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.