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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
Concept in theoretical computer science
"On Non-Computable Functions". One of the most consequential aspects of the busy beaver game is that, if it were possible to compute the functions Σ(n) and
Busy_beaver
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
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
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
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
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
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
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
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)
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)
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
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
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
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
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
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
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
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
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
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
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
Proof by Alan Turing
proof by Alan Turing, first published in November 1936 with the title "On Computable Numbers, with an Application to the Entscheidungsproblem". It was the
Turing's_proof
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
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
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
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
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
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
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
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
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)
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
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
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
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
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)
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
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
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
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
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
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)
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
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
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
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)
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
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
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
Method of comparing problems by transforming one into another in computability theory
A} is one-one reducible to B {\displaystyle B} if there is a computable one-to-one function f {\displaystyle f} with A ( x ) = B ( f ( x ) ) {\displaystyle
Reduction (computability theory)
Reduction_(computability_theory)
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
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
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
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
Relationship where 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
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
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
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
Branch of computational complexity theory
{\displaystyle f(k)\cdot {|x|}^{O(1)}} , where f is a computable function. Typically, this function is thought of as single exponential, such as 2 O ( k
Parameterized_complexity
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
Value indicating the relation of a proposition to truth
has only two possible values (true or false). Truth values are used in computing as well as various types of logic. In some programming languages, any
Truth_value
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
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
Mathematical operation with two operands
arity two. More specifically, a binary operation on a set is a binary function that maps every pair of elements of the set to an element of the set. Examples
Binary_operation
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
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
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
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)
Mathematical use of "there exists"
union of sets. A quantified propositional function is a statement; thus, like statements, quantified functions can be negated. The ¬ {\displaystyle \lnot
Existential_quantification
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
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
Collection of mathematical objects
symbols, points in space, lines, other geometric shapes, variables, functions, or even other sets. Mathematics typically does not define precisely what
Set_(mathematics)
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
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
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
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
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
Academic subfield of computer science
Walter A. Carnielli (2000). Computability: Computable Functions, Logic, and the Foundations of Mathematics, with Computability: A Timeline (2nd ed.). Wadsworth/Thomson
Theory_of_computation
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
Esoteric, minimalist programming language
Turing-complete language, Brainfuck is theoretically capable of computing any computable function or simulating any other computational model if given access
Brainfuck
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)
Problem-solving procedures with certain characteristics
number-theoretic function that is effectively calculable is recursively computable. As this is not a mathematical statement, it cannot be proven by a mathematical
Effective_method
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
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)
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 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
Characteristic of some logical systems
is a consistent theory. Gödel's incompleteness theorem shows that any computable system that is sufficiently powerful, such as Peano arithmetic, cannot
Completeness_(logic)
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)
Size of a possibly infinite set
strictly more real numbers than natural numbers. The cardinality function is a cardinal function that takes in a set A {\displaystyle A} and returns its cardinal
Cardinal_number
Branch of mathematical logic
coincide with a natural class of functions, such as the primitive recursive or polynomial-time computable functions. Functional interpretations have also
Proof_theory
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
computable function was primitive recursive. This was refuted by Gabriel Sudan and Wilhelm Ackermann — both his students — using different functions that
Sudan_function
Axioms for the natural numbers
model of PA in which either the addition or multiplication operation is computable. This result shows it is difficult to be completely explicit in describing
Peano_axioms
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
Branch of mathematics that studies sets
theory, introduces into mathematics methods and objects that are not computable even in principle. The feasibility of constructivism as a substitute foundation
Set_theory
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
COMPUTABLE FUNCTION
COMPUTABLE FUNCTION
Male
Celtic
, great justiciary, or functionary.
Surname or Lastname
English
English : topographic name for someone who lived by the gates of a medieval walled town. The Middle English singular gate is from the Old English plural, gatu, of geat ‘gate’ (see Yates). Since medieval gates were normally arranged in pairs, fastened in the center, the Old English plural came to function as a singular, and a new Middle English plural ending in -s was formed. In some cases the name may refer specifically to the Sussex place Eastergate (i.e. ‘eastern gate’), known also as Gates in the 13th and 14th centuries, when surnames were being acquired.Americanized spelling of German Götz (see Goetz).Translated form of French Barrière (see Barriere).In New England, Gates was the preferred English version of the name of an extensive French family, called Barrière dit Langevin.
Male
Egyptian
, an Egyptian functionary.
Biblical
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Girl/Female
Indian
Sacred, Pure, Comparable to the ganges, Another name for Durga, ***, Another name for Durga
Boy/Male
Afghan, Arabic, Celebrity, German, Indian, Muslim, Sindhi
Observer; Supervisor; Little; Insignificant; Warner; Similar; Comparable; Another Name for the Quran; One who Preaches
Male
Egyptian
, Functionary of the Interior.
