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تسجيل مستخدم جديدA composition theorem for randomized query complexity via max conflict complexity
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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.
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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 $g:{0,1}^m rightarrow {0,1}$, $R_{1/3}(fcirc
g^n) = Omega(R_{4/9}(f)cdot R_{1/2-1/n^4}(g))$, where $f circ g^n$ is the relation obtained by composing $f$ and $g$. We also show that $R_{1/3}left(f circ left(g^oplus_{O(log n)}right)^nright)=Omega(log n cdot R_{4/9}(f) cdot R_{1/3}(g))$, where $g^oplus_{O(log n)}$ is the function obtained by composing the xor function on $O(log n)$ bits and $g^t$.
Let $R(cdot)$ stand for the bounded-error randomized query complexity. We show that for any relation $f subseteq {0,1}^n times mathcal{S}$ and partial Boolean function $g subseteq {0,1}^n times {0,1}$, $R_{1/3}(f circ g^n) = Omega(R_{4/9}(f) cdot sqr
t{R_{1/3}(g)})$. Independently of us, Gavinsky, Lee and Santha cite{newcomp} proved this result. By an example demonstrated in their work, this bound is optimal. We prove our result by introducing a novel complexity measure called the emph{conflict complexity} of a partial Boolean function $g$, denoted by $chi(g)$, which may be of independent interest. We show that $chi(g) = Omega(sqrt{R(g)})$ and $R(f circ g^n) = Omega(R(f) cdot chi(g))$.
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