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In mathematics, a Boolean function is a function whose arguments and result assume values from a two-element set (usually {true, false}, {0,1} or {-1,1}). [1] [2] Alternative names are switching function, used especially in older computer science literature, [3] [4] and truth function (or logical function), used in logic. Boolean functions are the subject of Boolean algebra and switching theory. [5]
A Boolean function takes the form , where is known as the Boolean domain and is a non-negative integer called the arity of the function. In the case where , the function is a constant element of . A Boolean function with multiple outputs, with is a vectorial or vector-valued Boolean function (an S-box in symmetric cryptography). [6]
There are different Boolean functions with arguments; equal to the number of different truth tables with entries.
Every -ary Boolean function can be expressed as a propositional formula in variables , and two propositional formulas are logically equivalent if and only if they express the same Boolean function.
The rudimentary symmetric Boolean functions (logical connectives or logic gates) are:
An example of a more complicated function is the majority function (of an odd number of inputs).
A Boolean function may be specified in a variety of ways:
Algebraically, as a propositional formula using rudimentary Boolean functions:
Boolean formulas can also be displayed as a graph:
In order to optimize electronic circuits, Boolean formulas can be minimized using the Quine–McCluskey algorithm or Karnaugh map.
A Boolean function can have a variety of properties: [7]
Circuit complexity attempts to classify Boolean functions with respect to the size or depth of circuits that can compute them.
A Boolean function may be decomposed using Boole's expansion theorem in positive and negative Shannoncofactors (Shannon expansion), which are the (k-1)-ary functions resulting from fixing one of the arguments (to zero or one). The general (k-ary) functions obtained by imposing a linear constraint on a set of inputs (a linear subspace) are known as subfunctions. [8]
The Boolean derivative of the function to one of the arguments is a (k-1)-ary function that is true when the output of the function is sensitive to the chosen input variable; it is the XOR of the two corresponding cofactors. A derivative and a cofactor are used in a Reed–Muller expansion. The concept can be generalized as a k-ary derivative in the direction dx, obtained as the difference (XOR) of the function at x and x + dx. [8]
The Möbius transform (or Boole-Möbius transform) of a Boolean function is the set of coefficients of its polynomial (algebraic normal form), as a function of the monomial exponent vectors. It is a self-inverse transform. It can be calculated efficiently using a butterfly algorithm ("Fast Möbius Transform"), analogous to the Fast Fourier Transform. [9] Coincident Boolean functions are equal to their Möbius transform, i.e. their truth table (minterm) values equal their algebraic (monomial) coefficients. [10] There are 2^2^(k−1) coincident functions of k arguments. [11]
The Walsh transform of a Boolean function is a k-ary integer-valued function giving the coefficients of a decomposition into linear functions (Walsh functions), analogous to the decomposition of real-valued functions into harmonics by the Fourier transform. Its square is the power spectrum or Walsh spectrum. The Walsh coefficient of a single bit vector is a measure for the correlation of that bit with the output of the Boolean function. The maximum (in absolute value) Walsh coefficient is known as the linearity of the function. [8] The highest number of bits (order) for which all Walsh coefficients are 0 (i.e. the subfunctions are balanced) is known as resiliency, and the function is said to be correlation immune to that order. [8] The Walsh coefficients play a key role in linear cryptanalysis.
The autocorrelation of a Boolean function is a k-ary integer-valued function giving the correlation between a certain set of changes in the inputs and the function output. For a given bit vector it is related to the Hamming weight of the derivative in that direction. The maximal autocorrelation coefficient (in absolute value) is known as the absolute indicator. [7] [8] If all autocorrelation coefficients are 0 (i.e. the derivatives are balanced) for a certain number of bits then the function is said to satisfy the propagation criterion to that order; if they are all zero then the function is a bent function. [12] The autocorrelation coefficients play a key role in differential cryptanalysis.
