Oracle turing reducibility interface

To decide A from B when $A{\le }_{m}B$, run the total map f, put its canonical result on the query tape, make one query to B and output that bit. Both f and the single query terminate, and the iff in F1 gives the correct answer. Identity querying also decides any B from itself.

Suppose $A{\le }_{T}B$ and $B{\le }_{T}C$. A C-program simulates the A-decider, keeping its finite configuration in a protected work zone. When the simulated machine queries a canonical number b, run the total C-decider for B on b in separate work zones, and return its bit as the simulated B answer; malformed queries return zero immediately. Finite table simulation and work-zone copying do not depend on knowing B. Each subcall halts. On a fixed input the original A computation has finitely many transitions and queries; induction over those transitions shows the composite eventually reaches the same halt and bit. Thus $A{\le }_{T}C$, with no uniform bound on the lengths or number of subcalls required. Replacing the B-decider by a recognizer would invalidate this argument at a negative query, since that call might diverge.