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A Dual Version of Reimer's Inequality and a Proof of Rudich's Conjecture

By Clifford Smyth


Abstract We prove a dual version of the celebrated inequality of D. Reimer (a.k.a. the van den Berg-Kesten conjecture). We use the dual inequality to prove a combinatorial conjecture of S. Rudich motivated by questions in cryptographic complexity. One consequence of Rudich's Conjecture is that there is an oracle relative to which one-way functions exist but oneway permutations do not. The dual inequality has another combinatorial consequence which allows R. Impagliazzo and S. Rudich to prove that if P = NP then NP " co-NP ` i.o.AvgP relative to a random oracle. Keywords: correlation inequalities, Rudich's conjecture, Reimer's inequality, van den Berg-Kesten conjecture, BK conjecture. 1 Introduction Let V = {v1, v2,..., vn} be a set of Boolean ({0, 1}-valued) variables. Let T bea set of conjunctive terms over the variables and negations of variables in V; e.g., T = {v1v2v3, v3v4, v6v7v9v10,...}. We say s, t 2 T are dependent, s, t, if theyhave at least one variable in common (where we ignore negations). Otherwis

Year: 1999
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