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mathematics, a Boolean matrix is a matrix with entries from a Boolean algebra. When the two-element Boolean algebra is used, the Boolean matrix is called a
Boolean_matrix
Matrix of binary truth values
matrix, binary matrix, relation matrix, Boolean matrix, or (0, 1)-matrix is a matrix with entries from the Boolean domain B = {0, 1}. Such a matrix can
Logical_matrix
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
Relationship between two sets, defined by a set of ordered pairs
exists if and only if (x,y) ∈ R. Boolean matrix: The members of X are arranged in some fixed sequence x1, ..., xn; the matrix has dimensions n × n, with the
Relation_(mathematics)
Formal language generated by context-free grammar
Boolean matrix multiplication, thus inheriting its complexity upper bound of O(n2.3728596). Conversely, Lillian Lee has shown O(n3−ε) Boolean matrix multiplication
Context-free_language
Technique for speeding up algorithms involving Boolean matrices
may be applied include: computing the transitive closure of a graph, Boolean matrix multiplication, edit distance calculation, sequence alignment, index
Method_of_Four_Russians
matrices used in mathematics, science and engineering. A matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers called entries
List_of_named_matrices
Square matrix used to represent a graph or network
whether or not two vertices are connected (i.e., the connection matrix, which contains Boolean values), it gives the exact distance between them. The convention
Adjacency_matrix
Type of grammar for describing formal languages
parsing according to a general CFG is asymptotically equivalent to boolean matrix multiplication (thus likely between quadratic and cubic time). One classical
Parsing_expression_grammar
Matrix representation of a graph
Laplacian matrix for the latter. In the matrix notation, the adjacency matrix of the undirected graph could, e.g., be defined as a Boolean sum of the
Laplacian_matrix
Algebraic structure modeling logical operations
In mathematics, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties
Boolean_algebra_(structure)
Smallest transitive relation containing a given binary relation
reachable from node a. The data structure is typically stored as a Boolean matrix, so if matrix[1][4] = true, then it is the case that node 1 can reach node
Transitive_closure
Programming language designed 1942 to 1945
regarded as byte) is denoted by 8 × S 0 {\displaystyle 8\times S0} , and Boolean matrix of size m {\displaystyle m} by n {\displaystyle n} is described by
Plankalkül
Vector operation
51, Cambridge University Press ISBN 0-521-59916-4 Ki-Hang Kim (1982) Boolean Matrix Theory and Applications, page 37, Marcel Dekker ISBN 0-8247-1788-0 Carlen
Outer_product
Modelling technique in mechanical engineering
condition can be expressed using a Boolean matrix. Use is made of the transpose of the Boolean localisation matrix L {\displaystyle \mathbf {L} } that
Dynamic_substructuring
Topological model
the boolean domain {T,F}. The matrix, denoted with operators, can be expressed as The elements of the matrix can be named as shown below: Both matrix forms
DE-9IM
Rule system for formal languages
Boolean matrix multiplication, thus inheriting its complexity upper bound of O(n2.3728639). Conversely, Lillian Lee has shown O(n3−ε) Boolean matrix multiplication
Context-free_grammar
Matrix equal to its conjugate-transpose
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a square matrix with complex-valued entries that is equal to its own conjugate transpose
Hermitian_matrix
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
Problem in computational complexity theory
integer n {\displaystyle n} and an n × n {\displaystyle n\times n} Boolean matrix M {\displaystyle M} . The algorithm then runs for n {\displaystyle n}
Online matrix-vector multiplication problem
Online_matrix-vector_multiplication_problem
Relationship between elements of two sets
{\displaystyle X=Y} ) form a matrix semiring (indeed, a matrix semialgebra over the Boolean semiring) where the identity matrix corresponds to the identity
Binary_relation
Table for coordinating educational schedules at an institution
ISSN 1572-9338. S2CID 254227100. Gunther Schmidt and Thomas Ströhlein (1976) "A Boolean matrix iteration in timetable construction", Linear Algebra and Its Applications
School_timetable
Planning of tasks and events
Procedures for MPOs)". Gunther Schmidt and Thomas Ströhlein (1976) "A Boolean matrix iteration in timetable construction", Linear Algebra and Its Applications
Schedule
Symbolic boolean function representation, extension of BDDs
represented by a matrix according to its cofactors. For example, consider a function f ( x , y ) {\displaystyle f(x,y)} over Boolean variables x {\displaystyle
Algebraic_decision_diagram
Functional programming language for arrays
list stored in matrix X according to word length: X[⍋X+.≠' ';] The following function "life", written in Dyalog APL, takes a Boolean matrix and calculates
