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Concept in complexity theory
theory, a time-constructible function is a function f from natural numbers to natural numbers with the property that f(n) can be constructed from n by a
Constructible_function
Regular polygon that can be constructed with compass and straightedge
a constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible with
Constructible_polygon
Number constructible via compass and straightedge
coordinate system, a point is constructible if and only if its Cartesian coordinates are both constructible numbers. Constructible numbers and points have also
Constructible_number
Given more time, a Turing machine can solve more problems
notion of a time-constructible function. A function f : N → N {\displaystyle f:\mathbb {N} \rightarrow \mathbb {N} } is time-constructible if there exists
Time_hierarchy_theorem
Possible axiom for set theory in mathematics
{\displaystyle L} represents the constructible sets. In Zermelo–Fraenkel set theory (ZF), the property of being constructible is expressible as a single formula
Axiom_of_constructibility
Topics referred to by the same term
B over A Constructible universe, Kurt Gödel's model L of set theory, constructed by transfinite recursion Constructible function, a function whose values
Constructibility
Particular class of sets which can be described entirely in terms of simpler sets
In set theory, the constructible universe (or Gödel's constructible universe), denoted by L , {\displaystyle L,} is a particular class of sets that can
Constructible_universe
Both deterministic and nondeterministic machines can solve more problems given more space
common functions that we work with are space-constructible, including polynomials, exponents, and logarithms. For every space-constructible function f :
Space_hierarchy_theorem
Function in algebraic geometry
In algebraic geometry, the Behrend function of a scheme X, introduced by Kai Behrend, is a constructible function ν X : X → Z {\displaystyle \nu _{X}:X\to
Behrend_function
Extension of the factorial function
The gamma function then is defined in the complex plane as the analytic continuation of this integral function: it is a meromorphic function which is holomorphic
Gamma_function
Concept in theoretical computer science
Retrieved 7 July 2022. Green recursively constructs machines for any number of states and provides the recursive function that computes their score (computes
Busy_beaver
Memory space for a deterministic Turing machine
assumed. □ The above theorem implies the necessity of the space-constructible function assumption in the space hierarchy theorem. L = DSPACE(O(log n))
DSPACE
Mathematical relation assigning a probability event to a cost
optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one
Loss_function
Association of one output to each input
mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the
Function_(mathematics)
Function-Spacer-Lipid (FSL) Kode constructs (Kode Technology) are amphiphatic, water dispersible biosurface engineering constructs that can be used to
Function-spacer-lipid Kode construct
Function-spacer-lipid_Kode_construct
Quickly growing function
Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function that is not
Ackermann_function
Mapping arbitrary data to fixed-size values
A hash function is any function that can be used to map data of arbitrary size to fixed-size values, though there are some hash functions that support
Hash_function
Mathematical-logic system
as λ-calculus) is a formal system for expressing computation based on function abstraction and application using variable binding and substitution. Untyped
Lambda_calculus
Mathematical description of quantum state
In quantum mechanics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common
Wave_function
complexity functions, then f + g, fg, and 2f are also proper complexity functions. Similar notions include honest functions, space-constructible functions, and
Proper_complexity_function
Functions of an angle
mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of
Trigonometric_functions
Analytic function in mathematics
Riemann zeta function, or Euler–Riemann zeta function, denoted by the lowercase Greek letter ζ {\displaystyle \zeta } (zeta), is a function of a complex
Riemann_zeta_function
Infinite cardinal number
all prime numbers, the set of all rational numbers, the set of all constructible numbers (in the geometric sense), the set of all algebraic numbers,
Aleph_number
Psychological concept
In psychology, a construct, also called a hypothetical construct or psychological construct, is a sophisticated cognitive framework that individuals and
Construct_(psychology)
Complexity class
NTIME is also related to DSPACE in the following way. For any time constructible function t(n), we have N T I M E ( t ( n ) ) ⊆ D S P A C E ( t ( n ) ) {\displaystyle
NTIME
Probability that random variable X is less than or equal to x
cumulative distribution function (CDF) of a real-valued random variable X {\displaystyle X} , or just distribution function of X {\displaystyle X} ,
Cumulative distribution function
Cumulative_distribution_function
Mathematical function having a characteristic S-shaped curve or sigmoid curve
sigmoid function is any mathematical function whose graph has a characteristic S-shaped or sigmoid curve. A common example of a sigmoid function is the
Sigmoid_function
Mathematical set formed from two given sets
