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Lecture 2

1 Review

·
| |

∗

{
}^∗→ {
}

∗

{
}

{
|
}

∈

∈∈
⇒⇒

• 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).

∈ time
space
space

time
⊆ space

2 P, NP, and NP-Completeness

2.1 The Class P

def

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.)

2.2 The Classes NP and coNP

∈ NP

∈
P
∈ P

| |
∈
P
NP

∈
P
NP

NP
P ⊆ NP ⊆ _≥1time

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

| |

ntime
∈ ntime

| |
| |

nspace

| |

_≥1ntime

ntime
time

time
⊆ ntime

coNP

|
∈ C
{
}^∗\

coNP
∈ coNP

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

Karp reducible
many-to-one reducible

≤

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

def

0
0

≤

≤≤

∈ P
∈ P

∈ NP
∈ NP

NP NP

NP-hard

∈ NP
NP
≤

NP-complete

∈ NP
NP

coNP
coNP

coNP
∈ coNP

≤
coNP
∈ coNP

NP
NP
NP NP
NP
NP

∃
∈ {
}

References

Recommended publications

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