A Composition Theorem for Randomized Query Complexity via Max-Conflict Complexity

For any relation f subseteq {0,1}^n x S and any partial Boolean function g:{0,1}^m -> {0,1,*}, we show that R_{1/3}(f o g^n) in Omega(R_{4/9}(f) * sqrt{R_{1/3}(g)})where R_epsilon(*) stands for the bounded-error randomized query complexity with error at most epsilon, and f o g^n subseteq ({0,1}^m)^n x S denotes the composition of f with n instances of g. The new composition theorem is optimal, at least, for the general case of relational problems: A relation f_0 and a partial Boolean function g_0 are constructed, such that R_{4/9}(f_0) in Theta(sqrt n), R_{1/3}(g_0)in Theta(n) and R_{1/3}(f_0 o g_0^n) in Theta(n). The theorem is proved via introducing a new complexity measure, max-conflict complexity, denoted by bar{chi}(*). Its investigation shows that bar{chi}(g) in Omega(sqrt{R_{1/3}(g)}) for any partial Boolean function g and R_{1/3}(f o g^n) in Omega(R_{4/9}(f) * bar{chi}(g)) for any relation f, which readily implies the composition statement. It is further shown that bar{chi}(g) is always at least as large as the sabotage complexity of g

A composition theorem for randomized query complexity via max conflict complexity

Let $R_\epsilon(\cdot)$ stand for the bounded-error randomized query complexity with error $\epsilon > 0$. For any relation $f \subseteq \{0,1\}^n \times S$ and partial Boolean function $g \subseteq \{0,1\}^m \times \{0,1\}$, we show that $R_{1/3}(f \circ g^n) \in \Omega(R_{4/9}(f) \cdot \sqrt{R_{1/3}(g)})$, where $f \circ g^n \subseteq (\{0,1\}^m)^n \times S$ is the composition of $f$ and $g$. We give an example of a relation $f$ and partial Boolean function $g$ for which this lower bound is tight. We prove our composition theorem by introducing a new complexity measure, the max conflict complexity $\bar \chi(g)$ of a partial Boolean function $g$. We show $\bar \chi(g) \in \Omega(\sqrt{R_{1/3}(g)})$ for any (partial) function $g$ and $R_{1/3}(f \circ g^n) \in \Omega(R_{4/9}(f) \cdot \bar \chi(g))$; these two bounds imply our composition result. We further show that $\bar \chi(g)$ is always at least as large as the sabotage complexity of $g$, introduced by Ben-David and Kothari

Low-Sensitivity Functions from Unambiguous Certificates

We provide new query complexity separations against sensitivity for total Boolean functions: a power $3$ separation between deterministic (and even randomized or quantum) query complexity and sensitivity, and a power $2.22$ separation between certificate complexity and sensitivity. We get these separations by using a new connection between sensitivity and a seemingly unrelated measure called one-sided unambiguous certificate complexity ($UC_{min}$). We also show that $UC_{min}$ is lower-bounded by fractional block sensitivity, which means we cannot use these techniques to get a super-quadratic separation between $bs(f)$ and $s(f)$. We also provide a quadratic separation between the tree-sensitivity and decision tree complexity of Boolean functions, disproving a conjecture of Gopalan, Servedio, Tal, and Wigderson (CCC 2016). Along the way, we give a power $1.22$ separation between certificate complexity and one-sided unambiguous certificate complexity, improving the power $1.128$ separation due to G\"o\"os (FOCS 2015). As a consequence, we obtain an improved $\Omega(\log^{1.22} n)$ lower-bound on the co-nondeterministic communication complexity of the Clique vs. Independent Set problem.Comment: 25 pages. This version expands the results and adds Pooya Hatami and Avishay Tal as author