The Walsh coefficients of a Boolean function and its autocorrelation coefficients are related by the equivalent of the Wiener–Khinchin theorem, which states that the autocorrelation and the power spectrum are a Walsh transform pair. [8]
These concepts can be extended naturally to vectorial Boolean functions by considering their output bits (coordinates) individually, or more thoroughly, by looking at the set of all linear functions of output bits, known as its components. [6] The set of Walsh transforms of the components is known as a Linear Approximation Table (LAT) [13] [14] or correlation matrix; [15] [16] it describes the correlation between different linear combinations of input and output bits. The set of autocorrelation coefficients of the components is the autocorrelation table, [14] related by a Walsh transform of the components [17] to the more widely used Difference Distribution Table (DDT) [13] [14] which lists the correlations between differences in input and output bits (see also: S-box).
Any Boolean function can be uniquely extended (interpolated) to the real domain by a multilinear polynomial in , constructed by summing the truth table values multiplied by indicator polynomials:For example, the extension of the binary XOR function iswhich equalsSome other examples are negation (), AND () and OR (). When all operands are independent (share no variables) a function's polynomial form can be found by repeatedly applying the polynomials of the operators in a Boolean formula. When the coefficients are calculated modulo 2 one obtains the algebraic normal form (Zhegalkin polynomial).
Direct expressions for the coefficients of the polynomial can be derived by taking an appropriate derivative:this generalizes as the Möbius inversion of the partially ordered set of bit vectors:where denotes the weight of the bit vector . Taken modulo 2, this is the Boolean Möbius transform, giving the algebraic normal form coefficients:In both cases, the sum is taken over all bit-vectors a covered by m, i.e. the "one" bits of a form a subset of the one bits of m.
When the domain is restricted to the n-dimensional hypercube , the polynomial gives the probability of a positive outcome when the Boolean function f is applied to n independent random (Bernoulli) variables, with individual probabilities x. A special case of this fact is the piling-up lemma for parity functions. The polynomial form of a Boolean function can also be used as its natural extension to fuzzy logic.
Often, the Boolean domain is taken as , with false ("0") mapping to 1 and true ("1") to -1 (see Analysis of Boolean functions). The polynomial corresponding to is then given by:Using the symmetric Boolean domain simplifies certain aspects of the analysis, since negation corresponds to multiplying by -1 and linear functions are monomials (XOR is multiplication). This polynomial form thus corresponds to the Walsh transform (in this context also known as Fourier transform) of the function (see above). The polynomial also has the same statistical interpretation as the one in the standard Boolean domain, except that it now deals with the expected values (see piling-up lemma for an example).
Boolean functions play a basic role in questions of complexity theory as well as the design of processors for digital computers, where they are implemented in electronic circuits using logic gates.
The properties of Boolean functions are critical in cryptography, particularly in the design of symmetric key algorithms (see substitution box).
In cooperative game theory, monotone Boolean functions are called simple games (voting games); this notion is applied to solve problems in social choice theory.
Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable as a function of the time lag between them. The analysis of autocorrelation is a mathematical tool for finding repeating patterns, such as the presence of a periodic signal obscured by noise, or identifying the missing fundamental frequency in a signal implied by its harmonic frequencies. It is often used in signal processing for analyzing functions or series of values, such as time domain signals.
In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign =. The word equation and its cognates in other languages may have subtly different meanings; for example, in French an équation is defined as containing one or more variables, while in English, any well-formed formula consisting of two expressions related with an equals sign is an equation.
In mathematics, the term linear is used in two distinct senses for two different properties:
In mathematics, an algebra over a field is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear".
In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form where a0(x), ..., an(x) and b(x) are arbitrary differentiable functions that do not need to be linear, and y′, ..., y(n) are the successive derivatives of an unknown function y of the variable x.
A maximum length sequence (MLS) is a type of pseudorandom binary sequence.
Reed–Muller codes are error-correcting codes that are used in wireless communications applications, particularly in deep-space communication. Moreover, the proposed 5G standard relies on the closely related polar codes for error correction in the control channel. Due to their favorable theoretical and mathematical properties, Reed–Muller codes have also been extensively studied in theoretical computer science.
