Decision problems for tag systems

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Abstract
The aim of this paper is to study tag systems as defined by Post [Post 1943, pp. 203–205 and Post, 1965, pp. 370–373]. The existence of a tag system with unsolvable halting problem was proved by Minsky by constructing a universal tag system [Minsky 1961, see also Cocke and Minsky 1964, Wang 1963, and Minsky 1967, pp. 267–273]. Hence the halting problem of a tag system can be of the complete degree 0′. We shall prove that the halting problem for a tag system can have an arbitrary (recursively enumerable) degree of undecidability (Corollary III).A related problem arises when we ask if there exists a uniform procedure for determining, given a tag system, whether or not there is any word on which the tag system does not halt, an “immortal” word in the system. The alternative, of course, being that the system eventually halts on every (finite) word. It is shown here that this problem, the immortality problem for tag systems, is recursively unsolvable of degree 0″ (Corollary II).