lec3 - Summary Remaining TM details: big-oh necessary....

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1 CS151 Complexity Theory Lecture 3 April 5, 2011 April 5, 2011 2 Summary • Remaining TM details: big-oh necessary. • First complexity classes: L, P, PSPACE, EXP • First separations (via simulation and diagonalization): P EXP , L PSPACE • First major open questions: L = P P = PSPACE • First complete problems: – CVAL is P -complete – Succinct CVAL is EXP -complete ? ? April 5, 2011 3 Summary EXP PSPACE P L April 5, 2011 4 Nondeterminism: introduction A motivating question: • Can computers replace mathematicians? L = { (x, 1 k ) : statement x has a proof of length at most k } April 5, 2011 5 Nondeterminism: introduction • Outline: – nondeterminism – nondeterministic time classes NP , NP -completeness, P vs. NP coNP – NTIME Hierarchy – Ladner’s Theorem April 5, 2011 6 Nondeterminism • Recall deterministic TM Q finite set of states alphabet including blank: “_” q start , q accept , q reject in Q – transition function: δ : Q x ∑ ! Q x ∑ x {L, R, -}
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2 April 5, 2011 7 Nondeterminism nondeterministic Turing Machine: Q finite set of states alphabet including blank: “_” q start , q accept , q reject in Q – transition relation (Q x ∑) x (Q x ∑ x {L, R, -}) • given current state and symbol scanned, several choices of what to do next. April 5, 2011 8 Nondeterminism • deterministic TM: given current configuration, unique next configuration • nondeterministic TM: given current configuration, several possible next configurations q start x 1 x 2 x 3 …x n q start x 1 x 2 x 3 …x n q accept q reject x L x L April 5, 2011 9 Nondeterminism • asymmetric accept/reject criterion q start x 1 x 2 x 3 …x n q start x 1 x 2 x 3 …x n q accept q reject x L x L “guess” “computation path” April 5, 2011 10 Nondeterminism • all paths terminate • time used : maximum length of paths from root • space used : maximum # of work tape squares touched on any path from root April 5, 2011 11 Nondeterminism NTIME(f(n)) = languages decidable by a multi-tape NTM that runs for at most f(n) steps on any computation path , where n is the input length, and f : N ! N NSPACE(f(n)) = languages decidable by a multi-tape NTM that touches at most f(n) squares of its work tapes along any computation path , where n is the input length, and f : N ! N April 5, 2011 12 Nondeterminism • Focus on time classes first: NP = k NTIME(n k ) NEXP = k NTIME(2 n k )
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3 April 5, 2011 13 Poly-time verifiers Very useful alternate definition of NP: Theorem : language L is in NP if and only if it is expressible as: L = { x| 9 y, |y| ≤ |x| k , (x, y) R } where R is a language in P.
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lec3 - Summary Remaining TM details: big-oh necessary....

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