O problema MODn com composição de autômatos celulares unidimensionais: resolução e simplificações

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Martins, Claudio Luis de Meo
Oliveira, Pedro Paulo Balbi de
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Monteiro, Luiz Henrique Alves
Mendonça, José Ricardo Gonçalves de
Barbosa, Valmir Carneiro
Silva, Leandro Augusto da
Engenharia Elétrica
The understanding of how the composition of cellular automata rules can perform prede ned computations may contribute to the general notion of emerging computation by means of locally processing components. In this context, we propose a solution to the MODn Problem, which is the determination of whether the number of 1-bits in a binary string is perfectly divisible by the positive integer n > 1. The solution is a composition of one-dimensional cellular automata rules, i.e., the application of di erent rules on a lattice with periodic boundary conditions, which are replaced after some iterations, and all of them with maximum radius equal to n 􀀀 1. In this work, the (XU; LEE; CHAU, 2003) solution for MOD3 Problem (n = 3) is extended for any value of n, and the solution is given for any lattice size N that is co-prime to n. In this generalised solution, the number of iterations depends only on N, with O(N2). This solution relies upon two essential classes of rules, that have been de ned herein: the Replacement rules, that replace a certain amount of identical end bits on the lattice with the opposite value, and the Grouping rules, that group isolated strings of identical and consecutive bits on the lattice, to larger strings of the same bit value. Furthermore, we also show how the solution can be simpli- ed in terms of a reduction on the number of required rules, by de ning some operations that involve the rules' active state transitions, i.e., those that change the value of the centre cell of the neighbourhood. To this end, we de ned the operations of Partitioning (the separation of the active transitions of a rule in di erent rules), Joining (the union of the all active transitions of di erent rules in the same rule), and Merging (the joining of all active transitions of the rules involved, but removing some of them or even adding new active transitions to get the desired adjustments. Using the same concepts and methodology, we proposed a x for the only rule that had been reported in the literature for solving the MOD2 Problem, which is known as the Parity Problem.
autômatos celulares , computação emergente , composição de regras , problema módulo-n , transições de estado ativas
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MARTINS, Claudio Luis de Meo. O problema MODn com composição de autômatos celulares unidimensionais: resolução e simplificações. 2016. 93 f. Tese (Engenharia Elétrica) - Universidade Presbiteriana Mackenzie, São Paulo .