54 Slides--More on Mapping Reductions

54 Slides--More on Mapping Reductions - CS103 HO#54...

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HO#54 Slides--More on Mapping Reductions May 24, 2010 1 Mapping Reducibility Language A is mapping reducible to language B, which is written A M B, if there is a computable function f: *  * where for every w, w A f(w) B. The function f is called the reduction of A to B. AB f To test whether w A, we use the reduction f and test whether f(w) B. Mapping Reducibility Language A is mapping reducible to language B, which is written A M B, if there is a computable function f: *  * where for every w, w A f(w) B. w A f(w) B f(w) B w A f(w) B w A w A f(w) B To test whether w A, we use the reduction f and test whether f(w) B. Note that A M B A M B Mapping Reducibility Theorem 5.22: If A M B and B is decidable, then A is decidable. Corollary 5.23: If A M B and A is undecidable, then B is undecidable. B decidable means we can always tell whether f(w) B. That means we can tell whether w A, i.e., we can decide A. AB f f Proving Theorems Using Mapping Reductions maps to solves A B Solvable(B) Solvable(A) ¬ Solvable(A) ¬ Solvable(B) Map the problem to B Problem for A Solver for B Accept Solver for A Problem for B Reject Proving Theorems Using Mapping Reductions maps to solves A B ¬Dec idab le(A TM ) ¬ Decidable(HALT TM ) f M, w Decider for HALT TM Accept Decider for A TM M', w' Reject Proving Theorems Using Mapping Reductions To show that A TM M HALT TM we need a function f such that M, w  A TM if and only if M', w'  HALT TM where f( M, w ) = M', w' . f uses M, w to build M' and w' as follows: M' : On input x, run M on x. If M accepts, then M' accepts. If M rejects, then M' enters a loop. (and if M loops, M' loops) w' = w For improper inputs, f outputs M', w'  HALT TM . So if M accepts w, M' halts, and

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54 Slides--More on Mapping Reductions - CS103 HO#54...

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