On Formalisms for Turing Machines

Turing's original quintuple formalism for an abstract computing machine is compared with the quadruple approach of Post and with some new alterr~atives. In each case the possibility or nmipossibility of two-symbol or two-state ~miversal machines is demon. strated. The term "Turing machine" has been applied to several different characterizations of an abstract computing machine. Since each of the formalisms has been adequate for a development of recursive function theory, no serious trouble has arisen from the multiple use of the term. In this paper w~rious formal definitions for the notion of a general-purpose abstract computer are compared, and some new alternative deft-nitions are introduced. Particular attention is paid to one of Turing's original formalisms and to one by Post; the latter has been used extensively by Davis in [2]. Most of the theorems below assert that a certain kind of machine simulates am other kind of machine. However, the concept of simulation of one machine by another is extremely difficult to define precisely. Too stringent a definition excludes cases in which one intuitively feels a bona fide simulation is being performed. Too liberal a definition allows the use of encodings of input and output in which the real computational work is done by the encoding and decoding algorithms and not by the machine which is supposedly performing the simulation. The notion of simulation of one machine by another used here requires that intermediate results of the computations by the two machines be closely related as well as the outputs of the computations; i.e., the simulation is "step by step." An attempt at a precise definition is given in the Appendix, and it is hoped that the notion of simulation is correctly captured by the definition. However, the theorems of this paper clearly satisfy any reasonable definition of simulation, and the author invites suggestions for improving the definition. Theorem 2 is due jointly to S. Aanderaa and the auttmr [1], and Theorem 8 is due to P. K. Hooper [4]. The author is also indebted to the referee for his comments and for his suggestion of a way to strengthen the originally submitted version of Theorem 3. A Turing machine is usually regarded as a small computer with a finite number of states and a (potentially) infinite tape marked off into discrete squares. Upon each square of the tape is written one symbol selected from a finite alphabet; all but a …