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MAking Hard Problems Harder
, 2005
"... We present a general approach to the hoary problem of (im)proving circuit lower bounds. We define notions of hardness condensing and hardness extraction, in analogy to the corresponding notions from the computational theory of randomness. A hardness condenser is a procedure that takes in a Boolean f ..."
Abstract

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We present a general approach to the hoary problem of (im)proving circuit lower bounds. We define notions of hardness condensing and hardness extraction, in analogy to the corresponding notions from the computational theory of randomness. A hardness condenser is a procedure that takes in a Boolean function as input, as well as an advice string, and outputs a Boolean function on a smaller number of bits which has greater hardness when measured in terms of input length. A hardness extractor takes in a Boolean function as input, as well as an advice string, and outputs a Boolean function defined on a smaller number of bits which has close to maximum hardness. We prove several positive and negative results about these objects. First, we observe that hardnessbased pseudorandom generators can be used to extract deterministic hardness from nondeterministic hardness. We derive several consequences of this observation. Among other results, we show that if E has exponential nondeterministic hardness, then E with linear advice has close to maximum deterministic hardness. We demonstrate a rare downward closure result: there is δ> 0 such that E with subexponential advice is contained in nonuniform space 2 δn if and only if there is k> 0 such that P with quadratic advice can be approximated in nonuniform space n k. Next, we consider limitations on natural models of hardness condensing and extraction. We show lower bounds on the length of the advice required for hardness condensing in a very general model of “relativizing ” condensers. We show that nontrivial blackbox extraction of deterministic hardness from deterministic hardness is essentially impossible. Finally, we prove positive results on hardness condensing in certain special cases. We show how to condense hardness from a biased function without any advice, using a hashing technique. We also give a hardness condenser without advice from averagecase hardness to worstcase hardness. Our technique involves a connection between hardness condensing and certain kinds of explicit constructions of covering codes.
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"... (Note that even though we believe Σi ̸ = Πi, oracle access to Σi gives the same power as oracle access to Πi. Do you see why?) We show that this leads to an equivalent definition. For this section only, let ΣO i refer to the definition in terms of oracles. We prove by induction that Σi = ΣO i. (Sinc ..."
Abstract
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(Note that even though we believe Σi ̸ = Πi, oracle access to Σi gives the same power as oracle access to Πi. Do you see why?) We show that this leads to an equivalent definition. For this section only, let ΣO i refer to the definition in terms of oracles. We prove by induction that Σi = ΣO i. (Since ΠOi = coΣOi, this proves it for Πi, ΠO i as well.) For i = 1 this is immediate, as Σ1 = N P = N PP = ΣO 1. Assuming Σi = ΣO i, we prove that Σi+1 = ΣO i+1. Let us first show that Σi+1 ⊆ ΣO i+1 L ∈ Σi+1. Then there exists a polynomialtime Turing machine M such that x ∈ L ⇔ ∃w1∀w2 · · · Qi+1wi+1 M(x, w1,..., wi+1) = 1. In other words, there exists a language L ′ ∈ Πi such that x ∈ L ⇔ ∃w1 (x, w1) ∈ L ′.