as simply ×Xi. If f is a function from X to A and g is a function from Y to B, then their Cartesian product f × g is a function from X × Y to A × B with
Cartesian_product
Axiom of set theory
of choice is not a theorem of ZF by constructing an inner model (the constructible universe) that satisfies ZFC, thus showing that ZFC is consistent if
Axiom_of_choice
Concept in the analysis of dynamical systems
Lyapunov functions for linear systems, and conservation laws can often be used to construct Lyapunov functions for physical systems. A Lyapunov function for
Lyapunov_function
Real number uniquely specified by description
rational number, is constructible. The positive square root of 2 is constructible. However, the cube root of 2 is not constructible; this is related to
Definable_real_number
Generalized function whose value is zero everywhere except at zero
Dirac delta function (or δ {\displaystyle {\boldsymbol {\delta }}} distribution), also known as the unit impulse, is a generalized function on the real
Dirac_delta_function
Type of infinite number in set theory
{\displaystyle \Delta _{0}} -definable subsets of X {\displaystyle X} (see constructible universe). It is worth pointing out that the first claim can be weakened:
Inaccessible_cardinal
Theorem in computability theory
can be applied to construct fixed points of certain operations on computable functions, to generate quines, and to construct functions defined via recursive
Kleene's_recursion_theorem
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
Concept in mathematics
modification of Gödel's constructible hierarchy, L, that circumvents certain technical difficulties that exist in the constructible hierarchy. The J-Hierarchy
Jensen_hierarchy
Coding guidelines by Gerald J. Holzmann
about 60 lines of code per function. The code's assertions density should average to minimally two assertions per function. Assertions must be used to
The Power of 10: Rules for Developing Safety-Critical Code
The_Power_of_10:_Rules_for_Developing_Safety-Critical_Code
Limitative results in mathematical logic
numbering, but which are not strong enough to have multiplication as a function, and so fail to prove the second incompleteness theorem; that is to say
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Iterative optimization method
is an iterative optimization method which exploits the convexity of a function in order to find its maxima or minima. The MM stands for “Majorize-Minimization”
MM_algorithm
Mathematical concept
Recursion Theorem (version 2). Given a set g1, and class functions G2, G3, there exists a unique function F: Ord → V such that F(0) = g1, F(α + 1) = G2(F(α))
Transfinite_induction
Trigonometric values in terms of square roots and fractions
those that can be constructed with a compass and straight edge, and the values are called constructible numbers. The trigonometric functions of angles that
Exact_trigonometric_values
Proposition in mathematical logic
adopted, i.e. from ZFC. His proof shows that both CH and AC hold in the constructible universe L {\displaystyle L} , an inner model of ZF set theory, assuming
Continuum_hypothesis
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
Function computable with bounded loops
In computability theory, a primitive recursive function is, roughly speaking, a function that can be computed by a computer program whose loops are all
Primitive_recursive_function
Subfield of mathematics
set theory (with or without the axiom of choice), by developing the constructible universe of set theory in which the continuum hypothesis must hold.
Mathematical_logic
Relation between deterministic and nondeterministic space complexity
non-deterministic space complexity. It states that for any space-constructable function f ∈ Ω ( log ( n ) ) {\displaystyle f\in \Omega (\log(n))} , N
Savitch's_theorem
Thesis on the nature of computability
Church–Turing thesis is a thesis about the nature of computable functions. It states that a function on the natural numbers can be calculated by an effective
Church–Turing_thesis
topology and integral geometry that integrates constructible functions and more recently definable functions by integrating with respect to the Euler characteristic
Euler_calculus
Mathematical operation with two operands
mathematics, a binary operation or dyadic operation is a kind of binary function, i.e. given a pair of input values (called operands), it produces an output
Binary_operation
formula 7.2) extended the formula to constructible sheaves over a curve (Raynaud 1965). Suppose that F is a constructible sheaf over a genus g smooth projective
Grothendieck–Ogg–Shafarevich formula
Grothendieck–Ogg–Shafarevich_formula
Scale to rate how well one is meeting various problems in living
The Global Assessment of Functioning (GAF) is a numeric scale used by mental health clinicians and physicians to rate subjectively the social, occupational
Global Assessment of Functioning
Global_Assessment_of_Functioning
Software programming optimization technique
memoized function object in a decorator pattern. In pseudocode, this can be expressed as follows: function construct-memoized-functor (F is a function object
Memoization
Mathematical approximation of a function
of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the
Taylor_series
Mathematical function, inverse of an exponential function
to base b, written logb x = y, so log10 1000 = 3. As a single-variable function, the logarithm to base b is the inverse of exponentiation with base b.