APL_(programming_language)
Mathematical ring whose elements are matrices
called the matrix semiring. Similarly, if R is a commutative semiring, then Mn(R) is a matrix semialgebra. For example, if R is the Boolean semiring (the
Matrix_ring
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
Boolean_function
Method for finding patterns in networks
Boolean matrix A1 and A2 represent the connectivity of each pair of vertices in G1 and G2 by a colorful path, respectively, and let B be the matrix describing
Color-coding
d-or-fewer columns have the same boolean sum. A matrix is said to be d-disjunct if no set of d columns has a boolean sum which is a superset of any other
Disjunct_matrix
Parsing algorithm for context-free grammars
Lee, Lillian (2002). "Fast context-free grammar parsing requires fast Boolean matrix multiplication". J. ACM. 49 (1): 1–15. arXiv:cs/0112018. doi:10.1145/505241
CYK_algorithm
Method of deriving an ontology
order dimension (of the concept lattice) and has applications e.g. for Boolean matrix factorization. Given an object-attribute numerical data-table, the goal
Formal_concept_analysis
Copy of a directed graph with redundant edges removed
equivalence with Boolean matrix multiplication. They let A be the adjacency matrix of the given directed acyclic graph, and B be the adjacency matrix of its transitive
Transitive_reduction
Data buffer in graphics hardware
to represent numerical values in the range [0, 2n-1], and also as a Boolean matrix (n is the number of allocated bits), each of which may be used to control
Stencil_buffer
Russian-American mathematician (1938–2023)
number of "concepts" necessary to express the logical matrix of a binary relation as a Boolean sum, and where a concept is an outer product of logical
Boris_M._Schein
m is the number of non-zeros. Cycle rank of a graph, or a symmetric Boolean matrix, measures the minimum parallel time needed to perform Cholesky decomposition
Nested_dissection
Involutive change of basis in linear algebra
mathematician Joseph L. Walsh. The Hadamard transform Hm is a 2m × 2m matrix, the Hadamard matrix (scaled by a normalization factor), that transforms 2m real numbers
Hadamard_transform
give low power dissipation when synthesized. This algorithm uses a boolean matrix with rows corresponding to state codes and columns corresponding to
State_encoding_for_low_power
Property of operations
\vee )} and ( { 0 , 1 } , ∧ ) {\displaystyle (\{0,1\},\wedge )} of the Boolean domain with logical disjunction ∨ {\displaystyle \vee } and logical conjunction
Idempotence
Soviet and Israeli computer scientist
1970, Dinitz, Mikhail Kronrod, Arlazarov, and Faradjev published the Boolean matrix multiplication algorithm that would make them famous as the "Four Russians"
Yefim_Dinitz
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
Array data structure that compactly stores bits
called a logical matrix. In the calculus of relations, these arrays are composed with matrix multiplication where the arithmetic is Boolean, and such a composition
Bit_array
solves types of the Boolean satisfiability problem despite there being no known efficient algorithm in the general case. The Boolean satisfiability (or
Boolean satisfiability algorithm heuristics
Boolean_satisfiability_algorithm_heuristics
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
Algebraic_normal_form
Korean-American mathematician (1936–2009)
1)-matrix semigroupsof (0, 1)-matrix semigroups." Period. Math. Hung. 4, 303–305 (1973). Butler, K.KH. (1974). "A moore-penrose inverse for boolean relation
Ki-Hang_Kim
True when either but not both inputs are true
q&\Leftrightarrow &r=p+q{\pmod {2}}\\\end{matrix}}} The description of a Boolean function as a polynomial in F 2 {\displaystyle \mathbb {F} _{2}} , using
Exclusive_or
Finite field of two elements
GF(2) may be identified with the two possible values of a bit and to the Boolean values true and false. It follows that GF(2) is fundamental and ubiquitous
GF(2)
Degree of connectedness within a graph
powers of the graph's adjacency matrix gives the number of walks of length given by that power. Similarly, the matrix exponential is also closely related
Centrality
Algebraic ring that need not have additive negative elements
{\displaystyle Y} with entries in the Boolean semiring, matrix addition corresponds to union of relations, and matrix multiplication corresponds to composition
Semiring
Vector space equipped with a bilinear product
associative algebra over the field of real numbers under matrix addition and matrix multiplication since matrix multiplication is associative. Three-dimensional
Algebra_over_a_field
Any logic with four truth values
sixteen have zero in one corner only, so the output of vector-matrix multiplication with Boolean arithmetic is always D, except for C input. Nine further logical
Four-valued_logic
Recursive algorithm for matrix multiplication
semirings, such as min-plus or boolean algebra, where the naive algorithm still works, and so called combinatorial matrix multiplication. Volker Strassen
