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computability theory, a function f : R → R {\displaystyle f\colon \mathbb {R} \to \mathbb {R} } is sequentially computable if, for every computable sequence
Computable_real_function
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
Real number that can be computed within arbitrary precision
A real number is computable if and only if there is a computable Dedekind cut D corresponding to it. The function D is unique for each computable number
Computable_number
Study of mathematical analysis seen through computability theory
that not every function is computable. Every computable real function is continuous. The arithmetic operations on real numbers are computable. While the equality
Computable_analysis
Study of computable functions and Turing degrees
with the study of computable functions and Turing degrees. The field has since expanded to include the study of generalized computability and definability
Computability_theory
Function computable with bounded loops
exponential function, and the function which returns the nth prime are all primitive recursive. In fact, for showing that a computable function is primitive
Primitive_recursive_function
Models of computation
a Turing machine. Hypercomputers compute functions that a Turing machine cannot and which are, hence, not computable in the Church–Turing sense. Technically
Hypercomputation
Study of hardware and software systems that have a "real-time constraint"
Real-time computing (RTC) is the computer science term for hardware and software systems subject to a "real-time constraint", for example from event to
Real-time_computing
Quickly growing function
total computable function that is not primitive recursive. All primitive recursive functions are total and computable, but the Ackermann function illustrates
Ackermann_function
Thesis on the nature of computability
In computability theory, the Church–Turing thesis is a thesis about the nature of computable functions. It states that a function on the natural numbers
Church–Turing_thesis
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
Ability of a computing system to simulate Turing machines
Turing-equivalent if every function it can compute is also Turing-computable; i.e., it computes precisely the same class of functions as do Turing machines
Turing_completeness
Concept in probability theory and statistics
theory and statistics, the moment generating function of a real-valued random variable is a generating function that provides an alternative specification
Moment_generating_function
Typed functional language
science, Programming Computable Functions (PCF), or Programming with Computable Functions, or Programming language for Computable Functions, is a programming
Programming Computable Functions
Programming_Computable_Functions
Association of one output to each input
same functions. All the other models of practicably computable functions that have ever been proposed define the same set of computable functions or a
Function_(mathematics)
Mathematical logic concept
pairing function) are computably enumerable sets. The preimage of a computably enumerable set under a partial computable function is a computably enumerable
Computably_enumerable_set
Limit of a uniformly computable sequence of functions
computability theory, a function is called limit computable if it is the limit of a uniformly computable sequence of functions. The terms computable in
Computation_in_the_limit
Halting probability of a random computer program
recognize. The domain of any universal computable function is a computably enumerable set but never a computable set. The domain is always Turing equivalent
Chaitin's_constant
Sequence of program instructions invokable by other software
example in Pascal: function E(x: real): real; function F(y: real): real; begin F := x + y end; begin E := F(3) + F(4) end; The function F is nested within
Function (computer programming)
Function_(computer_programming)
Ordered listing of items in collection
arbitrary function with domain ω and only countably many computable functions. A specific example of a set with an enumeration but not a computable enumeration
Enumeration
Mathematics of real numbers and real functions
effective and computable constant that determines how well the linear approximation (or higher-order Taylor polynomial) approximates a function on an interval
Real_analysis
Set with algorithmic membership test
if it is not computable. A subset S {\displaystyle S} of the natural numbers is computable if there exists a total computable function f {\displaystyle
Computable_set
Concept in theoretical computer science
1962 paper, "On Non-Computable Functions". An implication of the busy beaver game is that, if it were possible to compute the functions Σ(n) and S(n) for
Busy_beaver
Point where function's value is zero
mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f {\displaystyle f} , is a member x {\displaystyle
Zero_of_a_function
Computation model defining an abstract machine
ideas leads to the author's definition of a computable function, and to an identification of computability with effective calculability. It is not difficult
Turing_machine
Sigmoid shape special function
applications, the function argument is a real number, in which case the function value is also real. In some older texts, the error function is defined without
Error_function
Mathematical model of analog computers