Logarithm
Function in mathematical logic
In mathematical logic, a Gödel numbering is a function that assigns to each symbol and well-formed formula of some formal language a unique natural number
Gödel_numbering
Shape containing unit line segments in all directions
approaching this problem is to consider a particular type of maximal function, which we construct as follows: Denote Sn−1 ⊂ Rn to be the unit sphere in n-dimensional
Kakeya_set
Circle with radius of one
Triangles constructed on the unit circle can also be used to illustrate the periodicity of the trigonometric functions. First, construct a radius OP
Unit_circle
Programming construct
computer programming, a function object is a construct allowing an object to be invoked or called as if it were an ordinary function, usually with the same
Function_object
Method of solution to differential equations
In mathematics, a Green's function (or Green function) is the impulse response of an inhomogeneous linear differential operator defined on a domain with
Green's_function
Every set is smaller than its power set
Y {\displaystyle Y} if and only if there is an injective function but no bijective function from X {\displaystyle X} to Y {\displaystyle Y} . It suffices
Cantor's_theorem
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)
Theorem for proving more complex theorems
Often, a theorem is broken into multiple cases (for example, a quadratic function may have no real roots, one double root, or two distinct roots), and each
Lemma_(mathematics)
Smooth and compactly supported function
kernels used to construct mollifiers. Some authors use the term more broadly for any compactly supported smooth function. Such functions are important examples
Bump_function
Probability distribution
real-valued random variable. The general form of its probability density function is f(x)=12πσ2exp(−(x−μ)22σ2). The parameter μ is the mean or expectation
Normal_distribution
Function used in signal processing
processing and statistics, a window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside
Window_function
General-purpose programming language
run-time polymorphism may be achieved using function pointers. Control flow is provided through constructs such as if, for, do, while, and switch. The
C_(programming_language)
pairing function, and π 1 , π 2 {\displaystyle \pi _{1},\pi _{2}} be its projection functions for inversion. Theorem: Any function constructible via the
Gödel's_β_function
Technique for creating lexically scoped first class functions
lexical closure or function closure, is a technique for implementing lexically scoped name binding in a language with first-class functions. Operationally
Closure (computer programming)
Closure_(computer_programming)
The function assigning to α {\displaystyle \alpha } the α {\displaystyle \alpha } th level L α {\displaystyle L_{\alpha }} of Godel's constructible hierarchy
Primitive recursive set function
Primitive_recursive_set_function
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
Set theory concept
obtained the notation from earlier sources such as Whitehead and Russell. Constructible universe Grothendieck universe Inaccessible cardinal S (set theory)
Von_Neumann_universe
Theorem in set theory
if there exist injective functions f : A → B and g : B → A between the sets A and B, then there exists a bijective function h : A → B. In terms of the
Schröder–Bernstein_theorem
Standard system of axiomatic set theory
particular inner models, such as in the constructible universe. However, some statements that are true about constructible sets are not consistent with hypothesized
Zermelo–Fraenkel_set_theory
Control flow construct for executing code repeatedly
< 5; i++) { printf("%d\n", i); } Assuming there is a properly declared function or method called do_work(), the following are equivalent in programming
Loop_(statement)
Function definition that is not bound to an identifier
higher-order functions or used for constructing the result of a higher-order function that needs to return a function. If the function is only used once
Anonymous_function
Natural number
967,295 the largest known odd number of sides of a constructible polygon, but since constructibility is related to factorization, the list of odd numbers
4,294,967,295
Yes-or-no question that cannot ever be solved by a computer
2019, Ben-David and colleagues constructed an example of a learning model (named EMX), and showed a family of functions whose learnability in EMX is undecidable
Undecidable_problem
Kind of mathematical function
In mathematics, and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves
Measurable_function
Size of a set in mathematics
