NP and NP completeness 姚鹏晖 Email: pyao@nju.edu.cn 助教: 刘明谋 liu.mingmou@smail.nju.edu.cn 答疑时间: 周四 2pm-4pm, 计算机科学与技术楼 502

What we have learnt Turing machines: a mathematical model of computational processes. Church-Turing thesis. The existence of universal Turing machines. The existence of uncomputable functions. Halting problem. The complexity class P, which captures all the problems that are efficiently solvable. Robustness of the definition of Turing machines. Strong Church-Turing thesis.

Definition of NP

Examples in NP

Examples in NP

Relation between NP and P

Deterministic Turing machines

Nondeterministic Turing machines

Nondeterministic Turing machines

Nondeterministic Turing machines

Reducibility and NP-Completeness

Reducibility and NP-Completeness

Reducibility and NP-Completeness

Reducibility and NP-Completeness

The Cook-Levin Theorem

The Cook-Levin Theorem

Computation is local ? It can be verified locally

Must accept within p(n) steps. Variables interpretation Numbers Expression Interpretation Numbers Initial contents of the cells Initial state of M 1 Initial position of the heads At most one symbol per cell At least one symbol per cell Cells remain unchanged unless written Only one state at a time Only one head position at a time Possible transitions step t when the head of i-th tape is at position j Must accept within p(n) steps.

The Cook-Levin Theorem

The Cook-Levin Theorem

Examples in NP

Examples in NP

Decision vs. Search

Must accept within p(n) steps. Variables interpretation Numbers Expression Interpretation Numbers Initial contents of the cells Initial state of M 1 Initial position of the heads At most one symbol per cell At least one symbol per cell Cells remain unchanged unless written Only one state at a time Only one head position at a time Possible transitions step t when the head of i-th tape is at position j Must accept within p(n) steps.

CoNP

NEXP

What if P=NP?

What if P≠NP?