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Function returning one of only two values
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})
Boolean_function
Order-preserving mathematical function
optimal provided that the heuristic they use is monotonic. In Boolean algebra, a monotonic function is one such that for all ai and bi in {0,1}, if a1 ≤ b1
Monotonic_function
Function that outputs either true or false
A Boolean-valued function (sometimes called a predicate or a proposition) is a function of the type f : X → B, where X is an arbitrary set and where B
Boolean-valued_function
Generalization of binary functions
pseudo-Boolean function is a function of the form f : B n → R , {\displaystyle f:\mathbf {B} ^{n}\to \mathbb {R} ,} where B = {0, 1} is a Boolean domain
Pseudo-Boolean_function
Algebraic manipulation of "true" and "false"
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
Boolean_algebra
Properties of mathematical relationships
above function is considered affine in linear algebra (i.e. not linear). A Boolean function is linear if one of the following holds for the function's truth
Linearity
Study of Boolean functions via discrete Fourier analysis
analysis of Boolean functions is the study of real-valued functions on {0,1}n or {−1,1}n (such functions are sometimes known as pseudo-Boolean functions) from
Analysis_of_Boolean_functions
Model of computation
Each gate corresponds to some Boolean function that takes a fixed number of bits as input and outputs a single bit. Boolean circuits provide a model for
Boolean_circuit
In mathematics and computer science, a balanced Boolean function is a Boolean function whose output yields as many 0s as 1s over its input set. This means
Balanced_Boolean_function
Mathematical topics based on the works of George Boole
Look up Boolean, Booleans, or boolean in Wiktionary, the free dictionary. Any kind of logic, function, expression, or theory based on the work of George
Boolean
Boolean function whose output depends only on the number of true inputs
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
Symmetric_Boolean_function
Algorithm for supervised learning of binary classifiers
called a linearly separable Boolean function, or threshold Boolean function. The sequence of numbers of threshold Boolean functions on n inputs is OEIS A000609
Perceptron
Problem of determining if a Boolean formula could be made true
the form R(l1,...,ln) for some Boolean function R and (ordinary) literals li. Different sets of allowed Boolean functions lead to different problem versions
Boolean satisfiability problem
Boolean_satisfiability_problem
Geometric property of a pair of sets of points in Euclidean geometry
whether a Boolean function given in disjunctive or conjunctive normal form is linearly separable. A linear threshold logic gate is a Boolean function defined
Linear_separability
Analysis of Boolean functions Balanced Boolean function Bent function Boolean algebras canonically defined Boolean function Boolean matrix Boolean-valued function
List of Boolean algebra topics
List_of_Boolean_algebra_topics
Function in Boolean algebra
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
Parity_function
In mathematics, an evasive Boolean function f {\displaystyle f} (of n {\displaystyle n} variables) is a Boolean function for which every decision tree
Evasive_Boolean_function
Data structure for Boolean functions
branching program is a data structure that is used to represent a Boolean function. On a more abstract level, BDDs can be considered as a compressed representation
Binary_decision_diagram
Expression in a computer program
True/False or Yes/No, Boolean-typed variables, Boolean-valued operators, and Boolean-valued functions. Boolean expressions correspond to propositional formulas
Boolean_expression
Model of computational complexity
decision trees by Steele and Yao. For Boolean decision trees, the task is to compute the value of an n-bit Boolean function f : { 0 , 1 } n → { 0 , 1 } {\displaystyle
Decision_tree_model
Boolean function
In Boolean logic, the majority function (also called the median operator) is the Boolean function that evaluates to false when half or more arguments are
Majority_function
Theorem about complexity measures of Boolean functions
theorem, proved by Hao Huang in 2019, states that the sensitivity of a Boolean function f : { 0 , 1 } n → { 0 , 1 } {\displaystyle f:\{0,1\}^{n}\to \{0,1\}}
Sensitivity_theorem
Special type of Boolean function
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
Bent_function
Discrete set of Boolean variables
A Boolean network consists of a discrete set of Boolean variables each of which has a Boolean function (possibly different for each variable) assigned
Boolean_network
Logic constructed only from NAND gates
The NAND Boolean function has the property of functional completeness. This means that any Boolean expression can be re-expressed by an equivalent expression
NAND_logic
Provides lower bounds on the circuit complexity of boolean functions
the circuit complexity of boolean functions. A natural proof shows, either directly or indirectly, that a boolean function has a certain natural combinatorial
Natural_proof
Model of computational complexity
