Fitness landscape of the cellular automata majority problem: View from the “Olympus”

Abstract

In this paper we study cellular automata (CAs) that perform the computational Majority task. This task is a good example of what the phenomenon of emergence in complex systems is. We take an interest in the reasons that make this particular fitness landscape a difficult one. The first goal is to study landscape as such, and thus it is ideally independent from the actual heuristics used to search the space. However, a second goal is to understand the features a good search technique for this particular problem space should possess. We statistically quantify in various ways the degree of difficulty of searching this landscape. Due to neutrality, investigations based on sampling techniques on the whole landscape are difficult to conduct. So, we go exploring the landscape from the top. Although it has been proved that no CA can perform the task perfectly, several efficient CAs for this task have been found. Exploiting similarities between these CAs and symmetries in the landscape, we define the Olympus landscape which is regarded as the "heavenly home" of the best local optima known (blok). Then we measure several properties of this subspace. Although it is easier to find relevant CAs in this subspace than in the overall landscape, there are structural reasons that prevents a searcher from finding overfitted CAs in the Olympus. Finally, we study dynamics and performances of genetic algorithms on the Olympus in order to confirm our analysis and to find efficient CAs for the Majority problem with low computational cost.

An Iterated Local Search to find many solutions of the 6-states Firing Squad Synchronization Problem

“…In this work, we prefer to use the fitness cloud which gives a better insight into the correlation of fitness between neighboring solutions [27]. The fitness cloud is the conditional bivariate probability density of reaching a fitness value from a solution of a given fitness value applying a local search operator [7]. Several scatter plots and metrics can be deduced from the fitness cloud.…”

“…Some optimization problems related to cellular automata are already known to be shaped as a neutral fitness landscape [6,7].…”

“…In [7], Verel et al gave a deeper insight in the fitness landscape structure of the search space associated to the majority problem, characterizing its neutral nature. Wolz et al [8] proposed an efficient evolutionary algorithm to find solutions as good as possible for the majority problem.…”

On the Landscape of Combinatorial Optimization Problems

“…Algebraic properties of the solutions in the landscape [16], [25], modality (number of local optima) [26] and the fractal dimension of the fitness landscape [27] are other examples. Some researchers try to explain when a problem becomes hard, by studying the area in the landscape called "Olympus", in which the better local optima are located [28], [29]. Fitness clouds is another method proposed to visualise the fitness landscape, which tries to represent some properties of the fitness landscape [30], [31], [32], reflecting the problem hardness.…”

Neutrality in the Graph Coloring Problem

“…The average or the distribution of neutral degrees over the landscape may be used to qualify the level of neutrality of a problem instance. This measure plays an important role in the dynamics of local search algorithms [25,27].…”