Infinite time Turing machine

Infinite time Turing machines (ITTMs) are a generalization of Turing machines to transfinite ordinals.

In addition to all the standard properties of ordinary TMs, ITTMs have a special state called the limit state (similar to start and halt). At each step, the ITTM proceeds normally with a timer \(t\) starting at zero and incrementing at each step. If it does not halt for any \(t < \omega\), at \(t = \omega\) each cell on the tape is set to the of that cell's symbol for steps \(t = 0, 1, 2, \ldots\). In addition, the head is set to the initial cell and the state is set to limit. The ITTM continues as before, and if it continues to \(t = \omega2\) without halting, the same thing happens with the limit superior of the tape for \(t = \omega, \omega + 1, \omega + 2, \ldots\). In general, if the ITTM reaches a countable limit ordinal \(\alpha\) without halting, at time \(t = \alpha\), it is set to the limit superior of the tape at \(t = \alpha[n]\) as \(n \rightarrow \omega\).

ITTMs are capable of outputting far more outputs than ordinary TMs, including ordinals:


 * An ordinal \(\alpha\) is writable iff an ITTM with trivial input can write \(\alpha\) and immediately halt.
 * An ordinal \(\alpha\) is eventually writable iff the output tape of an ITTM with trivial input ultimately converges to \(\alpha\).
 * An ordinal \(\alpha\) is accidentally writable iff an ITTM with trivial input writes \(\alpha\) at any step.
 * An ordinal \(\alpha\) is clockable iff an ITTM with trivial input halts at time \(\alpha\).

(The first three definitions can apply to many mathematical objects other than ordinals.) These definitions give rise to several large countable ordinals (the infinite time Turing machine ordinals):


 * \(\lambda\), the supremum of all writable ordinals
 * \(\zeta\), the supremum of all eventually writable ordinals
 * \(\Sigma\), the supremum of all accidentally writable ordinals
 * \(\gamma\), the supremum of all clockable ordinals

It has been shown that \( \ll \lambda = \gamma < \zeta < \Sigma\).

Fundamental sequences for these ordinals are reasonably simple to define, allowing use in FGH:

This idea won't work, however, for \(\Sigma\).
 * \(\lambda[n]\) is the supremum of all ordinals writable with ITTMs with at most \(n\) states. The function \(f(n) = \lambda[n]\) is in direct analogy with the busy beaver function.
 * \(\gamma[n]\) is the supremum of all ordinals clockable with ITTMs with at most \(n\) states.
 * \(\zeta[n]\) is the supremum of all ordinals eventually writable with ITTMs with at most \(n\) states.