An Introduction to Some Basic Concepts and Techniques in the Theory of Computable Functions

Abstract

We will talk about a class of function defined on subsets of the set of nonnegative integers, and assuming nonnegative integer values: the so-called “computable functions”. We define a function to be computable if there exists an “effective description” for it, that is a finite sentence unambiguously indicating (possibly only some explicit way) what is its “behaviour” in correspondence to each natural number. In other words, an effective description must allow us to deduce, for any given argument, whether the described function is defined on it and, if it is so, the value it assumes there. Perhaps the careful reader will observe that this definition is very different from the one usually found in the literature: in fact, it may seem strange to call “computable” those functions for which we didn’t explicitly prescribe the precise means for computing them, for example, some particular procedure or model of an abstract “computing device”. Moreover, even the general concepts of “effective procedure”, “algorithm”, or “abstract computing device” are not needed for the definition we gave of a computable function.