Strassen_algorithm
Graphical method to simplify Boolean expressions
Karnaugh map (KM or K-map) is a diagram that can be used to simplify a Boolean algebra expression. Maurice Karnaugh introduced the technique in 1953 as
Karnaugh_map
vectors. Vector logic is a direct translation into a matrix–vector formalism of the classical Boolean polynomials. This kind of formalism has been applied
Vector_logic
Measure of network community structure
to community 2, s v = − 1 {\displaystyle s_{v}=-1} . Let the adjacency matrix for the network be represented by A {\displaystyle A} , where A v w = 0
Modularity_(networks)
Mathematical theory on behavior of connected clusters in a random graph
structure Edge Loop Neighborhood Path Vertex Adjacency list / matrix Incidence list / matrix Types Bipartite Complete Directed Hyper Labeled Multi Random
Percolation_theory
Real-valued function that quantifies similarity between two objects
techniques that work with binary data such as presence/absence data or Boolean data; The Jaccard similarity is particularly useful for clustering techniques
Similarity_measure
Special type of Boolean function
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
Bent_function
Models replicating electronic behavior
transistor topology of the integrated circuit, and finally, with inline Boolean logic expressions. Exact representations are used mainly in the analysis
Electronic_circuit_simulation
Network whose degree distribution follows a power law
structure Edge Loop Neighborhood Path Vertex Adjacency list / matrix Incidence list / matrix Types Bipartite Complete Directed Hyper Labeled Multi Random
Scale-free_network
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
Topics referred to by the same term
describes Shannon cofactor, a term in Boole's (or Shannon's) expansion of a Boolean function Factor (disambiguation) This disambiguation page lists articles
Cofactor
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\colon \{0,1\}^{n}\to
Sensitivity_theorem
Approximate nearest neighbor search algorithm
Blockmodeling Maximum entropy Soft configuration LFR Benchmark Dynamics Boolean network agent based Epidemic/SIR Lists Categories Topics Software Network
Hierarchical navigable small world
Hierarchical_navigable_small_world
Standard hostname for a networked device's loopback interface
Blockmodeling Maximum entropy Soft configuration LFR Benchmark Dynamics Boolean network agent based Epidemic/SIR Lists Categories Topics Software Network
Localhost
Social structure made up of a set of social actors
Blockmodeling Maximum entropy Soft configuration LFR Benchmark Dynamics Boolean network agent based Epidemic/SIR Lists Categories Topics Software Network
Social_network
Standard model in theoretical computer science
differences between the study of arithmetic circuits and the study of Boolean circuits. In Boolean complexity, one is mostly interested in computing a function
Arithmetic_circuit_complexity
Study of graphs as a representation of relations between discrete objects
quantify degree correlations. The recurrence matrix of a recurrence plot can be considered as the adjacency matrix of an undirected and unweighted network
Network_theory
Process by which people befriend similar people
Blockmodeling Maximum entropy Soft configuration LFR Benchmark Dynamics Boolean network agent based Epidemic/SIR Lists Categories Topics Software Network
Homophily
Scale-free network generation algorithm
Blockmodeling Maximum entropy Soft configuration LFR Benchmark Dynamics Boolean network agent based Epidemic/SIR Lists Categories Topics Software Network
Barabási–Albert_model
Reasoning about equations with free variables
of two others is then represented by the logical matrix obtained by matrix multiplication using Boolean arithmetic. An example of calculus of relations
Algebraic_logic
Clustering and community detection algorithm
partition a graph. The equation for this metric is defined for an adjacency matrix, A, as: Q = 1 2 m ∑ i j ( A i j − k i k j 2 m ) δ ( c i , c j ) {\displaystyle
Leiden_algorithm
In mathematics, element that equals its square
decompositions of modules, and connected to homological properties of the ring. In Boolean algebra, the main objects of study are rings in which all elements are
Idempotent_(ring_theory)
Measure of centrality in a network based on nodal influence
{\displaystyle A^{T}} denotes the transposed matrix of A and ( I − α A T ) − 1 {\displaystyle (I-\alpha A^{T})^{-1}} denotes matrix inversion of the term ( I − α A
Katz_centrality
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
Set of rules defining correctly structured programs
right examples] (using the same / symbol, right example) demonstrate how Boolean values (0s and 1s) can be used as left arguments for the \ expand and /
APL_syntax_and_symbols
Stochastic process formalizing cumulative advantage
Blockmodeling Maximum entropy Soft configuration LFR Benchmark Dynamics Boolean network agent based Epidemic/SIR Lists Categories Topics Software Network
