analysis. We show that there is a remarkable difference between the two types. We consider that the latter is a suitable model for implementing lexical analysers. Various properties of fuzzy regular languages are reviewed and studied. A fuzzy lexical analyzer generator (FLEX) is proposed.

Finite degrees of ambiguity in pattern languages

The paper investigates nondeterminism and degrees of nondeterminism in representing words according to a pattern given a priori. The issues involved belong to the basic combinatorics of words. Our main results concern decidability and construction of finite degrees of nondeterminism

Contexts and the Concept of Mild Context-Sensitivity

We introduce and study a natural extension ofMarcus external contextual grammars. This mathematically simple mechanism whichgenerates a proper subclass of simple matrix languages,known to be mildly context-sensitive ones, is still mildlycontext-sensitive. Furthermore, we get an infinite hierarchy ofmildly context-sensitive families of languages.Then we attempt to fill a gap regarding the linguistic relevanceof these mechanisms which consists in defining a tree structure on thestrings generated by many-dimensional external contextual grammars,and investigate some related issues. Several open problemsare finally discussed.