Question: What is known about the role that binary representation of data plays in algorithmic complexity theory?
I define "analog representation of data" loosely as any method of representing data which is not a sequence of 1's and 0. A more precise definition would be nice to have obviously. The basic idea is that an analog representation of data contains additional information beyond what is contained in a binary representation.
A very trivial example of an "analog" representation of the natural numbers is to label all the primes $p_1, p_2, p_3 \cdots$ and then represent numbers by their prime factorization, instead of in binary form. It is obvious that with this representation of numbers, the problem of factoring large numbers is of polynomial complexity.
Less trivially, according the Shor's Alogirithm, if we represent numbers using an analog physical system (in this case, using quibits instead of bits), then it's possible to factor large numbers in polynomial time as well.
The natural conclusion to draw is that the choice to use binary or not for representing data likely plays an important role in determining if a given problem is in P or NP, and one would think that in order to prove P is not equal to NP it would be required to use the assumption that data is represented in a binary way at some point.
Sorry for the open ended question. Any and all feedback is very much appreciated. I am getting out of my comfort zone here and was unable to find much help using google.