<p>In the context of evolution and evolutionary algorithms it's called a <a href="http://en.wikipedia.org/wiki/Fitness_landscape">fitness landscape</a>.  In other areas such as statistical physics it's called an <a href="http://en.wikipedia.org/wiki/Energy_landscape">energy landscape</a>.</p>

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

<blockquote>
  <p>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 $\mathbb{R}^n$."</p>
</blockquote>

<p>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.</p>

<p>For a treatment of these discrete fitness landscapes that should satisfy you with regard to mathematical rigour I would direct you to <a href="http://www.jstor.org/pss/4148412">this survey paper</a> by Reidys and Stadler.</p>

CC BY-SA 3.0
2012-02-14T12:51:53.417Z
The fitness landscape is just a surface in the space that I look for. "Landscape" is also no mathematical term like "space".
Stephan
CC BY-SA 3.0
2012-02-14T13:51:18.560Z
I've now elaborated upon my original answer.
James King
CC BY-SA 3.0
2012-02-14T14:56:53.523Z
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?
Stephan
CC BY-SA 3.0
2012-02-14T15:41:23.187Z
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.
James King
CC BY-SA 3.0
2012-02-14T18:43:51.080Z
is specifying the neighbourhood function instead of a directed graph with edges weighted by likelihood of mutations (or 0-1 for impossible/possible) standard?
Artem Kaznatcheev
CC BY-SA 3.0
2012-02-14T19:24:22.320Z
I've seen this before, with the neighbourhood function replaced by a Markov chain on the set X of states.
James King
CC BY-SA 3.0
2012-02-15T10:36:23.510Z
The neighborhood is defined by euclidean distance of each gene to the others individual gene on the same locus.
Stephan
CC BY-SA 3.0
2012-02-16T10:54:04.313Z
Do you mean Hamming distance?  If you're not familiar with this you should look into Hamming distance, Hamming spaces, and Hamming graphs (short articles for all on Wikipedia) as they are all very relevant here.
James King
CC BY-SA 3.0
2012-02-14T12:48:52.093Z
2012-02-14T15:30:22.703Z
2012-02-14T15:30:22.703Z
James King
<p>I'm looking for a formal definition of the space that consists of the dimensions gene 1-N and the fitness.</p>

<p>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?</p>

<optimization><ne.neural-evol><genetic-algorithms><genetic-programming>
What is the correct name for the space of genotypes and fitness?
Stephan
# What is the correct name for the space of genotypes and fitness?

(Asked by: username_0 on 2012-02-14)

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?

**Tags:** optimization, ne.neural-evol, genetic-algorithms, genetic-programming


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## 1 answers

### Answer by username_1 on 2012-02-14. Score: 5 | Accepted answer

In the context of evolution and evolutionary algorithms it's called a [fitness landscape](http://en.wikipedia.org/wiki/Fitness_landscape). In other areas such as statistical physics it's called an [energy landscape](http://en.wikipedia.org/wiki/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](http://www.jstor.org/pss/4148412) by Reidys and Stadler.

> **Comments:**
> **username_0**: The fitness landscape is just a surface in the space that I look for. "Landscape" is also no mathematical term like "space".  
> **username_1**: I've now elaborated upon my original answer.  
> **username_0**: 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?  
> **username_1**: 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.  
> **username_2**: is specifying the neighbourhood function instead of a directed graph with edges weighted by likelihood of mutations (or 0-1 for impossible/possible) standard?  
> **username_1**: I've seen this before, with the neighbourhood function replaced by a Markov chain on the set X of states.  
> **username_0**: The neighborhood is defined by euclidean distance of each gene to the others individual gene on the same locus.  
> **username_1**: Do you mean Hamming distance?  If you're not familiar with this you should look into Hamming distance, Hamming spaces, and Hamming graphs (short articles for all on Wikipedia) as they are all very relevant here.  

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