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Topic: Primitive recursive function

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	Recursive function - Wikipedia, the free encyclopedia
		However, not every recursive function is a primitive recursive function — the most famous example of one which is not is the Ackermann function.
		The set of all recursive functions is known as R.
		The set of partial recursive functions is defined as the smallest set of partial functions of any arity from natural numbers to natural numbers which contains the zero, successor, and projection functions, and which is closed under composition, primitive recursion, and unbounded search.
	en.wikipedia.org /wiki/Recursive_function (417 words)

	PlanetMath: recursive function (Site not responding. Last check: 2007-11-07)
		Intuitively, a recursive function is a positive integer valued function of one or more positive integer arguments which may be computed by a definite algorithm.
		Recursive functions may be defined more rigorously as the largest class of partial functions from
		This is version 11 of recursive function, born on 2004-09-04, modified 2005-09-19.
	planetmath.org /encyclopedia/PrimitiveRecursiveFunction2.html (249 words)

	PlanetMath: primitive recursive (Site not responding. Last check: 2007-11-07)
		The class of primitive recursive functions is the smallest class of functions on the naturals (from
		The primitive recursive functions are Turing-computable, but not all Turing-computable functions are primitive recursive (see Ackermann's function).
		This is version 2 of primitive recursive, born on 2002-03-23, modified 2002-03-23.
	planetmath.org /encyclopedia/PrimitiveRecursive.html (82 words)

	CMPSCI 601 Q&A for HW#4, Spring 2004
		In the mu-operator, the function f was a primitive recursive function?
		It seems to me that if the definition of general recursive were closer to the definition for primitive recursive--replacing primitive recursion with unbound recursion, or something of the sort--then adapting the proof would be straightforward.
		Primitive recursive functions are f(x) = 0 the successor function, the projection function and any function that can be made from combining these elements with function composition or recursion.
	www.cs.umass.edu /~barring/cs601/qa/4.html (2067 words)

	math lessons - Ackermann function
		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 combination with the Ackermann function's applications in analysis of algorithms, discussed later, this debunks the theory that all useful or simple functions are primitive recursive functions.
		The Ackermann function, due to its definition in terms of extremely deep recursion, can be used as a benchmark of a compiler's ability to optimize recursion.
	www.mathdaily.com /lessons/Ackermann_function (1807 words)

	Peter Suber, "Recursive Function Theory"
		Recursive function theory, like the theory of Turing machines, is one way to make formal and precise the intuitive, informal, and imprecise notion of an effective method.
		Because recursive function theory was developed in part to capture the intuitive sense of effectiveness in a rigorous, formal theory, it is important to the theory that the class of recursive functions can be built up from intuitively effective simple functions by intuitively effective techniques.
		The use of the terms "recursive" and "primitive recursive" for components for the overall theory, and again for the the overall theory, is confusing and regrettable but that's the way the terminology has evolved.
	www.earlham.edu /~peters/courses/logsys/recursiv.htm (3359 words)

	Recursive Functions
		The recursive functions are characterized by the process in virtue of which the value of a function for some argument is defined in terms of the value of that function for some other (in some appropriate sense "smaller") arguments, as well as the values of certain other functions.
		Primitive recursion is a procedure that defines the value of a function at an argument n by using its value at the previous argument n − 1 (see Odifreddi, 1989, I.1.3).
		Primitive recursion can be used to define functions of many variables, but only by keeping all but one of them fixed.
	plato.stanford.edu /entries/recursive-functions (6936 words)

	ipedia.com: Primitive recursive function Article (Site not responding. Last check: 2007-11-07)
		Primitive recursive functions are a class of functions which form an important building block on the way to a full formalization of computability.
		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).
		Nevertheless they form an important class and many of the functions normally studied in number theory, and approximations to real-valued functions, are primitive recursive.
	www.ipedia.com /primitive_recursive_function.html (967 words)

	[No title]
		{rec_axiom} is the primitive recursion theorem for the concrete type in question; this must be a theorem obtained from {define_type}.
		This theorem is derived by formal proof from an instance of the general primitive recursion theorem given as the second argument.
		A theorem {th} of the form returned by {define_type} is a primitive recursion theorem for an automatically-defined concrete type {ty}.
	www.cs.utah.edu /~swalton/hol98/help/Docfiles/Prim_rec.new_recursive_definition.doc (743 words)

	Computability and Complexity
		The important idea is that the primitive recursive functions comprise a very large and powerful class of computable functions, all generated in an extremely simple way.
		Here composition is the natural way to combine functions, and primitive recursion is a restricted kind of recursion in which h with first argument n+1 is defined in terms of h with first argument n, and all the other arguments unchanged.
		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.
	plato.stanford.edu /entries/computability (5266 words)

