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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
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 {\displaystyle
Computable_set
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
Real number that can be computed within arbitrary precision
the recursive numbers, effective numbers, computable reals, or recursive reals. The concept of a computable real number was introduced by Émile Borel
Computable_number
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
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
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
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
One of several equivalent definitions of a computable function
recursive function, partial recursive function, or μ-recursive function is a partial function from natural numbers to natural numbers that is "computable" in
General_recursive_function
Type of computational algorithm
important property of logspace computability is that, if functions f , g {\displaystyle f,g} are logspace computable, then so is their composition g
Log-space_reduction
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
1970s automated theorem prover
Logic for Computable Functions (LCF) is an interactive automated theorem prover developed at Stanford and Edinburgh by Robin Milner and collaborators in
Logic for Computable Functions
Logic_for_Computable_Functions
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
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
Turing machine that halts for any input
partial function computable by a partial Turing machine be extended (that is, have its domain enlarged) to become a total computable function? Is it possible
Decider_(Turing_machine)
Theorem in computability theory
In computability theory, Kleene's recursion theorems are a pair of fundamental results about the application of computable functions to their own descriptions
Kleene's_recursion_theorem
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
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
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
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)
Topics referred to by the same term
function may refer to: Recursive function (programming), a function which references itself General recursive function, a computable partial function
Recursive_function
Affirms the existence of a computable universal function
numbering of the computable functions in terms of the smn theorem and the UTM theorem. The theorem states that a partial computable function u of two variables
UTM_theorem
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
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
Concept in computability theory
of all partial computable functions. Such enumerations are formally called computable numberings of the partial computable functions. An arbitrary numbering
Admissible_numbering
Ordinal-indexed family of rapidly increasing functions
a total function. If the fundamental sequences are computable (e.g., as in the Wainer hierarchy), then every fα is a total computable function. In the
Fast-growing_hierarchy
Classes of partial recursive functions
numbering of partial computable functions. Let φ e {\displaystyle \varphi _{e}} be a computable enumeration of all partial computable functions, and W e {\displaystyle
Index_set_(computability)
Mathematical-logic system
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
There are arbitrarily large computable gaps in the hierarchy of complexity classes
computable function that represents an increase in computational resources, one can find a resource bound such that the set of functions computable within
Gap_theorem
Generalization of Rice's theorem
total computable functions such that the index set of P {\displaystyle P} is decidable with a promise that the input is the index of a total computable function
Rice–Shapiro_theorem
Rules out assigning to arbitrary functions their computational complexity
parameters, then there exists a total computable predicate g {\displaystyle g} (a boolean valued computable function) so that for every program i {\displaystyle
Blum's_speedup_theorem
Number of arguments required by a function
science, arity (/ˈærɪti/ ) is the number of arguments or operands taken by a function, operation or relation. In mathematics, arity may also be called rank,
Arity
Programming language
Therefore, the set of functions computable by LOOP-programs is a proper subset of computable functions (and thus a subset of the computable by WHILE and GOTO
LOOP_(programming_language)
Axiom
total functions are computable functions. The similarly named Church–Turing thesis states that every effectively calculable function is a computable function
Church's thesis (constructive mathematics)
Church's_thesis_(constructive_mathematics)
Deductive system for computable functions by Dana Scott
Logic of Computable Functions (LCF) is a deductive system for computable functions proposed by Dana Scott in 1969 in a memorandum unpublished until 1993
Logic_of_Computable_Functions
Concept in computability theory
numbers to computable functions (given as Turing machines). This numbering must be sufficiently effective that, given an index of a computable function and an
Kleene's_T_predicate
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
Collection of efficiently-computable functions which emulate a random oracle
In cryptography, a pseudorandom function family, abbreviated PRF, is a collection of efficiently-computable functions which emulate a random oracle in
Pseudorandom_function_family
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
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
Statement in mathematical logic
the stronger assumption that the theory can represent all (total) computable functions, but all the theories mentioned have that capacity, as well. The
Diagonal_lemma
Axioms in computational complexity theory
arbitrarily difficult ways of computing any function: for any total computable f {\displaystyle f} , and any partial computable ϕ {\displaystyle \phi } ,
Blum_axioms
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
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
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
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 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
approximated either from above or from below by a computable function. More precisely a partial function f : Q → R {\displaystyle f:\mathbb {Q} \rightarrow
Semicomputable_function
Limitative results in mathematical logic
answer. Such a problem is said to be undecidable if there is no computable function that correctly answers every question in the problem set (see undecidable
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Theorem about natural numbers
hierarchies.) Goodstein's theorem can be used to construct a total computable function that Peano arithmetic cannot prove to be total. The Goodstein sequence
Goodstein's_theorem
Sequence of program instructions invokable by other software
In computer programming, a function (also procedure, method, subroutine, routine, or subprogram) is a callable unit of software logic that has a well-formed
Function (computer programming)
Function_(computer_programming)
Mathematical methods
intuitionist analysis of computable or computably enumerable elements of data structures that are not necessarily computable, such as computable operations on all
Realizability
Concept in complexity theory
{\displaystyle \ln n=o(n)} . For every computable function f {\displaystyle f} , there is a computable function g {\displaystyle g} that is time constructible
Constructible_function
Yes-or-no question that cannot ever be solved by a computer
