Randomness and Non-determinism

TL;DR

This paper surveys recent developments in understanding the power and relationships of randomness and non-determinism in computation, highlighting techniques like polynomial methods and Fourier analysis.

Contribution

It provides a comprehensive overview of recent results and a stronger form of a key finding relating randomness and non-determinism in computational complexity.

Findings

Emerging patterns link randomness and non-determinism in computation.

Techniques like polynomial methods and Fourier analysis are instrumental.

A significant result with a stronger formulation is presented.

Abstract

Exponentiation makes the difference between the bit-size of this line and the number (<< 2^{300}) of particles in the known Universe. The expulsion of exponential time algorithms from Computer Theory in the 60's broke its umbilical cord from Mathematical Logic. It created a deep gap between deterministic computation and -- formerly its unremarkable tools -- randomness and non-determinism. Little did we learn in the past decades about the power of either of these two basic "freedoms" of computation, but some vague pattern is emerging in relationships between them. The pattern of similar techniques instrumental for quite different results in this area seems even more interesting. Ideas like multilinear and low-degree multivariate polynomials, Fourier transformation over low-periodic groups seem very illuminating. The talk surveyed some recent results. One of them, given in a stronger form than previously published, is described below.