RecursiveFunctions

RecursiveFunctions - Outline Introduction Primitive...

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Outline Introduction Primitive Recursive Models Definition Examples Incomplete and Complete Models Ackermann’s Function μ-Recursive Functions Conclusion
Introduction A recursive function is one that calls upon itself to determine the solution Fibonacci Numbers F(0) = 1, F(1) = 1 F(n+1) = F(n) + F(n-1) for all n > 0 Can be utilized to resolve many problems (in fact, μ- Recursion is as powerful as a Turing Machine) Gödel developed general notion of recursive functions Made no claims on their strength, however

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Primitive Recursion First level of recursive functions Is an algorithm, has a finite number of steps to it Relies on simple axioms to build other functions Can be written very formally or informally Starting out there are no negative numbers We will define subtraction less than zero to be zero Term was said to be coined by Rózsa Péter.
Defined functions Constant Function - C a (x 1 ,…,x n ) = a No matter what the arguments, always return a Identity Function - I i n (x 1 ,…,x n ) = x i Given argument X1 through Xn, return Xi A projection function Successor Function – S(x) = x+1 Advance the argument x by 1

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Function Rules Composition - F(x 1 ,…,x n ) = G(H 1 (x 1 ,…,x n ), … , H k (x 1 ,…,x n )) A function can be defined as some function G acting on a composition of functions H1 to Hk. All carry the same input Iteration – The main Primitive Recursion Rule F(0, x 1 ,…,x n ) = G(x 1 ,…,x n ) F(y+1, x 1 ,…,x n ) = H(y, x
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This note was uploaded on 07/14/2011 for the course COT 4610 taught by Professor Dutton during the Fall '10 term at University of Central Florida.

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RecursiveFunctions - Outline Introduction Primitive...

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