Oracle programming in a higher level language (Q5967040)

The author considers a special concept of computability with oracles, in which oracles are considered as partial functions and it is possible to receive two types of answers for an arbitrary \(x\); the first answer is the value of \(F(x)\) in case the latter exists, otherwise the answer is ``refusal.'' Both kinds of answers are regarded as regular. The author considers the situation in which a machine proceeds the first \(k\) refusals but ``gets tired'' and ``hands up'' after the \((k+1)\)st refusal. Such specific questions as the selection principles, enumerability of some natural sets, etc. are examined. Although these properties are easy to prove for usual computability, they may fail to be true for the above computability. In the course of the study, a ``higher level language'' is defined and used.