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The Post Correspondence Problem
Fall 2006 Costas Busch - RPI
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Some undecidable problems for context-free languages:
Is ? are context-free grammars Is context-free grammar ambiguous? Fall 2006 Costas Busch - RPI
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The Post Correspondence Problem
Input: Two sets of strings Fall 2006 Costas Busch - RPI
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There is a Post Correspondence Solution
if there is a sequence such that: PC-solution: Indices may be repeated or omitted Fall 2006 Costas Busch - RPI
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Example: PC-solution: Fall 2006 Costas Busch - RPI
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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 Fall 2006 Costas Busch - RPI
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The Modified Post Correspondence Problem
Inputs: MPC-solution: Fall 2006 Costas Busch - RPI
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Example: MPC-solution: Fall 2006 Costas Busch - RPI
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Membership problem Input: Turing machine string Question: Undecidable
Fall 2006 Costas Busch - RPI
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Input: unrestricted grammar string
Membership problem Input: unrestricted grammar string Question: Undecidable Fall 2006 Costas Busch - RPI
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Suppose we have a decider for the MPC problem
String Sequences MPC solution? YES MPC problem decider NO Fall 2006 Costas Busch - RPI
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We will build a decider for the membership problem
YES Membership problem decider NO Fall 2006 Costas Busch - RPI
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Membership problem decider
The reduction of the membership problem to the MPC problem: Membership problem decider yes yes MPC problem decider no no Fall 2006 Costas Busch - RPI
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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 Fall 2006 Costas Busch - RPI
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Convert grammar and string
Reduction: Convert grammar and string to sets of strings and Such that: There is an MPC solution for generates Fall 2006 Costas Busch - RPI
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Membership problem decider
yes yes Construct MPC problem decider no no Fall 2006 Costas Busch - RPI
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Since the membership problem is undecidable,
The MPC problem is undecidable END OF PROOF Fall 2006 Costas Busch - RPI
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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 Fall 2006 Costas Busch - RPI
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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 Fall 2006 Costas Busch - RPI
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Suppose we have a decider for the intersection problem
Context-free grammars Empty- interection problem decider YES NO Fall 2006 Costas Busch - RPI
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We will build a decider for the PC problem
String Sequences PC solution? YES PC problem decider NO Fall 2006 Costas Busch - RPI
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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 Fall 2006 Costas Busch - RPI
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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 Fall 2006 Costas Busch - RPI
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Introduce new unique symbols:
Context-free grammar : Context-free grammar : Fall 2006 Costas Busch - RPI
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has a PC solution if and only if Fall 2006 Costas Busch - RPI
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Because are unique There is a PC solution: Fall 2006
Costas Busch - RPI
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PC problem decider no yes yes no Construct Intersection Context-Free
Grammars Intersection problem decider yes no Fall 2006 Costas Busch - RPI
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Since PC is undecidable, the Intersection problem is undecidable
END OF PROOF Fall 2006 Costas Busch - RPI
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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 Fall 2006 Costas Busch - RPI
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PC problem decider no yes yes no Construct Ambiguous Context-Free
Grammar Ambiguous problem decider yes no Fall 2006 Costas Busch - RPI
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start variable of start variable of start variable of Fall 2006
Costas Busch - RPI
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has a PC solution if and only if if and only if is ambiguous Fall 2006
Costas Busch - RPI
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