1. CSC 4170
Theory of Computation
Turing reducibility
Section 6.3

2. Oracle Turing machines
Giorgi Japaridze Theory of Computability
An oracle for a language B is an external device that is capable of
reporting whether any string w is a member of B.
An oracle Turing machine (OTM) is a modified Turing machine that
has the additional capability of querying an oracle.
Example: Construct an OTM with an oracle for ATM that decides ATM
O = “On input <M,w>, where M is a TM and w is a string:
Language A is Turing reducible (or decidable relative to) language B,
written ATB, iff there is an OTM with an oracle for B that decides A.

3. Using Turing reducibility for proving decidability/undecidability
Giorgi Japaridze Theory of Computability
Theorem 6.21:
a) If ATB and B is decidable, then A is decidable.
Consequently,
b) If ATB and A is undecidable, then B is undecidable.
Proof (a): If B is decidable, then we may replace the oracle for B by
an actual procedure that decides B. Thus we may replace the OTM
that decides A relative to B by an ordinary TM that decides A.