Recursive Unsolvability of Post's Problem of "Tag" and other Topics in Theory of Turing Machines

Abstract

The equivalence of the notions of effective computability as based (1) on formal systems (e.g., those of Post), and (2) on computing machines (e.g., those of Turing) has been shown in a number of ways. The main results of this paper show that the same notions of computability can be realized within (1) the highly restricted monogenic formal systems called by Post the Tag systems, and (2) within a peculiarly restricted variant of Turing machine which has two tapes, but can neither write on nor erase these tapes. From these, or rather from the arithmetization device used in their construction, we obtain also an interesting basis for recursive function theory involving programs of only the simplest arithmetic operations. We show first how Turing machines can be regarded as programmed computers. Then by defining a hierarchy of programs which perform certain arithmetic transformations, we obtain the representation in terms of the restricted two-tape machines. These machines, in turn, can be represented in terms of Post normal canonical systems in such a way that each instruction for the machine corresponds to a set of productions in a system which has the monogenic property (for each string in the Post system just one production can operate). This settles the questions raised