Non-erasing turing machines: A new frontier between a decidable halting problem and universality

Abstract

We define a new criterion which allows to separate cases when all non erasing Turing machines on {0, 1} have a decidable halting problem from cases where a universal non erasing machine can be constructed. It is the case of the number of left instructions in the machine program. In this paper we give the main ideas of the proof for both parts of the frontier result. We prove that there is a universal non-erasing Turing machine whose program has precisely 3 left instructions and that the halting problem is decidable for any non-erasing Turing machine on alphabet {0, 1}, the program of which contains at most 2 left instructions. For this latter result, we have a uniform decision algorithm.