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I'm looking for a formal definition of the space that consists of the dimensions gene 1-N and the fitness.

In literature there is often the search space mentioned, but it only contains all genes. If I extend the search space by the fitness dimension, how can I call this space?

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In the context of evolution and evolutionary algorithms it's called a fitness landscape. In other areas such as statistical physics it's called an energy landscape.

If you have a continuous landscape (in evolution you typically don't) things are fairly straightforward. From Wikipedia:

In mathematical terms, an energy landscape can be defined as a pair $(X, f)$ consisting of a topological space $X$ representing the physical states or parameters of a system together with a continuous function $f: X → \mathbb{R}$ representing the energies associated to these states or parameters such that the image of $f$ represents a hypersurface in $\mathbb{R}^n$."

In discrete landscapes things are more complicated since we need to specify which states neighbour each other, essentially giving us a graph underlying the set of solutions. What you end up with is not as clean as you'd like. It's a triple $(X,N, f)$, where $X$ is the set of solutions (i.e., possible genotypes), $N:X\to 2^X$ is a neighbourhood function, and $f:X\to\mathbb{R}$ is the fitness function.

For a treatment of these discrete fitness landscapes that should satisfy you with regard to mathematical rigour I would direct you to this survey paper by Reidys and Stadler.

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  • $\begingroup$ The fitness landscape is just a surface in the space that I look for. "Landscape" is also no mathematical term like "space". $\endgroup$ – Stephan Feb 14 '12 at 12:51
  • $\begingroup$ I've now elaborated upon my original answer. $\endgroup$ – James King Feb 14 '12 at 13:51
  • $\begingroup$ Thanks for your reply, that goes deep into the theory. But I just wanted to clarify: "The fitness function is a mapping from the genotype space to the ??? space". Or "The ??? space is the genotype space extended by the dimension of fitness". Can I say something like that? $\endgroup$ – Stephan Feb 14 '12 at 14:56
  • $\begingroup$ You can say that the fitness function is a mapping from the genotype space to the real numbers. Talking about the dimensionality of a discrete space is not straightforward. Unless you say which genotypes neighbour which other genotypes, all you have is a set of points to which f associates real values. If you do specify neighbourhoods, you have a graph with real values on the vertices. $\endgroup$ – James King Feb 14 '12 at 15:41
  • $\begingroup$ is specifying the neighbourhood function instead of a directed graph with edges weighted by likelihood of mutations (or 0-1 for impossible/possible) standard? $\endgroup$ – Artem Kaznatcheev Feb 14 '12 at 18:43

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