1. The Post Correspondence Problem
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2. Some undecidable problems for context-free languages:
Is ?
are context-free grammars
Is context-free grammar ambiguous?
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3. The Post Correspondence Problem
Input:
Two sets of strings
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4. There is a Post Correspondence Solution
if there is a sequence such that:
PC-solution:
Indices may be repeated or omitted
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5. Example:
PC-solution:
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6. Because total length of strings from
Example:
There is no solution
Because total length of strings from
is smaller than total length of strings from
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7. The Modified Post Correspondence Problem
Inputs:
MPC-solution:
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8. Example:
MPC-solution:
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9. Membership problem Input: Turing machine string Question: Undecidable
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10. Input: unrestricted grammar string
Membership problem
Input: unrestricted grammar
string
Question:
Undecidable
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11. Suppose we have a decider for the MPC problem
String Sequences
MPC solution?
YES
MPC problem
decider NO
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12. We will build a decider for the membership problem
YES
Membership
problem
decider NO
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13. Membership problem decider
The reduction of the membership problem
to the MPC problem:
Membership problem decider
yes
yes
MPC problem
decider no no
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14. Membership problem decider
We need to convert the input instance of
one problem to the other
Membership problem decider
yes
yes
MPC problem
decider
Reduction? no no
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15. Convert grammar and string
Reduction:
Convert grammar and string
to sets of strings and
Such that:
There is an MPC
solution for
generates
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16. Membership problem decider
yes
yes
Construct
MPC problem
decider no no
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17. Since the membership problem is undecidable,
The MPC problem is undecidable
END OF PROOF
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18. Some undecidable problems for context-free languages:
Is ?
are context-free grammars
Is context-free grammar
ambiguous?
We reduce the PC problem to these problems
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19. grammars. It is undecidable to determine if
Theorem:
Let be context-free
grammars. It is undecidable
to determine if
(intersection problem)
Proof:
Reduce the PC problem to this
problem
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20. Suppose we have a decider for the intersection problem
Context-free
grammars
Empty-
interection
problem
decider
YES NO
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21. We will build a decider for the PC problem
String Sequences
PC solution?
YES
PC problem
decider NO
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22. PC problem decider no yes yes no The reduction of the PC problem
to the empty-intersection problem:
PC problem decider no
yes
Intersection
problem
decider
yes no
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23. PC problem decider no yes Reduction? yes no
We need to convert the input instance of
one problem to the other
PC problem decider no
yes
Intersection
problem
decider
Reduction?
yes no
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24. Introduce new unique symbols:
Context-free grammar :
Context-free grammar :
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25. has a PC solution
if and only if
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26. Because are unique There is a PC solution: Fall 2006
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27. PC problem decider no yes yes no Construct Intersection Context-Free
Grammars
Intersection
problem
decider
yes no
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28. Since PC is undecidable, the Intersection problem is undecidable
END OF PROOF
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29. For a context-free grammar ,
Theorem:
For a context-free grammar ,
it is undecidable to determine
if G is ambiguous
Proof:
Reduce the PC problem
to this problem
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30. PC problem decider no yes yes no Construct Ambiguous Context-Free
Grammar
Ambiguous
problem
decider
yes no
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31. start variable of start variable of start variable of Fall 2006
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32. has a PC solution if and only if if and only if is ambiguous Fall 2006
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