See also
What shape is the fitness landscape explored by agents in an evolutionary process?
In genetic algorithms, as I have used them in recent times — fixed-length genomes having floating values in the range \([0,1]\) — the landscape is \([0,1]^n\) for genome of length \(n\). (In e.g. John-Holland-style classic GAs, of course, it’s \(\{0,1\}^n\), and in terrestrial life, something like \(\{G,A,T,C\}^n\).)
What do you do with non-fixed-length genomes when .e.g your crossover or mutation operators no longer conserve length? (as in, e.g. real DNA-based organisms) I suppose you pick a very large upper bound and claim that represents the dimensionality of the space you’re in. You don’t want to unbounded and thus in a general Hilbert space where I suspect that things break down horribly in a number of ways - the error threshold vanishes, for example.) But what am I, a measure theorist? Let’s gloss over that.
Consider more general evolutionary processes, say, genetic programming- is \(\mathbb{R}^n\) still the most natural space in which to embed our landscape? There is still that trivial mapping from the free monoid depicting the genome to \(\mathbb{R}^n\), but now there are problem-domain- and encoding-specific “folds”, cases where the symbols in the string can be swapped or substituted without changing the functional form of the algorithm, and which can be known a priori given said encoding and search-space. (Anyone who uses genetic programming for symbolic regression is used to, e.g. getting both \(\sin(x)\) and \(-\sin(-x)\) as solutions, or \(x+y\) and \(y+x\).) How do the likelihoods of these degenerate solutions increase with the genome length? Is it still exponential in genome length, as with the volume of space encompassed by a non-hierarchical GA?
Further, consider evolutionary processes that include significant levels of niche construction, where evolution becomes path-dependent. Is there still some notion of fitness landscape that can be made rigorous for these algorithms, or some mapping between phenotypic and genotypic fitnesses that captures the same function as the fitness landscape? I suspect this problem is well-explored, but I’m missing the keywords to find it. A jelly bean for you if you can tell me, so I can tell my old ecology lecture what to show on the slides for the lecture on path-dependence.
Now, going out on a limb, consider a problem domain that looks evolutionary if you squint at it: creating mathematical theorems. Certainly Gödel and Turing invite looking at the things as symbol strings. I saw a presentation by Greg Leibon suggesting that there was a natural embedding of mathematical field onto hyperbolic geometry. Sure, his data set was Wikipedia mathematical article links, and the whole idea was tongue-in-cheek. But it feels like there is something in there, if not a whole-cloth topological theory of human knowledge. Is there some process driving mathematical innovation that means that the links between fields sit so naturally in hyperbolic space? Is it some characteristic of the subject matter itself? If either of these are true, would they be true of other fields? Science in general? Philosophy? Engineering? Design? Biological fitnesses?
I know people must occasionally toy with these areas, but google scholar has not helped me thus far. Hints?
It is probably clear from my sloppy language here that I skipped the differential geometry elective in my undergraduate degree. Perhaps I should be couching this in terms of topology alone? Or is that importing too much theoretical machinery across the border with no reason to pay the tariff? Will it all fall out just as well if I think about marginal cost of exploring fitness space without needing exotic geometries to represent that cost as a volume measure on the space to explore?