In mathematics, the correlation immunity of a Boolean function is a measure of the degree to which its outputs are uncorrelated with some subset of its inputs. Specifically, a Boolean function is said to be correlation-immune of order m if every subset of m or fewer variables in is statistically independent of the value of .
In theoretical computer science, circuit complexity is a branch of computational complexity theory in which Boolean functions are classified according to the size or depth of the Boolean circuits that compute them. A related notion is the circuit complexity of a recursive language that is decided by a uniform family of circuits .
Zhegalkinpolynomials, also known as algebraic normal form, are a representation of functions in Boolean algebra. Introduced by the Russian mathematician Ivan Ivanovich Zhegalkin in 1927, they are the polynomial ring over the integers modulo 2. The resulting degeneracies of modular arithmetic result in Zhegalkin polynomials being simpler than ordinary polynomials, requiring neither coefficients nor exponents. Coefficients are redundant because 1 is the only nonzero coefficient. Exponents are redundant because in arithmetic mod 2, x2 = x. Hence a polynomial such as 3x2y5z is congruent to, and can therefore be rewritten as, xyz.
In algebra, a multilinear polynomial is a multivariate polynomial that is linear in each of its variables separately, but not necessarily simultaneously. It is a polynomial in which no variable occurs to a power of or higher; that is, each monomial is a constant times a product of distinct variables. For example is a multilinear polynomial of degree whereas is not. The degree of a multilinear polynomial is the maximum number of distinct variables occurring in any monomial.
In mathematics, a symmetric Boolean function is a Boolean function whose value does not depend on the order of its input bits, i.e., it depends only on the number of ones in the input. For this reason they are also known as Boolean counting functions.
In Boolean algebra, a parity function is a Boolean function whose value is one if and only if the input vector has an odd number of ones. The parity function of two inputs is also known as the XOR function.
In computational complexity theory, the decision tree model is the model of computation in which an algorithm can be considered to be a decision tree, i.e. a sequence of queries or tests that are done adaptively, so the outcome of previous tests can influence the tests performed next.
In the mathematical field of combinatorics, a bent function is a Boolean function that is maximally non-linear; it is as different as possible from the set of all linear and affine functions when measured by Hamming distance between truth tables. Concretely, this means the maximum correlation between the output of the function and a linear function is minimal. In addition, the derivatives of a bent function are balanced Boolean functions, so for any change in the input variables there is a 50 percent chance that the output value will change.
A locally decodable code (LDC) is an error-correcting code that allows a single bit of the original message to be decoded with high probability by only examining a small number of bits of a possibly corrupted codeword. This property could be useful, say, in a context where information is being transmitted over a noisy channel, and only a small subset of the data is required at a particular time and there is no need to decode the entire message at once. Locally decodable codes are not a subset of locally testable codes, though there is some overlap between the two.
In cryptography, SWIFFT is a collection of provably secure hash functions. It is based on the concept of the fast Fourier transform (FFT). SWIFFT is not the first hash function based on the FFT, but it sets itself apart by providing a mathematical proof of its security. It also uses the LLL basis reduction algorithm. It can be shown that finding collisions in SWIFFT is at least as difficult as finding short vectors in cyclic/ideal lattices in the worst case. By giving a security reduction to the worst-case scenario of a difficult mathematical problem, SWIFFT gives a much stronger security guarantee than most other cryptographic hash functions.
In theoretical computer science, multiparty communication complexity is the study of communication complexity in the setting where there are more than 2 players.
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values true and false, usually denoted 1 and 0, whereas in elementary algebra the values of the variables are numbers. Second, Boolean algebra uses logical operators such as conjunction (and) denoted as ∧, disjunction (or) denoted as ∨, and negation (not) denoted as ¬. Elementary algebra, on the other hand, uses arithmetic operators such as addition, multiplication, subtraction, and division. Boolean algebra is therefore a formal way of describing logical operations in the same way that elementary algebra describes numerical operations.