Natural Proofs
Alexander A. Razborov,Steven Rudich +1 more
- Vol. 55, Iss: 1, pp 24-35
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It is shown that the weaker class ofAC0-natural proofs which is sufficient to prove the parity lower bounds of Furst, Saxe, and Sipser, Yao, and Hastad is inherently incapable of proving the bounds of Razborov and Smolensky.Abstract:
We introduce the notion ofnaturalproof. We argue that the known proofs of lower bounds on the complexity of explicit Boolean functions in nonmonotone models fall within our definition of natural. We show, based on a hardness assumption, that natural proofs can not prove superpolynomial lower bounds for general circuits. Without the hardness assumption, we are able to show that they can not prove exponential lower bounds (for general circuits) for the discrete logarithm problem. We show that the weaker class ofAC0-natural proofs which is sufficient to prove the parity lower bounds of Furst, Saxe, and Sipser, Yao, and Hastad is inherently incapable of proving the bounds of Razborov and Smolensky. We give some formal evidence that natural proofs are indeed natural by showing that every formal complexity measure, which can prove superpolynomial lower bounds for a single function, can do so for almost all functions, which is one of the two requirements of a natural proof in our sense.read more
Citations
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Computational Complexity: A Modern Approach
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TL;DR: This beginning graduate textbook describes both recent achievements and classical results of computational complexity theory and can be used as a reference for self-study for anyone interested in complexity.
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TL;DR: The goal of this monograph is to survey the field of arithmetic circuit complexity, focusing mainly on what it finds to be the most interesting and accessible research directions, with an emphasis on works from the last two decades.
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TL;DR: A new construction of pseudo-random functions such that computing their value at any given point involves two multiple products, much more efficient than previous proposals.
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Derandomizing polynomial identity tests means proving circuit lower bounds
TL;DR: If Permanent requires superpolynomial-size arithmetic circuits, then one can test in subexponential time whether a given arithmetic circuit of polynomially bounded degree computes an identically zero polynomial.
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Pseudorandom generators without the XOR lemma
TL;DR: The main result of Impagliazzo and Wigderson as discussed by the authors is that if there exists a decision problem solvable in time 2/sup O(n)/ and having circuit complexity 2 /sup /spl Omega/(n)/ (for all but finitely many n) then P=BPP.
References
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