CSC 4170 Theory of Computation Turing reducibility Section 6.3

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.

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.