43370814
Boundary NLC Graph Grammars-Basic Definitions, Normal Forms, and Complexity
Node label controlled (NLC) grammars are graph grammars (operating on node labeled undirected graphs) which rewrite single nodes only and establish connections between the embedded graph and the neighbors of the rewritten node on the basis of the labels of the involved nodes only. They define (possibly infinite) languages of undirected node labeled graphs (or, if we just omit the labels, languages of unlabeled graphs). Here we consider a restriction of NLC grammars, so-called boundary NLC (BNLC) grammars , distinguished by the property that whenever in a graph already generated two nodes may be rewritten, then these nodes are not adjacent. The graph languages generated by this type of grammars are called BNLC languages . Although we show that this restriction leads to a smaller class of languages, still enough generative power remains to define interesting graph languages. For example, trees, complete bipartite graphs, maximal outerplanar graphs, k -trees, graphs of bandwidth ⩽ k , graphs of cyclic bandwidth ⩽ k , graphs of binary tree bandwidth ⩽ k , graphs of cutwidth ⩽ k (always for a fixed positive integer k ) turn out all to be BNLC languages. We prove a number of normal forms for BNLC grammars and then we indicate their usefulness by various applications. In particular, we show that for connected graphs of bounded degree the membership problem for BNLC languages is solvable in deterministic polynomial time.
17444444
The Bounded Degree Problem for NLC Grammars is Decidable
Abstract The degree of a graph H is the maximum among the degrees of its nodes. A set of graphs L is of bounded degree if there exists a positive integer n such that the degree of each graph in L does not exceed n . We demonstrate that it is decidable whether or not the (graph) language of an arbitrary node label controlled (NLC) grammar is of bounded degree. Moreover, it is shown that, given an arbitrary NLC grammar G generating the language L(G) of bounded degree, one can effectively compute the maximum integer which appears as the degree of a graph in L(G) .
38741370
Repetitions in Homomorphisms and Languages
39400225
On Subword Complexities of Homomorphic Images of Languages
— The subword complexity of a language K is the function nK on positive integer s such that nK (n) equals the number of different subwords of length n appearing in words of X. We investigate the relationship between nk(K) and nK where Kis a language and h is a homorphism. This study is aiso carried out for the special case when Kis a DOL language. Resumé. — La complexité des facteurs d'un langage K est la fonction nK définie sur les entiers positifs telle que nK (n) est le nombre de facteurs distincts de longueur n apparaissant dans les mots de K. Nous étudions les relations existant entre KH(K) et nKi où K est un langage et h est un homomorphisme. Cette étude est menée également dans le cas particulier où Kest un DOL langage.
1753130
Conditions Enforcing Regularity of Context-Free Languages
21626854
An Observation on Scattered Grammars
Scattered context-grammars were introduced in [ 1 j. They constitute one of the most interesting examples of string rewriting systems that are ‘context-dependent’. However they form one of the least understood types of grammars. Although the relationship of (some of the varier.-.._ _ -+i-~ of! the class of scattered context grammars to some classes of languages was studied (see e.g. [2,31) the main question remains open: Do scattered context grammars generate all context sensitive languages’! Clearly an answer to this question would shed light on the role that context plays in grammars. In this work we show that recursively enumerable languages possess a very strong representation in the class of scattered context languages. As a matter of fact it is well known th!at they possess the same representation in the class of context sensitive languages! Thus a result of this form supports in some sense the conjecture that the class of scattered context languages equals the class of context sensitive languages. In any case it says that the class of scattered context languages is ‘close to’ the class of context sensitive languages. Since we provide a direct proof of !2!r reslult (an explicit construction is given) it gives some insight iI& how scattered grammars work. We assume the reader to be familiar with Post Ilormal systems (e.g. in the scope of [4j). If H is a Post normal system, c4 is its axiom and al$ --) S&, ar*S --, S& are productions of l-l, then wr zlrssume that loill = ia21 SK Iwl and Ipli, l&l >O. The language of H consists of al! the words over its alphabet for which
13303483
Theory of Traces
Abstract The theory of traces , originated by A. Mazurkiewicz in 1977, is an attempt to provide a mathematical description of the behavior of concurrent systems. Its aim is to reconcile the sequential nature of observations of the system behavior on the one hand and the nonsequential nature of causality between the actions of the system on the other hand. One can see the theory of traces to be rooted in formal string language theory with the notion of partial commutativity playing the central role. Alternatively one can see the theory of traces to be rooted in the theory of labeled acyclic directed graphs (or even in the theory of labeled partial orders). This paper attempts to present a major portion of the theory of traces in a unified way. However, it is not a survey in the sense that a number of new notions are introduced and a number of new results are proved. Although traditionally most of the development in the theory of traces follows the string-language-theoretic line, we try to demonstrate to the reader that the graph-theoretic point of view may be more appropriate. The paper essentially consists of two parts. The first one (Sections 1 through 4) is concerned with the basic theory of traces. The second one (Section 5) presents applications of the theory of traces to the theory of the behavior of concurrent systems, where the basic system model we have chosen is the condition/event system introduced by C.A. Petri.
