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Generalization of binary functions
a 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
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
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
Combinatorial optimization method for pseudo-Boolean functions
Quadratic pseudo-Boolean optimisation (QPBO) is a combinatorial optimization method for minimizing quadratic pseudo-Boolean functions in the form f (
Quadratic pseudo-Boolean optimization
Quadratic_pseudo-Boolean_optimization
Class of mathematical functions
(supermodular) functions can be found in "Maximization of submodular functions: Theory and enumeration algorithms", B. Goldengorin. Pseudo-Boolean function Topkis's
Supermodular_function
Order-preserving mathematical function
efficiently when all involved functions and predicates are monotonic and Boolean. Monotone cubic interpolation Pseudo-monotone operator Spearman's rank
Monotonic_function
Combinatorial optimization method for a family of functions of discrete variables
equivalent to computing the maximum flow over the network. Given a pseudo-Boolean function f {\displaystyle f} , if it is possible to construct a flow network
Graph_cut_optimization
Type of polynomial
also the basis used in the Fourier analysis of (pseudo-)Boolean functions. Every (pseudo-)Boolean function can be uniquely expressed as a multilinear polynomial
Multilinear_polynomial
American mathematician (1936–2006)
research and applied discrete mathematics through the study of pseudo-Boolean functions and their connections to graph theory and data mining. Hammer was
Peter_L._Hammer
Function that returns cardinal numbers
b\ \vee \ b\leq a){\big )}{\big \}}} . Pseudo-weight π ( B ) {\displaystyle \pi (\mathbb {B} )} of a Boolean algebra B {\displaystyle \mathbb {B} } is
Cardinal_function
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
Mathematical model of ferromagnetism in statistical mechanics
The Ising Hamiltonian is an example of a pseudo-Boolean function; tools from the analysis of Boolean functions can be applied to describe and study it
Ising_model
Algebraic structure used in logic
In mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with least
Heyting_algebra
Algorithm for generating pseudo-randomized numbers
congruential generator (LCG) is an algorithm that yields a sequence of pseudo-randomized numbers calculated with a discontinuous piecewise linear equation
Linear_congruential_generator
Problem in graph theory
Crowston et al. proved the bound using linear algebra and analysis of pseudo-boolean functions. The Edwards-Erdős bound extends to the Balanced Subgraph Problem
Maximum_cut
Cryptographical test for pseudo-randomness
collection C = { C k } {\displaystyle C=\{C_{k}\}} is a collection of boolean circuits of size less than P C ( k ) {\displaystyle P_{C}(k)} . Let p k
Yao's_test
Elements in exactly one of two sets
symmetric distance between them diverges, or vice versa. Algebra of sets Boolean function Complement (set theory) Difference (set theory) Exclusive or Fuzzy
Symmetric_difference
Algebraic structure
on the Boolean space. Homomorphisms between interior algebras correspond to a class of continuous maps between the Boolean spaces known as pseudo-epimorphisms
Interior_algebra
Transistor switch output for integrated circuits
and will be low (false) otherwise, like Boolean AND. When treated as active-low logic, this behaves like Boolean OR, since the output is low (true) when
Open_collector
Simple Turing complete logic
Boolean logic can be implemented in terms of Booleans acting as if-then-else structures. Boolean NOT (which returns the opposite of a given Boolean)
SKI_combinator_calculus
Array data structure that compactly stores bits
module with assorted bitwise functions and operators, including shift and rotate operations and an "unboxed" array over Boolean values may be used to model
Bit_array
Historical programming language
+100.4, -4., .05E-2, -.05E2, 5E02, 5.E2); Boolean (1B for true and 0B for false); Statement Label, and Function Name written as a name followed by a period
MAD_(programming_language)
Algorithmic paradigm for constraint satisfaction or enumeration problems
