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Function returning one of only two values
vector-valued Boolean function (an S-box in symmetric cryptography). There are 2 2 k {\displaystyle 2^{2^{k}}} different Boolean functions with k {\displaystyle
Boolean_function
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
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
ones and is therefore a symmetric Boolean function. The n-variable parity function and its negation are the only Boolean functions for which all disjunctive
Parity_function
Elements in exactly one of two sets
becomes a Boolean ring, with symmetric difference as the addition of the ring and intersection as the multiplication of the ring. The symmetric difference
Symmetric_difference
Algebraic structure in mathematics
symmetric difference (not disjunction ∨, which would constitute a semiring). Conversely, every Boolean algebra gives rise to a Boolean ring. Boolean rings
Boolean_ring
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
Study of Boolean functions via discrete Fourier analysis
and theoretical computer science, analysis of Boolean functions is the study of real-valued functions on { 0 , 1 } n {\displaystyle \{0,1\}^{n}} or {
Analysis_of_Boolean_functions
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
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
Overview of and topical guide to logic
expression Boolean ring Boolean function Boolean-valued function Parity function Symmetric Boolean function Conditioned disjunction Field of sets Functional
Outline_of_logic
Algebraic structure modeling logical operations
ring addition to exclusive disjunction or symmetric difference (not disjunction ∨). However, the theory of Boolean rings has an inherent asymmetry between
Boolean_algebra_(structure)
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
Technical treatment of Boolean algebras
\mathbb {Z} } of integers and the symmetric group Sn of permutations of n objects, there are also basic examples of Boolean algebras such as the following
Boolean algebras canonically defined
Boolean_algebras_canonically_defined
English mathematician and philosopher (1815–1864)
equations and the study of the sum of residues of a rational function. In 1847, Boole developed Boolean algebra, a fundamental concept in binary logic, which
George_Boole
Mathematical set of all subsets of a set
both of these operations forms a Boolean ring. In set theory, XY is the notation representing the set of all functions from Y to X. As "2" can be defined
Power_set
Collection of mathematical objects
difference, symmetric difference and absolute complement (complement in U {\displaystyle U} ). The powerset is a Boolean ring that has symmetric difference
Set_(mathematics)
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
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 \{0
Sensitivity_theorem
True when either but not both inputs are true
description of a Boolean function as a polynomial in F 2 {\displaystyle \mathbb {F} _{2}} , using this basis, is called the function's algebraic normal
Exclusive_or
Relationship between two sets, defined by a set of ordered pairs
neither reflexive nor symmetric. "is sister of" is neither reflexive (e.g. Pierre Curie is not a sister of himself), nor symmetric, nor asymmetric; while
Relation_(mathematics)
Russian mathematician
Razborov, A. A. (December 1990). "Lower bounds of the complexity of symmetric boolean functions of contact-rectifier circuits". Mathematical Notes of the Academy
Alexander_Razborov
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
Graphical method to simplify Boolean expressions
while each cell value represents the corresponding output value of the Boolean function. Optimal groups of 1s or 0s are identified, which represent the terms
Karnaugh_map
Function that is its own inverse
(on the real numbers) is symmetric across the line y = x. This is due to the fact that the inverse of any general function will be its reflection over
Involution_(mathematics)
Theories in mathematical logic
relation symbol ~, no constants, and no functions. Equivalence relations satisfy the axioms: Reflexive ∀x x~x; Symmetric ∀x ∀y x~y → y~x; Transitive: ∀x ∀y
List_of_first-order_theories
Well-quasi-ordering of finite trees
application of the theorem gives the existence of a fast-growing TREE function. TREE(3) is one of the largest simply defined finite numbers, dwarfing
Kruskal's_tree_theorem
Boolean algebra with all operators and laws forming a complete logical system
mathematics, a complete Boolean algebra is a Boolean algebra in which every subset has a supremum (least upper bound). Complete Boolean algebras are used to
Complete_Boolean_algebra
Directed graph isomorphic to its own transpose graph
