{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# lec3 - CS151 Complexity Theory Lecture 3 April 5 2011...

This preview shows pages 1–14. Sign up to view the full content.

CS151 Complexity Theory Lecture 3 April 5, 2011

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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, -}
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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
April 5, 2011 8 Nondeterminism deterministic TM: given current configuration, unique next configuration nondeterministic TM: given current configuration, several possible next configurations q s tart x 1 x 2 x 3 …x n q s tart x 1 x 2 x 3 …x n q ac c e pt q re je c t x   L   L
April 5, 2011 9 Nondeterminism asymmetric accept/reject criterion q s tart x 1 x 2 x 3 …x n q s tart x 1 x 2 x 3 …x n q ac c e pt q re je c t   L   L “g ue s s ” “c o m putatio n  path”

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
April 5, 2011 12 Nondeterminism Focus on time classes first: NP = k NTIME(n k ) NEXP = k NTIME(2 n k )
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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}