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Say I define a turing machine with the following states $$Q \in \{q_0, q_1, \dots, q_n, q_{\text{halt}}\}$$ Here, $n$ is guaranteed to be finite.

Then, also have a set of rules where I do something like $$\langle q_n, \text{input}\rangle\mapsto\langle q_{n-1}, \text{output}, \text{movement}\rangle\quad n\neq 0$$ $$\langle q_0, \text{input}\rangle\mapsto\langle q_\text{halt}, \text{output}, \text{movement}\rangle\quad$$

I am wondering if this way of defining a turing machine is "proper", or if using an ambiguous amount of states $n$ and using stuff like $n\neq 0$ when defining rules of how the turing machine evolves is not allowed.

Thanks!

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  • $\begingroup$ I'm a bit lost. I don't understand what you mean by "proper" or what concerns you have about that. I wonder if it might help to add the broader context. Why do you think $n$ is ambiguous? Of course your Turing machine will depend on $n$, but I don't see anything that makes $n$ ambiguous. $\endgroup$
    – D.W.
    Commented Apr 22 at 6:55
  • $\begingroup$ @D.W. In all the examples I've seen, all the states of a turing machine are explicitely defined, so something like this $Q \in \{q_0, q_1, q_3, q_{\text{halt}}\}$, instead of having an variable $n$ number of states. I'm just wondering if what I did is a "proper" (sorry, idk a better word lol) way, since I've never seen turing machines defined like the way i did here $\endgroup$
    – Max0815
    Commented Apr 22 at 7:01

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It is fine, but by limiting your transitions to take the machine from $q_n$ to $q_{n-1}$ for every $n$, you severely restrict the power of your machines: they will take exactly $n+1$ steps and then halt, when starting from state $q_n$. You can only decide finite languages that way.

If you allow transitions from $q_m$ to $q_n$ for arbitrary $m,n$, you have unrestricted Turing machines.

You may also wish to consider intermediate cases, e.g. restricting transitions to be between states $q_m$ and $q_n$ where $m \geq n$. This will limit the Turing machines you can define, but not their power: you can still accept and decide all the same languages arbitrary Turing machines can accept and decide.

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