without any fixed points. Skew-symmetric graphs are identical to the double covering graphs of bidirected graphs. Skew-symmetric graphs were first introduced
Skew-symmetric_graph
Type of polynomial
is a symmetric hollow matrix. In particular, the Laplacian ∇ 2 f = 0 {\displaystyle \nabla ^{2}f=0} , so f {\displaystyle f} is a harmonic function. This
Multilinear_polynomial
Mathematical concept for comparing objects
mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in geometry
Equivalence_relation
Branch of mathematics
and Boolean algebras, which both introduce a new operation ~ called negation. Both structures play a role in mathematical logic and especially Boolean algebras
Order_theory
Set of the values of a function
In mathematics, for a function f : X → Y {\displaystyle f:X\to Y} , the image is a relation between inputs and outputs, used in three related ways: The
Image_(mathematics)
+ 1 return T else return rotateLeft(T) function joinLeftAVL(TL, k, TR) /* symmetric to joinRightAVL */ function join(TL, k, TR) if h(TL) > h(TR) + 1 return
Join-based_tree_algorithms
Axiom of set theory
of countable choice.) Stone's representation theorem for Boolean algebras needs the Boolean prime ideal theorem. The Nielsen–Schreier theorem, that every
Axiom_of_choice
Cryptographic hash function
BLAKE is a cryptographic hash function based on Daniel J. Bernstein's ChaCha stream cipher, but a permuted copy of the input block, XORed with round constants
BLAKE_(hash_function)
Diagram that shows all possible logical relations between a collection of sets
He also showed that such symmetric Venn diagrams exist when n is five or seven. In 2002, Peter Hamburger found symmetric Venn diagrams for n = 11 and
Venn_diagram
Mathematical function conceived as a crude model
y(t+1)=0} 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
Class of graph dynamical systems
{0,1}. For vertex functions use the symmetric, boolean function nor : K3 → K defined by nor(x,y,z) = (1+x)(1+y)(1+z) with boolean arithmetic. Thus, the
Sequential_dynamical_system
Topics referred to by the same term
in a state machine Implication graph, a skew-symmetric directed graph used for analyzing complex Boolean expressions Implication (information science)
Implication
System including an indeterminate value
tables. Philosophy portal Binary logic (disambiguation) Boolean algebra (structure) Boolean function Digital circuit Four-valued logic Homogeneity (linguistics)
Three-valued_logic
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 be
Logical_matrix
Computation model defining an abstract machine
'mechanical'" (Hodges p. 96). While at Princeton pursuing his PhD, Turing built a Boolean-logic multiplier (see below). His PhD thesis, titled "Systems of Logic
Turing_machine
Self-balancing binary search tree data structure
to form the left tree, and the right part is symmetric. For some applications, Split also returns a Boolean value denoting if x appears in the tree. The
Red–black_tree
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
Symbols representing logical operations
connectives within Boolean algebra. Truth functions are functions from sequences of truth values to truth values. A unary truth function, for example, takes
Logic_alphabet
Type of symmetric key cipher
parallel LFSRs into a non-linear Boolean function to form a combination generator. Various properties of such a combining function are critical for ensuring
Stream_cipher
Algebraic structure of set algebra
between two sets is defined as the measure of the symmetric difference of the two sets. The symmetric difference of two distinct sets can have measure
Σ-algebra
Use of functions that call themselves
evaluation of the Boolean || (OR) operator, to only check the right child if the left child fails. In fact, the entire control flow of these functions can be replaced
Recursion_(computer_science)
Symmetric monoidal closed category equipped with a dualizing object
In mathematics, a *-autonomous (read "star-autonomous") category is a symmetric monoidal closed category equipped with a dualizing object ⊥ {\displaystyle
*-autonomous_category
Set of elements in any of some sets
operations given by union, intersection, and complementation, is a Boolean algebra. In this Boolean algebra, union can be expressed in terms of intersection and
Union_(set_theory)
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
Mathematical ranking of a set
Incomparability with respect to < {\displaystyle \,<\,} is always a homogeneous symmetric relation on S . {\displaystyle S.} It is reflexive if and only if < {\displaystyle
Weak_ordering
Set whose pairs have minima and maxima
semilattices, and some notable subclasses of lattices are Heyting algebras, Boolean algebras, distributive lattices, and geometric lattices (matroids). These
Lattice_(order)
