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). If the ITTM keeps on running infinitely, at time \(\omega\), each cell on the tape is set to the of that cell's symbol for the steps approaching that limit. In addition, the head is set to the leftmost cell and the state is set to limit.

A mathematical object is writable if an ITTM, given trivial input, can halt with the object on the output tape. Similarly, it is eventually writable if the ITTM's output ultimately converges to the object, and accidentally writable if it occurs anywhere during the ITTM's computation. These definitions give rise to several large countable ordinals:


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

It has been shown that \( \leq \lambda < \zeta < \leq \Sigma\).