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CS5371 Theory of Computation
Homework 4 (Suggested Solution)
1.
Ans.
Suppose on the contrary that
T
is decidable. Let
R
be its decider. Then, the
following TM
Q
is a decider for
A
TM
:
Q
= “On input
h
M,w
i
,
1. Construct a TM
M
0
as follows:
M
0
= “On input
x
,
1. If
x
6
=
011
,
accept
.
2. Run
M
on
w
.
3. If
M
accepts
w
,
accept
.”
2. Run
R
to decide if
h
M
0
i
is in
T
.
3. If yes (i.e.,
R
accepts),
accept
.
4. Else,
reject
.”
It is easy to check that
Q
runs in ﬁnite steps. Also, in Step 1,
M
0
has the property that:
(i) If
M
accepts
w
,
L
(
M
0
) = Σ
*
, so that
h
M
0
i ∈
T
.
(ii) Else,
L
(
M
0
) = Σ
*
 {
011
}
, so that
h
M
0
i
/
∈
T
.
So, if
Q
accepts
h
M,w
i
, it must mean that
R
accepts
h
M
0
i
, which implies that
h
M
0
i ∈
T
,
which implies
M
accepts
w
. On the other hand, if
Q
rejects
h
M,w
i
,
R
rejects
h
M
0
i
, which
in turn implies that
M
does not accept
w
.
Thus,
Q
is a decider for
A
TM
, and a contradiction occurs. So, we conclude that
T
is
undecidable.
2. In the
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This note was uploaded on 02/22/2010 for the course CS 881 taught by Professor H.f. during the Spring '10 term at Shahid Beheshti University.
 Spring '10
 H.F.

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