Showing papers by "Grzegorz Rozenberg published in 1999"

The Theory of 2-Structures: A Framework for Decomposition and Transformation of Graphs

Abstract: The theory of 2-structures provides a convenient framework for decomposition and transformation of mathematical systems where one or several different binary relationships hold between the objects of the system. In particular, it forms a useful framework for decomposition and transformation of graphs.The decomposition methods presented in this book correspond closely to the top-down design methods studied in computer science. The transformation methods considered here have a natural interpretation in the dynamic evolution of certain kinds of communication networks. From the mathematical point of view, the clan decomposition method presented here, also known as modular decomposition or substitution decomposition, is closely related to the decomposition by quotients in algebra. The transformation method presented here is based on labelled 2-structures over groups, the theory of which generalizes the well-studied theory of switching classes of graphs.This book is both a text and a monograph. As a monograph, the results concerning the decomposition and transformation of 2-structures are presented in a unified way. In addition, detailed notes on references are provided at the end of each chapter. These notes allow the reader to trace the origin of many notions and results, and to browse through the literature in order to extend the material presented in the book.To facilitate its use as a textbook, there are numerous examples and exercises which provide an opportunity for the reader to check his or her understanding of the discussed material. Furthermore, the text begins with preliminaries on partial orders, semigroups, groups and graphs to the extent needed for the book.

Jewels are Forever

Abstract: ly we have the following setting for our problem. We are given a finite set S of generators and a finite set E of equations of the form a == b where a and b are words over the alphabet S. Let S* be the free monoid with identity 1 generated by S, and let T/ be the smallest semilattice congruence on S* containing E. Then the quotient semilattice S· = S* IT/ is the semilattice defined by the presentation (S, E). For any word u over S, let [U]17 be the T/-class of u. The multiplication. on S· is given by [U]17 • [V]17 = [UV]17' For a word u over S, let o:(u) be the set of elements of S appearing in u. If u and v are words over S such that o:(u) = o:(v), then u and v represent the same element of the free semilattice generated by S, hence also of S·. Thus, via 0:, the free semilattice with identity generated by S is isomorphic with the semilattice (28 , U, 0). For an equation a == b in E, let o:(a == b) be the equation o:(a) == o:(b); moreover, let o:(E) be the set of equations o:(a == b) where a == b is in E. Let f) be the smallest congruence relation on 28 containing o:(E). Then the semilattice S· is isomorphic with the quotient semilattice 28 If). Thus to compute the product of two elements in S·, it is equivalent to compute the product of the corresponding sets in 28 If). In general, there may be several different subsets of S representing the same element of 28 If). However, for every T ~ S one can choose a canonical representative as follows. Proposition 1. Let T ~ S. Then there is a unique set rep(T) in the f)-class [T]o of T such that T' ~ rep(T) for all T' E [T]o. Proof. The class [T]o is closed under union. Hence, the union of all sets T' E [T]o is the desired canonical representative. We now give an algorithm for constructing the canonical representative rep(T) of any set T ~ S. Given that 0: is a semilattice isomorphism, instead of words and equations on words, we consider only subsets of S and equations on sets. An equation ei = Ai == Bi is applicable to T if Ai ~ T or Bi ~ T. The set of equations will be represented by an array £ = (el,"" em). Variable £' represents equations that have not yet been eliminated. Variable T' represents the set obtained from T by the equations used so far. The size of the array £' is denoted by 1£'1. An application of an applicable equation ei to T' consists of Semilattices of Fault Semiautomata 5 replacing T' by T'UAiuBi and of deleting ei from £'. With these assumptions we have the algorithm shown in Fig. 1. function rep(e:array[1..m] of equations, T: subset of S): subset of S; var e': array[1..m] of equations, T': subset of S; e' +e; T' +T; while e' "I 0 do i +1; while ei not applicable do i +i + 1; if i > le'l then rep +T'; exit {rep}; e' +e' with ei deleted; T' +T' U Ai U Bi; rep +T'; Fig. 1. Function rep. Scoping is indicated by indentation. Let SV be the set of distinct representatives obtained by rep. Let V be defined as follows: For any sets A, BE SV, A V B = rep (A U B). TheoreDl 1. The algebra (SV, V, 0) is a semilattice with identity. The three semilattices S·, 28 If) and SV are isomorphic. Proof. The mapping rep from 28 to SV induces an isomorphism j..t from 28 If) onto 2v by j..t([T]Ii) = rep(T) for T ~ S. Note that Theorem 1 provides an algorithm for solving the word problem in finitely presented semilattices. The solvability of the word problem itself follows already from some very general theorems about finitely presented algebras [4]. Surprisingly, however, we failed to find our simple solution to this problem in the literature. As in any finite semilattice, one can derive a second semilattice operation A on SV such that (SV, V, A, 0, S) is a lattice, where TAT' = v u. UE8V ,U£TnT' It turns out that TAT' = TnT', that is, the set of canonical representatives is closed under intersection. Example 1. Let S = {80,81,to,td, and One verifies that S· = S*ITJ has the 7 elements 0, to, tl, tOtl' 80tOtl, 81tOtl, and 8081 tOtl. The Hasse diagram of S· is shown in Fig. 2. 6 J. A. Brzozowski and H. Jurgensen

Applications, languages and tools

Abstract: Term rewriting and functional languages visual and object-oriented languages applications to software engineering applications to engineering disciplines applications to pictures implemented specification languages and tools structuring and modularization concepts.

