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Topic: Computable function

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Lambda calculus

Functional programming

Pulse computer

Hypercomputation

Quantum computer

Computer science

Recursion

Universal Turing machine

Undecidable

Turing machine

Halting problem

Computational complexity theory

Oracle machine

Algorithm

Complexity classes P and NP

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	Computable function - Wikipedia, the free encyclopedia
		The notion of computability of a function can be relativized to an arbitrary set of natural numbers A, or equivalently to an arbitrary function f from the naturals to the naturals, by using Turing machines extended by an oracle for A or f.
		According to the Church–Turing thesis, the class of computable functions is equivalent to the class of functions defined by recursive functions, lambda calculus, or Markov algorithms.
		In computational complexity theory, the problem of determining the complexity of a computable function is known as a function problem.
	en.wikipedia.org /wiki/Computable_function (388 words)

	NationMaster.com - Encyclopedia: Lambda calculus
		The calculus can be used to cleanly define what a computable function is. The question of whether two lambda calculus expressions are equivalent cannot be solved by a general algorithm, and this was the first question, even before the halting problem, for which undecidability could be proved.
		In lambda calculus, every expression stands for a function with a single argument; the argument of the function is in turn a function with a single argument, and the value of the function is another function with a single argument.
		A function of two variables is expressed in lambda calculus as a function of one argument which returns a function of one argument (see currying).
	www.nationmaster.com /encyclopedia/Lambda-calculus (838 words)

	Lambda calculus - LearnThis.Info Enclyclopedia (Site not responding. Last check: 2007-11-05)
		Functions are anonymously defined by a lambda expression which expresses the function's action on its argument.
		Recursion is the definition of a function using the function itself; on the face of it, lambda calculus does not allow this.
		A function F : N → N of natural numbers is defined to be computable if there exists a lambda expression f such that for every pair of x, y in N, F(x) = y if and only if the expressions f x and y are equivalent.
	encyclopedia.learnthis.info /l/la/lambda_calculus.html (2254 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		Ackermann proved that A is a recursive function, a function a computer with infinite memory can calculate, but it is not a primitive recursive function, a class of functions including almost all familiar functions such as addition and factorial.
		In mathematical logic and computer science, the recursive functions are a class of functions from natural numbers to natural numbers which are "computable" in some intuitive sense.
		Ackermann function: in the theory of computation, a recursive function that is not primitive recursive.
	www.worldhistory.com /wiki/P/Primitive-recursive-function.htm (714 words)

	Computable function: Facts and details from Encyclopedia Topic (Site not responding. Last check: 2007-11-05)
		Computability theory is that part of the theory of computation dealing with which problems are solvable by algorithmalgorithms (equivalently, by turing machines),...
		In mathematics and computer science, a partial function from the domain x to the codomain y is a binary relation over x and y which...
		In mathematics a constant function is a function whose values do not vary and thus are constant....
	www.absoluteastronomy.com /encyclopedia/c/co/computable_function.htm (658 words)

	Recursively enumerable set - Wikipedia, the free encyclopedia
		In computability theory, often less suggestively called recursion theory, a countable set S is called recursively enumerable, computably enumerable, semi-decidable or provable if
		The set S is either empty or the range of a primitive recursive function.
		The preimage of a recursively enumerable set under a computable function is a recursively enumerable set.
	en.wikipedia.org /wiki/Recursively_enumerable (625 words)

	Primitive_recursive_function (Site not responding. Last check: 2007-11-05)
		In computability theory, '''primitive recursive functions''' are a class of functions which form an important building block on the way to a full formalization of computability.
		Many of the functions normally studied in number theory, and approximations to real-valued functions, are primitive recursive, such as addition, division, factorial, exponential, finding the ''n''th prime, and so on (Brainerd and Landweber, 1974).
		Many other familiar functions can be shown to be primitive recursive; some examples include conditionals, exponentiation, primality testing, and mathematical induction, and the primitive recursive functions can be extended to operate on other objects such as integers and rational numbers.
	q-basic.xodox.de /Primitive_recursive_function (971 words)

	Primitive recursive function (Site not responding. Last check: 2007-11-05)
		The primitive recursive functions are a strict subset of the recursive functions (which are exactly those functions which we call " computable "; see Church-Turing thesis).
		Certainly the initial set of functions are intuitively computable (in their very simplicity), and the two operations by which one can create new primitive recursive functions are also very straightforward.
		One can also explicitly exhibit a simple 1-ary computable function which is recursively defined for any natural number, but which is not primitive recursive, see Ackermann function.
	www.serebella.com /encyclopedia/article-Primitive_recursive_function.html (1375 words)

