Curiouser and Curiouser: The Link between Incompressibility and Complexity

Abstract

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 $\mbox{\rm BPP}$ denote the class of problems that can be solved with negligible error by probabilistic polynomial-time computations, and let $\mbox{\rm NEXP}$ denote the class of problems solvable in nondeterministic exponential time.

Conjecture 1. $\mbox{\rm NEXP} = \mbox{\rm NP}^R$.

Conjecture 2. $\mbox{\rm BPP}$ is the class of problems non-adaptively polynomial-time reducible to R.

These conjectures are not only audacious; they are obviously false! R is not a decidable set, and thus it is absurd to suggest that the class of problems reducible to it constitutes a complexity class.

The absurdity fades if, for example, we interpret “$\mbox{\rm NP}^R$” to be “the class of problems that are $\mbox{\rm NP}$-Turing reducible to R, no matter which universal machine we use in defining Kolmogorov complexity”. The lecture will survey the body of work (some of it quite recent) that suggests that, when interpreted properly, the conjectures may actually be true.