On Universal Complexity Measures
Abstract
We relate the computational complexity of finite strings to universal representations of their underlying symmetries. First, Boolean functions are classified using the universal covering topologies of the circuits which enumerate them. A binary string is classified as a fixed point of its automorphism group; the irreducible representation of this group is the string's universal covering group. Such a measure may be used to test the quasi-randomness of binary sequences with regard to first-order set membership. Next, strings over general alphabets are considered. The complexity of a general string is given by a universal representation which recursively factors the codeword number associated with a string. This is the complexity of the representation recursively decoding a Godel number having the value of the string; the result is a tree of prime numbers which forms a universal representation of the string's group symmetries.
Cite
@article{arxiv.1005.2254,
title = {On Universal Complexity Measures},
author = {John Scoville},
journal= {arXiv preprint arXiv:1005.2254},
year = {2011}
}
Comments
The paper has been withdrawn due to the poor predictive performance of circuit complexity vs. universal data compression. Given this fact, as well as the high computational complexity, the method seems to have little practical utility