hw9 - A L ={h M i M accepts some even-length string B L ={h...

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ECS 120: Introduction to the Theory of Computation Homework 9 Problem 1. In the movie The Matrix there is a TM called The Oracle which can solve problems that other TMs cannot (even some that the TM Neo couldn’t!), although, as she states herself, there are problems that are even beyond her reach. Here, we’ll try to understand why. Define Halting Oracles as regular TMs but with the added functionality of a “black box” which allows them to decide A TM = {h M,w i| w L ( M ) ,M is a TM } instanta- neously. Prove that the language O TM = {h O,w i| w L ( O ) ,O is a Halting Oracle } is not decidable by a Halting Oracle. Problem 2. Classify the following languages as decidable , acceptable (but not decid- able), co-acceptable (but not decidable), or neither acceptable nor co-acceptable. Prove all your answers, giving decision procedures or reductions.
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Unformatted text preview: A. L = {h M i : M accepts some even-length string } . B. L = {h M i : M accepts some palindrome } . C. L = {h M i : M never prints a “0” (regardless of the input) } . D. L = {h φ i : φ is a Boolean formula which has no satisfying assignment } . Problem 3. Suppose you are given a polynomial time algorithm which, on input of a Boolean formula φ , decides if φ is satisfiable. Describe an efficient procedure which finds a satisfying assignment for φ . Problem 4. Prove that if P = NP then ∀ A ∈ P \ {∅ , Σ * } , A is NP-complete. Problem 5. Prove that DOUBLE-SAT= {h φ i| φ has at least two satisfying assignments } is NP-complete. 1...
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