backtrack(P, s) s ← next(P, s) The reject procedure should be a Boolean-valued function that returns true only if it is certain that no possible extension
Backtracking
Logic gate
different kinds. As alternative, if different gates are available we can apply Boolean algebra to transform ( A ⋅ B ¯ ) + ( A ¯ ⋅ B ) ≡ ( A + B ) ⋅ ( A ¯ + B
XOR_gate
Unreproducible object used in digital security
primitives that provide unpredictable input/output behavior, such as pseudo-random functions. In some cases, PUFs can even be built from existing hardware with
Physical_unclonable_function
Process in computer graphics programming
Example logic is presented in the pseudo-code below: function HitTest(Rectangle r1, Rectangle r2) returns boolean { return not((r1.X + r1.Width < r2
Hit-testing
Programming virtual machine
P in P-code to mean pseudo more often than portable; they adopted a unique label for pseudo-code meaning instructions for a pseudo-machine. Although the
P-code_machine
assertion In computer programming, a statement that a predicate (Boolean-valued function, i.e. a true–false expression) is always true at that point in
Glossary_of_computer_science
Set whose pairs have minima and maxima
y=0.} If the pseudo-complement of every element of a Heyting algebra is in fact a complement, then the Heyting algebra is in fact a Boolean algebra. A chain
Lattice_(order)
Scripting language created in 1994
scalar (integer, float, string and boolean) types since PHP 7.0. PHP 7.0 also has type declarations for function return types, expressed by placing the
PHP
Involutive change of basis in linear algebra
2 × 2. Formally, the Hadamard transform is a Fourier transform on the Boolean group ( Z / 2 Z ) n {\displaystyle (\mathbb {Z} /2\mathbb {Z} )^{n}} .
Hadamard_transform
Algebraic ring that need not have additive negative elements
lattices. The smallest semiring that is not a ring is the two-element Boolean algebra, for instance with logical disjunction ∨ {\displaystyle \lor }
Semiring
American cryptographer (1890–1960)
(the "keystream") to generate the ciphertext, using the Boolean "exclusive or" (XOR) function. This is symbolised by ⊕ and is represented by the following
Gilbert_Vernam
Lightweight programming language
platforms. It supports only a small number of atomic data structures such as Boolean values, numbers (double-precision floating point and 64-bit integers by
Lua
Complexity class used in circuit complexity
feedforward network of Boolean functions. A binary language L ∈ 2 ∗ {\displaystyle L\in 2^{*}} is in the TC0 class if there exists a Boolean circuit family C
TC0
Function called at the end of an object's lifetime
cleaner = Cleaner.create(); static class State implements Runnable { private boolean cleaned = false; @Override public void run() { cleaned = true; System.out
Destructor (computer programming)
Destructor_(computer_programming)
Mathematical operator
In mathematics, a closure operator on a set S is a function cl : P ( S ) → P ( S ) {\displaystyle \operatorname {cl} :{\mathcal {P}}(S)\rightarrow {\mathcal
Closure_operator
Measure of algorithmic complexity
the pseudo-code: def generate_string2(): return "4c1j5b2p0cv4w1x8rx2y39umgw5q85s7" whereas the first string is output by the (much shorter) pseudo-code:
Kolmogorov_complexity
qucond Classical - int, real, complex, boolean, string, vector, matrix, tensor Function types qufunct - Pseudo-classic operators. Can only change the
Quantum_Computation_Language
Specific element of an algebraic structure
inverse Generalized inverse Identity (equation) Identity function Inverse element Monoid Pseudo-ring Quasigroup Unital (disambiguation) Weisstein, Eric
Identity_element
Logical principle
expressed, must be admitted to be spurious. It is also in the fragments of Pseudo-Archytas, to be found in Stobæus. [Hamilton LECT. V. LOGIC. 65] Hamilton
Law_of_excluded_middle
Standard library for the C programming language
declared in a number of header files. Each header file contains one or more function declarations, data type definitions, and macros. An implementation of a
C_standard_library
partitioned into two subsets that have equal sum. n ← |S| K ← sum(S) P ← empty boolean table of size ( ⌊ K / 2 ⌋ {\displaystyle \lfloor K/2\rfloor } + 1) by (n
Pseudopolynomial time number partitioning
Pseudopolynomial_time_number_partitioning
