Post Correspondence Problem for short words

We prove that the generalized Post Correspondence Problem is undecidable for instances where the lengths of the image words are at most 2 and the number of pairs of words is at most 30. The proof uses undecidability of the word problem of the Tzeitin semigroup. We also transform our constructions in order to achieve a proof for the undecidability of the (not generalized) Post Correspondence Problem with image words of length at most 2.