Athale-relativity - RELATIVIZATION CSE860 Vaishali Athale...

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RELATIVIZATION CSE860 Vaishali Athale
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Overview Introduction Idea behind “Relativization” Concept of “Oracle” Review of Diagonalization Proof Limits of Diagonalization method Proof idea Proof Implications of proof
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Introduction Revisiting question of NP=P? Diagonalization proof used to show that Halting Problem is undecidable Can we use it to prove that NP=P or NP P? Strong evidence against the possibility of solving the P Versus NP problem using Diagonalization technique.(BGS theorem, 1975)
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Idea behind “Relativization” Turing Machine provided with some information for “free” Concept of “Oracle” for a language Black box that answers membership of a string in the given language in one step Information affects the outcome of Turing Machine TM can solve some problems more easily
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Example of “Oracle” Consider an oracle for SAT Ability to solve SAT problem in a single step, for any size Boolean formula. With the help of an oracle for SAT, Turing Machine can solve any NP problem in polynomial time Regardless of whether NP=P, every NP problem is polynomial time reducible to SAT Such a machine is computing relative to the SAT problem – “Relativization”
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Oracle Turing Machine Consider an oracle for language A Oracle Turing Machine M A gets the result of question of whether the given string is in A in a single computation step.
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