34528489
Reductions for Primitive 2-Structures
A subset X of a 2-structure (a reversible edge-colored directed graph) g is a clan, if X cannot be distinguished by colors from outside of X. We show that if g is primitive, i.e. it has no nontrivial clans, then there exists an edge e or an end vertex x such that when e or x is removed from g, the resulting 2-structure g′ remains primitive. We also present a composition result for the unstable 2-structures, which are those primitive 2-structures where a removal of any edge results in a nonprimitive 2-structure.
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.
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.
34951842
Primitivity is Hereditary for 2-Structures
Abstract It is proved that each primitive 2-structure on n elements ( n ⩾3) contains a primitive substructure on either ( n −1) or ( n −2) elements.
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.
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).
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.
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.
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.
1753130
Conditions Enforcing Regularity of Context-Free Languages
206851807
Behaviour of Elementary Net Systems