Topological model
(Contains, Crosses, Intersects, Touches, etc.) as boolean functions, and the DE-9IM model, as a function that returns a string (the DE-9IM code), with domain
DE-9IM
Bound lattice in which every element has a complement
distributive lattice has a unique orthocomplementation and is in fact a Boolean algebra. A complemented lattice is a bounded lattice (with least element
Complemented_lattice
One-to-one correspondence
them. A bijective function from a set to itself is also called a permutation, and the set of all permutations of a set forms its symmetric group. Some bijections
Bijection
Mathematical proposition equivalent to the axiom of choice
lemma is strictly weaker than the axiom of choice; it is equivalent to the boolean prime ideal theorem. On the other hand, somehow surprisingly, Tychonoff's
Zorn's_lemma
mathematics, the notions of an absolutely monotonic function and a completely monotonic function are two very closely related concepts. Both imply very
Absolutely and completely monotonic functions and sequences
Absolutely_and_completely_monotonic_functions_and_sequences
Matrix representation of a graph
\end{cases}}} The symmetrically normalized Laplacian matrix is symmetric if and only if the adjacency matrix is symmetric. For a non-symmetric adjacency matrix
Laplacian_matrix
Theorem in mechanism design
Parikshit; Nisan, Noam; Roughgarden, Tim (2015). "Public projects, Boolean functions and the borders of Border's theorem". arXiv:1504.07687 [cs.GT].{{cite
Border's_theorem
restrictions (see below) this question was settled in the positive for Boolean domains by Schaefer's dichotomy theorem and for any finite domain by Andrei
Complexity of constraint satisfaction
Complexity_of_constraint_satisfaction
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
Heyting_algebra
Partially ordered set equipped with a rank function
S2CID 14857863. Butler, Lynne M. (1994), Subgroup Lattices and Symmetric Functions, Memoirs of the American Mathematical Society, vol. 539, American
Graded_poset
Lattice formed by all integer partitions
quantitative substitutional analysis, developed the representation theory of the symmetric group. In Young's theory, the objects now called Young diagrams and the
Young's_lattice
Cryptographic protocol for two-party computation
steps as follows: The underlying function (e.g., in the millionaires' problem, comparison function) is described as a Boolean circuit with 2-input gates. The
Garbled_circuit
algorithm: reduce the bandwidth of a symmetric sparse matrix Minimum degree algorithm: permute the rows and columns of a symmetric sparse matrix before applying
List_of_algorithms
Glossary of terms used in branch of mathematics
x R∗ y implies that x R y or not y R x. Symmetric relation. A homogeneous relation R on a set X is symmetric, if x R y implies y R x, for all elements
Glossary_of_order_theory
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
Relationship between elements of two sets
indexed by X {\displaystyle X} and Y {\displaystyle Y} with entries in the Boolean semiring (addition corresponds to OR and multiplication to AND) where matrix
Binary_relation
Mathematical operation
corresponding to compared objects. Working with such matrices involves the Boolean arithmetic with 1 + 1 = 1 {\displaystyle 1+1=1} and 1 × 1 = 1. {\displaystyle
Composition_of_relations
(with involution) Łukasiewicz–Moisil algebra Boolean algebra (structure) Boolean ring Complete Boolean algebra Orthocomplemented lattice Quantale Partially
List_of_order_theory_topics
Area of physical and philosophical debate
equations of quantum mechanics to be symmetric with respect to time reversal. (See Wheeler–Feynman time-symmetric theory.) This creates retrocausality:
Interpretations of quantum mechanics
Interpretations_of_quantum_mechanics
Technique for finding an extremum of a function
private static double[] gss(Function f, double a, double b, double tol, double h, boolean noC, double c, double fc, boolean noD, double d, double fd) {
Golden-section_search
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
System that regulates the formation of blocks on a blockchain
that implements a variant of WalkSAT, a local search algorithm to solve Boolean problems. Optimisable proof of work (OPoW) is a variant of proof of work
Proof_of_work
Reflexive and transitive binary relation
cases of a preorder: an antisymmetric preorder is a partial order, and a symmetric preorder is an equivalence relation. Moreover, a preorder on a set X {\displaystyle
Preorder
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
Generalization of the discrete Fourier transform
for example the symmetric group, by decomposing the matrix U {\displaystyle U} associated to a G {\displaystyle G} -invariant symmetric bilinear form as
