Grzegorz Rozenberg

Bio: Grzegorz Rozenberg is an academic researcher from Leiden University. The author has contributed to research in topics: Petri net & Formal language. The author has an hindex of 81, co-authored 679 publications receiving 31378 citations. Previous affiliations of Grzegorz Rozenberg include Åbo Akademi University & University of Warsaw.

On 0L-Languages

Abstract: Summary In 0 L -languages, words are produced from each other by the simultaneous transition of all letters according to a set of production rules; the context is ignored. (i) 0 L -languages are not closed under the operations usually considered. (ii) 0 L -languages over a one-letter alphabet are discussed separately; a characterization is given of a subclass. (iii) 0 L -languages are incomparable with regular sets, incomparable with context-free languages, and strictly included in context-sensitive languages.

Parallel Generation of Maps: Developmental Systems for Cell Layers

Abstract: In this paper we extend parallel string generating systems (string L-systems) to parallel map generating systems(BPMOL-systems). The main distinguishing feature of of these constructs is that they generate map patterns by rewriting the graph of the map directly, and not its dual graph. These systems and their languages and sequences are illustrated in particular by applying them to the development of epidermal cell layers in which hexagonal arrays of cells are generated from previous hexagonal arrays.

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

Membrane Computing with External Output

Abstract: A membrane computing system (also called P system) consists of computing cells which are organized hierarchically by the inclusion relation: cells may include cells, which again may include cells, etc Each cell is enclosed by its membrane Each cell is an independent computing agent with its own computing program, which produces objects The interaction between cells consists of the exchange of objects through membranes The output of a computation is a partially ordered set of objects which leave the system through its external membrane The fundamental properties of computations in such P systems with external output are investigated These include the computing power, normal forms, and basic decision problems

Petri nets: Properties, analysis and applications

Abstract: Starts with a brief review of the history and the application areas considered in the literature. The author then proceeds with introductory modeling examples, behavioral and structural properties, three methods of analysis, subclasses of Petri nets and their analysis. In particular, one section is devoted to marked graphs, the concurrent system model most amenable to analysis. Introductory discussions on stochastic nets with their application to performance modeling, and on high-level nets with their application to logic programming, are provided. Also included are recent results on reachability criteria. Suggestions are provided for further reading on many subject areas of Petri nets. >