computational complexity theory in which Boolean functions are classified according to the size or depth of the Boolean circuits that compute them. A related
Circuit_complexity
Book by Marvin Minsky and Seymour Papert
The order of a boolean function on R is the minimal order possible for a perceptron implementing the boolean function. A boolean function is conjunctively
Perceptrons_(book)
Method for increasing reliability
circuits (logic gates) are used to compute the same set of specified Boolean function. If there are no circuit failures, the outputs of the three circuits
Triple_modular_redundancy
Device performing a Boolean function
A logic gate is a device that performs a Boolean function, a logical operation performed on one or more binary inputs that produces a single binary output
Logic_gate
Set of rules defining correctly structured programs
the Boolean type, Mozilla recommends that the Boolean() function (without new) be used in preference to the Boolean object. const b = new Boolean(false);
JavaScript_syntax
Topics referred to by the same term
Boolean operation or Boolean operator may refer to: Boolean function, a function whose arguments and result assume values from a two-element set Boolean
Boolean_operation
Boolean algebra generated by a set with no relations beyond Boolean laws
a free Boolean algebra is a Boolean algebra with a distinguished set of elements, called generators, such that: Each element of the Boolean algebra can
Free_Boolean_algebra
Subject field of Boolean algebra discussing changes of Boolean variables and functions
of Boolean algebra discussing changes of Boolean variables and Boolean functions. Boolean differential calculus concepts are analogous to those of classical
Boolean_differential_calculus
Algebraic structure modeling logical operations
List of Boolean algebra topics Boolean domain Boolean function Boolean logic Boolean ring Boolean-valued function Canonical form (Boolean algebra) Complete
Boolean_algebra_(structure)
Special type of Boolean function
In mathematics, a read-once function is a special type of Boolean function that can be described by a Boolean expression in which each variable appears
Read-once_function
Symbolic boolean function representation, extension of BDDs
(MTBDD), is a data structure that is used to symbolically represent a Boolean function whose codomain is an arbitrary finite set S. An ADD is an extension
Algebraic_decision_diagram
Standard form of Boolean function
In Boolean logic, a formula for a Boolean function f is in Blake canonical form (BCF), also called the complete sum of prime implicants, the complete
Blake_canonical_form
Mathematical table used in logic
mathematical table used in logic—specifically in connection with Boolean algebra, Boolean functions, and propositional calculus—which sets out the functional
Truth_table
Argument that classification is not really possible without some sort of bias
completed features contain every Boolean function on k Boolean variables, with each one exactly once. Viewing these Boolean functions as polynomials in k variables
Ugly_duckling_theorem
Set of all things that may be the input of a mathematical function
In mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by dom ( f ) {\displaystyle \operatorname
Domain_of_a_function
True when either but not both inputs are true
The description of a Boolean function as a polynomial in 𝔽2, using this basis, is called the function's algebraic normal form. Disjunction
Exclusive_or
Device that selects between several analog or digital input signals
device per input signal. Multiplexers can also be used to implement Boolean functions of multiple variables. Conversely, a demultiplexer (or demux) is a
Multiplexer
Logical gate whose output is false if all its inputs are true
inverters followed by an OR gate. The NAND gate is significant because any Boolean function can be implemented by using a combination of NAND gates. This property
NAND_gate
Data having only values "true" or "false"
In computer science, the Boolean is a data type that has one of two possible values representing the two truth values of logic: true and false. It is
Boolean_data_type
Function that preserves distinctness
In mathematics, an injective function (also known as injection, or one-to-one function) is a function f that maps distinct elements of its domain to distinct
Injective_function
Basic component of symmetric key algorithms
property of confusion. Mathematically, an S-box is a nonlinear vectorial Boolean function. In general, an S-box takes some number of input bits, m, and transforms
S-box
Topics referred to by the same term
object-oriented programming Function (computer programming), a callable sequence of instructions Boolean function, used in hardware design Function (music), a relationship
Function
Logical connective OR
' Affirming a disjunct Boolean algebra (logic) Boolean algebra topics Boolean domain Boolean function Boolean-valued function Conjunction/disjunction
Logical_disjunction
Algorithm for the minimization of Boolean functions
implicants or the tabulation method, is a method used for minimization of Boolean functions that was developed by Willard V. Quine in 1952 and extended by Edward
Quine–McCluskey_algorithm
Concept in mathematical logic
terms of the functions fi. Since every Boolean function of at least one variable can be expressed in terms of binary Boolean functions, F is functionally
