Chapter 5: Higher-Order Procedures

Computability

The power of computational processes has led computer scientists to speculate on the following:

Q: Is it possible for any mathematically specified function to be computed by some procedure?

If the answer to this question is "yes", then the mathematically precise function h below should be computable:

Let P = the set of all procedures.
Let h: P → {true,false}.
For all p ∈ P, define:

h(p) = true if p eventually halts
= false otherwise

If all mathematically precise functions are computable, then there must be a procedure to compute h.

Call it halts?.

h(p)	=	true if p eventually halts
	=	false otherwise