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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
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
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
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
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
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
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
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
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
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
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
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
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)
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
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
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
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
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
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-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
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
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)
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)
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
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
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
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
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
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
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
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
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
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
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
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)
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
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
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
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
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
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
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)
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
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
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
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
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
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
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
of computable functions. The theorem states that there exists no largest complexity class, with computable boundary, which contains all computable functions
Compression_theorem
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
Characterization of hyperarithmetic sets
to Γ {\displaystyle \Gamma } . Moreover, there exists a (total) computable function S : N → N {\displaystyle S:\mathbb {N} \to \mathbb {N} } such that
Suslin–Kleene_theorem
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
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
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
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)
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
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
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
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)
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
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
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
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)
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
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 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
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
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
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
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
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
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
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
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
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)
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
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)
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
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
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
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)
Mathematical set of all subsets of a set
demonstrated below. An indicator function or a characteristic function of a subset A of a set S with the cardinality |S| = n is a function from S to the two-element
Power_set
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
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
Syntactically correct logical formula
constant symbols, predicate symbols, and function symbols of the theory at hand, along with the arities of the function and predicate symbols. The definition
Well-formed_formula
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)
computable function was primitive recursive. This was refuted by Gabriel Sudan and Wilhelm Ackermann — both his students — using different functions that
Sudan_function
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)
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
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
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
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 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
Estimate of time taken for running an algorithm
parameterized problems ( L , k ) {\displaystyle (L,k)} for which there is a computable function f : N → N {\displaystyle f:\mathbb {N} \to \mathbb {N} } with f ∈
Time_complexity
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
COMPUTABLE FUNCTION
COMPUTABLE FUNCTION
Biblical
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Girl/Female
Indian
Sacred, Pure, Comparable to the ganges, Another name for Durga, ***, Another name for Durga
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.
Boy/Male
Afghan, Arabic, Celebrity, German, Indian, Muslim, Sindhi
Observer; Supervisor; Little; Insignificant; Warner; Similar; Comparable; Another Name for the Quran; One who Preaches
Girl/Female
Tamil
Sacred, Pure, Comparable to the ganges, Another name for Durga, ***, Another name for Durga
Male
Egyptian
, a high Egyptian functionary.
Girl/Female
Indian
Sacred, Pure, Comparable to the ganges, Another name for Durga, ***, Another name for Durga
Male
Celtic
, great justiciary, or functionary.
Boy/Male
Muslim
Similar. Comparable.
Male
Egyptian
, an Egyptian functionary.
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.
Boy/Male
Arabic, Australian, Muslim
Similar; Comparable; One who Warns
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
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.
Boy/Male
Muslim
Similar. Comparable.
Male
Egyptian
, a great functionary.
Male
Egyptian
, Functionary of the Interior.
Girl/Female
Tamil
Sacred, Pure, Comparable to the ganges, Another name for Durga, ***, Another name for Durga
COMPUTABLE FUNCTION
COMPUTABLE FUNCTION
Female
French
French form of Latin Natalia, NATHALIE means "birthday," or in Church Latin "Christmas day."
Boy/Male
Tamil
Lord Shiva
Boy/Male
Bengali, Indian
Sun's Fire
Girl/Female
Assamese, Gujarati, Hindu, Indian, Kannada, Marathi, Telugu
Lit by Lamp
Boy/Male
Scottish American
Scottish surname and place name. From Leslie.
Boy/Male
Hindu, Indian
One who Gets Victory on Religion
Female
English
English variant spelling of Roman Latin Valerie, VALARIE means "to be healthy, to be strong."Â
Male
Czechoslovakian
, spring favor.
Girl/Female
Biblical
Silence, resemblance.
Girl/Female
Hindu, Indian
Queen
COMPUTABLE FUNCTION
COMPUTABLE FUNCTION
COMPUTABLE FUNCTION
COMPUTABLE FUNCTION
COMPUTABLE FUNCTION
a.
Capable of being commuted or interchanged.
a.
Suitable; consistent.
a.
Not confutable.
a.
Compatible; suitable; consistent.
a.
Not computable.
n.
The quality of being imputable; imputableness.
n.
Quality of being imputable.
a.
Capable of bending or yielding; apt to yield; compliant.
a.
Correspondent; conformable; hence, comparable.
n.
The quality of being commutable; interchangeableness.
a.
Capable of existing in harmony; congruous; suitable; not repugnant; -- usually followed by with.
a.
Capable of being computed, numbered, or reckoned.
a.
Comparable.
a.
Not compliable; not conformable.
n.
The quality of being commutable.
a.
Not commutable; not capable of being exchanged with, or substituted for, another.
a.
That may be confuted.
a.
Capable of being attributed; ascribable; imputable.
a.
Such as can be, or is liable to be, combated; as, combatable foes, evils, or arguments.
adv.
In a compatible manner.