Abstract |
The density classification problem on graphs consists in finding a local dynamics such that, given a graph and an initial configuration of 0's and 1's assigned to the nodes of the graph, the dynamics converge to the fixed point configuration of all 1's if the fraction of 1's is greater than the critical density (typically 1/2) and, otherwise, it converges to the all 0's fixed point configuration. To solve this problem, we follow the idea proposed in [R. Briceno, P. M. de Espanes, A. Osses and I. Rapaport, Physica D 261, 70 (2013)], where the authors designed a cellular automaton inspired by two mechanisms: diffusion and amplification. We apply this approach to different well-known graph classes: complete, regular, star, Erdos-Renyi and Barabasi-Albert graphs. |
Abstract |
Cellular automata are locally defined, homogeneous dynamical systems, discrete in space, time and state variables. Within the context of one-dimensional, binary, cellular automata operating on cyclic configurations of odd length, we consider the general decision problem: if the initial configuration satisfies a given property, the lattice should converge to the fixed-point of all 1s ((1) over right arrow), or to (0) over right arrow, otherwise. Two problems in this category have been widely studied in the literature, the parity problem [1] and the density classification task [4]. We are interested in determining all cellular automata rules with neighborhood sizes of 2, 3, 4 and 5 cells (i.e., radius r of 0.5, 1, 1.5 and 2.5) that solve decision problems of the previous type. We have demonstrated a theorem that, for any given rule in those spaces, ensures the non existence of fixed points other than (0) over right arrow and (1) over right arrow for configurations of size larger than 2(2r), provided that the rule does not support different fixed points for any configuration with size smaller than or equal to 2(2r). In addition, we have a proposition that ensures the convergence to only (0) over right arrow or (1) over right arrow of any initial configuration, if the rule complies with given conditions. By means of theoretical and computational approaches, we determined that: for the rule spaces defined by radius 0.5 and r = 1, only 1 and 2 rules, respectively, converge to (1) over right arrow or (0) over right arrow, to any initial configuration, and both recognize the same language, and for the rule space defined by radius r = 1.5, 40 rules satisfy this condition and recognize 4 different languages. Finally, for the radius 2 space, out of the 4,294,967,296 different rules, we were able to significantly filter it out, down to 40,941 candidate rules. We hope such an extensive mining should unveil new decision problems of the type widely studied in the literature. |
Address |
[Lobos, Fabiola; Goles, Eric; Montealegre, Pedro] Univ Adolfo Ibanez, Fac Ingn & Ciencias, Ave Diagonal Torres 2640, Santiago, Chile, Email: pp.balbi@gmail.com |