Lecture 2

1 Review

·
| |

1

·
| |

∗

• {
• }^∗ → {
• }

∗

• {
• }

• {
• |
• }

∈

∈∈
⇒⇒

• • time
• ∈ time

• ·
• ·
• | |

• space

¹Note that we do not count the space used on the input or output tapes; this allows us to meaningfully speak of sub-linear space machines (with linear- or superlinear-length output).

2

0

• ∈ time
• space
• space

0

• time
• ⊆ space

2 P, NP, and NP-Completeness

2.1 The Class P

P

def

P

time

≥1

P
| |
P

•

•

100
100 8

P
P

²This decision is also motivated by “speedup theorems” which state that if a language can be decided in time
(resp., space) ( ) then it can be decided in time (resp., space) ( ) for any constant . (This assumes that ( ) is a “reasonable” function, but the details need not concern us here.)

P

2.2 The Classes NP and coNP

• NP
• NP

∈ NP

3

• | |
• ∈

• ∈
• P
• ∈ P

| |
∈
P
NP

∈
P
NP

NP
P ⊆ NP ⊆ _≥1 time

• P ⊆ NP
• ∈ NP

| |

• | |
• ∈

| |
| |

³It is essential that the running time of is to require the length of to be at most (| |) in condition (2). be measured in terms of the length of alone. An alternate approach

≤ (| |)

• ∈ {
• }

(| |)

| | ·

∈ time

NP

• 0
• 1

• 0
• 1

• ∈
• ⇔

| |

ntime
∈ ntime

| |
| |

nspace

| |

• NP
• P

NP

_≥1 ntime

?

• P
• NP

• P
• NP

• P
• NP

• P
• NP

• ntime
• time

• 2
• 3

• time
• ⊆ ntime

coNP

• def
• def

C

• coC
• coC

NP

• |
• ∈ C
• {
• }^∗ \

coNP
∈ coNP

4

• | |
• ∈

⁴See footnote 3.

∈

• coNP
• NP

• coNP
• ∈

• NP
• coNP

SAT { |

}

SAT
SAT ∈ coNP

SAT ∈ NP

TAUT
TAUT

• {
• }

TAUT

coNP coNP

• P ⊆ NP ∩ coNP
• NP
• coNP

SAT ∈ NP
NP coNP

2.3 NP-Completeness

NP

0

5

• 0
• 0

0

• Karp reducible
• many-to-one reducible

0

• ∈
• ∈

0

≤

⁵Technically speaking, I mean “at least as hard as”.

0

• 0
• 0

def

0
0

≤

• ≤
• ≤
• ≤

≤≤

• ∈ P
• ∈ P

• ∈ NP
• ∈ NP

NP NP

• NP
• NP

NP

• 0
• 0
• 0

NP-hard

∈ NP
NP
≤

• 0
• 0

NP-complete

• ∈ NP
• NP

• 0
• 0

• NP
• ≤

0

• coNP
• coNP

coNP
∈ coNP

• 0
• 0
• 0
• 0

• ≤
• coNP
• ∈ coNP

NP
NP
NP NP
NP
NP

• ∃
• ∈ {
• }

NP

References