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Unformatted text preview: 1 Post Correspondence Problem (PCP): Given a finite set of ordered pairs ( x 1 , y 1 ), . . . , ( x n , y n ) of strings over Σ, determine whether there is a finite sequence of integers ( i 1 , i 2 , . . ., i m ), with each i j ∈ { 1 , . . ., n } , such that x i 1 x i 2 · · · x i m = y i 1 y i 2 · · · y i m . (1) For a particular instance { ( x 1 , y 1 ) , . . ., ( x n , y n ) } of the problem PCP, if there exists a sequence ( i 1 , i 2 , . . ., i m ) satisfying (1), then we say the string x i 1 x i 2 · · · x i m is a solution to this instance. An easy way to understand the problem PCP is to treat each pair ( x i , y i ) as a domino with string x i at the top and string y i at the bottom: x i y i . The question here then is to select, from the given dominoes x 1 y 1 , x 2 y 2 , · · · , x n y n , with unlimited supply for each type, some dominoes and arrange them into a row so that the top part of the dominoes spells the same word as the bottom part of the dominoes. For instance, we can obtain a solution baaaaa from the following given dominoes aa a , ba baaa as follows: ba aa aa baaa a a . ⋆ Example 0.1 Prove that the problem PCP is undecidable (with respect to some alphabet Σ ). Proof . Let M be a fixed DTM, with a one-way tape (i.e., the original one-tape DTM defined in Section 4.1), such that the problem of determining whether M halts on a given string x ∈ { , 1 } ∗ is undecidable. (I.e., L ( M ) is a nonrecursive set.) We construct a reduction from the halting problem of this fixed DTM M to the problem PCP. That is, for each string x , we need to produce an instance P x = { ( x 1 , y 1 ) , . . ., ( x m , y m ) } such that M halts on x if and only if P x has a solution z ....
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