The Turing Machine

Abstract

The Turing machine convincingly formalized the concepts of “algorithm,” “computation,” and “computable.” It convinced researchers by its simplicity, generality, mechanical operation, and resemblance to human activity when solving computational problems, and by Turing’s reasoning and analysis of “computable” functions and his argumentation that partial “computable” functions are exactly Turing computable functions. Turing considered several variants that are generalizations of the basic model of his machine. But he also proved that they add nothing to the computational power of the basic model. This strengthened the belief in the Turing machine as an appropriate model of computation. Turing machines can be encoded and consequently enumerated. This enabled the construction of the universal Turing machine that is capable of computing anything that can be computed by any other Turing machine. This seminal discovery laid the theoretical grounds for several all important practical consequences, the general-purpose computer and the operating system being the most notable. The Turing machine is a versatile model of computation: it can be used to compute values of a function, or to generate elements of a set, or to decide about the membership of an object in a set. The last led to the notions of decidable and semi-decidable sets that would later prove to be very important in solving general computational problems.

A machine is a mechanically, electrically, or electronically operated device for performing a task. A program is a sequence of coded instructions that can be inserted into a machine to control its operation.