A Tight Composition Theorem for the Randomized Query Complexity of Partial Functions

• URL: http://arxiv.org/abs/2002.10809v2
• Date: Mon, 7 Dec 2020 09:52:03 GMT
• Title: A Tight Composition Theorem for the Randomized Query Complexity of Partial Functions
• Authors: Shalev Ben-David, Eric Blais
• Abstract summary: We prove two new results about the randomized query complexity of composed functions. We show that for all $f$ and $g$, $(Rcirc g)=Omega(mathopnoisyR(f)cdot R(g))$, where $mathopnoisyR(f)$ is a measure describing the cost of computing $f$ on noisy inputs.
• Score: 1.2284934135116514
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $R(f\circ g)\ll R(f) R(g)$. In fact, we show that the left hand side can be polynomially smaller than the right hand side (though in our construction, both sides are polylogarithmic in the input size of $f$). Second, we show that for all $f$ and $g$, $R(f\circ g)=\Omega(\mathop{noisyR}(f)\cdot R(g))$, where $\mathop{noisyR}(f)$ is a measure describing the cost of computing $f$ on noisy oracle inputs. We show that this composition theorem is the strongest possible of its type: for any measure $M(\cdot)$ satisfying $R(f\circ g)=\Omega(M(f)R(g))$ for all $f$ and $g$, it must hold that $\mathop{noisyR}(f)=\Omega(M(f))$ for all $f$. We also give a clean characterization of the measure $\mathop{noisyR}(f)$: it satisfies $\mathop{noisyR}(f)=\Theta(R(f\circ gapmaj_n)/R(gapmaj_n))$, where $n$ is the input size of $f$ and $gapmaj_n$ is the $\sqrt{n}$-gap majority function on $n$ bits.

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