Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, August 1999. / Includes bibliographical references. Also available on the Internet.

Lindenmayer (L) systems form a class of interesting computational formalisms due to their parallel nature, the various circumstances under which they operate, the restrictions imposed on language acceptance, and other attributes. These systems have been extensively studied in the Formal Languages literature. In the past decade a new type of Lindenmayer system had been proposed: Watson-Crick Lindenmayer Systems. These systems are essentially a marriage between Developmental systems and DNA Computing. At their heart they are Lindenmayer systems augmented with a complementary relation amongst elements in the system just as the base pairs of DNA strands can be complementary with respect to one another. When conditions and a mechanism for 'switching' the state of a computation to it's complementary version are provided then these systems can become surprisingly more powerful than the L systems which form their backbone. This dissertation explores the computational power of new variants of Watson-Crick L systems. It is found that many of these systems are Computationally-Complete. These investigations differ from prior ones in that the systems under consideration have extended alphabets and usually Regular Triggers for complementation are considered as opposed to Context-Free Triggers investigated in previous works. / Thesis (Master, Computing) -- Queen's University, 2010-12-06 18:29:23.584

In the last decade, some researches on the star problem in trace monoids (is the iteration of a recognizable language also recognizable?) has pointed out the interest of the finite power property to achieve partial solutions of this problem. We prove that the star problem is decidable in some trace monoid if and only if in the same monoid, it is decidable whether a recognizable language has the finite power property. Intermediary results allow us to give a shorter proof for the decidability of the two previous problems in every trace monoid without C4-submonoid. We also deal with some earlier ideas, conjectures, and questions which have been raised in the research on the star problem and the finite power property, e.g. we show the decidability of these problems for recognizable languages which contain at most one non-connected trace.

In the last decade, some researches on the star problem in trace monoids (is the iteration of a recognizable language also recognizable?) has pointed out the interest of the finite power property to achieve partial solutions of this problem. We prove that the star problem is decidable in some trace monoid if and only if in the same monoid, it is decidable whether a recognizable language has the finite power property. Intermediary results allow us to give a shorter proof for the decidability of the two previous problems in every trace monoid without C4-submonoid. We also deal with some earlier ideas, conjectures, and questions which have been raised in the research on the star problem and the finite power property, e.g. we show the decidability of these problems for recognizable languages which contain at most one non-connected trace.

Cette thèse est dédiée à l’étude de deux familles de problèmes de partition de graphe. Nous considérons tout d’abord le problème de sommet-partitionner un graphe en sous-graphesconnexes. Plus précisément, étant donnés p entiers positifs n1; n2; :::; np dont la somme vautl’ordre d’un graphe G, peut-on partitionner V (G) en p parts V1; V2; :::; Vp de sorte que chaque Vi induise un sous-graphe connexe d’ordre ni ? Nous nous intéressons ensuite à des questions plus fortes. Que peut-on dire si l’on souhaite que G soit partitionnable de cette manière quels que soient p et n1; n2; :::; np ? Si l’on souhaite que des sommets particuliers de G appartiennent à des sous-graphes particuliers de la partition ? Et si l’on souhaite que les sous-graphes induits soient plus que connexes ? Nous considérons toutes ces questions à la fois du point de vue structurel (sous quelles conditions structurelles une partition particulière existe-t-elle nécessairement ?) et algorithmique (est-il difficile de trouver une partition particulière ?).Nous nous intéressons ensuite à la 1-2-3 Conjecture, qui demande si tout graphe G admet une 3-pondération voisin-somme-distinguante de ses arêtes, i.e. une 3-pondération par laquelle chaque sommet de G peut être distingué de ses voisins en comparant uniquement leur somme de poids incidents. Afin d’étudier la 1-2-3 Conjecture, nous introduisons notamment la notionde coloration localement irrégulière d’arêtes, qui est une coloration d’arêtes dont chaque classe de couleur induit un sous-graphe dans lequel les sommets adjacents sont de degrés différents.L’intérêt principal de cette coloration est que, dans certaines situations, une pondération d’arêtes voisin-somme-distinguante peut être déduite d’une coloration d’arêtes localement irrégulière. Nospréoccupations dans ce contexte sont principalement algorithmiques (est-il facile de trouver une pondération d’arêtes voisin-somme-distinguante ou une coloration d’arêtes localement irrégulière utilisant le plus petit nombre possible de poids ou couleurs ?) et structurelles (quel est le plus petit nombre de couleurs d’une coloration d’arêtes localement irrégulière ?). Nous considérons également ces questions dans le contexte des graphes orientés. / This thesis is dedicated to the study of two families of graph partition problems.First, we consider the problem of vertex-partitioning a graph into connected subgraphs.Namely, given p positive integers n1; n2; :::; np summing up to the order of some graph G, canwe partition V (G) into p parts V1; V2; :::; Vp so that each Vi induces a connected subgraph withorder ni? We then consider stronger questions. Namely, what if we want G to be partitionablewhatever are p and n1; n2; :::; np? What if we also want specific vertices of G to belong to somespecific subgraphs induced by the vertex-partition? What if we want the subgraphs induced bythe vertex-partition to be more than connected? We consider all these questions regarding boththe structural (are there structural properties ensuring that a specific vertex-partition necessarilyexists?) and algorithmic (is it hard to deduce a specific vertex-partition?) points of view.Then, we focus on the so-called 1-2-3 Conjecture, which asks whether every graph G admitsa neighbour-sum-distinguishing 3-edge-weighting, i.e. a 3-edge-weighting by which all adjacentvertices of G get distinguished by their sums of incident weights. As a tool to deal with the1-2-3 Conjecture, we notably introduce the notion of locally irregular edge-colouring, which isan edge-colouring in which every colour class induces a subgraph whose adjacent vertices havedistinct degrees. The main point is that, in particular situations, a neighbour-sum-distinguishingedge-weighting of G can be deduced from a locally irregular edge-colouring of it. Our concernsin this context are mostly algorithmic (can we easily find a neighbour-sum-distinguishing edgeweightingor locally irregular edge-colouring using the least number of weights or colours?) andstructural (what is the least number of colours in a locally irregular edge-colouring?). We alsoconsider similar matters in the context of oriented graphs.