User blog:LittlePeng9/ITTM galore

In this blog post I'll post programs for infinite time Turing machines.

Preliminaries
Machines I'll post here are going to have only one tape. These machines don't have full power of infinite-time computability.

Programs are going to be written in style similar to that used in this simulator. Because ITTM's investigated by Hamkins et al. are only one way infinite, every input will have left end marker #, and every program will have this mandatory transition: * # * r * Which follows convention that, if machine tries to move left of leftmost cell, it just doesn't move. No machine can have any other transition involving #. Also, moving right from this character doesn't count as a step. If I write "empty input" I will mean "only # on input tape".

In every program let L denote limit state. When machine enters limit stage, it enters L state and moves to left end marker (and instantly moves right). There is no other transition rule which allows entering L state. In my convention, L state doesn't count towards state count, just like halt state doesn't, because it's mandatory for this type of machines. At one point I might start making machines related to writable ordinals. I'll already make a convention on pairing function: \(\langle a,b\rangle=2^a\cdot(2b+1)-1\) (one may easily verify this is a one-to-one \(\mathbb{N}^2\rightarrow\mathbb{N}).

Last but not least, machines will use only 3 characters: _ (empty cell), 1 (filled cell) and # (unique left end marker). I believe everything else can be found on this article or in Hamkins's paper.

Programs
This is where everything will happen!

Machine halting in exactly \(\omega\) steps
We have to start somewhere, right? Machine halts whenever it enters limit state. * # * r * 0 _ _ r 0 L _ _ r halt

Machine halting in exactly \(\omega^2\) steps
I believe this is longest lasting machine with one state, estabilishing \(\gamma[1]=\omega^2\). After every limit stage machine flashes a 1 on the first cell, so it is first 1 at limit stage at \(\omega^2\), then machine halts. * # * r * 0 _ _ r 0 0 1 _ r 0 L _ 1 l 0 L 1 1 r halt