Lecture16 - 15-453 FORMAL LANGUAGES AUTOMATA AND...

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FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY 15-453
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NP = NTIME(n k ) k N
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Theorem: L NP if there exists a poly-time Turing machine V( erifier ) with L = { x | 5 y ( witness ) |y| = poly(|x|) and V(x,y) accepts } Proof: (1) If L = { x | 5 y |y| = poly(|x|) and V(x,y) accepts } then L NP Because we can guess y and then run V (2) If L NP then L = { x | 5 y |y| = poly(|x|) and V(x,y) accepts } Let N be a non-deterministic poly-time TM that decides L and define V(x,y) to accept if y is an accepting computation history of N on x
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A language is in NP if and only if there exist polynomial-length certificates for membership to the language SAT is in NP because a satisfying assignment is a polynomial-length certificate that a formula is satisfiable
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NP = all the problems for which once you have the answer it is easy (i.e. efficient) to verify
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P = NP?
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If P = NP… Cryptography as we know it would not be possible (e.g. RSA) Mathematicians would be out of a job AI program become perfect as exhaustive search is efficient
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If P = NP… Writing symphonies is as easy as listening to them. Being a chef is as easy as eating. Writing Shakespeare is as easy as recognizing Shakespeare. Generation is as easy as recognition :
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POLY-TIME REDUCIBILITY A language A is polynomial time reducible to language B, written A P B , if there is a polynomial time computable function f : Σ* Σ*, where for every w, w A f(w) B f is called a polynomial time reduction of A to B
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NP Complete Problem: hardest problem in NP
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