45+Slides--Decidability,+Rice_s+Theorem (3)

45+Slides--Decidability,+Rice_s+Theorem (3) - CS103 HO#45

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Unformatted text preview: CS103 HO#45 Slides--Decidablility, Rice's Theorem 11/8/10 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 . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . How do we get started? Theorem 4.11: A TM = { M, w | M is a Turing machine and M accepts w } is undecidable. Suppose H is a decider for A TM . Build the machine D. Run D on its own description. H: A TM D D D accepts D D D Accept Reject If D accepts D , it rejects it, and if D rejects D , it accepts it. Thus D and H cannot exist, and A TM is undecidable . D rejects or loops on D Proving Theorems Using Reductions reduces to decides A TM HALT TM Decidable(A TM ) Decidable(HALT TM ) M, w Accepts Decider for A TM Rejects Decider for HALT TM Proving Theorems Using Reductions reduces to decides A TM HALT TM Decidable(A TM ) Decidable(HALT TM ) M, w Accepts S: Decider for A TM Rejects Decider for HALT TM R Simulate M on w Accepts Rejects Accepts Theorem 5.1 HALT TM is not decidable. R e je c ts Proving Theorems Using Reductions reduces to solves Multiplication Addition Reduce the problem to addition 3 4 4 + 4 + 4 Adder 12 Multiplier CS103 HO#45 Slides--Decidablility, Rice's Theorem 11/8/10 2 Proving Theorems Using Reductions reduces to decides A TM HALT TM Decidable(A TM ) Decidable(HALT TM ) M, w Accepts Decider for A TM Rejects Decider for HALT TM Reduce the problem to a Halting problem M', w' Proving Theorems Using Reductions M, w Accepts Decider for A TM Rejects Decider for HALT TM Reduce the problem to a Halting problem M', w' Simulate M on w M, w Accepts Rejects Loops Halt TM groups the results like this Proving Theorems Using Reductions M, w Accepts Decider for A TM Rejects Decider for HALT TM Reduce the problem to a Halting problem M', w' Simulate M on w M, w Accepts Rejects Loops We would like them grouped like this Proving Theorems Using Reductions M, w Accepts Decider for A TM Rejects Decider for HALT TM Reduce the problem to a Halting problem M', w' Simulate M on w M, w Accepts Rejects Loops We just change what happens when M rejects, so that it loops instead Proving Theorems Using Reductions M, w Accepts Decider for A TM Rejects Decider for HALT...
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45+Slides--Decidability,+Rice_s+Theorem (3) - CS103 HO#45

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