
## 6 Conclusions

Cellular automata are capable of universal computation and their time evolution can be complex and unpredictable. We have studied CAs that perform the computational Majority task. This task is a good example of the **phenomenon of emergence in complex systems is. In this paper we have taken an**
interest in the reasons that make this particular fitness landscape **a difficult** one. The first goal was to study the landscape as such, and thus it **is ideally** independent from the actual heuristics used to search the space. However, a second goal was to understand the features a good search technique for this particular problem space should possess. We have statistically quantified in various ways some features of the landscape and the degree of difficulty of optimizing. The neutrality of the landscape is high, and the neutral network topology is not completely random. The main observation was that the landscape has a considerable number of points with fitness 0 or 0.**5 which means**
that investigations based on sampling techniques on the whole landscape are unlikely to give good results. In the second part we have studied the landscape from the top**. Although it**
has been proved that no CA can perform the task perfectly, six efficient CAs for the majority task have been found either by hand or by using heuristic methods, especially evolutionary computation. Exploiting similarities between these CAs and symmetries in the landscape, we have defined the **Olympus**
landscape as a subspace of the Majority problem landscape which is regarded as the "heavenly home" of the six symmetric of best local optima known (**blok**′).

Then, we have measured several properties of the Olympus landscape and we have compare with those of the full landscape, finding that there are less solutions with fitness 0. FDC shows that fitness is a reliable guide to drive a searcher toward the blok′and its centroid. An ARMA(2, **1) model has been**
used to describe the fitness/fitness correlation structure. The **model indicates**
that local search heuristics are adequate for finding good rules. Fitness clouds and nsc confirm that it is easy to reach solutions with fitness higher than 0.**5. Although it is easier to find relevant CAs in this subspace than in the**
complete landscape, there are structural reasons that prevents a searcher from finding overfitted GAs in the Olympus. Finally, we have studied the dynamics and performances of three Genetic Algorithms on the Olympus in order to confirm our analysis and to find efficient CAs for the Majority problem **with**
low computational effort. Beyond this particular optimization problem, the method presented **in this** paper could be generalized. Indeed, in many optimization problems, several efficient solutions are available, and we can make good use of this set to design an "Olympus subspace" in the hope of finding better solutions or finding good solutions more quickly.