On the Impact of Turing Machines

Abstract

Turing contributed a simple model of computation that has become the definition of computable. A function is considered to be computable if and only if it is computable on Turing's model of computation. Since our notion of computable is informal and Turing's model gives a precise definition of computable, we cannot prove the two equivalent. However, for every mathematical definition of computable that has been proposed, a proof has been developed that any function computable by the proposed model is also computable by Turing's model. Turing's model is very simple, it consists of an infinite tape made up of cells, each cell capable of holding one of a finite set of symbols, along with a control with a finite number of states and a tape head by which the finite control can scan the tape and read the content of the cell scanned. A move consists of reading the contents of the scanned cell and depending on the internal state of the finite state control, writing a new symbol in the cell, moving the read head one cell right or one cell left, and changing the internal state of the finite control to a new state. Although logicians had their own models of computable, Turing's model made the notion of computable accessible to a larger community. The impact of a mathematical model that corresponded to a physical device and allowed one to picture and more fully understand the notion of computability accelerated the science of computability in a way which many do not appreciate and is the function of this talk. One of the major advances came when Hartmanis and Stearns used the Turing model to define complexity classes. This then gave a formal definition to the intuitive notion of polynomial time algorithms. It helped led to asymptotic complexity as a way to compare performance of algorithms. Another major advance was that the Turing model lead to the notion of an instantaneous description of a computation and a valid computation. An instantaneous description is a string that completely describes a computation at one instance of time. A valid computation is a sequence of successive instantaneous description. Once a valid computation was represented by a string of symbols it was quickly recognized that a valid computation of a Turing machine could be expressed as the intersection of two context-free languages and hence the question whether the intersection was empty was undecidable. Many other problems arising in computer science were quickly shown to be undecidable. In the mid sixties the language ALGOL was invented and was described by a context-free grammar. However, as people soon noticed that what a program did sometimes depended on the compiler used. It was quickly discovered that the context-free grammar describing ALGOL was ambiguous. When researcher set out to write an algorithm to determine if a context-free grammar was ambiguous they quickly discovered that this problem was also undecidable. One of the major discoveries of this century was when Stephan Cook proved that every problem in polynomial time could be reduced to the problem of satisfying a formula in conjunctive normal form. This lead to the notion of NP-complete problems and that many problems such as integer programming, finding the maximal clique, and many others were really all equivalent. Although Turing's model was very simple it was that simplicity that lead to major advances in computer science.