Programming language standard
for example, all symbols in the keyword package are self-evaluating. Boolean values in Common Lisp are represented by the self-evaluating symbols T
Common_Lisp
Transformation of one computational problem to another
quite possible to reduce a difficult-to-solve NP-complete problem like the boolean satisfiability problem to a trivial problem, like determining if a number
Reduction_(complexity)
System of digitally encoding numbers
tetrade) while the unused, don't care-states are named pseudo-tetrad(e)s[de], pseudo-decimals, or pseudo-decimal digits. BCD's main virtue, in comparison to
Binary-coded_decimal
Object-oriented programming language
a Boolean object, passing as an argument the block of code to be executed if and only if the Boolean receiver is true. The two subclasses of Boolean both
Smalltalk
constraints. ATTENTION NEEDED: This link is no longer valid! Conjunctive and Boolean Grammars (Okhotin) Conjunctive grammars, first introduced by Okhotin, introduce
Syntactic_predicate
Statement that is taken to be true
mathematicians of the 19th century and the developers of systems such as Boolean algebra made elaborate efforts to derive them from traditional arithmetic
Axiom
CPU instruction to set a memory location to a flag value and return its prior value
creates a new value (not just copying a reference). function TestAndSet(boolean_ref lock) { boolean initial = lock; lock = true; return initial; } Not
Test-and-set
Search algorithm
permit such a modification. The pseudo-code for depth limited minimax with alpha–beta pruning is as follows: function alphabeta(node, depth, α, β, maximizingPlayer)
Alpha–beta_pruning
Axiomatic set theories based on the principles of mathematical constructivism
C^{A}} is also written f : A → C {\displaystyle f\colon A\to C} . The Boolean-valued χ B : A → { 0 , 1 } {\displaystyle \chi _{B}\colon A\to \{0,1\}}
Constructive_set_theory
Glossary of terms used in branch of mathematics
is monotone and idempotent under function composition. Projections play an important role in domain theory. Pseudo-complement. In a Heyting algebra,
Glossary_of_order_theory
Testing method for testing the randomness of pseudo-random number generators
cryptography and the theory of computation, the next-bit test is a test against pseudo-random number generators. We say that a sequence of bits passes the next
Next-bit_test
property but has no infinite pseudo-intersection. P 1. The powerset function 2. A poset pairing function A pairing function is a bijection from X×X to
Glossary_of_set_theory
Programming paradigm focused on difficult search problems
smodels, used backtracking to find solutions. As the theory and practice of Boolean SAT solvers evolved, a number of ASP solvers were built on top of SAT solvers
Answer_set_programming
Programming language
wide (ASCII), but J also supports other literals (Unicode). Numeric and Boolean operations are not supported on literals, but collection-oriented operations
J_(programming_language)
Logic problem, AND of pairwise ORs
"Optimal cell flipping to minimize channel density in VLSI design and pseudo-Boolean optimization", Discrete Applied Mathematics, 90 (1–3): 69–88, doi:10
2-satisfiability
Components of a mathematical or logical formula
t1,...,tn, and every n-ary function symbol f ∈ Fn, a larger term f(t1, ..., tn) can be built. Using an intuitive, pseudo-grammatical notation, this is
Term_(logic)
Input where a function output does not matter
states or logical remainders) for a boolean function is an input-sequence (a series of bits) for which the function output does not matter. An input that
Don't-care_term
Set of rules defining correctly structured programs
main function, which behaves similarly to an entry point in most compiled languages, using the entry point idiom described as follows. A pseudo-entry
Python_syntax_and_semantics
General-purpose programming language
specified, e.g., as (x: Int) => x < 2 or even (x: Int) => (x < 2): Boolean. Anonymous functions behave as true closures in that they automatically capture any
Scala_(programming_language)
System for reasoning about vagueness
may range between completely true and completely false. By contrast, in Boolean logic, the truth values of variables may only be the integer values 0 or