	CS 5315, Homework #2 (Site not responding. Last check: 2007-11-07)
		In the class, we proved that if P(n) is a primitive recursive predicate, and f(n) and g(n) are primitive recursive functions, then the function "if P(n) then f(n) else g(n)" is also primitive recursive.
		Use the ideas from this proof to prove that if P1(n) and P2(n) are primitive recursive predicates, and f(n), g(n), and h(n) are primitive recursive functions, then the function "if P1(n) then f(n) else if P2(n) then g(n) else h(n)" is also primitive recursive.
		Show that the function pow(a,n)=a^n (a to the power n) is a primitive recursive function.
	www.cs.utep.edu /vladik/cs5315.05/home02sol.html (210 words)

	Non-primitive Recursive Function Definitions (Site not responding. Last check: 2007-11-07)
		This paper presents an approach to the problem of introducing non-primitive recursive function definitions in higher order logic.
		A recursive specification is translated into a domain theory version, where the recursive calls are treated as potentially non-terminating.
		Hence, the derivation of a domain theory specification has been automated completely, and for well-founded recursive function specifications the process of deriving the original specification from the domain theory one has been automated as well, though a user must supply a well-founded relation and prove certain termination properties of the specification.
	www.brics.dk /RS/95/36 (122 words)

	Recursive definitions (from foundations of mathematics) -- Encyclopædia Britannica (Site not responding. Last check: 2007-11-07)
		N that can be defined with the help of such a recursion scheme (and with the help of 0, S, and substitution) are called primitive recursive.
		in logic and mathematics, a type of function or expression predicating some concept or property of one or more variables, which is specified by a procedure that yields values or instances of that function by repeatedly applying a given relation or routine operation to known values of the function.
		The theory of recursive functions was developed by the 20th-century...
	www.britannica.com /eb/article-35461 (755 words)

	[No title]
		The significance of this function is that it can be proved to be a recursive function of two variables.
		(The difference between recursive and primitive recursive is that computation of a recursive function may, in general, involve a finite but unbounded search for a solution to an equation involving recursive functions.
		In the case of primitive recursive functions there must be an a priori primitive recursive upper bound on the number of steps in the search.) Ackermann's function grows so quickly, especially in the y variable, that it is completely infeasible to calculate it for more than a few small values of y.
	barnyard.syr.edu /quickies/ackermann.c (697 words)

	Hilbert Levitz's Research Interests (Site not responding. Last check: 2007-11-07)
		A recursive universal function for the primitive recursive functions is constructed.
		Ackermann's function is the classical example of a total computable function which is not primitive recursive.
		A proof of the existence of a recursive universal function for the class of primitive recursive functions is given.
	www.cs.fsu.edu /~levitz/research.html (432 words)

	CS 323 - Lab 12 - Fall 2004 (Site not responding. Last check: 2007-11-07)
		The primitive recursive functions are all basic functions and all functions that can be obtained from them by any number of successive applications of composition or recursive definition.
		Explain why plus is a primitive recursive function by identifying g and h in the definition of primitive recursive functions.
		Write the predicates in terms of known primitive recursive functions and predicates: greaterthanorequal(m,n) (use sub), lessthanorequal(m,n), equals(m,n) and(p(m,n),q(m,n)), or(p(m,n), q(m,n)), where p and q are predicates.
	www.snc.edu /compsci/cs323/lab12/lab12.html (395 words)

	Re: C equal to partial recursive function
		I was able to > >build a function being equal to what should be a function build with > >the recursion principle up to any integer, and being equal to zero > >aftern.
		The minimisation principle (as defined above) is not used for primitive recursive functions.
		And one can prove that primitive recursive function is strictly included in the set of partial recursive functions.
	www.usenet.com /newsgroups/sci.logic/msg03408.html (386 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		When we run the C++ compiler, after specifying a partial recursive function $f$ and a natural number $x$ as C++ source code, the error message typed out by the compiler will be $f(x)$.
		Then the function $f$ with % \[\begin{array}{lcl} f(x_1, \ldots, x_n, 0) & = & g(x_1, \ldots, x_n) \: \mbox{ and}\\ f(x_1, \ldots, x_n, y+1) & = & h(x_1, \ldots, x_n, y, f(x_1, \ldots, x_n, y)) \end{array} \] is also primitive recursive.
		Let $f$ be a unary partial recursive function, let $h$ be a binary primitive recursive function such that $f(x) = \minop\, y [h(x,y)=0]$, and let \cppid{H} be a function type that computes $h$.
	www.cogs.susx.ac.uk /users/vs/beatcs/conSample.tex.txt (1752 words)

	Test 1 - CS 5315 Spring 2005 (Site not responding. Last check: 2007-11-07)
		(a) Prove, from scratch, that the functions ab, a mod b, and a div b are primitive* recursive.
		(c) The notion of a primitive recursive function is a formalization of the for-loop.
		(a) Prove that the functions a*b, a mod b, and a div b are mu-recursive.
	www.cs.utep.edu /vladik/cs5315.05/test1.html (253 words)

