Computation with External Help

Abstract

According to the Computability Thesis, all models of computation, including those yet to be discovered, are equivalent to the Turing machine and formalize the intuitive notions of computation. In other words, what cannot be solved on a Turing machine cannot be solved in nature. But what if Turing machines could get external help from a supernatural assistant? In this chapter we will describe the birth of this idea, its development, and the formalization of it in the concept of the oracle Turing machine. We will then briefly describe how external help can be added to other models of computation, in particular to partial recursive functions. We will conclude with the Relative Computability Thesis, which asserts that all such models are equivalent, one to the other, thus formalizing the intuitive notion of the “computation with external help.” Based on this, we will adopt the oracle Turing machine as the model of computation with external help.

An oracle was an ancient priest who made statements about future events or about the truth.