FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY

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15-453
FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY
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THE POST CORRESPONDENCE PROBLEM
THURSDAY OCT 6
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THE PCP GAME
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GENERAL RULE 1
If every top string is longer than the
corresponding bottom one, there cant be a match
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GENERAL RULE 2
If there is a domino with the same string on the
top and on the bottom, there is a match
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POST CORRESPONDENCE PROBLEM
Given a collection of dominos, is there a match?
PCP P P is a set of dominos with a match
PCP is undecidable!
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THE FPCP GAME
is just like the PCP game except that a match
has to start with the first domino
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FPCP
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FPCP
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Theorem FPCP is undecidable
Proof
Assume machine C decides FPCP
We will show how to use C to decide ATM
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Given (M,w)
we will construct a set of dominos P where a
match is an accepting computation history for M
on w
P
C
P has a match?
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x ? x, L
2n
0 ? 0, L
0 n 0
q2
? ? ?, R
? ? ?, L
x ? x, R
x ? x, R
q0
q1
q3
0 ? ?, R
0 ? x, R
x ? x, R
0 ? 0, R
? ? ?, R
0 ? x, R
? ? ?, R
qaccept
q4
x ? x, R
? ? ?, R
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q00000
?q1000
?xq300
?x0q40
?x0xq3
?x0q2x
?xq20x
?q2x0x
q2?x0x
q00000?q1000?xq300?x0q40?x0xq3 ...

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Given (M,w), we will construct an instance P of
FPCP in 7 steps
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STEP 1
Put
into P
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STEP 2
If ?(q,a) (p,b,R) then add
STEP 3
If ?(q,a) (p,b,L) then add
for all c ? G
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x ? x, L
2n
0 ? 0, L
0 n 0
q2
? ? ?, R
? ? ?, L
x ? x, R
x ? x, R
q0
q1
q3
0 ? ?, R
0 ? x, R
x ? x, R
0 ? 0, R
? ? ?, R
0 ? x, R
? ? ?, R
qaccept
q4
x ? x, R
? ? ?, R
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STEP 4
add
for all a ? G
STEP 5
add
STEP 6
add
for all a ? G
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STEP 7
add
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0 ? 0, R
? ? ?, R
q0
q1
qaccept
0 ? 0, R
? ? ?, R
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Given (M,w), we can construct an instance of FPCP
that has a match if and only if M accepts w
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Can convert an instance of FPCP into one of PCP
Let u u1u2un, define
?u ? u1 ? u2 ? u3 ? ? un
u? u1 ? u2 ? u3 ? ? un ?
?u? ? u1 ? u2 ? u3 ? ? un ?

FPCP

PCP
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Given (M,w), we can construct an instance of PCP
that has a match if and only if M accepts w
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WWW.FLAC.WS
Read chapters 5.2 and 5.3 of the book for next
time