Let $n$ be a member of $\omega$, which the omnipotent provers $Y$ and $N$ know. A Turing machine will be run starting with the $n$ already inputted, and the machine can have a natural number inputted from the prover of its choice, with no limit on the number of times it can do so.


Let $S$ be a subset of $\omega$.

Definition: S is decidable from competing provers if and only if

there exists a Turing machine such that, for all members $n$ of $\omega$, when run as above,
(1) if $n\in S$ then $Y$ has a strategy that will force the Turing machine to accept
(0) else $N$ has a strategy that will force the Turing machine to reject



What subsets $S$ of $2^{\omega}$ are decidable from competing provers?
What if the Turing machine is required to always halt?
In each case, is the resulting class low for itself?

Let $n$ be a member of $\omega$, which the omnipotent provers $Y$ and $N$ know. A Turing machine will be run starting with the $n$ already inputted, and the machine can have a natural number inputted from the prover of its choice, with no limit on the number of times it can do so.


Let $S$ be a subset of $\omega$.

Definition: S is decidable from competing provers if and only if

there exists a Turing machine such that, for all members $n$ of $\omega$, when run as above,
(1) if $n\in S$ then $Y$ has a strategy that will force the Turing machine to accept
(0) else $N$ has a strategy that will force the Turing machine to reject



What subsets $S$ of $\omega$ are decidable from competing provers?
What if the Turing machine is required to always halt?
In each case, is the resulting class low for itself?