An incompressibility theorem for automatic complexity

Kolmogorov Complexity and the Incompressibility Method

1. Introduction. What makes one object more complex than another? Kolmogorov complexity, or program-size complexity, provides one of many possible answers to this fundamental question. In this theory, whose foundations have been developed independently by R. the complexity of an object is defined as the length of its shortest effective description, which is the minimum number of symbols that mu...

An Isomorphism Theorem for Circuit Complexity

We show that all sets complete for NC1 under AC0 reductions are isomorphic under AC0computable isomorphisms. Although our proof does not generalize directly to other complexity classes, we do show that, for all complexity classes C closed under NC1-computable many-one reductions, the sets complete for C under NC0 reductions are all isomorphic under AC0-computable isomorphisms. Our result showin...

Curiouser and Curiouser: The Link between Incompressibility and Complexity

This talk centers around some audacious conjectures that attempt to forge firm links between computational complexity classes and the study of Kolmogorov complexity. More specifically, let R denote the set of Kolmogorov-random strings. Let BPP denote the class of problems that can be solved with negligible error by probabilistic polynomial-time computations, and let NEXP denote the class of pro...

An Operator Embedding Theorem for Complexity Classes of Recursive Functions

Looking for an Analogue of Rice's Theorem in Complexity Theory

Rice's Theorem says that every nontrivial semantic property of programs is undecidable. It this spirit we show the following: Every nontrivial absolute (gap, relative) counting property of circuits is UP-hard with respect to polynomial-time Turing reductions.