On Universal Complexity Measures

TL;DR

This paper introduces a universal framework linking string complexity to their symmetries, using topological and group-theoretic methods to classify and measure randomness and structure in binary and general strings.

Contribution

It develops a novel universal representation-based complexity measure for strings, connecting symmetry groups, topologies, and recursive factorizations, extending to general alphabets.

Findings

Classifies Boolean functions via universal covering topologies.

Uses automorphism groups to assess string randomness.

Provides a recursive prime-based representation for general strings.

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.