	Primitive recursive function (Site not responding. Last check: 2007-11-05)
		Many other familiar functions can be shown to be primitive recursive; some examples include conditionals, exponentiation, primality testing, and course-of-values induction, and the primitive recursive functions can be extended to operate onother objects such as integers and rational numbers.
		Primitive recursive functions tend to correspond very closely with our intuition of what a computable function must be.Certainly the initial set of functions are intuitively computable (in their very simplicity), and the two operations by which onecan create new primitive recursive functions are also very straightforward.
		This function is computable (by the above), but clearly noprimitive recursive function exists which computes it as it differs from each possible primitive recursive function by at leastone value.
	www.therfcc.org /primitive-recursive-function-45957.html (893 words)

	Lambda calculus - Wikipedia, the free encyclopedia
		A function is anonymously defined by a lambda expression which expresses the function's action on its argument.
		Lisp uses a variant of lambda notation for defining functions, but only its purely functional subset is really equivalent to lambda calculus.
		Actually implementing the lambda calculus on a computer involves treating "functions" as first-class objects, which turns out to be rather difficult to accomplish using stack-based computer languages.
	en.wikipedia.org /wiki/Lambda_calculus (2201 words)

	Primitive recursive function - InfoSearchPoint.com (Site not responding. Last check: 2007-11-05)
		The primitive recursive functions are a strict subset of the recursive functions (which are exactly those functions which we call "computable"; see equivalence of models of computation) and Church-Turing thesis.
		Many other familiar functions can be shown to be primitive recursive; some examples include conditionals, exponentiation, primality testing, and course-of-values induction, and the primitive recursive functions can be extended to operate on other objects such as integer and rational numbers.
		The numbering is computable in the sense that it can be defined under format models of computability such as recursive functions or Turing machines; but an appeal to the Church-Turing thesis is likely sufficient.
	www.infosearchpoint.com /display/Primitive_recursive_function (929 words)

	PlanetMath: computable real function
		A real function is computable if it is both sequentially computable and effectively uniformly continuous.
		It is not hard to generalize these definitions to functions of more than one variable or functions only defined on a subset of
		This is version 3 of computable real function, born on 2004-09-29, modified 2004-09-29.
	planetmath.org /encyclopedia/EffectiveUniformContinuity.html (147 words)

	Primitive recursive function (Site not responding. Last check: 2007-11-05)
		Certainly the initial set of are intuitively computable (in their very simplicity) the two operations by which one can new primitive recursive functions are also very However the set of primitive recursive functions not include every possible computable function --- can be seen with a variant of Cantor's diagonal argument.
		This function is computable (by the but clearly no primitive recursive function exists computes it as it differs from each primitive recursive function by at least one Thus there must be computable functions which not primitive recursive.
		One can also explicitly exhibit a simple computable function which is recursively defined for natural number but which is not primitive see Ackermann function.
	www.freeglossary.com /Primitive_recursive_function (1132 words)

	On computable numbers, with an application to the Entscheidungsproblem - A. M. Turing, 1936
		The “computable” numbers may be described briefly as the real numbers whose expressions as a decimal are calculable by finite means.
		Although the subject of this paper is ostensibly the computable numbers, it is almost equally easy to define and investigate computable functions of an integral variable or a real or computable variable, computable predicates, and so forth.
		It would be true if we could enumerate the computable sequences by finite means, but the problem of enumerating computable sequences is equivalent to the problem of finding out whether a given number is the D.N of a circle-free machine, and we have no general process for doing this in a finite number of steps.
	www.abelard.org /turpap2/tp2-ie.asp (3956 words)

	Functions
		A computable function may be expressed in many forms, yet to be reasonable, a description of a function as a rule form computation must be finite.
		Functions are some times categorized by the relations of the domain and range.
		The computer approximation of the trigonometric sine function, y is a member of the Range -1.0..
	cs.wwc.edu /~aabyan/Logic/Book/book/node154.html (314 words)

