Limits on the provable consequences of one-way permutations (invited talk)
CRYPTO '88 Proceedings on Advances in cryptology
Reimer's inequality and tardos' conjecture
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
A Dual Version of Reimer's Inequality and a Proof of Rudich's Conjecture
COCO '00 Proceedings of the 15th Annual IEEE Conference on Computational Complexity
Proof of the Van den Berg–Kesten Conjecture
Combinatorics, Probability and Computing
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Let be a set of terms over an arbitrary (but finite) number of Boolean variables. Let U() be the set of truth assignments that satisfy exactly one term in . Motivated by questions in computational complexity, Rudich conjectured that there exist ∊, δ 0 such that, if is any set of terms for which U() contains at least a (1−∊)-fraction of all truth assignments, then there exists a term t ∈ such that at least a δ-fraction of assignments satisfy some term of sharing a variable with t [8]. We prove a stronger version: for any independent assignment of the variables (not necessarily the uniform one), if the measure of U() is at least 1 − ∊, there exists a t ∈ such that the measure of the set of assignments satisfying either t or some term incompatible with t (i.e., having no satisfying assignments in common with t) is at least $\Gd = 1-\Ge-\frac{4\Ge}{1-\Ge}$. (A key part of the proof is a correlation-like inequality on events in a finite product probability space that is in some sense dual to Reimer's inequality [11], a.k.a. the BKR inequality [5], or the van den Berg–Kesten conjecture [3].)