Modified post correspondence problem 113 formal

Modified Post Correspondence Problem
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FORMAL LANGUAGES AND AUTOMATA THEORY 10CS56
• We have seen an undecidable problem, that is, given a Turing machine M and
an input w, determine whether M will accept w (universal language problem).
• We will study another undecidable problem that is not related to Turing machine
directly.
Given two lists A and B:
A = w1, w2, …, wk B = x1, x2, …, xk
The problem is to determine if there is a sequence of one or more
integers i1, i2, …, im such that:
w1wi1wi2…wim = x1xi1xi2…xim
(wi, xi) is called a corresponding pair.
Example
A B
i
w
i
x
i
1 11 1
2 1 111
3 0111 10
4 10 0
This MPCP instance has a solution: 3, 2, 2, 4:
w
1
w
3
w
2
w
2
w
4
= x
1
x
3
x
2
x
2
x
4
= 1101111110
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FORMAL LANGUAGES AND AUTOMATA THEORY 10CS56
8.2: a un decidable problem that is RE
Undecidability of PCP
To show that MPCP is undecidable, we will
reduce the universal language problem
(ULP) to MPCP:
Universal
A mapping MPCP Language
Problem (ULP)
If MPCP can be solved, ULP can also be solved.
Since we have already shown that ULP is un-
decidable, MPCP must also be undecidable.
Mapping ULP to MPCP
• Mapping a universal language problem instance to an MPCP instance
is not as easy.
• In a ULP instance, we are given a Turing machine M and an input w, we
want to determine if M will accept w. To map a ULP instance to an MPCP
instance success-fully, the mapped MPCP instance should have a
solution if and only if M accepts w.
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FORMAL LANGUAGES AND AUTOMATA THEORY 10CS56
Mapping ULP to MPCP
ULP instance
MPCP instance
Given:
(T,w)
Construct an
MPCP instance
Two lists:
A and B
If T accepts w, the two lists can be matched.
Otherwise, the two lists cannot be matched.
Mapping ULP to MPCP
• We assume that the input Turing machine T:
– Never prints a blank
– Never moves left from its initial head position.
• These assumptions can be made because:
– Theorem (p.346 in Textbook): Every language accepted by a TM
M2 will also be accepted by a TM M1 with the following restrictions:
(1) M1 ‟s head never moves left from its initial position. (2) M1 never
writes a blank.
Mapping ULP to MPCP
Given T and w, the idea is to map the transition function of T to strings in
the two lists in such a way that a matching of the two lists will correspond
to aconcatenation of the tape contents at each time step .
We will illustrate this with an example first.
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FORMAL LANGUAGES AND AUTOMATA THEORY 10CS56
Example of ULP to MPCP
• Consider the following Turing machine:
T = ({q
0
, q
1
},{0,1},{0,1,#}, , q
0
, #, {q
1
})
q
0
0/0, L q
1
1/0, R
(q
0
,1)=(q
0
,0,R) (q
0
,0)=(q
1
,0,L)
• Consider input w=110.
Example of ULP to MPCP
• Now we will construct an MPCP
instance from T and w. There are five
types of strings in list A and B:
• Starting string (first pair ):
List A
List B
# #q
0
110#

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FORMAL LANGUAGES AND AUTOMATA THEORY 10CS56
Example of ULP to MPCP
• Strings from the transition function :
List A List B
q
0
1 0q
0
(from
(q
0
,1)=(q
0
,0,R))
0q
0
0 q
1
00
(from
(q
0
,0)=(q
1
,0,L))
1q
0
0 q
1
10
(from
(q
0
,0)=(q
1
,0,L))
Example of ULP to MPCP
• Strings for copying:
List A List B
# #
0 0
1 1
Example of ULP to MPCP
• Strings for consuming the tape symbols at the end:
List A List B List A List B
0q1 q1 0q11 q1
1q1 q1 1q10 q1
q10 q1 0q10 q1
q11 q1 1q10 q1