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 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 “$W \cap S \ne \emptyset$?” 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.

Appendices

Appendix A: Proof of Lemma 1

In order to apply Yao’s minimax principle [8], we express $m(\mathcal W,\mathcal Q)$ in terms of a particular matrix M. Let $\mathcal {F}$ be the set of functions {⊤,⊥}^m→{0,1}ⁿ. Let $\mathcal {A} := \mathcal {Q}^{m} \times \mathcal {F}$ (representing the set of deterministic witness finding algorithms). Let $\mathcal {W}_{0} := \mathcal {W} \setminus \{\emptyset \}$. Finally, let M be the $\mathcal {A} \times \mathcal {W}_{0}$-matrix defined by

$$M_{(Q_{1},\dots,Q_{m};f),W} := \left\{\begin{array}{ll} 1 &\text{if } f(Q_{1}(W),\dots,Q_{m}(W)) \in W,\\ 0 &\text{otherwise}. \end{array}\right. $$

In this context, Yao’s minimax principle states that for all random variables W on $\mathcal W_{0}$ and $(\mathbf {Q}_{1},\dots ,\mathbf {Q}_{m};\mathbf {f})$ on $\mathcal {A}$,

$$\min_{(Q_{1},\dots,Q_{m};f) \in \mathcal A} \mathbb{E}[M_{(Q_{1},\dots,Q_{m};f),\mathbf{W}}] \le \max_{W \in \mathcal W_{0}} \mathbb{E}[M_{(\mathbf{Q}_{1},\dots,\mathbf{Q}_{m};\mathbf f),W}]. $$

It follows that, if $\mathbb {P}[f(Q_{1}(\mathbf {W}),\dots ,Q_{m}(\mathbf {W})) \in \mathbf {W}] \le 1/2$ for all $Q_{1},\dots ,Q_{m} \in \mathcal Q$ and every function f:{⊤,⊥}^m→{0,1}ⁿ, then for all $(\mathbf {Q}_{1},\dots ,\mathbf {Q}_{m};\mathbf {f}) \in \mathcal {A}$ (including the special case where f is deterministic, as in the definition of witness finding procedures), there exists $W \in \mathcal {W}_{0}$ such that $\mathbb {P}[\mathbf {f}(\mathbf {Q}_{1}(W),\dots ,\mathbf {Q}_{m}(W)) \in W] \le 1/2$. Therefore, the $\mathcal {Q}$-query complexity of $\mathcal {W}$-witness finding is greater than m.

B: Proof of Lemma 2

For inequality (1), let $\mathbf Y_{1},\dots ,\mathbf Y_{2^{i}}$ be independent copies of W _𝜃−i . For each x∈{0,1}ⁿ, we have

$$\mathbb{P}[x \in (\mathbf Y_{1} \cup {\dots} \cup \mathbf Y_{2^{i}})] = 1 - (1 - 2^{\theta-i-n})^{2^{i}} < 2^{\theta-n} = \mathbb{P}[x \in \mathbf{W}_{\theta}]. $$

Moreover, events $\{x \in \mathbf Y_{1} \cup {\dots } \cup \mathbf Y_{2^{i}}\}$ are independent over x∈{0,1}ⁿ. Thus, $\mathbf {Y}_{1} \cup {\dots } \cup \mathbf {Y}_{2^{i}}$ and W _𝜃 are both product distributions on the lattice of subsets of {0,1}ⁿ with biases $p = 1 - (1 - 2^{\theta -i-n})^{2^{i}}$ and $q = 2^{\theta _{n}}$ respectively. Since $p < q, \mathbf {Y}_{1} \cup {\dots } \cup \mathbf {Y}_{2^{i}}$ is stochastically dominated by W _𝜃 . (That is, $\mathbb {E}[f(\mathbf {Y}_{1} \cup {\dots } \cup \mathbf {Y}_{2^{i}})] \le \mathbb {E}[f(\mathbf {W}_{\theta })]$ for every monotone increasing function f of subsets of {0,1}ⁿ). Since Q is a monotone query, we have

$$\mathbb{P}[Q(\mathbf Y_{1}) \vee {\dots} \vee Q(\mathbf Y_{2^{i}})] \le \mathbb{P}[Q(\mathbf Y_{1} \cup {\dots} \cup \mathbf Y_{2^{i}})] \le \mathbb{P}[Q(\mathbf{W}_{\theta})]. $$

By independence of $\mathbf Y_{1},\dots ,\mathbf Y_{2^{i}}$, it follows that

$$\begin{array}{@{}rcl@{}} 1/2 \ge \mathbb{P}[Q(\mathbf{W}_{\theta})] \ge \mathbb{P}[\textstyle\bigvee_{j=1}^{2^{i}} Q(\mathbf Y_{j})] = 1 - \mathbb{P}[\neg Q(\mathbf{W}_{\theta-i})]^{2^{i}} = 1 - (1 - p_{\theta-i})^{2^{i}}. \end{array} $$

Therefore, $p_{\theta -i} \le 1 - (1/2)^{1/2^{i}} < (\ln 2)/2^{i}$.

For inequality (2), let $\mathbf Z_{1},\dots ,\mathbf Z_{2^{i}}$ be independent copies of W _𝜃+1. By a similar argument, we have

$$p_{\theta+i+1} = \mathbb{P}[Q(\mathbf{W}_{\theta+i+1})] \ge \mathbb{P}[{\textstyle\bigvee_{j=1}^{2^{i}}} Q(\mathbf Z_{j})] = 1 - \mathbb{P}[\neg Q(\mathbf{W}_{\theta+1})]^{2^{i}} > 1 - \frac{1}{2^{2^{i}}}. $$

Finally, for inequality (3), note that for all p, q∈[0,1],

$$0 \le \min(p,1-p) \le q \le 1/2 \,\Longrightarrow\, H(p) \le H(q) \le 2q\log(1/q). $$

By this observation, together with (1) and (2), we have

$$H(p_{\theta-i-1}) \le 2\frac{\ln 2}{2^{i+1}}\log(\frac{2^{i+1}}{\ln 2}) < \frac{i+2}{2^{i}},\! \quad H(p_{\theta+i+1}) \le 2\frac{1}{2^{2^{i}}}\log(2^{2^{i}}) = \frac{1}{2^{2^{i} - i - 1}}. $$

From these two inequalities, it follows that H(p _k )≤(|𝜃−k|+1)/2^{|𝜃−k|−1}.