Solutions for Problem Set 8
CS 373: Theory of Computation
Assigned: October 26, 2010
Due on: November 2, 2010 at 10am
Homework Problems
Problem 1
. [Category: Proof] Read about Post’s Correspondence Problem from section 5.2. Solve problem
5.18.
Solution:
Given an instance of PCP over alphabet Σ =
{
a
1
,...a
n
}
, we will reduce it to an instance of PCP
over the binary alphabet. The idea will be to apply the homomorphism
h
that maps the symbol
a
i
to string
10
i
. The crucial property about this mapping the following: for
u,v
∈
Σ
*
,
u
=
v
iﬀ
h
(
u
) =
h
(
v
). Let
P
be
an instance of PCP over Σ. Take
h
(
P
) to be the PCP instance over the binary alphabet, where we replace
any string
w
appearing in any tile of
P
by
h
(
w
). Given that
u
=
v
iﬀ
h
(
u
) =
h
(
v
), we can conclude that
P
is in PCP iﬀ
h
(
P
) is in PCP. Since we have reduced PCP over Σ to PCP over
{
0
,
1
}
and PCP over Σ is
undecidable, we conclude PCP over
{
0
,
1
}
is undecidable.
Problem 2
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 Fall '08
 Viswanathan,M

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