Physics:The Entropy Influence Conjecture

This article describes The Entropy Influence Conjecture.

The Conjecture

For a function [math]\displaystyle{ f: \{-1,1\}^n \to \{-1,1\} \! }[/math] the Entropy-Influence relates the following two quantities, both of which may be expressed in terms of the Fourier Expansion of the functions [math]\displaystyle{ f = \sum_{S \subset [n]} \hat{f}(S) x_S \! }[/math], where [math]\displaystyle{ x_S = \prod_{i \in S} x_i \! }[/math]. The first expression is the total influence of the function defined by [math]\displaystyle{ I(f) = \sum_S |S| \hat{f}^2(S) \! }[/math]. The second terms is the Entropy (of the spectrum) of the function defined by [math]\displaystyle{ H(f) = - \sum_S \hat{f}^2(S) \log \hat{f}^2(S) \! }[/math] (where [math]\displaystyle{ x \log x = 0 \! }[/math] when [math]\displaystyle{ x=0 \! }[/math]).

The conjecture states that there exists an absolute constant C such that for all Boolean functions [math]\displaystyle{ f: \{-1,1\}^n \to \{-1,1\} \! }[/math] it holds that [math]\displaystyle{ H(f) \leq C I(f) \! }[/math].

See also

• Hilbert's 23 problems

References

• Cameron–Erdős conjecture (Ben J. Green, 2003, conjectured by Paul)

References

• Unsolved Problems in Number Theory, Logic and Cryptography
• The Open Problems Project, discrete and computational geometry problems