Fourier transform on finite groups
Fourier_transform_on_finite_groups
Type of lattice in mathematical order theory
dually M-symmetric or M*-symmetric if its dual is M-symmetric. It can be shown that a finite lattice is modular if and only if it is M-symmetric and M*-symmetric
Modular_lattice
introduced by Yao (1979), two players, P1 and P2 attempt to compute a Boolean function f ( x 1 , x 2 ) : { 0 , 1 } n → { 0 , 1 } , x 1 , x 2 ∈ { 0 , 1 }
Multiparty communication complexity
Multiparty_communication_complexity
Measure of dependence between two variables
1], but are not necessarily equal. This measure is not symmetric. If one desires a symmetric measure, one may consider the following redundancy measure:
Mutual_information
Subfield of cryptography
is a Boolean predicate), and in generality (for any feasible computation) in 1986 by Andrew Yao. The area is also referred to as Secure Function Evaluation
Secure multi-party computation
Secure_multi-party_computation
Binary relation over a set and itself
kind of) quasi-reflexivity. Symmetric for all x, y ∈ X, if xRy then yRx. For example, "is a blood relative of" is a symmetric relation, because x is a blood
Homogeneous_relation
Mathematical property of subsets in order theory
directed set, which is a preordered set with additional properties. Final functions A map f : X → A {\displaystyle f:X\to A} between two directed sets is
Cofinal_(mathematics)
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
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
Size of a set in mathematics
place, it is called injective. If a function covers every member in the output set, it is called surjective. If a function is both injective and surjective
Cardinality
Branch of mathematics that studies sets
formula embodying the membership relation is not simply True or False. The Boolean-valued models of ZFC are a related subject. An enrichment of ZFC called
Set_theory
Algebraic object with an ordered structure
topology. The product is a Boolean space (compact, Hausdorff and totally disconnected), and XF is a closed subset, hence again Boolean. A fan on F is a preordering
Ordered_field
Collection of sets in mathematics that can be defined based on a property of its members
"classes". In ZF, the concept of a function can also be generalised to classes. A class function is not a function in the usual sense, since it is not
Class_(set_theory)
Relationship between certain categories
they form a natural generalization of Stone's representation theorem for Boolean algebras. These concepts are named in honor of Marshall Stone. Stone-type
Stone_duality
of the entire expression under the bar, particularly when dealing with Boolean algebra. For example, one of De Morgan's laws says that A ∧ B ¯ = A ¯
Glossary of mathematical symbols
Glossary_of_mathematical_symbols
Theory of generalized measures in mathematics
cases, a symmetric fuzzy measure will result in the ordered weighted averaging (OWA) operator. Submodular fuzzy measures result in convex functions, while
Fuzzy_measure_theory
Type of cipher
cipher Cusick, Thomas W.; Stanica, Pantelimon (2009). Cryptographic Boolean functions and applications. Academic Press. pp. 158–159. ISBN 9780123748904
Block_cipher
Axioms for the natural numbers
all natural numbers x and y, if x = y, then y = x. That is, equality is symmetric. For all natural numbers x, y and z, if x = y and y = z, then x = z. That
Peano_axioms
Unsolved problem in computer science
in NP can be transformed mechanically into a Boolean satisfiability problem in polynomial time. The Boolean satisfiability problem is one of many NP-complete
P_versus_NP_problem
Class in computational complexity theory
the NC-hierarchy. The smallest class, NC0, is the class of functions definable by Boolean circuits with constant depth and bounded fan-in. The next-smallest
NC_(complexity)
Reversal of the order of elements of a binary relation
a sibling of B {\displaystyle B} " is its own converse, since it is a symmetric relation. In the monoid of binary endorelations on a set (with the binary
Converse_relation
Number that, when added to the original number, yields the additive identity
≡ 0 (mod 11). In a Boolean ring, which has elements { 0 , 1 } {\displaystyle \{0,1\}} addition is often defined as the symmetric difference. So 0 + 0
Additive_inverse
SYMMETRIC BOOLEAN-FUNCTION
SYMMETRIC BOOLEAN-FUNCTION
Boy/Male
Hindu
Symmetry, Harmony
Boy/Male
Tamil
Symmetry, Harmony
Boy/Male
Irish
Puppy.
Surname or Lastname
English
English : topographic name for someone who lived on a curved or irregularly shaped piece of land, from Old English wÅh ‘curved’, ‘crooked’ + land ‘land’, ‘estate’, or a habitational name from Woolland in Dorset, named from an Old English winn, wynn ‘meadow’, ‘pasture’ + land ‘land’, ‘estate’.
