2
April 7, 2011
7
Sparse languages and
NP
• Sparse language
: one that contains at
most
poly(n)
strings of length
≤ n
• not very expressive – can we show this
cannot be
NP
-complete (assuming
P
≠
NP
) ?
– yes: Mahaney ‟82 (homework problem)
• Unary language
: subset of 1* (at most n
strings of length ≤ n)
April 7, 2011
8
Sparse languages and
NP
Theorem
(Berman ‟78): if a unary language
is
NP
-complete then
P
=
NP
.
• Proof:
– let U
1* be a unary language and assume
SAT ≤ U via reduction R
– φ(x
1
,x
2
,…,x
n
) instance of SAT
April 7, 2011
9
Sparse languages and
NP
– self-reduction tree for φ:
. . .
φ
(x
1
,x
2
,…,x
n
)
φ
(1,x
2
,…,x
n
)
φ
(0,x
2
,…,x
n
)
φ(0,0,…,0)
φ(1,1,…,1)
.
.
.
satisfying assignment
April 7, 2011
10
Sparse languages and
NP
– applying reduction R:
. . .
R(
φ
(x
1
,x
2
,…,x
n
))
R(
φ
(1,x
2
,…,x
n
))
R(
φ
(0,x
2
,…,x
n
))
R(
φ(0,0,…,0))
R(
φ(1,1,…,1))
.
.
.
satisfying assignment
April 7, 2011
11
Sparse languages and
NP
• on input of length
m = |φ(x
1
,x
2
,…,x
n
)|,
R
produces string of length
≤ p(m)
• R‟s different outputs are “colors”
– 1 color for strings not in 1
*
– at most p(m) other colors
• puzzle solution
can solve SAT in
poly(p(m)+1, n+1) = poly(m)
time!
April 7, 2011
12
Summary
• nondeterministic time classes:
NP, coNP, NEXP
• NTIME Hierarchy Theorem:
NP
≠
NEXP
• major open questions:
P
=
NP
NP
=
coNP
?
?