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hw9-sol

# hw9-sol - ECS 120 Introduction to the Theory of Computation...

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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. Solution: The proof is very similar to the diagonalization proof that A TM is unde- cidable, just substitute a Halting Oracle for a TM in the original proof. This also implies that there is a hierarchy of a sort among Oracle machines (e.g. imagine a Halting Oracle with added functionality which allows it to decide O TM , etc.) 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. A. L = {h M i : M accepts some even-length string } .

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