CSci 5403, Spring 2010
Homework 3
due: March 2, 2010
1.
PH lower bounds.
Show that for every
k >
0,
PH
contains languages whose circuit
complexity is Ω(
n
k
).
Hint:
Recall that by the size hierarchy theorem, there exist languages with circuit
complexity Ω(
n
k
). Write a
PH
sentence that ensures that a language is one of these.
2. Compressed circuits
Prove that every language has circuits of size
O
(2
n
/n
).
Hint:
The circuit for input size
n
might take a very long time to build. One approach
to this problem treats the first lg
n

1 bits separately from the rest.
3.
Shallow circuits.
Besides
NC
, two other important classes of circuits that are often
discussed in complexity theory are
AC
and
TC
.
A language is in
AC
i
if it is decided by
a polynomialsize,
O
((log
n
)
i
)depth circuit family, with unbounded fanin gates in the set
{∧
,
∨
,
¬}
; and a language is in
TC
i
if it is decided by a polynomial size,
O
((log
n
)
i
)depth
circuit family using unbounded fanin
threshold gates
— gates with a specified threshold
k
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 Spring '08
 Sturtivant,C
 Computational complexity theory, Tci, threshold gates, unbounded fanin gates, fanin threshold gates, arbitrary threshold gates

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