	Computation Theory
		The aim of this course is to introduce several apparently different formalisations of the informal notion of algorithm; to show that they are equivalent; and to use them to demonstrate that there are uncomputable functions and algorithmically undecidable problems.
		Primitive recursive partial function are computable and total.
		Alternative characterisations of recursively enumerable sets as the images and the domains of definition of partial recursive functions.
	www.cl.cam.ac.uk /DeptInfo/CST02/node112.html (248 words)

	Prim_rec.new_recursive_definition (Site not responding. Last check: 2007-11-07)
		automatically proves the existence of a function fn that satisfies the defining equations supplied as the fourth argument, and then declares a new constant in the current theory with this definition as its specification.
		A call to new_recursive_definition fails if the supplied theorem is not a primitive recursion theorem of the form returned by define_type; if the term argument supplied is not a well-formed primitive recursive definition; or if any other condition for making a constant specification is violated (see the failure conditions for new_specification).
		Note that the equations defining a (recursive or non-recursive) function on binary trees by cases can be given in either order.
	hol.sourceforge.net /kananaskis-1-helpdocs/help/Docfiles/HTML/Prim_rec.new_recursive_definition.html (814 words)

	An arguable inconsistency in ZF
		Classical theory proves that every primitive recursive function is strongly representable in PA; that PA and PRA can both be interpreted in ZF; and that if ZF is consistent, then PA+PRA is consistent.
		PA, and any finite segment of PRA (say PRA), are, both, recursively* enumerable languages, and both can be translated into ZF - in the sense that the axioms of PA, and of PRA*, correspond to theorems of ZF under a non-standard interpretation, and their rules of inference are preserved under the translation.
		Since all the added functions and relations are primitive recursive, they are strongly represented in PA ([Me64], Proposition 3.23, p134).
	alixcomsi.com /An_arguable_inconsistency_in_ZF.htm (1199 words)

	PRF - TheBestLinks.com - Frequency, Primitive recursive function, Radar, Pulse, ... (Site not responding. Last check: 2007-11-07)
		PRF - TheBestLinks.com - Frequency, Primitive recursive function, Radar, Pulse,...
		PRF, Frequency, Primitive recursive function, Radar, Pulse, Disambig, Message...
		This is a disambiguation page, i.e., a navigational aid which lists other pages that might otherwise share the same title.
	www.thebestlinks.com /PRF.html (141 words)

	Ackermann Function -- from MathWorld
		The Ackermann function is the simplest example of a
		primitive recursive, providing a counterexample to the belief in the early 1900s that every
		It grows faster than an exponential function, or even a multiple exponential function.
	www.dia.fi.upm.es /~ledesma/AckermannFunction.html (148 words)

	Chapter 2 : Every primitive recursive relation is not expressible formally
		Theorem VI : For every w-consistent primitive recursive class k of FORMULAS, there exists a primitive recursive CLASS EXPRESSION r such that neither vGenr nor Neg(vGenr) belongs to Flg(k) (where v is the FREE VARIABLE of r).
		x) is the primitive recursive function of A that yields the length of the sequence of numbers correlated with
		Theorem V [Gödel 1931 : p22], that every primitive recursive relation and function is constructively both expressible formally, and decidable, in the formal systems considered by him.
	alixcomsi.com /F_Fall02a.htm (782 words)

	PRF - the free encyclopedia (Site not responding. Last check: 2007-11-07)
		PRF is an acronym and can stand for:
		Primitive recursive function - a class of functions which form an important building block on the way to a full formalization of computability
		If an article link referred you here, you might want to go back and fix it to point directly to the intended page.
	www.free-web-encyclopedia.com /?t=PRF (81 words)

	Prim_rec.prove_rec_fn_exists (Site not responding. Last check: 2007-11-07)
		Proves the existence of a primitive recursive function over a concrete recursive type.
		The first argument is a theorem of the form returned by define_type, and the second is a user-supplied primitive recursive function definition.
		The theorem which is returned asserts the existence of the recursively-defined function in question (if it is primitive recursive over the type characterized by the theorem given as the first argument).
	www.cs.utah.edu /~swalton/hol98/help/Docfiles/HTML/Prim_rec.prove_rec_fn_exists.html (224 words)

	primitive recursive (Site not responding. Last check: 2007-11-07)
		Definition: A total function which can be written using only nested conditional (if-then-else) statements and fixed iteration (for) loops.
		Note: Ackermann's function is computable, but is not primitive recursive.
		Paul E. Black, "primitive recursive", from Dictionary of Algorithms and Data Structures, Paul E. Black, ed., NIST.
	www.nist.gov /dads/HTML/primitivrecr.html (129 words)

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