Athale-relativity

# 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
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
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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