Functional_completeness
Identities and relationships involving sets
relations. Any set of sets closed under the set-theoretic operations forms a Boolean algebra with the join operator being union, the meet operator being intersection
Algebra_of_sets
In logic, a statement which is always true
is defined as a propositional formula that is true under any possible Boolean valuation of its propositional variables. A key property of tautologies
Tautology_(logic)
Overview of and topical guide to logic
form (Boolean algebra) Boolean conjunctive query Boolean-valued model Boolean domain Boolean expression Boolean ring Boolean function Boolean-valued
Outline_of_logic
are an efficient way to represent and manipulate boolean functions. The value of a boolean function can be determined by following a path in its BDD down
Binary_decision
Boolean term that guarantees a function is true whenever the term is true
product term (i.e., a conjunction of literals) P is an implicant of a Boolean function F, denoted P ≤ F {\displaystyle P\leq F} , if P implies F (i.e., whenever
Implicant
is a computational problem of constructing the dual of a monotone Boolean function. Equivalent problems can also be formulated as constructing the transversal
Monotone_dualization
Function in logic
Proposition 5.101 Bitwise operation Binary function Boolean domain Boolean logic Boolean-valued function List of Boolean algebra topics Logical constant Modal
Truth_function
Standard forms of Boolean functions
In Boolean algebra, any Boolean function can be expressed in the canonical disjunctive normal form (CDNF), minterm canonical form, or Sum of Products
Canonical_normal_form
Boolean polynomials as sums of monomials
Algebraic normal form (ANF) is a representation of functions in boolean algebra. Formulas written in ANF are also known as ring sum normal form (RSNF or
Algebraic_normal_form
Cryptographic attack
(LFSRs) using a Boolean function. Correlation attacks exploit a statistical weakness that arises from the specific Boolean function chosen for the keystream
Correlation_attack
Logical connective AND
holds Boolean domain – Concept in mathematical logic Boolean function – Function returning one of only two values Boolean-valued function – Function that
Logical_conjunction
graph (PDAG) is a data structure that is used to represent a Boolean function. A Boolean function can be represented as a rooted, directed acyclic graph of
Propositional directed acyclic graph
Propositional_directed_acyclic_graph
Subfield of mathematics
study the semantics of formal logics. A fundamental example is the use of Boolean algebras to represent truth values in classical propositional logic, and
Mathematical_logic
Process in digital electronics and integrated circuit design
simplifies) a Boolean function. The Boolean function carried out by the circuit is directly related to the algebraic expression from which the function is implemented
Logic_optimization
Mathematical function such that every output has at least one input
surjective function (also known as surjection, or onto function /ˈɒn.tuː/) is a function f such that, for every element y of the function's codomain, there
Surjective_function
Expression language for XML documents
in 1999, and can be used to compute values (e.g., strings, numbers, or Boolean values) from the content of an XML document. Support for XPath exists in
XPath
Canadian computer scientist
Carnegie Mellon University. He is known for his work on the analysis of Boolean functions and for authoring the textbook on this subject. He is also known for
Ryan O'Donnell (computer scientist)
Ryan_O'Donnell_(computer_scientist)
Input to a mathematical function
of a function is a value provided to obtain the function's result. It is also called an independent variable. For example, the binary function f ( x
Argument_of_a_function
Symbol connecting formulas in logic
Psychology portal Boolean domain Boolean function Boolean logic Boolean-valued function Catuṣkoṭi Dialetheism Four-valued logic List of Boolean algebra topics
Logical_connective
Data operation used in computer graphics
computer graphics in which several bitmaps are combined into one using a boolean function. A two-bitmap form is called block transfer (BLT) in UEFI Graphics
Bit_blit
Logical operation
In Boolean functions and propositional calculus, the Sheffer stroke denotes a logical operation that is equivalent to the negation of the conjunction
Sheffer_stroke
Logical connective
reasoning normatively according to nonclassical laws. Boolean domain Boolean function Boolean logic Conditional quantifier Implicational propositional
Material_conditional
Topic in computer science
determine whether some combinatorial structure S (such as a graph or a boolean function) satisfies some property P, or is "far" from having this property (meaning
Property_testing
Type of logical system
second argument. Equivalently, predicate symbols may be assigned Boolean-valued functions from Dn to { t r u e , f a l s e } {\displaystyle \{\mathrm {true
First-order_logic
Combinatorial sequence of numbers
Dedekind number M ( n ) {\displaystyle M(n)} is the number of monotone Boolean functions of n {\displaystyle n} variables. Equivalently, it is the number of
Dedekind_number
