Post Correspondence Problem [Section 5.2] Suppose we have dominos of strings, e.g.: The question: is it possible to arrange the dominos in line (repetitions of dominos are allowed) in such a way so that the top forms the same string as the bottom? b ca abc c ca a a ab

Post Correspondence Problem [Section 5.2] Formally, given is a collection P of dominos: P = { (t 1,b 1 ), (t 2,b 2 ), …, t k,b k ) } A match is a sequence i 1,i 2,…,i s, where t i1 t i2 …t is = b i1 b i2 …b is. The Post Correspondence Problem (PCP) asks if there is a match for P. Thm 5.15: PCP is undecidable.

Post Correspondence Problem [Section 5.2] First, we’ll consider MPCP where we are looking for instances that have a match that starts with the first domino. MPCP = { | P = { (t 1,b 1 ), (t 2,b 2 ), …, t k,b k ) } is a PCP that has match starting with (t_1,b_1) } Claim: PCP is equivalent to MPCP.

Post Correspondence Problem [Section 5.2] Thm 5.15: PCP is undecidable.

Post Correspondence Problem [Section 5.2] Thm: Ambiguity of CFGs is undecidable.