Fourier Analysis of Boolean Functions
نویسنده
چکیده
This paper introduces the technique of Fourier analysis applied to Boolean functions. We use this technique to illustrate proofs of both Arrow’s Impossibility Theorem and H̊astad’s 3-bit Theorem.
منابع مشابه
Boolean Functions: Influence, threshold and noise
This lecture studies the analysis of Boolean functions and present a few ideas, results, proofs, and problems. We start with the wider picture of expansion in graphs and then concentrate on the graph of the n-dimensional discrete cube Ωn. Boolean functions are functions from Ωn to {0, 1}. We consider the notion of the influence of variables on Boolean functions. The influence of a variable on a...
متن کاملComments and Corrections Comments on “Canalizing Boolean Functions Maximize Mutual Information”
In their recent paper “Canalizing Boolean Functions Maximize Mutual Information,” Klotz et al. argued that canalizing Boolean functions maximize certain mutual informations by an argument involving Fourier analysis on the hypercube. This note supplies short new proofs of their results based on a coupling argument and also clarifies a point on the necessity of considering randomized functions.
متن کاملCharacterizing short-term stability for Boolean networks over any distribution of transfer functions
We present a characterization of short-term stability of Kauffman's NK (random) Boolean networks under arbitrary distributions of transfer functions. Given such a Boolean network where each transfer function is drawn from the same distribution, we present a formula that determines whether short-term chaos (damage spreading) will happen. Our main technical tool which enables the formal proof of ...
متن کاملCanalizing Boolean Functions Maximize the Mutual Information
The ability of information processing in biologically motivated Boolean networks is of interest in recent information theoretic research. One measure to quantify this ability is the well known mutual information. Using Fourier analysis we show that canalizing functions maximize the mutual information between an input variable and the outcome of the function. We proof our result for Boolean func...
متن کاملTesting Fourier Dimensionality and Sparsity
We present a range of new results for testing properties of Boolean functions that are defined in terms of the Fourier spectrum. Broadly speaking, our results show that the property of a Boolean function having a concise Fourier representation is locally testable. We give the first efficient algorithms for testing whether a Boolean function has a sparse Fourier spectrum (small number of nonzero...
متن کامل