On the Composition of Randomized Query Complexity and Approximate Degree

For any Boolean functions f and g, the question whether R(f?g) = ??(R(f) ? R(g)), is known as the composition question for the randomized query complexity. Similarly, the composition question for the approximate degree asks whether deg?(f?g) = ??(deg?(f)?deg?(g)). These questions are two of the most important and well-studied problems in the field of analysis of Boolean functions, and yet we are far from answering them satisfactorily. It is known that the measures compose if one assumes various properties of the outer function f (or inner function g). This paper extends the class of outer functions for which R and deg? compose. A recent landmark result (Ben-David and Blais, 2020) showed that R(f?g) = ?(noisyR(f)? R(g)). This implies that composition holds whenever noisyR(f) = ??(R(f)). We show two results: 1. When R(f) = ?(n), then noisyR(f) = ?(R(f)). In other words, composition holds whenever the randomized query complexity of the outer function is full. 2. If R composes with respect to an outer function, then noisyR also composes with respect to the same outer function. On the other hand, no result of the type deg?(f?g) = ?(M(f) ? deg?(g)) (for some non-trivial complexity measure M(?)) was known to the best of our knowledge. We prove that deg?(f?g) = ??(?{bs(f)} ? deg?(g)), where bs(f) is the block sensitivity of f. This implies that deg? composes when deg?(f) is asymptotically equal to ?{bs(f)}. It is already known that both R and deg? compose when the outer function is symmetric. We also extend these results to weaker notions of symmetry with respect to the outer function

On the Composition of Randomized Query Complexity and Approximate Degree

For any Boolean functions $f$ and $g$, the question whether $R(f\circ g) = \tilde{\Theta}(R(f)R(g))$, is known as the composition question for the randomized query complexity. Similarly, the composition question for the approximate degree asks whether $\widetilde{deg}(f\circ g) = \tilde{\Theta}(\widetilde{deg}(f)\cdot\widetilde{deg}(g))$. These questions are two of the most important and well-studied problems, and yet we are far from answering them satisfactorily. It is known that the measures compose if one assumes various properties of the outer function $f$ (or inner function $g$). This paper extends the class of outer functions for which $\text{R}$ and $\widetilde{\text{deg}}$ compose. A recent landmark result (Ben-David and Blais, 2020) showed that $R(f \circ g) = \Omega(noisyR(f)\cdot R(g))$. This implies that composition holds whenever noisyR(f) = \Tilde{\Theta}(R(f)). We show two results: (1)When $R(f) = \Theta(n)$, then $noisyR(f) = \Theta(R(f))$. (2) If $\text{R}$ composes with respect to an outer function, then $\text{noisyR}$ also composes with respect to the same outer function. On the other hand, no result of the type $\widetilde{deg}(f \circ g) = \Omega(M(f) \cdot \widetilde{deg}(g))$ (for some non-trivial complexity measure $M(\cdot)$) was known to the best of our knowledge. We prove that $\widetilde{deg}(f\circ g) = \widetilde{\Omega}(\sqrt{bs(f)} \cdot \widetilde{deg}(g)),$ where $bs(f)$ is the block sensitivity of $f$. This implies that $\widetilde{\text{deg}}$ composes when $\widetilde{\text{deg}}(f)$ is asymptotically equal to $\sqrt{\text{bs}(f)}$. It is already known that both $\text{R}$ and $\widetilde{\text{deg}}$ compose when the outer function is symmetric. We also extend these results to weaker notions of symmetry with respect to the outer function

The Power of Many Samples in Query Complexity

The randomized query complexity $R(f)$ of a boolean function $f\colon\{0,1\}^n\to\{0,1\}$ is famously characterized (via Yao's minimax) by the least number of queries needed to distinguish a distribution $D_0$ over $0$-inputs from a distribution $D_1$ over $1$-inputs, maximized over all pairs $(D_0,D_1)$. We ask: Does this task become easier if we allow query access to infinitely many samples from either $D_0$ or $D_1$? We show the answer is no: There exists a hard pair $(D_0,D_1)$ such that distinguishing $D_0^\infty$ from $D_1^\infty$ requires $\Theta(R(f))$ many queries. As an application, we show that for any composed function $f\circ g$ we have $R(f\circ g) \geq \Omega(\mathrm{fbs}(f)R(g))$ where $\mathrm{fbs}$ denotes fractional block sensitivity.Comment: 16 page

A Majority Lemma for Randomised Query Complexity

We show that computing the majority of n copies of a boolean function g has randomised query complexity R(Maj?g?) = ?(n?R ?_{1/n}(g)). In fact, we show that to obtain a similar result for any composed function f?g?, it suffices to prove a sufficiently strong form of the result only in the special case g = GapOr