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.
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.
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.
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).
40876494
Theory of 2-Structures
The process according to the present invention relates to the production of ferrovanadium alloys. The process comprises melting a charge containing iron and vanadium with a subsequent reduction of iron and vanadium to give a metal melt. In accordance with the present invention, iron and vanadium are reduced substantially completely. The resulting metal melt is poured into a converter and then purged, through the converter bottom, by means of oxygen blowing at a supply rate of oxygen and for a period of time sufficient to produce a slag containing at least 35% by weight of vanadium pentoxide. Thereafter, the metal melt is discharged from the converter, while vanadium and iron are reduced from the resulting slag by the metallothermal method.
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.
33584848
A combinatorial property of EOL languages
AbstractLet Δ be an alphabet and II its nontrivial binary partition. Each word over Δ can uniquely be decomposed in subwords (called blocks) consisting of letters of IIi only,i ∈ {1,2}. LetK
$$ \subseteq$$
 Δ*.K has a long block property (with respect to II), abbreviated asLB-property, if there exists a functionf:N+ →N+ such that for everyw ∈K and every positive integerm the number of blocks of length at mostm inw is bounded byf(m). K has a clustered block property (with respect to II), abbreviated asCB-property, if there exists a positive integerno and a growing functiong:N+ →N+ such that for everyw ∈K and for every positive integerm the blocks of length at mostm can be covered by at mostno segments of length at mostg(m).It is proved that aCB-property always implies aLB-property but not necessarily other way around. It is proved that an EOL language has aLB-property if and only if it has aCB-property.
39682467
On the Active and Full Use of Memory in Right-Boundary Grammars and Push-Down Automata
Abstract A coordinated pair system ( cp system for short) consists of a pair of grammars, the first of which is right-linear ( rl ) and the second is right-boundary ( rb ). A right-boundary grammar is like a right-linear grammar except that one does not distinguish between terminal and nonterminal symbols—still, the rewriting is applied to the last symbol of a string only (and erasing productions are allowed). A rewriting in a cp system consists of a pair of rewritings: one in the first and one in the second grammar—such a rewriting is possible if the pair of productions involved is in the finite set of rewrites given with the system. Is is easily seen that cp systems correspond very closely to (are another formulation of) push-down automata: the right-linear component models the input and the finite state control while the rb component models the push-down store. An rb grammar G transforms (rewrites) strings which are stored in a one-way (potentially infinite) tape. If one observes during a derivation δ the use of a fixed n th cell of the tape and one notes the symbol stored there, each time that (the contents of) the cell is rewritten, then one gets the n-active record of δ; the set of all n -active records for all successful derivations δ forms the n -active language of G , denoted ACT n ( G ). It is proved that, for each rb grammar G and each n ϵ N + , ACT n ( G ) is regular and moreover, for each M ⊆ N + , ∪ nϵM ACT n ( G ) is regular. Another way to register the use of memory during a derivation δ is to record the contents of (a fixed) n th cell during all consecutive steps of δ—in this way one gets the n-full record of δ. The set of all n -full records for all successful derivations δ forms the n -full record language of G , denoted FR n ( G ). It is proved that, as in the case of active records, ∪ nϵM FR n ( G ) does not have to be regular even if M = N + (actually, one can get arbitrarily complex languages in this way). Then we provide a representation theorem allowing one to represent a cp system by an rb grammar and using this theorem we transfer the above results on the use of memory to cp systems.
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.
840154
Graph theoretic closure properties of the family of boundary NLC graph languages
SummaryNode 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). Boundary NLC (BNLC) grammars are NLC grammars with 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.
The present paper continues the investigations of basic properties of BNLC grammars and languages where the central question is the following: “If L is a BNLC language and P is a graph theoretic property, is the set of all graphs from L satisfying P again a BNLC language?” We demonstrate that the class of BNLC languages is very “stable” in the sense that for almost all properties we consider the resulting languages are BNLC. In particular, the above question gets an affirmative answer, if the property P is: being k-colorable, being connected, having a subgraph homeomorphic to a given graph, and being nonplanar.
38741370
Repetitions in Homomorphisms and Languages