In this paper we study a variant of search problems - which we call witness finding problems - in which there is no input and no underlying binary relation. Rather, there is a hidden set W from a fixed family W of possible sets. The objective is to output any element of W, where information about W is obtained by asking yes/no questions from a fixed family Q of permitted "queries". As the measure of complexity, we study the number of randomized queries required to find a witness in any nonempty W with high probability. By varying W and Q, this framework allows for some interesting upper and lower bounds. One classic upper bound for search problems - which translates naturally into a witness finding algorithm - is the search-to-decision reduction of Ben-David, Chor, Goldreich and Luby. This algorithm solves the witness finding problem for arbitrary subsets of {0,1}^n using O(n^2) non-adaptive queries from the family Q_<NP> of queries characterized by NP machines with an oracle to W. Our main result is a matching lower bound showing that Ω(n^2) queries are necessary in this setting. We also present results and raise an intriguing question concerning the query complexity of witness finding with respect to affine witness sets and monotone queries.
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