33163861
Finitary and infinitary interpretations of languages
The main result of the paper can be viewed both as a generalization of a classical results concerning colorings of graphs, as well as a very useful over-all theorem from which various specific results concerning grammar and language families can be deduced. Essentially our main result says that, when considering infinitary families obtainable by inverse morphisms, it suffices to consider their finitary parts. Various applications are given, in particular, to grammar forms and context-free languages over infinite alphabets.
23304788
On non context-free grammar forms
A grammar formF defines via a so-calledinterpretation mechanism, a family of languages(F). In this paper we establish that for many grammar formsF, (the family of context-free languages) implies(F)=RE (the family of recursively enumerable sets). We conjecture that this is also true in general. Because of this, it seems necessary to use restricted interpretations for non context-free grammar forms, a form giving then rise to a family We compare the obvious alternatives for restricting interpretations and focus attention on two promising alternatives,Q(F) andst(F) and their combinationQ, st(F). Using st-interpretations, surprising families can be generated and strong normal form results can be obtained. Closure results and decidability results are also given. UnderQ, st-interpretation, it is possible to characterize a number of well-known families of languages betweenCF andRE, including the families of EOL, ETOL, matrix and scattered context languages.
11947788
Context-Dependent L Forms
The notions of an EIL form and its interpretations are introduced. The resulting families of L systems and languages are studied from a number of different points of view. Some of the main results are: the family of context-free languages is generated by an EIL form (contrary to the earlier results concerning context-free L forms) and, as regards reducibility in the amount of context, EIL forms lie between IL and EIL systems.
39074040
Pure Grammars
30874363
Synchronized EOL Forms Under Uniform Interpretation
— The aim ofthis paper is thejurther study oj EOLjorms under uniform interprétation. We are naturally led to the considération oj synchronized EOLjorms, which become the central thème of this paper. We prove: a ""strongest"" possible normaljorm resuit, the spanning normalJorm theorem; the non-existence oj uni-generatorsjor "reasonable" language amilies, which include thefinite, regular and context-jree j amilies; and a char act erization oj those ynchronized jorms which, under uniform interprétation, yield all regular languages. In closing we remark on three topics which are worthy oj separate investigation. Résumé. — Le but de cet article est de poursuivre l'étude des EOL-jormes sous des interprétations unijormes. Nous sommes conduits naturellement à considérer les EOL-jormes synchronisées qui deviennent le thème central de cet article. Nous établissons : un résultat sur la forme normale la plus « serrée » possible, le théorème sur la forme normale recouvrante, la non-existence d'uni-générateurs pour des jamilles « raisonnables » de langages qui comprennent les langages finis, rationnels et algébriques; et une caractérisation des formes synchronisées qui donnent, par interprétation unijorme, tous les langages rationnels. Pour terminer, nous effleurons trois domaines qui méritent une étude séparée.
205885836
On Predecessors of Finite Languages
We continue the study of finite grammatical families by characterizing when a family has a predecessor. This characterization implies that (i) the existence of predecessors for finite families is decidable and (ii) the collection of context-free grammatical families, each family of which contains all regular languages, is not maximal dense.
1092042
Really Straight Graph Drawings
We study straight-line drawings of graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2-trees, and planar 3-trees. We prove that every 3-connected plane graph on n vertices has a plane drawing with at most 5n/2 segments and at most 2n slopes, and that every cubic 3-connected plane graph has a plane drawing with three slopes (and three bends on the outerface). Drawings of non-planar graphs with few slopes are also considered. For example, it is proved that graphs of bounded degree and bounded treewidth have drawings with $\mathcal{O}({\rm log} n)$ slopes.
6424947
Queue Layouts, Tree-Width, and Three-Dimensional Graph Drawing
A three-dimensional (straight-line grid) drawing of a graph represents the vertices by points in Z3 and the edges by noncrossing line segments. This research is motivated by the following open problem due to Felsner, Liotta, and Wismath [Graph Drawing '01, Lecture Notes in Comput. Sci., 2002]: does every n-vertex planar graph have a three-dimensional drawing with O(n) volume? We prove that this question is almost equivalent to an existing one-dimensional graph layout problem. A queue layout consists of a linear order ? of the vertices of a graph, and a partition of the edges into queues, such that no two edges in the same queue are nested with respect to ?. The minimum number of queues in a queue layout of a graph is its queue-number. Let G be an n-vertex member of a proper minor-closed family of graphs (such as a planar graph). We prove that G has a O(1) × O(1) × O(n) drawing if and only if G has O(1) queue-number. Thus the above question is almost equivalent to an open problem of Heath, Leighton, and Rosenberg [SIAM J. Discrete Math., 1992], who ask whether every planar graph has O(1) queue-number? We also present partial solutions to an open problem of Ganley and Heath [Discrete Appl. Math., 2001], who ask whether graphs of bounded tree-width have bounded queue-number? We prove that graphs with bounded path-width, or both bounded tree-width and bounded maximum degree, have bounded queue-number. As a corollary we obtain three-dimensional drawings with optimal O(n) volume, for series-parallel graphs, and graphs with both bounded tree-width and bounded maximum degree.
341729
Lower Bounds for the Number of Bends in Three-Dimensional Orthogonal Graph Drawings
In this paper we present the first non-trivial lower bounds for the total number of bends in 3-D orthogonal drawings of maximum degree six graphs. In particular, we prove lower bounds for the number of bends in 3-D orthogonal drawings of complete simple graphs and multigraphs, which are tight in most cases. These result are used as the basis for the construction of infinite classes of c-connected simple graphs and multigraphs (2 ≤ c ≤ 6) of maximum degree Δ (3 ≤ Δ ≤ 6) with lower bounds on the total number of bends for all members of the class. We also present lower bounds for the number of bends in general position 3-D orthogonal graph drawings. These results have significant ramifications for the '2-bends' problem, which is one of the most important open problems in the field.
7215582
Light edges in degree-constrained graphs
Let @a denote the average degree, and @d denote the minimum degree of a graph. An edge is light if both its endpoints have degree bounded by a constant depending only on @a and @d. A graph is degree-constrained if @a<2@d. The primary result of this paper is that every degree-constrained graph has a light edge. Most previous results in this direction have been for embedded graphs. This result is extended in a variety of ways. First it is proved that there exists a constant c(@a,@d) such that for every 0=<@e
8780088
Dimension-Exchange Algorithms for Load Balancing on Trees
This paper considers dimension-exchange algorithms for load balancing on trees with finitely-divisible loads (token distribution). We present improved analysis of an existing protocol, and in particular, establish a logarithmic upper bound on the discrepancy of the final distribution. Our second contribution is a new algorithm, which assuming each node has knowledge of the total number of nodes, determines a perfectly balanced distribution.