Post correspondence problem

The Post correspondence problem is an undecidable decision problem that was introduced by Emil 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, ..., u_n and v₁, ..., v_n

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

u_i₁...u_{i_k} = v_i₁...v_{i_k}.

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 = aba + bb + aab + aba = ababbaababa = a + babba + abab + a = 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.