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Introduction
Primitive Recursive Models
Definition
Examples
Incomplete and Complete Models
Ackermann’s Function
μRecursive 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(n1) 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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View Full Document 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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View Full Document 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.
 Fall '10
 Dutton

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