Evolving two-dimensional cellular automata to perform density classification: A report on work in progress
Introduction
In many natural systems, simple, locally interacting components give rise to coordinated global information processing. In both natural and human-constructed information-processing systems, allowing global coordination to emerge from a decentralized collection of simple components has important potential advantages – e.g., speed, robustness, and evolvability – as compared with explicit central control. However, it is difficult to design a collection of individual components and their interactions in a way that will give rise to useful global information processing or “emergent computation”. The term “emergent computation” refers to the appearance in a system's temporal behavior of information-processing capabilities that are neither explicitly represented in the system's elementary components.
In order to understand the mechanisms by which an evolutionary process can discover methods of emergent computation, a simplified framework was proposed and studied by Crutchfield, Mitchell, and their colleagues [2], [3], [8] in which a genetic algorithm (GA) evolved 1D cellular automata (CA) to perform computations. In their work the GA was able to discover CAs with high performance on tasks requiring cooperative collective behavior. In this paper we describe extending this work to 2D CAs.
Section snippets
A density-classification task for cellular automata
In [8], 1D CAs were evolved by a GA to perform a density classification task called the “ρc=1/2” task. (This built on earlier work by Packard [9]). A successful CA for this task will decide whether or not the initial configuration (IC) contains more than half 1s. More formally, let ρ0 denote the density of 1s in the IC. If ρ0>0.5 then within M time steps the CA should go to the fixed-point configuration of all 1s; otherwise within M time steps the CA should go to the fixed-point configuration
Details of experiments
We used a GA to evolve 2D, binary state CAs to perform the ρc=1/2 task. GAs are search methods inspired by biological evolution. In a typical GA, candidate solutions to a given problem are encoded as bit strings (“chromosomes”). A population of such strings is chosen at random and evolves over several generations under selection, crossover and mutation. At each generation, the fitness of each bit string is calculated according to some externally imposed fitness function, and the highest-fitness
Results
We performed more than 100 different runs of the GA with the following parameters: for each CA in the population P=100; E=10; I=100; m=0.016; G=100 (in some runs G was set to 400), each with a different random-number seed. The dynamics of a typical run is shown in Fig. 6, which plots the best fitness rule, the elite mean fitness and the population mean fitness versus the generation for the first 50 generations. Before the GA discovers high fitness rules, the fitness of the best CA rule
Conclusion
To date, the best 2D CA rules evolved by the GA in our experiments implement unsophisticated block-expanding strategies. Although they have higher performance than that of simple hand-designed rules such as the majority or the anneal-rule, their performance is lower than that of the 2D GKL rule and the non-local majority rule, and that of the high-performance rules evolved by the GA in the 1D case [2], [3]. Why did the GA not find higher-performance rules with much better performance? Some of
Acknowledgements
Many thanks to Rajarshi Das and Wim Hordijk for their assistance and suggestions. This research was supported by the Santa Fe Institute, under grants NSF-IRI-9705830 and ONR N00014-95-1-1000. FJM was also supported by the Spanish Ministerio of Educacion y Ciencia project no. PB97-0741.
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