Fuzzy_logic
Database code that executes after certain events on a table
variables to access row states during data modifications. It also provides the boolean context flags INSERTING, UPDATING, and DELETING flags to programmatically
Database_trigger
American-Danish computer scientist
online algorithms, cryptology, and the computational complexity of the Boolean functions used in cryptology. She is a professor in the Department of Mathematics
Joan_Boyar
Branch of mathematical logic
non-finitary meaning of their existential theorems, regarding these as pseudo-meaningful stipulations of the existence of ideal entities. The failure
Proof_theory
cryptography Proof-of-work algorithms Boolean minimization Espresso heuristic logic minimizer: a fast algorithm for Boolean function minimization Petrick's method:
List_of_algorithms
Type of error-correcting code using convolution
that generates parity symbols via the sliding application of a boolean polynomial function to a data stream. The sliding application represents the 'convolution'
Convolutional_code
Type of cipher
cipher Cusick, Thomas W.; Stanica, Pantelimon (2009). Cryptographic Boolean functions and applications. Academic Press. pp. 158–159. ISBN 9780123748904
Block_cipher
Branch of statistics focusing on spatial data sets
probabilities Markov chain geostatistics Support vector machine Boolean simulation Genetic models Pseudo-genetic models Cellular automata Multiple-Point Geostatistics
Geostatistics
Design pattern in functional programming to build generic types
file really exists, and as a result, outputs a Boolean value within the IO monad. The second function, in contrast, is only concerned with acting on the
Monad (functional programming)
Monad_(functional_programming)
Variable used in a concurrent system
manipulated using a test-and-set-lock command. Consider a variable A and a Boolean variable S. A is only accessed when S is marked true. Thus, S is a semaphore
Semaphore_(programming)
ETL tool developed by Microsoft
2016, when Power Query was included in Excel natively. In Excel 2016, the function was renamed "Get & Transform" for a short time, but has since been changed
Power_Query
Interactive programming language
the Boolean value could also be stored in a variable directly, or one could convert true to 1 and false to 0 using the tv (truth value) function. In addition
JOSS
Clustering and community detection algorithm
connecting our aggregate nodes. This process is modeled in the pseudo-code, where the function aggregateGraph returns a new graph whose vertices are the partition
Louvain_method
General-purpose programming language
composition. C# has support for strongly-typed function pointers via the keyword delegate. Like the Qt framework's pseudo-C++ signal and slot, C# has semantics
C Sharp (programming language)
C_Sharp_(programming_language)
Broad concept generalizing scalars in mathematics and physics
takes its values in a vector space. Logical vector, a vector of 0s and 1s (Booleans). Look up vector in Wiktionary, the free dictionary. Vector (disambiguation)
Vector (mathematics and physics)
Vector_(mathematics_and_physics)
Algorithm for generating prime numbers
use of a boolean array to mark integers as being prime, or not. public static List<int> sieveOfAtkin(int limit) { boolean[] arr = new boolean[limit + 1];
Sieve_of_Atkin
Optimization algorithms using quantum computing
phrased as a maximization of an objective function which is a sum of Boolean functions. Each Boolean function C α : { 0 , 1 } n → { 0 , 1 } {\displaystyle
Quantum optimization algorithms
Quantum_optimization_algorithms
Algebraic ring without a multiplicative identity
abstract algebra, a rng (pronounced "rung" /rʌŋ/) or non-unital ring or pseudo-ring is an alternative name for a ring that does not assume the existence
Rng_(algebra)
Method for generating pseudo-random numbers
The Marsaglia polar method is a pseudo-random number sampling method for generating a pair of independent standard normal random variables. Standard normal
Marsaglia_polar_method
Complexity of sending information in a distributed algorithm
approximate degree of a Boolean function is the minimal degree of a polynomial that approximates the function on all Boolean points up to an additive
