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Let $f:{0,1}^n rightarrow {0,1}$ be a Boolean function. The certificate complexity $C(f)$ is a complexity measure that is quadratically tight for the zero-error randomized query complexity $R_0(f)$: $C(f) leq R_0(f) leq C(f)^2$. In this paper we study a new complexity measure that we call expectational certificate complexity $EC(f)$, which is also a quadratically tight bound on $R_0(f)$: $EC(f) leq R_0(f) = O(EC(f)^2)$. We prove that $EC(f) leq C(f) leq EC(f)^2$ and show that there is a quadratic separation between the two, thus $EC(f)$ gives a tighter upper bound for $R_0(f)$. The measure is also related to the fractional certificate complexity $FC(f)$ as follows: $FC(f) leq EC(f) = O(FC(f)^{3/2})$. This also connects to an open question by Aaronson whether $FC(f)$ is a quadratically tight bound for $R_0(f)$, as $EC(f)$ is in fact a relaxation of $FC(f)$. In the second part of the work, we upper bound the distributed query complexity $D^mu_epsilon(f)$ for product distributions $mu$ by the square of the query corruption bound ($mathrm{corr}_epsilon(f)$) which improves upon a result of Harsha, Jain and Radhakrishnan [2015]. A similar statement for communication complexity is open.
In this work we introduce, both for classical communication complexity and query complexity, a modification of the partition bound introduced by Jain and Klauck [2010]. We call it the public-coin partition bound. We show that (the logarithm to the ba
We prove two new results about the randomized query complexity of composed functions. First, we show that the randomized composition conjecture is false: there are families of partial Boolean functions $f$ and $g$ such that $R(fcirc g)ll R(f) R(g)$.
Let the randomized query complexity of a relation for error probability $epsilon$ be denoted by $R_epsilon(cdot)$. We prove that for any relation $f subseteq {0,1}^n times mathcal{R}$ and Boolean function $g:{0,1}^m rightarrow {0,1}$, $R_{1/3}(fcirc
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
We study the composition question for bounded-error randomized query complexity: Is R(f o g) = Omega(R(f) R(g)) for all Boolean functions f and g? We show that inserting a simple Boolean function h, whose query complexity is only Theta(log R(g)), in