Call a real $x$ limit computable iff there is a Turing machine $T$ such that, for any $i\in\omega$, there is $t(i)\in\omega$ such that the $i$th entry on the tape is not changed after time $t(i)$ and is from that time on equal to $1$ if $i\in x$ and otherwise equal to $0$. Intuitively, this means that the output of $T$ converges to $x$, though not at any effective rate.

Let us now consider transfinite iterations of this process: Define the Turing calcuation of a machine $T$ along an ordinal time axis by letting $T$ work classically at successor stages and at limit steps $\lambda$, let the tape content $B_{i\lambda}$ of cell $i$ be the eventually constant value of $(B_{i\alpha}|\alpha<\lambda)$ if that sequence is eventually constant. If it isn't for some $i\in\omega$, the calculation is undefined. The machine state and the head position at time $\lambda$ are taken to be the limes inferiors of the earlier states/head positions.

These calculations are similar to infinite time Turing machine calculations with arbitrary time and tape length $\omega$, but probably much weaker, as the 'eventually constant' rule for the tape content is much more restrictive than the liminf-rule of $ITTM$s.

My question is: Has this computability concept been studied? If yes, I would be thankful for references. If no, are there obvious reasons why not? (E.g. it is trivial, or obviously equivalent to well-studied notion XY...)


1 Answer 1


Yes, this notion has been studied. It is considerably weaker than the model of infinite time Turing machines. To see this, observe that your machines can never do anything after time $\omega^2$, because whatever local configuration the machine is in at time $\omega^2$ must have arisen many times earlier, and so whatever bit it would change then it must have changed earlier, which would have caused it to crash at $\omega^2$.

Meanwhile, if you are careful in arranging your computation, you can compute all arithmetic assertions in time uniformly less than $\omega^2$.

  • $\begingroup$ Thanks. Can you point me to a reference where this concept is considered? In particular, which reals can these machines compute? $\endgroup$
    – M Carl
    Nov 9, 2012 at 17:55

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