Surname or Lastname
Irish
Irish : Anglicized form of Gaelic Ó Baoighealláin. It was the name of a sept of Dartry, County Monaghan.English : variant of Boyland.
Surname or Lastname
English
English : variant of Wool.Americanized form of Jewish Wollman or German Wollmann (see Wollman).
Girl/Female
African, Arabic, Muslim, Swahili
Symmetry
Surname or Lastname
English
English : variant of Bowerman.
Boy/Male
American, British, English
Lives at the Buck Meadow
Surname or Lastname
English
English : variant spelling of Woolen.
Girl/Female
Assamese, Gujarati, Hindu, Indian, Kannada, Telugu, Traditional
Flowering
Surname or Lastname
English
English : possibly a variant of Woolen.
Surname or Lastname
English
English : variant of Bullen.
Girl/Female
Indian
Flowering, Blooming, Flower
Surname or Lastname
English
English : variant of Boland.Irish : Anglicized form of Gaelic Ó Beólláin, ‘descendant of Bjolan’, a Norse personal name.
Boy/Male
English American German
Cuts the nap of woolen cloth. 'Shireman' In medieval times the shireman served as governor-judge...
Surname or Lastname
Czech
Czech : from a pet form of the personal names Boleslav or Bolebor.Polish (Boleń) : from a pet form of the personal name Bolesław.Variant spelling of German Bohlen.Swedish (Bolén) : ornamental name composed of an unexplained first element + the common surname suffix -én, a derivative of Latin -enius ‘descendant of’.English : variant of Bullen.
Boy/Male
Indian, Punjabi, Sikh
God's Spoken Word
Surname or Lastname
English
English : variant of Bullen.
Boy/Male
Sikh
Symmetry, Harmony
SYMMETRIC BOOLEAN-FUNCTION
SYMMETRIC BOOLEAN-FUNCTION
Boy/Male
Hindu
Lord of God, Lord Ram, Ragavender God
Girl/Female
Scandinavian Polish
Archer.
Female
Egyptian
, a daughter of Takelot II.
Boy/Male
Indian, Tamil
King of One Kind of
Male
Swedish
Pet form of Swedish Nils, NISSE means "victor of the people."
Girl/Female
Bengali, Hindu, Indian
Purified Gem
Boy/Male
Arabic, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Muslim, Mythological, Sindhi, Tamil, Telugu
Rain Clouds
Boy/Male
Tamil
Boy/Male
Bengali, Hindu, Indian
Power of Lord Indra
Boy/Male
Basque Latin Greek Spanish
Ardent.
SYMMETRIC BOOLEAN-FUNCTION
SYMMETRIC BOOLEAN-FUNCTION
SYMMETRIC BOOLEAN-FUNCTION
SYMMETRIC BOOLEAN-FUNCTION
SYMMETRIC BOOLEAN-FUNCTION
a.
Having the organs or parts of one side corresponding with those of the other; having the parts in two or more series of organs the same in number; exhibiting a symmetry. See Symmetry, 2.
imp. & p. p.
of Symmetrize
a.
Not symmetrical; being without symmetry, as the parts of a flower when similar parts are of different size and shape, or when the parts of successive circles differ in number. See Symmetry.
a.
Exhibiting pseudo-symmetry.
a.
Not symmetrical; wanting proportion; esp., not bilaterally symmetrical.
v. t.
To make proportional in its parts; to reduce to symmetry.
n.
Same as Symmetrian.
a.
Of or pertaining to wool or woolen cloths; as, woolen manufactures; a woolen mill; a woolen draper.
a.
Alt. of Asymmetrical
a.
Symmetrical.
a.
Well-proportioned; symmetrical.
n.
The law of likeness; similarity of structure; regularity in form and arrangement; orderly and similar distribution of parts, such that an animal may be divided into parts which are structurally symmetrical.
a.
Made of wool; consisting of wool; as, woolen goods.
p. pr. & vb. n.
of Symmetrize
a.
Commensurable; symmetrical.
a.
Symmetrical.
n.
One eminently studious of symmetry of parts.
n.
One eminently studious of symmetry of parts.
n.
A kind of symmetry characteristic of certain crystals which from twinning, or other causes, come to resemble forms of a system other than that to which they belong, as the apparently hexagonal prisms of aragonite.
a.
Involving or exhibiting symmetry; proportional in parts; having its parts in due proportion as to dimensions; as, a symmetrical body or building.