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specifically computability theory, a function f : R → R {\displaystyle f:\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
numbers (or gaussian computable numbers) - whose both real and imaginary components are computable. The computable numbers form a real closed field and can
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
Set with algorithmic membership test
subset S {\displaystyle S} of the natural numbers is computable if there exists a total computable function f {\displaystyle f} such that f ( x ) = { 1 if
Computable_set
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
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
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
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
Concept in theoretical computer science
by Tibor Radó in his 1962 paper, "On Non-Computable Functions". If it were possible to compute the functions S(n) and Σ(n) for all n, then this would
Busy_beaver
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
Real-time_computing
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
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)
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
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
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
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
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
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
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)
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
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
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
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
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)
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
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
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
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
Inverse of the gamma function
^{-1}(24)=5} . Usually, the inverse gamma function refers to the principal branch with domain on the real interval [ β , + ∞ ) {\displaystyle \left[\beta
Inverse_gamma_function
C standard library header file
function that is defined for only real numbers encapsulates a total of 3 different functions: float, double and long double variants of the function.
C_mathematical_functions
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
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
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
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
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
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
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
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)
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
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
Fundamental trigonometric functions
the sine and cosine functions are denoted as sin(θ) and cos(θ). The definitions of sine and cosine have been extended to any real value in terms of the
Sine_and_cosine
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
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
Arctangent function with two arguments
In computing and mathematics, the function atan2 is the 2-argument arctangent. By definition, θ=atan2(y,x) is the angle measure between the positive
Atan2
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
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
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
Integral expressing the amount of overlap of one function as it is shifted over another
'shape' of one function is modified by the other. Some features of convolution are similar to cross-correlation: for real-valued functions, of a continuous
Convolution
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
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
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
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
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
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
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
Mathematical concept
an example, consider the real-valued function of a real variable given by f(x) = 5x − 7. One can think of f as the function which multiplies its input
Inverse_function
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 concept
Arithmetical hierarchy Computable set Computable number Hartley Rogers Jr. (1967). Theory of recursive functions and effective computability. McGraw-Hill. OCLC 527706
Arithmetical_set
Proof by Alan Turing
N of a Turing computing ... machine that ever prints a given symbol (0 say). 1 computable number — a number whose decimal is computable by a machine (i
Turing's_proof
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
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
Mathematical operation in calculus
f′ scaled by the current value of f. When f is a function f(x) of a real variable x, and takes real, strictly positive values, this is equal to the derivative
Logarithmic_derivative
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
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
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
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
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
Approximating an arbitrary function with a well-behaved one
to compute. First, for known target functions approximation theory is the branch of numerical analysis that investigates how certain known functions (for
Function_approximation
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
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
Subset of a function's codomain
used in real analysis (that is, as a function that inputs a real number and outputs its square). In this case, its codomain is the set of real numbers
Range_of_a_function
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)
Mathematical set that can be enumerated
standard model includes all the algebraic numbers and all effectively computable transcendental numbers, as well as many other kinds of numbers. Countable
Countable_set
Logarithm to the base of the mathematical constant e
multi-valued function: see complex logarithm for more. The natural logarithm function, if considered as a real-valued function of a positive real variable
Natural_logarithm
Mathematical transform that expresses a function of time as a function of frequency
Gaussian function is another Gaussian function. Joseph Fourier introduced sine and cosine transforms (which correspond to the imaginary and real components
Fourier_transform
Mapping arbitrary data to fixed-size values
regardless of the number of keys. In most applications, the hash function should be computable with minimum latency and secondarily in a minimum number of
Hash_function
Value approached by a mathematical object
(2014). Recursively enumerable sets and degrees : a study of computable functions and computably generated sets. Berlin: Springer. ISBN 978-3-540-66681-3
Limit_(mathematics)
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
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
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)
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
Function Real-Time Scheduling and Resource Management, Proc. Eighth IEEE International Symposium on Object-Oriented Real-Time Distributed Computing,
Time-utility_function
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)
Branch of mathematics studying functions of a complex variable
holomorphic functions, which are differentiable functions of a complex variable. In contrast with a differentiable real function, a holomorphic function is always
Complex_analysis
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
Counterexample to the converse of the intermediate value theorem
is not continuous. Conway's base 13 function is an example of a simple-to-define function which takes on every real value in every interval, that is, it
Conway's_base_13_function
that there exists a function on the unit square whose iterated integrals are not equal — the function is simply the indicator function of an ordering of
List of statements independent of ZFC
List_of_statements_independent_of_ZFC
Polynomial function of degree at most one
and related areas of mathematics, a linear function from the real numbers to the real numbers is a function whose graph (in Cartesian coordinates) is a
Linear_function_(calculus)
Mathematical function, inverse of an exponential function
as a function from the reals to the positive reals. Let b be a positive real number not equal to 1 and let f(x)=bx. It is a standard result in real analysis
Logarithm
Operation in 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
Types of special mathematical functions
\Gamma (s)\,\gamma ^{*}(s,z),} extends the real lower incomplete gamma function as a holomorphic function, both jointly and separately in z and s. It
Incomplete_gamma_function
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
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
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
Mathematical function characterizing set membership
Dirichlet function is the indicator function of the rational numbers as a subset of the real numbers. Given an arbitrary set X, the indicator function of a
Indicator_function
Distance from zero to a number
|t|=max(t,−t). The real absolute value function has a derivative for every x ≠ 0, given by a step function equal to the sign function except at x = 0 where
Absolute_value
Study of mathematical algorithms for optimization problems
or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization
Mathematical_optimization
Instantaneous rate of change (mathematics)
variables. For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector. A function of a real variable f(x) is
Derivative
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COMPUTABLE REAL-FUNCTION
COMPUTABLE REAL-FUNCTION
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