Abstract

The sensitivity of a Boolean function f of n Boolean variables is the maximum over all inputs x of the number of positions i such that flipping the i-th bit of x changes the value of f(x). Permitting to flip disjoint blocks of bits leads to the notion of block sensitivity, known to be polynomially related to a number of other complexity measures of f , including the decision-tree complexity, the polynomial degree, and the certificate complexity. A long-standing open question is whether sensitivity also belongs to this equivalence class. A positive answer to this question is known as the Sensitivity Conjecture. We present a selection of known as well as new variants of the Sensitivity Conjecture and point out some weaker versions that are also open. Among other things, we relate the problem to Communication Complexity via recent results by Sherstov (QIC 2010). We also indicate new connections to Fourier analysis.

Highlights

  • The sensitivity of a Boolean function f of n Boolean variables is the maximum over all inputs x of the number of positions i such that flipping the i-th bit of x changes the value of f (x)

  • Nisan introduced the notion of block sensitivity and proved that CREW( f ) = Θ(log bs( f )) for every Boolean function f [23]

  • Block sensitivity turned out to be polynomially related to a number of other complexity measures; to this day it remains unknown whether block sensitivity is bounded above by a polynomial in sensitivity

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Summary

The Sensitivity Conjecture

For Boolean strings x, y ∈ {0, 1}n let x ⊕ y ∈ {0, 1}n denote the coordinate-wise exclusive or of x and y. The block sensitivity of a Boolean function f , denoted by bs( f ), is the maximum possible value of bs( f , x) over all choices of x. The study of sensitivity of Boolean functions originated from Stephen Cook and Cynthia Dwork [10] and Rudiger Reischuk [29] They showed an Ω(log s( f )) lower bound on the number of steps required to compute a Boolean function f on a CREW PRAM. Block sensitivity turned out to be polynomially related to a number of other complexity measures (see Section 2); to this day it remains unknown whether block sensitivity is bounded above by a polynomial in sensitivity This problem was first stated by Nisan and Mario Szegedy [24]. To the best of our knowledge the results stated in this paper without attribution have not appeared previously in the literature

Measures related to block sensitivity
Progress on the Sensitivity Conjecture
Communication complexity
Parity decision trees
Fourier-analytic setting
We can express
Fourier entropy
Shi’s characterization of sensitivity
Subgraphs of the n-cube
Two-colorings of integer lattices
Rubinstein’s function
AND-of-ORs
Kushilevitz’s function
Chakraborty’s function
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