The query complexity of witness finding
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 x∈ {0,1} n such that x∈ W with probability >∈1/2? Motivated by questions in complexity theory, we prove tight lower bounds with respect to a few different classes of queries: We show that the monotone query complexity of witness finding is Ω(n 2). This matches an O(n 2) upper bound from the Valiant-Vazirani Isolation Lemma [8]. We also prove a tight Ω(n 2) 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. Finally, we consider the setting where W is an affine subspace of {0,1} n and prove an Ω(n 2) lower bound for the class of intersection queries (queries of the form "W∈∈S∈∈ ?" 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 . © 2014 Springer International Publishing Switzerland.
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- Artificial Intelligence & Image Processing
- 46 Information and computing sciences
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Published In
DOI
EISSN
ISSN
Publication Date
Volume
Start / End Page
Related Subject Headings
- Artificial Intelligence & Image Processing
- 46 Information and computing sciences