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• W3100540831 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}." @default.
• W3100540831 created "2020-11-23" @default.
• W3100540831 creator A5002208831 @default.
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• W3100540831 date "2020-11-12" @default.
• W3100540831 modified "2023-09-23" @default.
• W3100540831 title "Comparing computational entropies below majority (or: When is the dense model theorem false?)" @default.
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• W3100540831 hasPublicationYear "2020" @default.
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