
## 5 Olympus Landscape

We have seen that there are many similarity inside the blok**. In this section** we will use this feature to define the Olympus Landscape **and, to show and** exploit, the relevant properties of this subspace.

## 5.1 Definition

The Olympus Landscape is a subspace of the Majority problem landscape. It takes its name from the Mount Olympus **which is traditionally regarded as** the heavenly home of the ancient Greek gods. Before defining this subspace we study the two natural symmetries of the majority problem. The states 0 and 1 play the same role in the computational task ; so flipping bits in the entry of a rule and in the result have no effect on performance. In the same way, CAs can compute the majority problem according to right **or left**
direction without changing performance. We denote S01 and Srl **respectively**
the corresponding operator of 0/1 symmetry and right/lef t **symmetry. Let**
x = (x0, . . . , xλ−1) ∈ {0, 1}
λ be a solution with λ = 22r+1. The 0/**1 symmetric**
of x is S01(x) = y where for all i, yi = 1 − xλ−i. The right/lef t **symmetric of**
x is Srl(x) = y where for all i, yi = xσ(i) **with** σ(Pλ−1 j=0 2 nj ) = Pλ−1 j=0 2 λ−1−nj.

The operators are commutative: SrlS01 = S01Srl**. From the 128 bits, 16 are** invariant by Srl symmetry and none by S01**. Symmetry allows to introduce**
diversity without losing quality ; so evolutionary algorithm could be improved using the operators S01 and Srl.

We have seen that some bit values could be useful to reach a good solution
(see subsection 4.4.2), and some of those are among the 29 joint bits of the blok (see subsection 4.4.1). Nevertheless, two optima from the blok **could be**
distant whereas some of theirs symmetrics are closer. Here the idea is to choose for each blok **one symmetric in order to broadly maximize the number of joint**
bits. The optima GLK, Das, Davis and ABK have only 2 symmetrics because symmetrics by S01 and Srl **are equal. The optima Coe1 and Coe2 have 4 symmetrics. So, there are 2**4.4 2 **= 256 possible sets of symmetrics. Among these**
sets, we establish the maximum number of joint bits which is possible to **obtain** is 51. This "optimal" set contains the six **Symmetrics of Best Local Optima**
Known (blok′**) presented in table 5. The Olympus Landscape is defined from**
the blok′**as the schemata** S
′
**with the 51 fixed bits above:**
000*0*0* 0****1** 0***00** **0**1** 000***** 0*0**1** ******** 0*0**1*1