
Equivalence of Intuitionistic Inductive Definitions and Intuitionistic Cyclic Proofs under Arithmetic
A cyclic proof system gives us another way of representing inductive def...
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A Theory of Consciousness from a Theoretical Computer Science Perspective: Insights from the Conscious Turing Machine
The quest to understand consciousness, once the purview of philosophers ...
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Formalising perfectoid spaces
Perfectoid spaces are sophisticated objects in arithmetic geometry intro...
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Classical System of MartinLof's Inductive Definitions is not Equivalent to Cyclic Proofs
A cyclic proof system, called CLKIDomega, gives us another way of repre...
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An exposition of the false confidence theorem
A recent paper presents the "false confidence theorem" (FCT) which has p...
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The strong approximation theorem and computing with linear groups
We obtain a computational realization of the strong approximation theore...
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Injective Objects and Fibered Codensity Liftings
Functor lifting along a fibration is used for several different purposes...
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Comparing computational entropies below majority (or: When is the dense model theorem false?)
Computational pseudorandomness studies the extent to which a random variable Z looks like the uniform distribution according to a class of tests F. Computational entropy generalizes computational pseudorandomness by studying the extent which a random variable looks like a 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 F is closed under taking majorities. This equivalence constitutes (essentially) the socalled dense model theorem of Green and Tao (and later made explicit by TaoZeigler, 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 F is not closed under majority, this equivalence fails. This in turn provides examples where the dense model theorem is false.
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