=========================================================================== CSC 363H Lecture Summary for Week 5 Spring 2007 =========================================================================== Each of the following language is undecidable: - E_TM = { | M is a TM such that L(M) = {} }: Assume R decides E_TM, and construct S as follows: S = "On input : - Compute , the description of the following TM: Q = "On input x: - if x != w reject - if x == w then simulate M on x; accept if M accepts; reject if M rejects." - Run R on input and do the opposite (if R accepts, reject; if R rejects, accept)." Then S decides A_TM, a contradiction. - EQ_TM = { | M_1 and M_2 are TMs such that L(M_1) = L(M_2) }: Assume R decides EQ_TM and construct S as follows: S = "On input : - Compute , the description of the following TM: M' = "On input x: reject." - Run R on input and do the same." Then S decides E_TM, a contradiction. -------------------- Mapping Reducibility -------------------- General structure of proof of undecidability of some language A: Assume R decides A. Construct S to decide B (some undecidable language): S = "On input x: - Compute y such that y in A iff x in B. - Run R on y and do the same." Then, S decides B because y in A iff x in B. Since structure always the same, concentrate on "core" part: construction of y from x such that y in A iff x in B. Definition 5.20 (1st ed: 5.15): "mapping reducibility"; "reduction". Example 5.26 (1st ed: 5.20): E_TM <=m EQ_TM. Theorem 5.22 (1st ed: 5.16): If A <=m B and B is decidable, then A is decidable. Proof: (see textbook -- done in lecture) Theorem 5.28 (1st ed: 5.22): If A <=m B and B is recognizable, then A is recognizable. Proof: (see textbook -- done in lecture) Corollary 5.23 (1st ed: 5.17): If A <=m B and A is undecidable, then B is undecidable. Corollary 5.29 (1st ed: 5.17): If A <=m B and A is unrecognizable, then B is unrecognizable.