1
CS151
Complexity Theory
Lecture 5
April 12, 2011
April 12, 2011
2
Introduction
Power from an unexpected source?
•
we know
P
≠
EXP
, which implies no poly-
time
algorithm
for Succinct CVAL
•
poly-size Boolean
circuits
for Succinct
CVAL ??
Does
NP
have
linear-size, log-depth
Boolean circuits ??
April 12, 2011
3
Outline
•
Boolean circuits and formulas
•
uniformity and advice
•
the
NC
hierarchy and parallel computation
•
the quest for circuit lower bounds
•
a lower bound for formulas
April 12, 2011
4
Boolean circuits
•
C computes function
f:{0,1}
n
{0,1}
in
natural way
–
identify C with function
f
it computes
•
circuit
C
–
directed acyclic graph
–
nodes: AND (
); OR (
);
NOT (
); variables x
i
x
1
x
2
x
3
…
x
n
April 12, 2011
5
Boolean circuits
•
size
= # gates
•
depth
= longest path from input to output
•
formula
(or expression)
: graph is a tree
•
every function
f:{0,1}
n
{0,1}
computable
by a circuit of size at most
O(n2
n
)
–
AND
of n literals for each x such that f(x) = 1
–
OR
of up to 2
n
such terms
April 12, 2011
6
Circuit families
•
circuit works for specific input length
•
we‟re used to
f:∑
*
!
{0,1}
•
circuit
family
: a circuit for each input
length
C
1
, C
2
, C
3
, … = “{C
n
}”
• “{C
n
} computes f” iff for all x
C
|x|
(x) = f(x)
• “{C
n
} decides L”, where L is the language
associated with f

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