7149620
Pumping Lemmas for Regular Sets
It is well known that regularity of a language implies certain properties known as pumping lemmas or iteration theorems. However, the question of a converse result has been open. We show that the usual form of pumping is very far from implying regularity but that a strengthened pumping property, the block pumping property, is equivalent to regularity. The proof involves use of the finite version of Ramsey’s theorem. We compare our results with recent results of Jaffe and Beauquier and state some open questions.
38741370
Repetitions in Homomorphisms and 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
43629107
On the Separating Power of Eol Systems
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) .
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.
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.
12269815
Angular 2-Structures
The theory of 2-structures forms a convenient framework for investigating graphs. In this paper we investigate the class of angular 2-structures which results by requiring that the primitivity is forbidden on the lowest possible level, i.e. it is required that no substructure on three elements is primitive. The paper presents the basic theory of angular 2-structures and, in particular, the theory of primitive angular 2-structures. We demonstrate that the notion of an angular 2-structure is a well-chosen generalization of the notions of a symmetric graph and a partial order.
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.
33004065
Pancyclicity in switching classes
Abstract Switching classes of graphs were introduced by van Lint and Seidel in the context of equiangular lines in elliptic geometry. We show that every switching class, except the class of all complete bipartite graphs, contains a pancyclic graph. This implies that deciding whether a switching class contains a hamiltonian graph can be done in polynomial time (as was noted by Kratochvil et al. (1992)) although this problem is NP-complete for graphs.
17739753
Finite Languages for the Representation of Finite Graphs
We introduce a new way of specifying graphs: through languages, i.e., sets of strings. The strings of a given (finite, prefix-free) language represent the vertices of the graph; whether or not there is an edge between the vertices represented by two strings is determined by the pair of symbols at the first position in these strings where they differ. With this new, “positional” or lexicographic, method, classical and well-understood ways of specifying languages can now be used to specify graphs, in a compact way; thus, (small) finite automata can be used to specify (large) graphs. Since (prefix-free) languages can be viewed as trees, our method generalizes the hierarchical specification of particular types of graphs such as cographs and VSP graphs. Our main results demonstrate an intrinsic relationship between the fundamental operations of language concatenation and graph substitution.