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48+Slides--Mapping+Reductions

48+Slides--Mapping+Reductions - CS103 HO#48 Slides-Mapping...

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CS103 HO#48 Slides--Mapping Reductions 5/18/11 1 Proving Theorems Using Reductions reduces to solves A B Solvable(B) Solvable(A) ¬ Solvable(A) ¬ Solvable(B) Problem for A Solver for B Solution Solver for A Modify Problem Problem for 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. The function f is called the reduction of A to B. A B f To test whether w A, we use the reduction f and test whether f(w) B. Mapping Reducibility w A f(w) B f(w) B w A f(w) B w A w A f(w) B 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 i.e., w A if and only if f(w) B Mapping Reducibility w A f(w) B f(w) B w A f(w) B w A f(w) B w A w A f(w) B w A f(w) B So A m B A m B 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 i.e., w A if and only if f(w) 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. A B 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

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CS103 HO#48 Slides--Mapping Reductions 5/18/11 2 Proving Theorems Using Mapping Reductions maps to solves A B ¬ Decidable(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 computable 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 .
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