and E. Hainry. Polynomial differential equations compute all real computable functions on computable compact intervals. Journal of Complexity, 23:317–335
General purpose analog computer
General_purpose_analog_computer
Philosphical view that existence proofs must be constructive
more technically, the computable functions), or even left unspecified. If, for instance, the algorithmic view is taken, then the reals as constructed here
Constructivism (philosophy of mathematics)
Constructivism_(philosophy_of_mathematics)
Extension of the factorial function
{\displaystyle n} . The gamma function can be defined via a convergent improper integral for complex numbers with positive real part: Γ ( z ) = ∫ 0 ∞ t z
Gamma_function
Concept in computability theory
Hainry (Jun 2007). "Polynomial differential equations compute all real computable functions on computable compact intervals". Journal of Complexity. 23 (3):
Real_computation
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
Branch of mathematical logic
where "recursive" means "computable", as in computable function. This name is used because RCA0 corresponds informally to "computable mathematics". In particular
Reverse_mathematics
Infinite cardinal number
the set of all algebraic numbers, the set of all computable numbers, the set of all computable functions, the set of all binary strings of finite length
Aleph_number
Mathematical function such that every output has at least one input
surjective function (also known as surjection, or onto function /ˈɒn.tuː/) is a function f such that, for every element y of the function's codomain, there
Surjective_function
Inverse of the gamma function
{\displaystyle b} are real numbers with b ≧ 0 {\displaystyle b\geqq 0} . To compute the branches of the inverse gamma function one can first compute the Taylor series
Inverse_gamma_function
Computer operating system for applications with critical timing constraints
A real-time operating system (RTOS) is an operating system (OS) for real-time computing applications that processes data and events that have critically
Real-time_operating_system
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
Mathematical function, denoted exp(x) or e^x
In mathematics, the exponential function is the unique real function which maps zero to one and has a derivative everywhere equal to its value. It is denoted
Exponential_function
Axiomatic set theories based on the principles of mathematical constructivism
are computable trees K {\displaystyle K} for which no computable such path through it exists. To prove this, one enumerates the partial computable sequences
Constructive_set_theory
Generalized function whose value is zero everywhere except at zero
delta function (or δ {\displaystyle {\boldsymbol {\delta }}} distribution), also known as the unit impulse, is a generalized function on the real numbers
Dirac_delta_function
Type of mathematical function
elementary function is a function of a single variable (real or complex) that is typically encountered by beginners. The basic elementary functions are polynomial
Elementary_function
Smooth approximation of one-hot arg max
The softmax function, also known as softargmax or normalized exponential function, converts a tuple of K real numbers into a probability distribution
Softmax_function
Sequence of rational numbers
sequences has consequences for computable analysis. The fact that such sequences exist means that the collection of all computable real numbers does not satisfy
Specker_sequence
Fourier transform of the probability density function
In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution. If
Characteristic function (probability theory)
Characteristic_function_(probability_theory)
Parameterized complexity Process calculi Pi-calculus Hypercomputation Real computation Computable analysis Weihrauch reducibility List of algorithm general topics
List of computability and complexity topics
List_of_computability_and_complexity_topics
Numbers expressible as integrals of algebraic functions
possible to construct artificial examples of computable numbers which are not periods. However there are no computable numbers proven not to be periods, which
Period_(number_theory)
Model of thermodynamic properties
internal energy. Departure functions are used to calculate real fluid extensive properties (i.e. properties which are computed as a difference between two
Departure_function
Measure of algorithmic complexity
2^{*}} be a computable function mapping finite binary strings to binary strings. It is a universal function if, and only if, for any computable f : 2 ∗ →
Kolmogorov_complexity
Mathematical concept
Arithmetical hierarchy Computable set Computable number Hartley Rogers Jr. (1967). Theory of recursive functions and effective computability. McGraw-Hill. OCLC 527706
Arithmetical_set
Type of mathematical function
function is a real-valued function of a real variable, whose graph is composed of straight-line segments. A piecewise linear function is a function defined
Piecewise_linear_function
Ability to solve a problem by an effective procedure
Turing-computable and μ-recursive functions, and the lambda calculus, all of which have computationally equivalent power. Other forms of computability are
Computability
Whether a decision problem has an effective method to derive the answer
can be given either in terms of effective methods or in terms of computable functions. These are generally considered equivalent per Church's thesis. Indeed