6414809
Developmental systems and languages
Developmental systems were introduced (Lindenmayer, 1968, 1971) in order to model morphogenetic (pattern-generating) processes in growing, multicellular, filamentous organisms. These systems were originally conceived as linear arrays of interconnected finite automata, each automaton corresponding to a living cell, with the possibility that new automata can be added to the array (cells divide) or be deleted from the array (cells die).
 Each cell in the array is supposed to have the same state-transition and output functions. As required by biological considerations these functions must be applied to all cells in the array simultaneously at each time step. Thus one obtains infinite sequences of arrays once the functions and the initial arrays are specified. Simplified constructs are defined (and used in this paper) by considering the states and outputs to be identical and thus omitting the output functions. Such filamentous developmental systems have been called “Lindenmayer models” Herman, 1969, 1970) or “L-systems” (Van Dalen, 1971).
28657554
Hypergraph Systems and Their Extensions
The notion of a graph is a natural generalization of the notion of a string. Using one graph (the transition graph ofafinite automaton) one can define a language {the language of a given automaton). This well-known idea is generalized as follows: the notion of a hypergraph is a natural generalization of the notion of a graph. Using one hypergraph (equipped with an additional graph structure) one can define a graph language (a set ofgraphs). Several variants of graph grammars based on this idea are introduced in this paper together withformalism needed to investigate them and illustrating examples. The generating power of various graph-language-generating Systems is compared. Résumé. — La notion de graphe est une généralisation naturelle de la notion de mot. En utilisant un graphe {le graphe de transition d'un automate fini) on définit un langage {le langage accepté par l'automate). Cette idée est généralisée comme suit ; la notion de hypergraphe est une généralisation naturelle de la notion de graphe. Avec un hypergraphe (muni d'une structure de graphe supplémentaire) on peut définir un langage (un ensemble) de graphes. Dans cet article on introduit diverses variantes des grammaires de graphes basées sur cette idée; on donne le formalisme nécessaire pour les étudier, et on les illustre par des exemples. La puissance générative de divers systèmes engendrant des langages de graphes est comparée.
9443958
Dynamic change within workflow systems
Dynamic change is a large and pervasive unsolved problem which surfaces within office systems as well as within software engineering, manufacturing, and numerous other domains. Procedural changes, performed in an ad hoc manner, can cause inefficiencies, inconsistencies, and catastrophic breakdowns within offices. This paper is concerned with dynamic change to procedures in the context of workflow systems. How can we make workflow systems more flexible and open? We believe that part of the answer lies in the study and solution of the dynamic change problem. In this paper, we use a Petri net formalism to analyze structural change within office procedures. As an example, we define a class of change called “synthetic cut-over change”, and apply our formalism to prove that this class maintains correctness when downsizing occurs.
39431424
Incremental construction of 2-structures
Abstract A labeled 2-structure g is an edge-labeled directed graph with a reversibility condition on the labels. Each labeled 2-structure g can be represented as a tree, the prime tree of g. We characterize a labeled 2-structure h in terms of the prime tree of a labeled 2-structure g which results from h by removing one element from h. This leads to an O(n3) algorithm for the construction of the prime tree.