6 - Recursive Functions

Summary

The intuitive notion of an effectively computable function is the notion of a function for which there are definite, explicit rules, following which one could in principle compute its value for any given arguments. This chapter studies an extensive class of effectively computable functions, the recursively computable, or simply recursive, functions. According to Church's thesis, these are in fact all the effectively computable functions. Evidence for Church's thesis will be developed in this chapter by accumulating examples of effectively computable functions that turn out to be recursive. The subclass of primitive recursive functions is introduced in section 6.1, and the full class of recursive functions in section 6.2. The next chapter contains further examples. The discussion of recursive computability in this chapter and the next is entirely independent of the discussion of Turing and abacus computability in the preceding three chapters, but in the chapter after next the three notions of computability will be proved equivalent to each other.

Primitive Recursive Functions

Intuitively, the notion of an effectively computable function f from natural numbers to natural numbers is the notion of a function for which there is a finite list of instructions that in principle make it possible to determine the value f(x₁, …, x_n) for any arguments x₁, …, x_n. The instructions must be so definite and explicit that they require no external sources of information and no ingenuity to execute.