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Unformatted text preview: Chapter 3 Recursive Function Theory In this chapter, we will take a brief look at the recursive function theory. We will only consider functions that map positive integers to positive integers, and in particular, only functions which produce a single output i.e., f : N n N . Let us first define some operations that we can do on functions: Definition 4 Composition: Given the following functions, f 1 ( x 1 , x 2 , , x n ) , f 2 ( x 1 , x 2 , , x n ) , . . . f k ( x 1 , x 2 , , x n ) , and g ( x 1 , x 2 , , x k ) we define the function h ( x 1 , x 2 , , x n ) to be the composition of f 1 , f 2 , , f k with g as follows: h (( x 1 , x 2 , , x n ) = g ( f 1 ( x 1 , x 2 , , x n ) , , f k ( x 1 , x 2 , , x n )) . A function which has output values defined for each input value is called a Total function. An example of such a function is f ( x ) = 2 x . A function is not Total, if for some input values, no output value is defined. The following function is not a Total function:output value is defined....
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This note was uploaded on 01/28/2012 for the course CS 220 taught by Professor Ibarra,o during the Winter '08 term at UCSB.
- Winter '08