disproved from the axioms of ZFC by showing that both CH and AC hold in his constructible universe: an inner model of ZFC. The existence of a model of ZFC in
Cardinality
Swiss mathematician (1707–1783)
mathematical terminology and notation, including the notion of a mathematical function. He is known for his work in mechanics, fluid dynamics, optics, astronomy
Leonhard_Euler
Sociological theory regarding shared understandings
ontology, and communication theory. The term can serve somewhat different functions in each field; however, the foundation of this theoretical framework suggests
Social_constructionism
Symbolic description of a mathematical object
mathematical notation. Symbols can denote numbers, variables, operations, and functions. Other symbols include punctuation marks and brackets, used for grouping
Expression_(mathematics)
Continuous function that is not absolutely continuous
In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. It is a notorious counterexample in
Cantor_function
Mathematical concept
In mathematics, the inverse function of a function f (also called the inverse of f) is a function that undoes the operation of f. The inverse of f exists
Inverse_function
Syntactically valid part of a program formed from lexical tokens
language constructs, not functions. So while (true) is a language construct, while add(10) is a function call. In PHP print is a language construct. <?php
Language_construct
Number, approximately 3.14
positive number at which the cosine function equals 0. π is also the smallest positive number at which the sine function equals zero, and the difference between
Pi
Type of logical system
discourse (over which the quantified variables range), finitely many functions from that domain to itself, finitely many predicates defined on that domain
First-order_logic
Algebraic manipulation of "true" and "false"
complement function, the dual function and the contradual function (complemented dual). These four functions form a group under function composition
Boolean_algebra
Morphological form of a noun
marking (a, the) like in the construct state). In some non-Semitic languages, the construct state has various additional functions besides marking the head
Construct_state
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
Base of natural logarithms
mathematical constant that is the base of the natural logarithm and exponential function. It is approximately equal to 2.718281828459045235360287471352 e plays
E_(mathematical_constant)
trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both
List of trigonometric identities
List_of_trigonometric_identities
Mathematical use of "for all"
found in the Quantifier article. The negation of a universally quantified function is obtained by changing the universal quantifier into an existential quantifier
Universal_quantification
Fundamental trigonometric functions
In mathematics, sine and cosine are trigonometric functions. The sine and cosine of an acute angle are defined in the context of a right triangle: the
Sine_and_cosine
Theorem in algebraic geometry
cohomology lie not in a field but instead in a constructible sheaf. They prove that for a constructible sheaf F {\displaystyle {\mathcal {F}}} on an affine
Lefschetz_hyperplane_theorem
Linear combination of indicator functions of real intervals
mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of intervals
Step_function
Function in thermodynamics and statistical physics
terms of the partition function or its derivatives. The partition function is dimensionless. Each partition function is constructed to represent a particular
Partition function (statistical mechanics)
Partition_function_(statistical_mechanics)
Measure of algorithmic complexity
more involved. It shows that given a Kolmogorov complexity function, we can construct a function p {\displaystyle p} , such that p ( n ) ≥ B B ( n ) {\displaystyle
Kolmogorov_complexity
Intentionally devised human language
While most auxiliary languages are a posteriori due to their intended function as a medium of communication, many artistic languages are fully a posteriori
Constructed_language
Mathematical theory of data types
from a d d {\displaystyle \mathrm {add} } 's type. New function terms may be constructed using lambda expressions, and are called lambda terms. These
Type_theory
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CONSTRUCTIBLE FUNCTION
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CONSTRUCTIBLE FUNCTION
CONSTRUCTIBLE FUNCTION
CONSTRUCTIBLE FUNCTION
CONSTRUCTIBLE FUNCTION
CONSTRUCTIBLE FUNCTION
CONSTRUCTIBLE FUNCTION
CONSTRUCTIBLE FUNCTION
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