# fall 2006costas busch - rpi1 the post correspondence problem

Post on 15-Jan-2016

213 views

Category:

## Documents

0 download

Embed Size (px)

TRANSCRIPT

• The Post Correspondence Problem

Costas Busch - RPI

• Some undecidable problems forcontext-free languages: Is context-free grammar ambiguous? Is ?are context-free grammars

Costas Busch - RPI

• We need a tool to prove that the previousproblems for context-free languagesare undecidable:The Post Correspondence Problem

Costas Busch - RPI

• The Post Correspondence ProblemInput:Two sets of strings

Costas Busch - RPI

• There is a Post Correspondence Solutionif there is a sequence such that:PC-solution:Indices may be repeated or omitted

Costas Busch - RPI

• Example:PC-solution:

Costas Busch - RPI

• Example:There is no solutionBecause total length of strings from is smaller than total length of strings from

Costas Busch - RPI

• The Modified Post Correspondence ProblemInputs:MPC-solution:

Costas Busch - RPI

• Example:MPC-solution:

Costas Busch - RPI

• 1. The MPC problem is undecidable2. The PC problem is undecidable(by reducing MPC to PC)(by reducing the membership to MPC)We will show:

Costas Busch - RPI

• Theorem: The MPC problem is undecidableProof: We will reduce the membership problem to the MPC problem

Costas Busch - RPI

• Membership problemInput: Turing machine string Question:Undecidable

Costas Busch - RPI

• Membership problemInput: unrestricted grammar string Question:Undecidable

Costas Busch - RPI

• Suppose we have a decider for the MPC problemMPC solution?YESNOString SequencesMPC problemdecider

Costas Busch - RPI

• We will build a decider for the membership problemYESNOMembershipproblemdecider

Costas Busch - RPI

• MPC problemdeciderMembership problem deciderThe reduction of the membership problemto the MPC problem:yesyesnono

Costas Busch - RPI

• MPC problemdeciderMembership problem deciderWe need to convert the input instance ofone problem to the otheryesyesnonoReduction?

Costas Busch - RPI

• Convert grammar and stringReduction:to sets of strings and Such that:generatesThere is an MPCsolution for

Costas Busch - RPI

• : special symbolFor every symbolGrammar: start variableFor every variable

Costas Busch - RPI

• GrammarFor every production: special symbolstring

Costas Busch - RPI

• Example:Grammar : String

Costas Busch - RPI

• Costas Busch - RPI

• Costas Busch - RPI

• Grammar :

Costas Busch - RPI

• Derivation:

Costas Busch - RPI

• Derivation:

Costas Busch - RPI

• Derivation:

Costas Busch - RPI

• Derivation:

Costas Busch - RPI

• Derivation:

Costas Busch - RPI

• has an MPC-solution if and only if

Costas Busch - RPI

• MPC problemdeciderMembership problem deciderConstruct

yesyesnono

Costas Busch - RPI

• Since the membership problem is undecidable,The MPC problem is undecidableEND OF PROOF

Costas Busch - RPI

• Theorem: The PC problem is undecidableProof: We will reduce the MPC problem to the PC problem

Costas Busch - RPI

• Suppose we have a decider for the PC problemPC solution?YESNOString SequencesPC problemdecider

Costas Busch - RPI

• We will build a decider for the MPC problemMPC solution?YESNOString SequencesMPC problemdecider

Costas Busch - RPI

• PC problemdeciderMPC problem deciderThe reduction of the MPC problemto the PC problem:yesyesnono

Costas Busch - RPI

• PC problemdeciderMPC problem decideryesyesnonoReduction?We need to convert the input instance ofone problem to the other

Costas Busch - RPI

• : input to the MPC problem: input to the PC problemTranslated to

Costas Busch - RPI

• For eachFor eachreplace

Costas Busch - RPI

• PC-solutionHas to start with These strings

Costas Busch - RPI

• PC-solutionMPC-solution

Costas Busch - RPI

• has a PC solutionhas an MPC solutionif and only if

Costas Busch - RPI

• PC problemdeciderMPC problem decideryesyesnonoConstruct

Costas Busch - RPI

• Since the MPC problem is undecidable,The PC problem is undecidableEND OF PROOF

Costas Busch - RPI

• Some undecidable problems forcontext-free languages: Is context-free grammar ambiguous? Is ?are context-free grammarsWe reduce the PC problem to these problems

Costas Busch - RPI

• Theorem:Proof:Let be context-free grammars. It is undecidableto determine ifReduce the PC problem to thisproblem(intersection problem)

Costas Busch - RPI

• Suppose we have a decider for theintersection problemEmpty-interectionproblemdeciderYESNOContext-free grammars

Costas Busch - RPI

• We will build a decider for the PC problemPC solution?YESNOString SequencesPC problemdecider

Costas Busch - RPI

• PC problem deciderThe reduction of the PC problemto the empty-intersection problem:noyesyesnoIntersectionproblemdecider

Costas Busch - RPI

• PC problem decidernoyesyesnoReduction?We need to convert the input instance ofone problem to the otherIntersectionproblemdecider

Costas Busch - RPI

• Context-free grammar :Introduce new unique symbols:Context-free grammar :

Costas Busch - RPI

• if and only ifhas a PC solution

Costas Busch - RPI

• Because are uniqueThere is a PC solution:

Costas Busch - RPI

• PC problem decidernoyesyesnoIntersectionproblemdeciderConstructContext-FreeGrammars

Costas Busch - RPI

• Since PC is undecidable,the Intersection problem is undecidableEND OF PROOF

Costas Busch - RPI

• For a context-free grammar ,Theorem:it is undecidable to determineif G is ambiguousProof:Reduce the PC problem to this problem

Costas Busch - RPI

• PC problem decidernoyesyesnoAmbiguousproblemdeciderConstructContext-FreeGrammar

Costas Busch - RPI

• start variable of start variable of start variable of

Costas Busch - RPI

• if and only ifis ambiguousif and only ifhas a PC solution

Costas Busch - RPI

Recommended