Higher-Order Feedback Computation

Feedback Turing machines are Turing machines which can query a halting oracle which has information on the convergence or divergence of feedback computations. To avoid a contradiction by diagonalization, feedback Turing machines have two ways of not converging: they can diverge as standard Turing machines, or they can freeze. A natural question to ask is: what about feedback Turing machines which can ask if computations of the same type converge, diverge, or freeze? We define alpha th order feedback Turing machines for each computable ordinal alpha. We also describe feedback computable and semi-computable sets using inductive definitions and Gale-Stewart games.