Dictator

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Definition

A function [math]f:\{-1,1\}^n \to \{-1,1\}[/math] is called a dictator function if there exists an index [math]i[/math] such that one of the following two is true: Either [math]f(x) = x_i[/math], or [math]f(x) = -x_i[/math]. In other words, the function is a dictator if its output is controlled by only single bit from the input.

Sometimes, the term "dictator" refers only to the function [math]f(x) = x_i[/math], and the function [math]f(x) = -x_i[/math] is called an "anti-dictator".

Properties

  • The Fourier representation of a dictator function is either [math]f(x) = x_i[/math] or [math]f(x) = -x_i[/math].
  • In a dictatorship, the [math]i[/math]-th variable has influence 1, while all others have influence 0. Conversely, if a Boolean function [math]f[/math] has total influence 1, then it is a dictator.[1]
  • Dictators are a special case of Perceptrons.
  • Dictators are locally testable: Given query access to a function [math]f[/math], only a constant number of queries is needed in order to determine if it is a dictator with high probability. [2]
  • Dictators maximize mutual information: If [math]X,Y[/math] are two correlated random vectors in [math]\{-1,1\}^n[/math] with iid entries, and [math]f,g[/math] are two Boolean functions, then the mutual information [math]I(f(X), g(Y))[/math] is maximized when [math]f[/math] is a dictator and [math]g = \pm f[/math]. [3]
  • FKN theorem: If a Boolean function [math]f[/math]'s Fourier representation is close to a degree-1 polynomial, then [math]f[/math] is close to the dictator function. [4]
  • TODO: someone more knowledgeable should add something about the Unique Games Conjecture.

References

  1. https://math.stackexchange.com/questions/64449/a-boolean-function-with-total-influence-1-must-be-a-dictatorship
  2. Ryan O'Donnell, Analysis of Boolean functions, Theorem 7 in section 7.1/
  3. Georg Pichler, Pablo Piantanida, Gerald Matz, Dictator Functions Maximize Mutual Information.
  4. Friedgut, Ehud; Kalai, Gil; Naor, Assaf (2002). "Boolean functions whose Fourier transform is concentrated on the first two levels". Adv. Appl. Math. 29 (3): 427–437. doi:10.1016/S0196-8858(02)00024-6.