Decidability_(logic)
Proof by Alan Turing
to practical computation... (Hodges p. 124) 1 computable number — a number whose decimal is computable by a machine (i.e., by finite means such as an
Turing's_proof
Yes/no problem in computer science
into the function problem of computing the characteristic function of the set associated to the decision problem. If this function is computable then the
Decision_problem
Family of solutions to related differential equations
(1998), "ZEBEC: A mathematical software package for computing simple zeros of Bessel functions of real order and complex argument", Computer Physics Communications
Bessel_function
Analytic function in mathematics
applied statistics. Leonhard Euler first introduced and studied the function over the reals in the first half of the eighteenth century. Bernhard Riemann's
Riemann_zeta_function
Function returning one of only two values
switching function, used especially in older computer science literature, and truth function (or logical function), used in logic. Boolean functions are the
Boolean_function
Fundamental trigonometric functions
Zero is the only real fixed point of the sine function; in other words the only intersection of the sine function and the identity function is sin ( 0 )
Sine_and_cosine
Number with a real and an imaginary part
complex numbers are often used to compute certain real-valued improper integrals, by means of complex-valued functions. Several methods exist to do this;
Complex_number
Subfield of mathematics
also called computability theory, studies the properties of computable functions and the Turing degrees, which divide the uncomputable functions into sets
Mathematical_logic
Symbol representing a mathematical object
arguments of the functions. This is typically the case in sentences like "function of a real variable", "x is the variable of the function f : x ↦ f(x)"
Variable_(mathematics)
Function with a repeating pattern
functions. Functions that map real numbers to real numbers can display periodicity, which is often visualized on a graph. An example is the function f
Periodic_function
Number representing a continuous quantity
but an uncountable number of reals, almost all real numbers fail to be computable. Moreover, the equality of two computable numbers is an undecidable problem
Real_number
Impossible task in computing
intuitive notion of "effectively calculable" is captured by the functions computable by a Turing machine (or equivalently, by those expressible in the
Entscheidungsproblem
Mathematical function
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: ψ ( z ) = d d z ln Γ ( z ) = Γ ′ ( z ) Γ ( z )
Digamma_function
Axiom of set theory
and a function whose domain is strictly smaller than its range. In fact, this is the case in all known models. There is a function f from the real numbers
Axiom_of_choice
Used to count, measure, and label
testing the equality of two computable numbers. More precisely, there cannot exist any algorithm which takes any computable number as an input, and decides
Number
Type of mathematical relation
the real and imaginary parts of any complex function that is analytic in the upper half-plane. The relations are often used to compute the real part
Kramers–Kronig_relations
Piecewise function: is defined by different expressions on different intervals. Computable function: an algorithm can do the job of the function. Also semicomputable
List_of_types_of_functions
Real number uniquely specified by description
arithmetical number is computable. For example, the limit of a Specker sequence is an arithmetical number that is not computable. The definitions of arithmetical
Definable_real_number
Operation in mathematical calculus
integral computes the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line. Conventionally
Integral
3-volume treatise on mathematics, 1910–1913
edition's distinction between real and apparent variables, and it eliminates "the primitive idea 'assertion of a propositional function'. To add to the complexity
Principia_Mathematica
Set of all things that may be the input of a mathematical function
definition of a function rather than a property of it. In the special case that X and Y are both sets of real numbers, the function f can be graphed
Domain_of_a_function
Subset of a function's codomain
a function may refer either to the codomain of the function, or the image of the function. In some cases the codomain and the image of a function are
Range_of_a_function
Concept of complex analysis
tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well. It
Residue_theorem
Algorithms for zeros of functions
continuous functions. A zero of a function f is a number x such that f(x) = 0. As, generally, the zeros of a function cannot be computed exactly nor
Root-finding_algorithm
Conjecture on zeros of the zeta function
problem in mathematics Do all non-trivial zeros of the Riemann zeta function have a real part equal to one half? More unsolved problems in mathematics In
Riemann_hypothesis
Multivalued function in mathematics
for computing the real parts of the principal and secondary branches of the W function. The first does not need to evaluate any trancendental function and
Lambert_W_function
Logical connective AND
concept of vacuous truth, when conjunction is defined as an operator or function of arbitrary arity, the empty conjunction (AND-ing over an empty set of