Communication_complexity
Theory within consciousness research
our behaviour or cognitive functions would "not be conscious, but act as if they were", a condition the authors term pseudo-consciousness. In a 2024 preprint
Integrated_information_theory
Concept in machine learning
(June 1989). "Linear and Nonlinear Extension of the Pseudo-Inverse Solution for Learning Boolean Functions". Europhysics Letters. 9 (4): 315. Bibcode:1989EL
Double_descent
Method of modeling the metabolism of cells or microbes
stoichiometry constraints while maximizing the value of a pseudo-reaction (the objective function) representing the conversion of biomass precursors into
Flux_balance_analysis
Type of random mathematical object
by W, needs to be simulated, which is done by using a (pseudo)-random number generating function capable of simulating Poisson random variables. For the
Poisson_point_process
apartness relation of the real numbers is then the disjunction of its natural pseudo-order. The complex numbers, real vector spaces, and indeed any metric space
Apartness_relation
JavaScript and WebAssembly engine
representing the function, JägerMonkey instead operated by iterating linearly forward through SpiderMonkey bytecode, the internal function representation
SpiderMonkey
Early object-oriented programming language
features and its coroutine concept. Simula provided functions to do random number generation of pseudo-random numbers. Sam, Sally, and Andy are shopping
Simula
Jacobus; Goffman, W; Belzer, Jack (1961). "Inefficiency of the use of boolean functions for information retrieval systems". Communications of the ACM. 4 (12)
Jacobus_Verhoeff
Algebraic structure with addition and multiplication
of sets and multiplication to be intersection. This is an example of a Boolean ring. For any ring R and any natural number n, the set of all square n-by-n
Ring_(mathematics)
List of concepts in artificial intelligence
structures, or other components (e.g. "string", "array of float", "function returning boolean"). The main purpose of a type system is to reduce possibilities
Glossary of artificial intelligence
Glossary_of_artificial_intelligence
Computer programming concept
1970s. LISP 1.5 (1961) allowed exceptions to be raised by the ERROR pseudo-function, similarly to errors raised by the interpreter or compiler. Exceptions
Exception handling (programming)
Exception_handling_(programming)
Functional programming language for arrays
according to word length: X[⍋X+.≠' ';] The following function "life", written in Dyalog APL, takes a Boolean matrix and calculates the new generation according
APL_(programming_language)
Object-oriented computer programming language
true conditional operator, "If(condition as Boolean, truepart, falsepart)", to replace the "IIf" function. Anonymous types Support for LINQ Lambda expressions
Visual_Basic_(.NET)
given certain premises. logic gate A physical device implementing a Boolean function, used in digital circuits to perform logical operations on one or more
Glossary_of_logic
Signal processing computational method
(xj=1). Formally we have: xi=⋁j=1n(gij∧yj),i=1,2,…,m, where ∧ is Boolean AND and ∨ is Boolean OR. Noise is not explicitly modelled, rather, can be treated
Independent component analysis
Independent_component_analysis
Self-balancing binary search tree data structure
the right part is symmetric. For some applications, Split also returns a Boolean value denoting if x appears in the tree. The cost of Split is O(logn)
Red–black_tree
his Vorlesungen (1890–1905). Peirce (1880) showed how all the Boolean elective functions could be expressed by the use of a single primitive binary operation
History_of_logic
Overview of and topical guide to algebraic structures
ring: a commutative nonassociative ring that respects the Jordan identity Boolean ring: a commutative ring with idempotent multiplication operation. Kleene
Outline of algebraic structures
Outline_of_algebraic_structures
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PSEUDO BOOLEAN-FUNCTION
PSEUDO BOOLEAN-FUNCTION
PSEUDO BOOLEAN-FUNCTION
PSEUDO BOOLEAN-FUNCTION
PSEUDO BOOLEAN-FUNCTION
PSEUDO BOOLEAN-FUNCTION
PSEUDO BOOLEAN-FUNCTION
PSEUDO BOOLEAN-FUNCTION
PSEUDO BOOLEAN-FUNCTION
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