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Post correspondence problem

The Post correspondence problem is an undecidable decision problem that was introduced by Emile Post[?]. Because it is simpler than the Halting problem and the Entscheidungsproblem it is often used in proofs of undecidability.

Informally the problem can be described as follows. Given a dictionary that contains pairs of phrases, i.e., a list of words, that mean the same, decide if there is a sentence that means the same in both languages.

Definition of the problem

The input of the problem consists of two finite lists:

u1, ..., un and v1, ..., vn

of words over some alphabet Σ with at least two symbols. A solution to this problem is a sequence of indexes i1, ..., ik, 1 <= ij <= n, such that

ui1...uik = vi1...vik.

The decision problem then is to decide whether such a solution exists or not.

Example of an instance of the problem

Consider the following two lists:

        u1    u2    u3    u4     v1    v2     v3     v4
      "aba" "bbb" "aab" "bb"    "a" "aaa" "abab" "babba"
A solution to this problem would be the sequence 1, 4, 3, 1 because

u1u4u3u1 = "ababbaababa" = v1v4v3v1

If the two lists would have consisted of, for example, only u1, u2, u3 and v1, v2, v3 then there would have been no solution.



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