The Immortality Problem for Non-Erasing Turing Machines

TLDR: In this article, it was shown that the immortality problem for non-erasing TM-s is decidable, if the tape is allowed to contain ultimately periodic words, and if only a finite number of non-blanks are allowed.

Abstract: Publisher Summary In this chapter M is considered a Turing machine (TM). An instantaneous description (ID) of M is a triple 〈q,X,n〉 where q∈K, X∈ ∑∞ and n≥1. describes that M is in state q with the read-write head scanning square no. n and that the tape T contains X. M is to stop if M tries to go off the tape at the left end. M is called a non-writing TM if it contains no write-instructions. The immortality problem (IP) associated with a set of TM-s is the problem of deciding, for a given TM in the set, whether or not there exists an immortal ID. It is shown that IP for non-erasing TM-s is decidable, if the tape is allowed to contain ultimately periodic words. If, however, the tape is restricted to contain only a finite number of non-blanks, then the IP for the set of nonerasing TM-s is recursively undecidable (of degree 0").