The Query Complexity of Witness Finding

Abstract

We study the following information-theoretic witness finding problem: for a hidden nonempty subset W of {0,1}n, how many non-adaptive randomized queries (yes/no questions about W) are needed to guess an element xź{0,1}n such that xźW with probability >1/2? Motivated by questions in complexity theory, we prove tight lower bounds with respect to a few different classes of queries: źWe show that the monotone query complexity of witness finding is Ω(n2). This matches an O(n2) upper bound from the Valiant-Vazirani Isolation Lemma [8].źWe also prove a tight Ω(n2) lower bound for the class of NP queries (queries defined by an NP machine with an oracle to W). This shows that the classic search-to-decision reduction of Ben-David, Chor, Goldreich and Luby [3] is optimal in a certain black-box model.źFinally, we consider the setting where W is an affine subspace of {0,1}n and prove an Ω(n2) lower bound for the class of intersection queries (queries of the form WźSźź$W \cap S \ne \emptyset $? where S is a fixed subset of {0,1}n). Along the way, we show that every monotone property defined by an intersection query has an exponentially sharp threshold in the lattice of affine subspaces of {0,1}n.