Logical_conjunction
thesis Computable function Algorithm Recursion Primitive recursive function Mu operator Ackermann function Turing machine Halting problem Computability theory
List of mathematical logic topics
List_of_mathematical_logic_topics
Mathematical function, inverse of an exponential function
a function from the reals to the positive reals. Let b be a positive real number not equal to 1 and let f(x) = b x. It is a standard result in real analysis
Logarithm
Symbolic description of a mathematical object
powerful definition of 'well-defined' that is able to capture both computable and 'non-computable' statements. All statements characterised in modern programming
Expression_(mathematics)
Descriptive set theory concept
computable unions of them. That is, a set is lightface Σ 1 0 {\displaystyle \Sigma _{1}^{0}} , also called effectively open, if there is a computable
Pointclass
Mathematical set containing no elements
when considered as a subset of the real numbers, with its usual ordering, represented by the real number line, every real number is both an upper and lower
Empty_set
Complexity class consisting of all recursive languages
total computable functions in the sense that: a decision problem is in R if and only if its indicator function is computable, a total function is computable
R_(complexity)
Mathematical theory of data types
to compute the value. The axiom of choice is less powerful in type theory than most set theories, because type theory's functions must be computable and
Type_theory
Proposition in mathematical logic
between that of the integers and the real numbers. The name of the hypothesis comes from the term continuum for the real numbers. In Zermelo–Fraenkel set
Continuum_hypothesis
S-shaped curve
x_{0}} is the x {\displaystyle x} value of the function's midpoint. The logistic function has domain the real numbers, the limit as x → − ∞ {\displaystyle
Logistic_function
Number of arguments required by a function
(parenthesis) of the registers BX and CX. The arithmetic mean of n real numbers is an n-ary function: x ¯ = 1 n ( ∑ i = 1 n x i ) = x 1 + x 2 + ⋯ + x n n {\displaystyle
Arity
Form of mathematical proof
natural number. The successor function s of every natural number yields a natural number (s(x) = x + 1). The successor function is injective. 0 is not in
Mathematical_induction
Smallest fixed point of a function from a poset
fixed point is effectively computable, the optimal fixed point of a computable function may be a non-computable function. Knaster–Tarski theorem Fixed-point
Least_fixed_point
Mathematical function
the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial
Beta_function
Components of a mathematical or logical formula
and the number of i-ary function symbols as fi, the number θh of distinct ground terms of a height up to h can be computed by the following recursion
Term_(logic)
All numbers between two given numbers
continuous function is an interval; integrals of real functions are defined over an interval; etc. For example, interval arithmetic consists of computing with
Interval_(mathematics)
Basic framework of mathematics
defined either, but people were more accustomed to them). Real numbers, continuous functions, derivatives were not formally defined before the 19th century
Foundations_of_mathematics
programming language concepts such as function types. It turns out that restricting expression to the set of computable functions is not sufficient either if the
Function_type
Target set of a mathematical function
mathematics, a codomain or set of destination of a function is a set into which all of the outputs of the function are constrained to fall. It is the set Y in
Codomain
Size of a possibly infinite set
A . {\displaystyle \#A.} Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, there is a one-to-one
Cardinal_number
Mathematical-logic system based on functions
usual for such a proof, computable means computable by any model of computation that is Turing complete. In fact computability can itself be defined via
Lambda_calculus
Type of Turing machine
Turing machine capable of computing any computable sequence, as described by Alan Turing in his seminal paper "On Computable Numbers, with an Application
Universal_Turing_machine
COMPUTABLE REAL-FUNCTION
COMPUTABLE REAL-FUNCTION
Girl/Female
English
The bird teal; also the blue-green color.
Female
English
English name derived from the vocabulary word, TEAL means "blue-green" or "teal duck."
Male
English
Variant spelling of English Neil, NEAL means "champion."
Boy/Male
Hindu
Real
Girl/Female
Indian
Real
Surname or Lastname
English
English : nickname for a person with red hair or a ruddy complexion, from Middle English re(a)d ‘red’.English : topographic name for someone who lived in a clearing, from an unattested Old English rīed, r̄d ‘woodland clearing’.English : Read in Lancashire, the name of which is a contracted form of Old English rǣghēafod, from rǣge ‘female roe deer’, ‘she-goat’ + hēafod ‘head(land)’; Rede in Suffolk, so called from Old English hrēod ‘reeds’; or Reed in Hertfordshire, so called from an Old English ryhð ‘brushwood’.English : A family called Read were established in America in the early 18th century by John Read, who was born in Dublin, sixth in descent from Sir Thomas Read of Berkshire, England. His son, George Read (1733–98), was one of the signers of the Declaration of Independence, and as a lawyer helped frame the Constitution.
