L9 - Lecture 9 Undecidability and Rice’s Th David Dill Department of Computer Science 1 Outline • A language that is RE but not recursive •

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Unformatted text preview: Lecture 9: Undecidability and Rice’s Th David Dill Department of Computer Science 1 Outline • A language that is RE but not recursive • Rice’s theorem 2 A Language that is RE but not Recursive L u = { ( ( M ) ,w ) | M accepts w } . Theorem: L u is RE. L u is accepted by the “Universal Turing Machine” M U . Idea: To check ( ( M ) ,w ) , simulate M on w using one tape to hold second tape to simulate the tape of M (so the second tape starts o it). 3 Proofs by Reduction This is one of the most important proof techniques in computer scien To prove that problem P 2 is hard, show that there is an “easy” reduc known hard problem P 1 to P 2 . Then P 2 is at least as hard as P 1 . Example: (Suppose it is well-known that ( a student ) cannot lift a ca Thm ( a student ) cannot lift a loaded truck ( P 2 ). Proof By reduction from the car-lifting problem. Suppose ( a student ) could lift a loaded truck. Then, ( a student ) could solve the car-lifting problem by putting the c and lifting the truck. But, it is known that ( a student ) cannot lift a car. IMPORTANT: Make sure you are doing the reduction in the right dire hard problem → new problem . 4 A wrong proof Thm: ( a student ) cannot lift an anvil. proof: By reduction to the car-lifting problem. We can reduce anvil lifting to car-lifting by putting the anvil in a car. It is known that ( a student ) cannot lift a car. Therefore, ( a student ) cannot lift an anvil. It may be true, but this is not a proof because the reduction goes the Substitute “feather” for anvil.) 5 Theorem: L u is not RE, Proof sketch: We can reduce the problem of recognizing...
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L9 - Lecture 9 Undecidability and Rice’s Th David Dill Department of Computer Science 1 Outline • A language that is RE but not recursive •

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