Target set of a mathematical function
mathematics, a codomain or set of destination of a function is a set into which all of the outputs of the function are constrained to fall. It is the set Y in
Codomain
Mathematical function that can be computed by a program
Computable functions are the basic objects of study in computability theory. Informally, a function is computable if there is an algorithm that computes
Computable_function
Cryptanalytic attacks using a system of multivariate equations
a set of algebraic equations can be used to solve a cryptographic Boolean function that has a low degree or a high degree of non linearity. The main objective
Algebraic_attack
Yes/no problem in computer science
characterize complexity classes of decision problems. For example, the Boolean satisfiability problem is complete for the class NP of decision problems
Decision_problem
Complexity class used to classify decision problems
in NP. The Boolean satisfiability problem (SAT), where we want to know whether or not a certain formula in propositional logic with Boolean variables is
NP_(complexity)
Complexity class of problems
monotone Boolean functions, do they represent the same function? Monotone self-duality: given a CNF formula for a Boolean function, is the function invariant
NP-intermediate
Relationship in which one statement follows from another
penguin}. Abstract algebraic logic Ampheck Boolean algebra (logic) Boolean domain Boolean function Boolean logic Causality Deductive reasoning Logic gate
Logical_consequence
Symbol representing a mathematical concept
systems particularly mathematical logic, a function symbol is a non-logical symbol which represents a function or mapping on the domain of discourse, though
Function_symbol
Number of arguments required by a function
science, arity (/ˈærɪti/ ) is the number of arguments or operands taken by a function, operation or relation. In mathematics, arity may also be called rank,
Arity
Model of computation
sequence of gates, each of which computes a function. Circuits of this kind provide a generalization of Boolean circuits and a mathematical model for digital
Circuit_(computer_science)
Lattice in universal algebra
by the positive integers, and all finite intersections of these. A Boolean function, or logical connective, is an n-ary operation f: 2n → 2 for some n
Post's_lattice
Standard system of axiomatic set theory
exists a function f {\displaystyle f} from X {\displaystyle X} to the union of the members of X {\displaystyle X} , called a "choice function", such that
Zermelo–Fraenkel_set_theory
Computational input that relies on the length but not content of the input
each input of length n is contained in the language. Therefore, any boolean function is computable with an advice of length 2n and advice of more than exponential
Advice_(complexity)
Subset of a function's codomain
the range of a function may refer to either of two closely related concepts: the codomain of the function, or the image of the function. In some cases
Range_of_a_function
Logical operation
difference. Negation is a linear logical operator. In Boolean algebra, a self dual function is a function such that: f ( a 1 , … , a n ) = ¬ f ( ¬ a 1 , …
Negation
Set of problems in computational complexity theory
problems and function problems) and using other models of computation (e.g. probabilistic Turing machines, interactive proof systems, Boolean circuits, and
Complexity_class
Computer operation which manipulates individual bits of data
16 possible truth functions of two binary variables; this defines a truth table, termed a LUT2 lookup table, a.k.a. a Boolean function order k=2 (2 inputs)
Bitwise_operation
Mathematical function conceived as a crude model
firing; otherwise. It can be used to represent linearly separable boolean functions (for example, AND, OR, NOR) but not, for example, XOR. Each output
Artificial_neuron
Mathematical-logic system
that AND TRUE FALSE is equivalent to FALSE. A predicate is a function that returns a Boolean value. The most fundamental predicate is ISZERO, which returns
Lambda_calculus
Early British cryptanalysis computer
plaintext characters with a stream of key characters using the XOR Boolean function to produce the ciphertext.[citation needed] In August 1941, a blunder
Colossus_computer
Problem in computer science
often in discussions of computability since it demonstrates that some functions are mathematically definable but not computable. A key part of the formal
Halting_problem
Infinite cardinal number
defined either as an extreme limit of the real number line (applied to a function or sequence that "diverges to infinity" or "increases without bound"),
Aleph_number
Programming language construct
or McCarthy evaluation (after John McCarthy) is the semantics of some Boolean operators in some programming languages in which the second argument is
Short-circuit_evaluation
Value indicating the relation of a proposition to truth
of two values is also called the Boolean domain. Corresponding semantics of logical connectives are truth functions, whose values are expressed in the
Truth_value
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BOOLEAN FUNCTION
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BOOLEAN FUNCTION
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