Comparing computational entropies below majority (or: When is the dense model theorem false?)

TL;DR

This paper investigates the conditions under which the dense model theorem, relating different notions of computational entropy, holds or fails, revealing that it does not always apply when certain test classes are not closed under majority.

Contribution

It demonstrates that the equivalence of computational entropy notions fails when the test class is not closed under majority, providing explicit counterexamples and clarifying the theorem's limitations.

Findings

The dense model theorem does not hold for certain test classes.

Counterexamples show failure of equivalence in non-closed classes.

Implications for cryptography and complexity theory.

Abstract

Computational pseudorandomness studies the extent to which a random variable $\bf{Z}$ looks like the uniform distribution according to a class of tests $\cal{F}$. Computational entropy generalizes computational pseudorandomness by studying the extent which a random variable looks like a \emph{high entropy} distribution. There are different formal definitions of computational entropy with different advantages for different applications. Because of this, it is of interest to understand when these definitions are equivalent. We consider three notions of computational entropy which are known to be equivalent when the test class $\cal{F}$ is closed under taking majorities. This equivalence constitutes (essentially) the so-called \emph{dense model theorem} of Green and Tao (and later made explicit by Tao-Zeigler, Reingold et al., and Gowers). The dense model theorem plays a key role in Green and Tao's proof that the primes contain arbitrarily long arithmetic progressions and has since been connected to a surprisingly wide range of topics in mathematics and computer science, including cryptography, computational complexity, combinatorics and machine learning. We show that, in different situations where $\cal{F}$ is \emph{not} closed under majority, this equivalence fails. This in turn provides examples where the dense model theorem is \emph{false}.