answer. Such a problem is said to be undecidable if there is no computable function that correctly answers every question in the problem set. The connection
Undecidable_problem
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
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
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
O {\displaystyle {\mathcal {O}}} are exactly the computable ordinals. (The fact that every computable ordinal has a notation follows from the closure of
Kleene's_O
a computable function that is not primitive recursive. Dirac delta function: everywhere zero except for x = 0; total integral is 1. Not a function but
List of mathematical functions
List_of_mathematical_functions
Topics referred to by the same term
In computer science, a universal function is a computable function capable of calculating any other computable function. It is shown to exist by the UTM
Universal_function
Problem-solving procedures with certain characteristics
effectively calculable is recursively computable. Decidability (logic) Decision problem Effective results in number theory Function problem Model of computation
Effective_method
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
Theorem in computability theory
natural number b ∉ P {\displaystyle b\notin P} . Define the total computable function Q {\displaystyle Q} of e {\displaystyle e} and x {\displaystyle x}
Rice's_theorem
Concept in computability theory
complement. Every computable set is Turing reducible to every other set. Because any computable set can be computed with no oracle, it can be computed by an oracle
Turing_reduction
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
Input to a mathematical function
of a function is a value provided to obtain the function's result. It is also called an independent variable. For example, the binary function f ( x
Argument_of_a_function
computable function was primitive recursive. This was refuted by Gabriel Sudan and Wilhelm Ackermann — both his students — using different functions that
Sudan_function
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
Value indicating the relation of a proposition to truth
Boolean domain. Corresponding semantics of logical connectives are truth functions, whose values are expressed in the form of truth tables. Logical biconditional
Truth_value
In computability theory, the assignment of natural numbers to a set of objects
transfer the idea of computability and related concepts, which are originally defined on the natural numbers using computable functions, to these different
Numbering (computability theory)
Numbering_(computability_theory)
Axiomatic logical system
theorem does not apply to Q, and it has computable non-standard models. For instance, there is a computable model of Q consisting of integer-coefficient
Robinson_arithmetic
Type of Turing reduction
problem (whether an instance is in L 2 {\displaystyle L_{2}} ) using a computable function. The reduced instance is in the language L 2 {\displaystyle L_{2}}
Many-one_reduction
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)
linear flow constraints but the invariants and guards can be any computable function. This computational model was proposed by Manindra Agrawal and P
Lazy_linear_hybrid_automaton
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
Process of repeating items in a self-similar way
where a function being defined is applied within its own definition. While this apparently defines an infinite number of instances (function values),
Recursion
Cloud computing model
Backend as a service Cloud computing Function as a service "ISO/IEC 22123-2:2023 (E) - Information technology — Cloud computing — Part 2: Concepts". International
Serverless_computing
Topics referred to by the same term
(magazine), a Filipino fashion and lifestyle magazine Preview (computing), a computing function to display something before it is produced in final form Preview
Preview
Attempts to formalize the concept of algorithms
notions of algorithm and computable function are intimately related: by definition, a computable function is a function computable by an algorithm. . .
Algorithm_characterizations
Subset of a function's codomain
the range of a function may refer to either of two closely related concepts: the codomain of the function, or the image of the function. In some cases
Range_of_a_function
Mathematical theory
probability to any computable theory. Solomonoff proved that this induction is incomputable (or more precisely, lower semi-computable), but noted that "this
Solomonoff's theory of inductive inference
Solomonoff's_theory_of_inductive_inference
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)
in the degree computes a completion of PA.) Because there are no computable completions of PA, the degree 0 consisting of the computable sets of natural
PA_degree
Theorem that arithmetical truth cannot be defined in arithmetic
being a formula, being a sentence, etc.), these sets are computable. Moreover, any computable set of numbers can be defined by some arithmetical formula
Tarski's undefinability theorem
Tarski's_undefinability_theorem
Countable ordinal that is the order type of a computable well-ordering of natural numbers
specifically computability and set theory, a computable (or recursive) ordinal is an ordinal number that can be represented as a computable well-ordering
Computable_ordinal
Relationship in which one statement follows from another
Basic Papers on Undecidable Propositions, Unsolvable Problems And Computable Functions, New York: Raven Press, ISBN 9780486432281. Papers include those
Logical_consequence
Method of comparing problems by transforming one into another in computability theory
is linear reducible to B {\displaystyle B} if and only if a computable function computes for each x {\displaystyle x} a finite set F ( x ) {\displaystyle
Reduction (computability theory)
Reduction_(computability_theory)
Paradox in set theory
the function F(fx) could be its own argument: in that case there would be a proposition F(F(fx)), in which the outer function F and the inner function F
Russell's_paradox
Concept in computer science
large amounts of "garbage" history. RTMs compute precisely the set of injective (one-to-one) computable functions. They are not strictly universal in the
Reversible_computing
Determination of whether a given program halts for each input
difficult than the halting problem. Now as the question whether a computable function is total is not semi-decidable, each sound termination analyzer (i
Termination_analysis
Computational theorem
constant factor. Blum's speedup theorem, which provides speedup by any computable function (not just linear, as in the previous theorem). Amdahl's law, the
Speedup_theorem
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
Symbol representing a mathematical object
primarily for the argument of a function, in which case its value could be thought of as varying within the domain of the function. This is the motivation for
Variable_(mathematics)
Algebraic manipulation of "true" and "false"
Parkes, Alan (2002). Introduction to languages, machines and logic: computable languages, abstract machines and formal logic. Springer. p. 276. ISBN 978-1-85233-464-2
Boolean_algebra
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
Collection of mathematical objects
symbols, points in space, lines, other geometric shapes, variables, functions, or even other sets. Sets cannot be mathematically defined, since they
Set_(mathematics)
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
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
Theorem in mathematical logic
non-primitive recursive functions such as the Ackermann function. It dominates every computable function provably total (see partial function) in Peano arithmetic
Paris–Harrington_theorem
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