Surname or Lastname
English (chiefly Kent and Sussex)
English (chiefly Kent and Sussex) : occupational name for a designer or engineer, from a Middle English reduced form of Old French engineor ‘contriver’ (a derivative of engaigne ‘cunning’, ‘ingenuity’, ‘stratagem’, ‘device’). Engineers in the Middle Ages were primarily designers and builders of military machines, although in peacetime they might turn their hands to architecture and other more pacific functions.German : from the Latin personal name Januarius (see January 1). Jänner is a South German word for ‘January’, and so it is possible that this is one of the surnames acquired from words denoting months of the year, for example by converts who had been baptized in that month, people who were born or baptized in that month, or people whose taxes were due in January.
Boy/Male
Buddhist, Indian, Japanese
Mysterious Function
Boy/Male
Muslim
Similar. Comparable.
Surname or Lastname
English
English : occupational name for a dresser of cloth, Old English fullere (from Latin fullo, with the addition of the English agent suffix). The Middle English successor of this word had also been reinforced by Old French fouleor, foleur, of similar origin. The work of the fuller was to scour and thicken the raw cloth by beating and trampling it in water. This surname is found mostly in southeast England and East Anglia. See also Tucker and Walker.In a few cases the name may be of German origin with the same form and meaning as 1 (from Latin fullare).Americanized version of French Fournier.Samuel Fuller (1589–1633), born in Redenhall, Norfolk, England, was among the Pilgrim Fathers who sailed on the Mayflower in 1620. He was a deacon of the church and until his death functioned as Plymouth Colony’s physician.
Male
Egyptian
, an Egyptian functionary.
Girl/Female
Tamil
Sacred, Pure, Comparable to the ganges, Another name for Durga, ***, Another name for Durga
Male
Egyptian
, a high Egyptian functionary.
Boy/Male
Arabic, Australian, Muslim
Similar; Comparable; One who Warns
Boy/Male
Muslim
Similar. Comparable.
Girl/Female
Tamil
Sacred, Pure, Comparable to the ganges, Another name for Durga, ***, Another name for Durga
Girl/Female
Indian
Sacred, Pure, Comparable to the ganges, Another name for Durga, ***, Another name for Durga
Surname or Lastname
English
English : nickname from the animal, Middle English catte ‘cat’. The word is found in similar forms in most European languages from very early times (e.g. Gaelic cath, Slavic kotu). Domestic cats were unknown in Europe in classical times, when weasels fulfilled many of their functions, for example in hunting rodents. They seem to have come from Egypt, where they were regarded as sacred animals.English : from a medieval female personal name, a short form of Catherine.Variant spelling of German and Dutch Katt.
Male
Egyptian
, a great functionary.
COMPUTABLE FUNCTION
COMPUTABLE FUNCTION
Boy/Male
Hindu, Indian
Desired; Goddess
Male
Italian
Italian form of Roman Latin Quirinus, QUIRINO means "men together."
Girl/Female
Arabic, Bengali, Gujarati, Hindu, Indian, Kannada, Marathi, Muslim, Tamil, Telugu
Brilliant; Brilliant Origin Islamic
Girl/Female
Danish, German, Greek, Swedish
Pure
Boy/Male
Arabic
Servant of the Comforter
Girl/Female
Hindu
Name of a Raga
Boy/Male
American, Australian, Chinese, French, German, Latin, Portuguese, Spanish
Famous Ruler; Similar to Roderick Famous Ruler
Girl/Female
Hindu
Soft or tender
Boy/Male
Hindu
Wealthy, Happy
Boy/Male
Hindu
Wealth
COMPUTABLE FUNCTION
COMPUTABLE FUNCTION
COMPUTABLE FUNCTION
COMPUTABLE FUNCTION
COMPUTABLE FUNCTION
adv.
In a compatible manner.
a.
Suitable; consistent.
a.
Compatible; suitable; consistent.
a.
Capable of being attributed; ascribable; imputable.
n.
The quality of being commutable; interchangeableness.
a.
Such as can be, or is liable to be, combated; as, combatable foes, evils, or arguments.
a.
Not computable.
n.
The quality of being imputable; imputableness.
a.
That may be confuted.
a.
Comparable.
a.
Capable of being computed, numbered, or reckoned.
a.
Correspondent; conformable; hence, comparable.
n.
Quality of being imputable.
a.
Capable of being commuted or interchanged.
a.
Not compliable; not conformable.
a.
Capable of existing in harmony; congruous; suitable; not repugnant; -- usually followed by with.
a.
Not commutable; not capable of being exchanged with, or substituted for, another.
n.
The quality of being commutable.
a.
Not confutable.
a.
Capable of bending or yielding; apt to yield; compliant.