R. Verraedt

Subset Languages of Petri Nets

Formal language theory is used to some extent in the investigation of properties of Petri nets (see, e.g. [H], [JV] and [P]). In most applications of language theory for Petri nets one considers completely sequentialized versions of Petri nets only. That is one assumes that a Petri net has one central run place which allows only a single transition to fire at a time; any sequence of such firings is called a firing sequence and the language of a Petri net consists of the set of all firing sequences (or only of those firing sequences that lead to one of the finite number of final markings).

SUBSET LANGUAGES OF PETRI NETS PART I: THE RELATIONSHIP TO STRING LANGUAGES AND NORMAL FORMS

Abstract 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).

Basic formulas and languages: PART II.Applications to E0L systems and forms

Abstract This paper (the second of two parts) settles the decidability status of several properties of derivations in E0L systems (forms). In particular we show that the so called “one-to-many simulation” among E0L forms is decidable, solving in this way an open problem from [6]. We use a more general mathematical framework, based on the theory of well-quasi orders, developed in Part I of this paper [2].

Vompleteness of E0L forms is decidable

SummaryIt is proved that given an E0L form F it is decidable whether or not F is vomplete. The concept of n-completeness, n a positive integer, is introduced and it is shown that for an E0L form F and an integer n≧2 vompleteness is equivalent with n-completeness.

On pure, terminal invariant and nonterminal invariant interpretations of EOL forms

Abstract This paper (re) considers some basic axioms of the theory of EOL forms. Two new kinds of interpretations are introduced; they form natural steps in-between ordinary and pure interpretations. Various properties of those interpretations are proved and several open problem concerning pure interpretations are solved. In particular certain results are proved which show that synchronized EOL forms have a natural place in the theory of EOL forms (independently of their role as the normal form for EOL systems on language level).

Basic formulas and languages Part I. The theory

Abstract A general mathematical framework to deal with (the decidability status of) properties of derivations in E0L systems (forms) is developed. It is based on the theory of well-quasi-orders. This paper (the first of two parts) deals with the mathematical theory of the proposed approach.

On fixed, terminal fixed and nonterminal fixed interpretations of E0L forms

This paper presents a systematic investigation of the basic axioms of the theory of E0L forms, in particular it continues the research from Rozenberg and Verraedt. Three new interpretation mechanisms—fixed, terminal fixed and nonterminal fixed interpretations—are introduced and extensively studied. When compared with terminal invariant and nonterminal invariant interpretations from Rozenberg and Verraedt these interpretations constitute a natural “next step” towards increasing the similarity of the “master” E0L form with its interpretations. In particular it is shown that each (ordinary) E0L family is a nonterminal fixed E0L family. It is also proved that for each of the new defined interpretation mechanisms the form equivalence problem is undecidable.

Synchronized and desynchronized E0L forms

Abstract In this paper we investigate some basic properties of synchronized and desynchronized E0L forms. Several open problems concerning those forms (and their connection to grammar forms) are solved.

On inherently ambiguous E0L languages

Abstract An E0L system G is called ambiguous if its language contains a word with (at least) two different derivations in G. An E0L language is called inherently E0L-ambiguous if every E0L system generating it is ambiguous. It is demonstrated that there exist inherently ambiguous E0L languages and in particular that the language {amb2n : 1 ⩽ m ⩽ n} ∪ {am2b2n : 1 ⩽ m ⩽ n} is inherently E0L-ambiguous.

Synchronized, desynchronized and coordinated E0L systems

This paper presents a systematic study of the phenomenon of synchronization in E0L systems. By abstracting various essential features of derivations in those systems the notions of a coordinated E0L system and a desynchronized E0L system are formulated and properties of the classes of languages they generate studied.