	[No title]
		In computability theory on the set \NN\ of natural numbers, the class of semicomputable sets is closed under taking projections, but this is not true in the general case of algebras, even with \Whilex\ computability.
		For each leaf \lam\ of the computation tree \TTT, there is a {\it boolean} $\bbb_\lam$, which expresses the conjunction of results of all the successive tests, that (the current values of) the variables in $P$ must satisfy in order for the computation to ``follow'' the finite path from the root to \lam.
		Moreover, the sequence $(t_n)$ is {\it computable in} $n$, by use of the BCP to effectivise the transformation of the tree \TTT\ to \TTTp\ in the construction given by the proof of Lemma 3 of \S8.3.
	www.cas.mcmaster.ca /~zucker/Pubs/Top/text (7561 words)

	PHIL 411: Advanced Logic
		Functions are just 2-place relations whose extensions are sets of ordered pairs, where no two pairs have the same first member (input/argument) but different second members (output/value).
		The Range of a function is the set of all values for the function.
		Computability: A function ¦ is effectively computable if there is an algorithm (mechanical procedure) that will yield ¦(m) for any argument m at which ¦ is defined and give no result for any argument at which ¦ is undefined.
	www.siue.edu /~wlarkin/teaching/PHIL411/undecidability.html (704 words)

	MANIFOLD-10: Mathematics of the 70s
		A computable (recursive) function from N, the class of natural numbers, into N is, roughly speaking, one whose value for any given argument is determined by an ideal computer with unlimited memory (a Turing machine).
		The computable complex numbers form a field, and the roots of a polynomial with computable complex coefficients are computable complex numbers.
		This result can be extended to general computable metric spaces which are totally bounded, this generalization being the constructive analogue of the theorem asserting the continuity of the inverse function of a continuous injective function defined on a compact space.
	www.jaworski.co.uk /m10/10_70s.html (1663 words)

	All Programs = All Partially Computable Functions (Site not responding. Last check: 2007-11-05)
		In order to check to see if a function is PC (Partially Computable), we have to find or write a program that computes this function.
		A (refined) Class of functions that are bound to be Computable without writing a program.
		Note that this THM uses a PR function denoted as a(X).
	hal.lamar.edu /~KOH/5315/Prerec.htm (1491 words)

	Computability and Complexity
		In addition, there is an extensive classification of computable problems into computational complexity classes according to how much computation — as a function of the size of the problem instance — is needed to answer that instance.
		He finally used the primitive recursive functions to compute properties of the represented formulas including that a formula was well formed, a sequence of formulas was a proof, and that a formula was a theorem.
		With this definition, the Recursive functions are exactly the same as the set of partial functions computable by the Lambda calculus, by Kleene Formal systems, by Markov algorithms, by Post machines, and by Turing machines.
	plato.stanford.edu /entries/computability (5283 words)

	COCS5330TEST-1:1999 (Site not responding. Last check: 2007-11-05)
		SUCC (X,Y) This function is total as it is defined even when the first argument, X, may not be a real snapshot of a program represented by the second argument, Y. The complement of the set EMPTY is R.E. No m-complete set is recursive.
		A function is computable if and only if it can be obtained from the initial functions by a finite number of applications of composition, recursion, and bounded minimalization.
		Let a function g(x) be computable and let a set B be the range of this function.
	hal.lamar.edu /~KOH/5330/5330T1.HTML (643 words)

	Citations: Classical Recursion Theory - Odifreddi (ResearchIndex) (Site not responding. Last check: 2007-11-05)
		All functions are meant to be total if not explicitly attributed as being partial; in particular, a computable function is a partially computable function that is total.
		The notion of relative computability allows for a definition of relative randomness: a sequence is random in A if its initial segments can have essentially the same size as the shortest programs needed to generate them in a computer equipped with and oracle A. If a set A is computable from B,....
		The notion of relative computability induces a definition of relative randomness: A sequence is random in A if its initial segments can not be algorithmically compressed in a computer equipped with an oracle A. If a set A is computable from B, then randomness in B implies randomness in A. Thus,....
	citeseer.ist.psu.edu /context/71698/0 (4129 words)

	Turing machines
		The next move function says that, when the machine is in state q and scans symbol a on a cell, then the machine writes a symbol b instead of a, moves one cell in the direction left or right, and enters state p.
		Moreover, these functions grow faster than any computable function (that is, for any computable function f, there is an integer N such that, for all n > N, we have S(n,2) > f(n)).
		The definitions of functions S(n,2) and Sigma(n,2) are explicit enough to allow their computations for small n.
	www.logique.jussieu.fr /~michel/tmi.html (901 words)

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