Majority

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Definition

A function [math]f:\{-1,1\}^n \to \{-1,1\}[/math] is called a majority function if [math]f(x)[/math] returns the most common bit in the input:

[math] f(x) = \begin{cases} 1, & if ~ \sum_i x_i \geq 0 \\ -1 & otherwise \end{cases}[/math]

For even [math]n[/math], the above definition breaks ties in favor of 1, although any arbitrary rule may be used instead.

Majority is a special case of the perceptron function.

Properties

  • Among all monotone functions, majority has the largest influence: [math]\mathrm{Inf}(f) \leq \mathrm{Inf}(\mathrm{Maj})[/math] for all monotone [math]f[/math] [1].
  • TODO: a description of Majority's Fourier Transform. See http://www.contrib.andrew.cmu.edu/~ryanod/?p=877 for details.
  • Majority is the unique function that is symmetric, monotone and odd function. TODO May's theorem, credit.
  • Majority is not in AC0, even if we allow using mod q functions as gates for prime [math]q[/math]. [2]
  • For every [math]\varepsilon \gt 0 [/math], Majority can be [math]\varepsilon[/math]-approximated by a DNF of size [math]2^{O(\sqrt{n})}[/math]. [3]

References

  1. Ryan O'Donnell, Analysis of Boolean functions, Theorem 32 in section 2.3
  2. A. Razborov, Lower bounds on the size of bounded-depth networks over a complete basis with logical addition (Russian), in Matematicheskie Zametki, Vol. 41, No 4, 1987, pages 598-607. English translation in Mathematical Notes of the Academy of Sci. of the USSR, 41(4):333-338, 1987.
  3. O’Donnell R., Wimmer K. (2007) | Approximation by DNF: Examples and Counterexamples. In: Arge L., Cachin C., Jurdziński T., Tarlecki A. (eds) Automata, Languages and Programming. ICALP 2007. Lecture Notes in Computer Science, vol 4596. Springer, Berlin, Heidelberg