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chapter3 - Chapter 3 Recursive Function Theory In this...

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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: f ( x ) = x 2 if x even undefined if x odd Composition preserves Totality, i.e. if
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