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##### The Post Correspondence Problem

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**The Post Correspondence Problem**Prof. Busch - LSU**Some undecidable problems for**context-free languages: • Is ? are context-free grammars • Is context-free grammar ambiguous? Prof. Busch - LSU**We need a tool to prove that the previous**problems for context-free languages are undecidable: The Post Correspondence Problem Prof. Busch - LSU**The Post Correspondence Problem**Input: Two sets of strings Prof. Busch - LSU**There is a Post Correspondence Solution**if there is a sequence such that: PC-solution: Indices may be repeated or omitted Prof. Busch - LSU**Example:**PC-solution: Prof. Busch - LSU**Example:**There is no solution Because total length of strings from is smaller than total length of strings from Prof. Busch - LSU**The Modified Post Correspondence Problem**Inputs: MPC-solution: Prof. Busch - LSU**Example:**MPC-solution: Prof. Busch - LSU**We will show:**1. The MPC problem is undecidable (by reducing the membership to MPC) 2. The PC problem is undecidable (by reducing MPC to PC) Prof. Busch - LSU**Theorem: The MPC problem is undecidable**Proof: We will reduce the membership problem to the MPC problem Prof. Busch - LSU**Membership problem**Input: Turing machine string Question: Undecidable Prof. Busch - LSU**Membership problem**Input: unrestricted grammar string Question: Undecidable Prof. Busch - LSU**Suppose we have a decider for**the MPC problem String Sequences MPC solution? YES MPC problem decider NO Prof. Busch - LSU**We will build a decider for**the membership problem YES Membership problem decider NO Prof. Busch - LSU**The reduction of the membership problem**to the MPC problem: Membership problem decider yes yes MPC problem decider no no Prof. Busch - LSU**We need to convert the input instance of**one problem to the other Membership problem decider yes yes MPC problem decider Reduction? no no Prof. Busch - LSU**Reduction:**Convert grammar and string to sets of strings and Such that: There is an MPC solution for generates Prof. Busch - LSU**Grammar**: start variable : special symbol For every symbol For every variable Prof. Busch - LSU**Grammar**string : special symbol For every production Prof. Busch - LSU**Example:**Grammar : String Prof. Busch - LSU**Grammar :**Prof. Busch - LSU**Derivation:**Prof. Busch - LSU**Derivation:**Prof. Busch - LSU**Derivation:**Prof. Busch - LSU**Derivation:**Prof. Busch - LSU**Derivation:**Prof. Busch - LSU**has an MPC-solution**if and only if Prof. Busch - LSU**Membership problem decider**yes yes Construct MPC problem decider no no Prof. Busch - LSU**Since the membership problem is undecidable,**The MPC problem is undecidable END OF PROOF Prof. Busch - LSU**Theorem: The PC problem is undecidable**Proof: We will reduce the MPC problem to the PC problem Prof. Busch - LSU**Suppose we have a decider for**the PC problem String Sequences PC solution? YES PC problem decider NO Prof. Busch - LSU**We will build a decider for**the MPC problem String Sequences MPC solution? YES MPC problem decider NO Prof. Busch - LSU**The reduction of the MPC problem**to the PC problem: MPC problem decider yes yes PC problem decider no no Prof. Busch - LSU**We need to convert the input instance of**one problem to the other MPC problem decider yes yes PC problem decider Reduction? no no Prof. Busch - LSU**: input to the MPC problem**Translated to : input to the PC problem Prof. Busch - LSU**replace**For each For each Prof. Busch - LSU**PC-solution**Has to start with These strings Prof. Busch - LSU**PC-solution**MPC-solution Prof. Busch - LSU**has a PC solution**if and only if has an MPC solution Prof. Busch - LSU**MPC problem decider**yes yes Construct PC problem decider no no Prof. Busch - LSU**Since the MPC problem is undecidable,**The PC problem is undecidable END OF PROOF Prof. Busch - LSU**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 Prof. Busch - LSU**Theorem:**Let be context-free grammars. It is undecidable to determine if (intersection problem) Reduce the PC problem to this problem Proof: Prof. Busch - LSU**Suppose we have a decider for the**intersection problem Context-free grammars Empty- interection problem decider YES NO Prof. Busch - LSU**We will build a decider for**the PC problem String Sequences PC solution? YES PC problem decider NO Prof. Busch - LSU**The reduction of the PC problem**to the empty-intersection problem: PC problem decider no yes Intersection problem decider no yes Prof. Busch - LSU**We need to convert the input instance of**one problem to the other PC problem decider no yes Intersection problem decider Reduction? no yes Prof. Busch - LSU