Preferential_attachment
Collection of electronic switches arranged in a matrix
mechanical Boolean AND function of telephony crossbar switches, but other models had individual relays (one coil per crosspoint) in matrix arrays, connecting
Crossbar_switch
Concept in network science
C_{r}} , called communities; a symmetric r × r {\displaystyle r\times r} matrix P {\displaystyle P} of edge probabilities. The edge set is then sampled
Stochastic_block_model
Method of representing systems
represented as an N×N matrix where N is the number of nodes, and whose entries tell you if two nodes share an edge. Typically, this matrix will have entries
Biological_network
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)
Graph where most nodes are reachable in a small number of steps
Blockmodeling Maximum entropy Soft configuration LFR Benchmark Dynamics Boolean network agent based Epidemic/SIR Lists Categories Topics Software Network
Small-world_network
Function that is its own inverse
instance, involutive negation characterizes Boolean algebras among Heyting algebras. Correspondingly, classical Boolean logic arises by adding the law of double
Involution_(mathematics)
algebra AW*-algebra Azumaya algebra Banach algebra Birman–Wenzl algebra Boolean algebra Borcherds algebra Brauer algebra C*-algebra Central simple algebra
List_of_algebras
Arrangement of a communication network
Blockmodeling Maximum entropy Soft configuration LFR Benchmark Dynamics Boolean network agent based Epidemic/SIR Lists Categories Topics Software Network
Network_topology
Electronic communication subsystem on an integrated circuit
Blockmodeling Maximum entropy Soft configuration LFR Benchmark Dynamics Boolean network agent based Epidemic/SIR Lists Categories Topics Software Network
Network_on_a_chip
Network for communications over distance
Blockmodeling Maximum entropy Soft configuration LFR Benchmark Dynamics Boolean network agent based Epidemic/SIR Lists Categories Topics Software Network
Telecommunications_network
Unsolved problem in theoretical computer science
communication complexity of a two-party Boolean function is polynomially related to the logarithm of the rank of its input matrix. Let D ( f ) {\displaystyle D(f)}
Log-rank_conjecture
Data organization and storage formats
running times for a subset of this list see comparison of data structures. Boolean, true or false. Character Floating-point representation of a finite subset
List_of_data_structures
or false. A Boolean expression may be composed of a combination of the Boolean constants true or false, Boolean-typed variables, Boolean-valued operators
Glossary_of_computer_science
Specific element of an algebraic structure
matrices Matrix addition Zero matrix n-by-n square matrices Matrix multiplication In (identity matrix) m-by-n matrices ○ (Hadamard product) Jm, n (matrix of
Identity_element
Method of generating random small-world graphs
Blockmodeling Maximum entropy Soft configuration LFR Benchmark Dynamics Boolean network agent based Epidemic/SIR Lists Categories Topics Software Network
Watts–Strogatz_model
Data structure representing a graph
adjacency list is the adjacency matrix, a matrix whose rows and columns are indexed by vertices and whose cells contain a Boolean value that indicates whether
Adjacency_list
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
In mathematics, vector subspace
a result, this operation does not turn the lattice of subspaces into a Boolean algebra (nor a Heyting algebra).[citation needed] Most algorithms for dealing
Linear_subspace
Calculator designed to calculate problems in science, engineering, and mathematics
entered.) Hexadecimal, binary, and octal calculations, including basic Boolean mathematics Complex numbers Fractions calculations Statistics and probability
Scientific_calculator
Knowledge base that represents semantic relations between concepts in a network
Blockmodeling Maximum entropy Soft configuration LFR Benchmark Dynamics Boolean network agent based Epidemic/SIR Lists Categories Topics Software Network
Semantic_network
Concept in computer science
In computer programming, a guard is a Boolean expression that must evaluate to true if the execution of the program is to continue in the branch in question
Guard_(computer_science)
Branch of logic
Higher-order logic Boolean algebra (logic) Boolean algebra (structure) Boolean algebra topics Boolean domain Boolean function Boolean-valued function Categorical
Propositional_logic
Property involving two mathematical operations
numbers, polynomials, matrices, rings, and fields. It is also encountered in Boolean algebra and mathematical logic, where each of the logical and (denoted
Distributive_property
Example of a Semigroup
with zero of order two, also a monoid, and ultimately the two-element Boolean algebra; this is also isomorphic to (Z2, ·2), the multiplicative group
Semigroup_with_two_elements
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