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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