6.841 Advanced Complexity Theory
Feb 11, 2009
Lecture 3
Lecturer: Madhu Sudan
Scribe: Debmalya Panigrahi
In today’s lecture, we will focus on
nonuniform models of computation
. In nonuniform computation, we
have a
different gadget/program/machine for each input size
for a given problem. Specifically, in this lecture,
we will focus on
circuit families
(which are equivalent to deterministic Turing Machines (DTMs) with
advice
strings),
branching programs
and
formulas
, and compare them.
Finally, we will study
Neciporuk’s lower
bound
on the succinctness of branching programs.
1
Circuits and Circuit Families
A
circuit
is a
directed acyclic graph
(DAG), where the nodes are of three kinds:
input nodes
,
logic gates
and an
output node
. The goal of a circuit with
n
input nodes is to compute a boolean function
f
:
{
0
,
1
}
n
→ {
0
,
1
}
1
in the following way: if a boolean vector ¯
x
= (
x
1
, x
2
, . . . , x
n
) is fed to the
n
input nodes, then the logic gates
compute the function
f
(¯
x
) and output it at the output node. A
basis
is a set of logic gates that can be used
to compute
any
boolean function.
A circuit
C
has two main complexity measures:
•
The
size
of circuit
C
, denoted by SIZE(
C
) is the number of edges in
C
.
2
For a function
f
, C

SIZE(
f
) =
min
C
computes
f
SIZE(
C
). C

SIZE(
f
) depends on the basis of gates used in the circuit, but only upto
a constant factor.
•
The
depth
of circuit
C
, denoted by DEPTH(
C
) is the longest path in
C
.
The following is a set of simple facts about circuits:
•
Given a circuit
C
of size
S
, evaluating
C
(¯
x
) for an input vector ¯
x
takes
O
(
S
2
) time.
•
To simulate a
DTIME
(
t
(
n
)) TM on an input of length
n
, we need a circuit of size
O
(
t
(
n
)
2
).
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 Spring '09
 MadhuSudan
 Logic gate, Computational complexity theory, Boolean function, size, Circuit Families

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