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CS 151
Complexity Theory
Spring 2011
Problem Set 6
Out: May 12
Due:
May 19
Reminder: you are encouraged to work in groups of two or three; however you must turn in your
own writeup and note with whom you worked. You may consult the course notes and the text
(Papadimitriou). Please attempt all problems.
To facilitate grading, please turn in each
problem on a separate sheet of paper and put your name on each sheet. Do not staple
the separate sheets.
1. The following problem comes from Learning Theory, where the VCdimension gives impor
tant information about the diﬃculty of learning a given concept. Given a collection
S
=
{
S
1
, S
2
, . . . , S
m
}
of subsets of a ﬁnite set
U
, the
VC dimension
of
S
is the size of the largest
set
X
⊆
U
such that for every
X
′
⊆
X
, there is an
i
for which
S
i
∩
X
=
X
′
(we say that
X
is
shattered
by
S
). A Boolean circuit
C
that computes a function
f
:
{
0
,
1
}
m
×{
0
,
1
}
n
→ {
0
,
1
}
succinctly represents a collection
S
of 2
m
subsets of
U
=
{
0
,
1
}
n
as follows: the set
S
i
consists
of exactly those elements
x
for which
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