A Tight Composition Theorem for the Randomized Query Complexity of Partial Functions: Extended 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 \mathrm{R}(f\circ g)\ll \mathrm{R}(f)\mathrm{R}(g) . In fact, we show that the l...

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Vydáno v:Proceedings / annual Symposium on Foundations of Computer Science s. 240 - 246
Hlavní autoři: Ben-David, Shalev, Blais, Eric
Médium: Konferenční příspěvek
Jazyk:angličtina
Vydáno: IEEE 01.11.2020
Témata:
Boolean functions
Complexity theory
Computer science
Function composition
Gap Majority function
Loss measurement
Noise measurement
Noisy oracle
Query complexity
Randomized computation
Sensitivity
Upper bound
ISSN:2575-8454
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Popis
Shrnutí: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 \mathrm{R}(f\circ g)\ll \mathrm{R}(f)\mathrm{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,\ \mathrm{R}(f\circ g) = \Omega(\text{noisyR}(f)\ \mathrm{R}(g)) , where \text{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 \mathrm{R}(f\circ g)=\Omega(M(f)\mathrm{R}(g)) for all f and g , it must hold that \text{noisyR}(f)=\Omega(M(f)) for all f . We also give a clean characterization of the measure \text{noisyR}(f) : it satisfies \text{noisyR}(f)=\Theta(\mathrm{R}(f\circ\text{GapMaj}_{n})/\mathrm{R}(\text{GapMaj}_{n})) , where n is the input size of f and \text{GapMaj}_{n} is the \sqrt{n} -gap majority function on n bits.
ISSN:2575-8454
DOI:10.1109/FOCS46700.2020.00031