Application of Petri Nets to Communication Networks: Advances in Petri Nets

Abstract: From the Publisher: Petri nets offer a mathematically defined technique for the specification, design, analysis, verification and performance evaluation of concurrent distributed systems. Communications networks, ranging from traditional telecommunication systems to advanced Internet-based information services, are inherently distributed and comprise systems with concurrently operating components. This volume presents a selection of the latest advances in the use of Petri nets for the modeling, analysis and management of communication networks and systems in the broadest sense of these terms.

Jewels Are Forever Contributions On Theoretical Computer Science in Honor of Arto Salomaa

Abstract: Dedicated to Arto Salomaa, a towering figure of theoretical computer science, on the occasion of his 65th birthday, this book is a tribute to him on behalf of the theoretical computer science community. The contributions are written by internationally recognized scientists and cover most of Salomaa's many research areas. Due to its representative selection of classic and cutting edge trends in theoretical computer science, the book constitutes a comprehensive state-of-the-art survey. The contributions are in such central areas as automata theory, algorithms and complexity, and combinatorics of words. But not only that, they take up new areas such as regular sets and biocomputing. While some are survey articles of fundamental topics, most are original research papers.

Concurrency, parallelism, and distribution

Abstract: Graph grammars originated in the late 60s, motivated by considerations about pattern recognition and compiler construction. Since then, the list of areas which have interacted with the development of graph grammars has grown quite impressively. Besides the aforementioned areas, it includes software specification and development, VLSI layout schemes, database design, modeling of concurrent systems, massively parallel computer architectures, logic programming, computer animation, developmental biology, music composition, visual languages, and many others.The area of graph grammars and graph transformations generalizes formal language theory based on strings and the theory of term rewriting based on trees. As a matter of fact, within the area of graph grammars, graph transformation is considered as a fundamental computation paradigm where computation includes specification, programming, and implementation. Over the last three decades, graph grammars have developed at a steady pace into a theoretically attractive and important-for-applications research field.Volume 3 of the indispensable Handbook of Graph Grammars and Computing by Graph Transformations presents the research on concurrency, parallelism, and distribution — important paradigms of modern computer science. The topics considered include semantics for concurrent systems, modeling of concurrency, mobile and coordinated systems, algebraic specifications, Petri nets, visual design of distributed systems, and distributed algorithms. The contributions have been written in a tutorial/survey style by the top experts.

DNA Computing: New Ideas and Paradigms

Abstract: DNA computing is one of the most exciting new developments in computer science, from both technological and theoretical point of view. We begin by observing how the structure of DNA molecules and the technics available for manipulating them are very suitable for computing. We then establish a link with certain fairly old results from computability theory which essentially explain why the main feature of DNA molecules, the Watson-Crick complementarity, gives rise to the Turing-universality of DNA computations. Selected areas of DNA computing, interesting from a theoretical point of view but offering also practical potential, will be briefly examined.

Cross-fertilization between evolutionary computation and DNA-based computing

Abstract: The potential for cross-fertilization between the fields of DNA based computing and evolutionary computation is outlined both from a principal point of view and by means of an experimental investigation concerning the NP-hard maximum clique problem. A simple evolutionary approach to maximum clique is introduced and the hypothesis is tested whether the increase in population size possible by realizing evolutionary computation with DNA yields the expected improvement in solution quality. Results obtained for a limited range of population sizes up to 10/sup 4/ indicate that the hypothesis holds for about two-third of the investigated problem instances (which were taken from the DIMACS library).

Contexts on trajectories

Abstract: We introduce and investigate a new way of generating mildly context-sensitive languages The main idea is that the contexts are adjoined by shuffling them on certain trajectories In this way we obtain also a very general class of contextual grammars such that most of the fundamental classes of contextual gram-mars, for instance, internal contextual grammars, external contextual grammarsn-contextual grammars, are particular cases of contextual grammars with contexts shuffled on trajectories The approach is very flexible, able to model various aspects from linguistics

Arto Salomaa

Abstract: Arto Salomaa received his doctorate from the University of Turku in 1960. His studies built on the academic tradition of Finland, being of a fifth generation of dissertations in that country, in an academic genealogy with chains of direct supervision to Leibniz, Huygens, Mersenne, and in 20 generations to Nicolaus Copernicus. Professor Salomaa has supervised 25 doctoral students, giving rise to at least 138 academic descendants in some five further academic generations.

X-Families: An Approach to the Study of Families of Syntactically Similar Languages

Abstract: All classes of grammars investigated in formal language theory generate a language by starting from nite sets of axioms and iteratively applying certain production rules which transform \correct" strings into \correct" strings. If the set of rules is xed and the axiom set is varying over the family of nite languages, then to any grammar we associate a family of languages. When using grammars of certain type X, we call this family an X-family. The aim of this paper is to propose the investigation of such families of languages. We only formulate here some of the basic problems and we start the study of M-families, those obtained when using Marcus contextual grammars as starting point. Several properties of M-families are given, examples and counterexamples are produced, and some decidability results are proven.