Relative computability and relative enumerability

Fix the canonical binary numeral for each natural number (with zero encoded as $0$). For an oracle $X\subseteq \mathbb{N}$, an oracle program is a finite deterministic machine program with ordinary transitions as in The one-step configuration relation, together with query instructions. A query instruction reads a finite binary numeral $q$ delimited on the tape and selects one of two specified successor states according as $q\in X$ or $q\mathrm{\notin }X$; it leaves the tape and head unchanged. Malformed queries select the negative successor. On input $n$, start in the initial state with its numeral at the left end of an otherwise blank tape and the head at position zero. Iterate these transitions until an accepting or rejecting state is entered, if that ever happens.

A set $A\subseteq \mathbb{N}$ is $X$-computable when some such program halts on every numeral input and accepts exactly the members of $A$. It is $X$-c.e. when some such program accepts exactly the members of $A$; on nonmembers it may reject or run forever. Write $A{\le }_{T}X$ for $X$-computability. A function $f:\mathbb{N}\to \mathbb{N}$ is total $X$-computable when some such oracle program halts on every numeral input $n$ with the numeral of $f(n)$ as its delimited output.

An oracle program still makes finitely many transitions on any halting run; it does not read an infinite oracle at once.