Dynamics of automata networks: theory and numerical experiments

• Autor(a) principal: Gonzalez, Fabiola Lobos
• Data de Publicação: 2020
• Tipo de documento: Tese
• Idioma: eng
• Título da fonte: Biblioteca Digital de Teses e Dissertações do Mackenzie
• Texto Completo: https://dspace.mackenzie.br/handle/10899/28596
• topic: sistemas dinamicos discretos; automatas celulares unidimensionales; problemas de decision; complejidad computacional; problema de estabilidad; redes booleanas; CNPQ::CIENCIAS EXATAS E DA TERRA::CIENCIA DA COMPUTACAO

Automata Networks are discrete dynamical systems that have been used to model diverse complex systems such as the study of the evolution and self-organization. Automata Networks are composed of a graph, where each node acquires di erent states (from a nite set) and evolves in units of discrete time,according to a certain function { known as the local transition rule { that depends on the neighboring states of the network. Cellular Automata (CA) and Boolean Networks (BN) are particular cases of Automata Networks. In the CA, the neighborhood structure and the transition rules are the same for all nodes. On the other hand, Boolean Networks are non-uniform, binary systems, meaning that each node can take only two possible states and evolves according to its own local Boolean transition rule,on an arbitrary nite graph (orientated or not oriented).The thesis consists of three parts. The rst two study problems related to Cellular Automata: a class of decision problems with binary, one-dimensional CAs, and the complexity analysis of a speci c decision problem for elementary CA, the prediction of the so-called stability problem. The third part is focused on the dynamics of Boolean Networks with Memory (RBM) and their applications. In the rst one we are interested in determining all CAs rules of a xed radius that solve decision problems associated to the unique xed points !1 and !0 and no further attractor for any initial con guration. To do this, rst we look for the necessary conditions that the rules must meet to solve this problem and then we study di erent types of equivalences of the rules, in order to decrease the search space; the latter is important because as the radius of the CAs increases, the amount of rules grows exponentially. Later on we show all the rules that solve the decision problem for CAs with radius 0.5, 1 and 1.5, and the languages they recognize; furthermore, we show some results of searches for these problems for CAs with radius 2. The second problem we analyse is the complexity of the stability problem for rules in the ECA rule space. The stability problem can be stated as follows: given any nite con guration of a given length with periodic boundary condition, and a cell in such a con guration, the problem consists of determining whether or not the state of such a cell will ever change at some point during the (in nite) time evolution. Representatives of each of the 88 dynamically non-equivalent rule classes of the ECA em. This is carried out by grouping rules according to some simple and recurrent aspects of their transition tables. The analyses are also 4 made for more complex cases that must be studied particularly and some conditions for stability are given for rules for which the stability problem remains open.In the third problem we study the Boolean network model with memory from the theoretical and applied point of view by following a constructive approach. We develop an equivalent intermediate representation, merging gene and protein vertices, that simplify substantially the phase space. This representation is referred to as Memory Boolean Networks (MBN). The theoretical part of our account is followed by applications to two real biological systems: the immune control of the -phage and the genetic control of the oral morphogenesis of the plant Arabidopsis thaliana.