Owen J. Brison

Spatial updating, spatial transients, and regularities of a complex automaton with nonperiodic architecture

We study the dynamics of patterns exhibited by rule 52, a totalistic cellular automaton displaying intricate behaviors and wide regions of active/inactive synchronization patches. Systematic computer simulations involving 2(30) initial configurations reveal that all complexity in this automaton originates from random juxtaposition of a very small number of interfaces delimiting active/inactive patches. Such interfaces are studied with a sidewise spatial updating algorithm. This novel tool allows us to prove that the interfaces found empirically are the only interfaces possible for these periods, independently of the size of the automata. The spatial updating algorithm provides an alternative way to determine the dynamics of automata of arbitrary size, a way of taking into account the complexity of the connections in the lattice.

Complete sets of initial vectors for pattern growth with elementary cellular automata

Abstract Computer simulations of complex spatio-temporal patterns using cellular automata may be performed in two alternative ways, the better choice depending on the relative size between the spatial width W of the expected patterns and their corresponding temporal period T . While the traditional timewise updating algorithm is very efficient when W ≪ T , the complementary spacewise algorithm wins whenever T ≪ W . Independently of the algorithm used, the key to obtaining exhaustive answers, not just statistical estimates, is to have explicit knowledge of the complete sets of initial conditions that need to be individually tested as sizes grow. This paper reports an efficient algorithm for generating complete sets (without redundancy) of k -vectors of initial conditions allowing one to perform definitive classifications of patterns in systems with a minimal characteristic length k , either spatial or temporal.

Exact quantification of the complexity of spacewise pattern growth in cellular automata

We analyze the two possible ways of simulating complex systems with cellular automata: by using the familiar timewise updating or by using the complementary spacewise updating. Both updating algorithms operate on identical sets of initial conditions defining the state of the automaton. While timewise growth generally probes just vanishingly small sets of initial conditions producing statistical samples of the asymptotic attractors, spacewise growth operates with much restricted sets which allow one to simulate them all, exhaustively. Our main result is the derivation of an exact analytical formula to quantify precisely one of the two sources of algorithmic complexity of spacewise detection of the complete set of attractors for elementary 1D cellular automata with generic non-periodic architectures of any arbitrary size. The formula gives the total number of initial conditions that need to be investigated to locate rigorously all possible patterns for any given rule. As simple applications, we illustrate how this knowledge may be used (i) to uncover missing patterns in previous classifications in the literature and (ii) to obtain surprisingly novel patterns that are totally unreachable with the time-honored technique of artificially imposing spatially periodic boundary conditions.

Standard sequence subgroups in finite fields

In previous work, the authors describe certain configurations which give rise to standard and to non-standard subgroups for linear recurrences of order k=2, while in subsequent work, a number of families of non-standard subgroups for recurrences of order k>=2 are described. Here we exhibit two infinite families of standard groups for k>=2.

Extendible elements of the alternating groups

An element σ of An , the Alternating group of degree n, is extendible in Sn , the Symmetric group of degree n, if there exists a subgroup H of Sn but not An whose intersection with An is the cyclic group generated by σ. A simple number-theoretic criterion, in terms of the cycle-decomposition, for an element of An to be extendible in Sn is given here.