Study of Boolean functions via discrete Fourier analysis
In mathematics and theoretical computer science, analysis of Boolean functions is the study of real-valued functions on or (such functions are sometimes known as pseudo-Boolean functions) from a spectral perspective.[1] The functions studied are often, but not always, Boolean-valued, making them Boolean functions. The area has found many applications in combinatorics, social choice theory, random graphs, and theoretical computer science, especially in hardness of approximation, property testing, and PAC learning.
^O'Donnell, Ryan (2014). Analysis of Boolean functions. Cambridge University Press. arXiv:2105.10386. ISBN 978-1-107-03832-5.
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mathematics and theoretical computer science, analysisofBooleanfunctions is the study of real-valued functions on { 0 , 1 } n {\displaystyle \{0,1\}^{n}}...
switching function, used especially in older computer science literature, and truth function (or logical function), used in logic. Booleanfunctions are the...
is the dual of the function's labelled Venn diagram, which is the more common representation for n ≤ 3.) The monotonic Booleanfunctions are precisely...
describes numerical operations. Boolean algebra was introduced by George Boole in his first book The Mathematical Analysisof Logic (1847), and set forth...
Sensitivity is related to the notion of total influence from the analysisofBooleanfunctions, which is equal to average sensitivity over all x {\displaystyle...
changes of Boolean variables and Booleanfunctions. Boolean differential calculus concepts are analogous to those of classical differential calculus,...
latter being of the form R(l1,...,ln) for some Booleanfunction R and (ordinary) literals li. Different sets of allowed Booleanfunctions lead to different...
terms of binary Booleanfunctions, F is functionally complete if and only if every binary Booleanfunction can be expressed in terms of the functions in...
arrangements of relays to solve Boolean algebra problems. The utilization of the binary properties of electrical switches to perform logic functions is the...
functions defined on these spaces and suitably respecting these structures. The historical roots of functional analysis lie in the study of spaces of...
domain of a function, although functions may be defined on more general sets. The two concepts are sometimes conflated as in, for example, the study of partial...
of surjective functions is always surjective. Any function can be decomposed into a surjection and an injection. A surjective function is a function whose...
involving these operations and relations. Any set of sets closed under the set-theoretic operations forms a Boolean algebra with the join operator being union...
O'Donnell (2008), "Some topics in analysisofbooleanfunctions", Proceedings of the fortieth annual ACM symposium on Theory of computing - STOC 08, pp. 569–578...
For some functions, the image and the codomain coincide; these functions are called surjective or onto. For example, consider the function f ( x ) =...
synthesis is tractable only for small Booleanfunctions. Recent approaches map the optimization problem to a Boolean satisfiability problem. This allows...
would imply that function and its derivatives are alternately monotonically increasing and monotonically decreasing functions. Such functions were first studied...
other notations (such as integers, booleans, pairs, lists, and tagged unions) are mapped to higher-order functions under Church encoding. The Church–Turing...
conjecture is a statement about Booleanfunctions originally conjectured by Ehud Friedgut and Gil Kalai in 1996. For a function f : { − 1 , 1 } n → { − 1 ...
correspondence that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain...
same way that Booleanfunctions can be composed, allowing the construction of a physical model of all ofBoolean logic, and therefore, all of the algorithms...
A Boolean network consists of a discrete set ofboolean variables each of which has a Booleanfunction (possibly different for each variable) assigned...
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