Girl/Female
Tamil
Existence, Real
Surname or Lastname
English
English : variant of Dale (from the Old Kentish form del) or a habitational name from Deal in Kent, named with this word.Americanized spelling of German Diel or Diehl.Dutch (de Ruyter) : variant spelling (17th century) of De Ruiter
Boy/Male
Muslim
Similar. Comparable.
Boy/Male
Tamil
Real
Boy/Male
Muslim
Similar. Comparable.
Female
Greek
Variant spelling of Greek Rhea, REAH means "ease, flow."
Boy/Male
Tamil
Real
Boy/Male
Hindu
Real
Surname or Lastname
English, Spanish, and Portuguese
English, Spanish, and Portuguese : nickname for a loyal or trustworthy person, from Old French leial, Spanish and Portuguese leal ‘loyal’, ‘faithful (to obligations)’, Latin legalis, from lex, ‘law’, ‘obligation’ (genitive legis).
Male
English
English surname transferred to forename use, derived from an Old English byname, Red, READ means "red-headed or ruddy-complexioned."Â
Boy/Male
Tamil
Existence, Real
Girl/Female
Tamil
Real
Girl/Female
Gujarati, Hindu, Indian, Kannada, Muslim
Real
Girl/Female
Tamil
Existence, Real
COMPUTABLE REAL-FUNCTION
COMPUTABLE REAL-FUNCTION
Girl/Female
Gujarati, Hindu, Indian, Jain, Kannada, Malayalam, Marathi, Sindhi, Telugu
Pretty Face
Girl/Female
Arabic, Muslim
Sword
Boy/Male
Muslim
Sender of truth, Student
Girl/Female
Indian
Trust; Faith
Girl/Female
English American Greek
Melody.
Girl/Female
Australian, Finnish
Snow
Girl/Female
Christian & English(British/American/Australian)
Small
Boy/Male
Hindu, Indian, Marathi
Sky
Girl/Female
American, Australian, Christian, French, Greek, Hebrew
Weary; Tired; Delicate; A Combination of Leah and Beatrice; Voyager through Life
Male
Egyptian
, a surname of king Rameses III.
COMPUTABLE REAL-FUNCTION
COMPUTABLE REAL-FUNCTION
COMPUTABLE REAL-FUNCTION
COMPUTABLE REAL-FUNCTION
COMPUTABLE REAL-FUNCTION
a.
Comparable.
a.
Such as can be, or is liable to be, combated; as, combatable foes, evils, or arguments.
a.
Pertaining to things fixed, permanent, or immovable, as to lands and tenements; as, real property, in distinction from personal or movable property.
a.
Actually being or existing; not fictitious or imaginary; as, a description of real life.
v. t.
To close by means of a seal; as, to seal a drainpipe with water. See 2d Seal, 5.
a.
True; genuine; not artificial, counterfeit, or factitious; often opposed to ostensible; as, the real reason; real Madeira wine; real ginger.
v. t.
To breed and raise; as, to rear cattle.
v. t.
To go over, as characters or words, and utter aloud, or recite to one's self inaudibly; to take in the sense of, as of language, by interpreting the characters with which it is expressed; to peruse; as, to read a discourse; to read the letters of an alphabet; to read figures; to read the notes of music, or to read music; to read a book.
v. i.
To affix one's seal, or a seal.
n.
See Rial, an old English coin.
v. t.
To set or affix a seal to; hence, to authenticate; to confirm; to ratify; to establish; as, to seal a deed.
v. t.
To place in the rear; to secure the rear of.
a.
Not computable.
imp. & p. p.
of Read
a.
Royal; regal; kingly.
n.
A Spanish coin. See Real.
a.
Compatible; suitable; consistent.
a.
Not confutable.
v. t.
To sprinkle with, or as with, meal.