## Tuesday, September 16, 2014

### Adleman's K-potency and Kauffman's Atoms

One of the motivations for the sequence-space probability dispersion matrix is that as a model it might estimate the computational depth of nucleotide sequences, or the relative depth between two nucleotide sequences.   How deep is a given nucleotide sequence?

Kauffman writes in a recent foreword that the universe has produced every kind of atom it could produce (an ergodic process, whereas enumerating the realizable proteins is a non-ergodic process), but Leonard Adleman has elsewhere written in "The Rarest Things in the Universe" (among "entropy and information" here) that atoms with higher counts of protons than we've thus far encountered could be considered to have larger depth (and thus be somewhat analogous to Kauffman's sequences).
I am not a physicist, but I suppose it is possible to theorize about an atomic nucleus with a million protons. But what if I want to create one? It appears that producing transuranic elements takes huge amounts of time/energy and the greater the number of protons, the more time/energy it takes. It is even conceivable (to me at least) that there is not enough time/energy available (at least on earth) to actually produce one. Like the prime factorization of $2^{1,000,000}-1$, it may exist in theory but not in reality. On the other hand, physicists from Russia and America, using lots of time/energy, have created an atomic nucleus with 118 protons called Ununoctium. Ununoctium is analogous to Childers’ prime factorization; both exist in reality; both were very costly to create.
In his 1979 paper "Time, Space, and Randomness," Adleman develops an idea about "K-potency" motivated by an analogy with thermodynamics, specifically chemical reactions that take much less time going in one direction than the other.

This approach to "one-way functions" is important to crytography, incidentally.  His definition for K-potency follows here: