Post_Correspondence

Post_Correspondence - ThePostCorrespondenceProblem...

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The Post Correspondence Problem  
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Some undecidable  problems for context-free languages:  Is context-free grammar        ambiguous? G  Is                                     ? = ) ( ) ( 2 1 G L G L 2 1 , G G are context-free grammars
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We need a tool to prove that the previous problems for context-free languages are undecidable : The Post Correspondence Problem
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The Post Correspondence Problem Input: Two  s e ts  o f     s tring s n w w w A , , , 2 1 = n v v v B , , , 2 1 = n
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The re  is  a  P o s t  C o rre s po nde nc e  S o lutio n if the re  is  a s e q ue nc e                    s uc h tha t: k j i , , , k j i k j i v v v w w w = PC -s o lutio n: Indic e s  m ay b e  re pe ate d o r o m itte d
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Exa m ple : 11 100 111 001 111 11 1 w 2 w 3 w 1 v 2 v 3 v : A : B PC -s o lutio n: 3 , 1 , 2 3 1 2 3 1 2 v v v w w w = 11100111
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Exa m ple : 001 00 1000 0 11 011 1 w 2 w 3 w 1 v 2 v 3 v : A : B The re  is  no  s o lutio n Be c a us e  to tal le ng th o f s tring s  fro m   is   s m a lle r tha n to ta l le ng th o f s tring s  fro m    B A
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The   M o difie d  P o s t  C o rre s po nde nc e  Pro b le m Inputs : n w w w A , , , 2 1 = n v v v B , , , 2 1 = MPC -s o lutio n: k j i , , , , 1 k j i k j i v v v v w w w w 1 1 =
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Exa m ple : 11 100 111 001 111 11 1 w 2 w 3 w 1 v 2 v 3 v : A : B MPC -s o lutio n: 2 , 3 , 1 2 3 1 2 3 1 v v v w w w = 11100111
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1. The  MPC  pro b le m  is  unde c ida b le 2.  The  PC  pro b le m  is  unde c ida b le (by reducing MPC to PC) (by reducing the membership to MPC) We will show:
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Theorem:   The  MPC  pro b le m  is  unde c idab le Proof:  We  will re duc e  the  m e m b e rs hip              pro b le m  to  the  MPC  pro b le m
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Input:   Turing  m a c hine             s tring   M w Q ue s tio n: ? ) (M L w Unde c idab le
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This note was uploaded on 12/02/2011 for the course AR 107 taught by Professor Gracegraham during the Fall '11 term at Montgomery College.

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Post_Correspondence - ThePostCorrespondenceProblem...

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