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?


1 Answer 1


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.

  • $\begingroup$ 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, 2012 at 12:51
  • $\begingroup$ I've now elaborated upon my original answer. $\endgroup$
    – James King
    Feb 14, 2012 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, 2012 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, 2012 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? $\endgroup$ Feb 14, 2012 at 18:43

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.