Creating Difficult Instances of the Post Correspondence Problem

Computational aspects of the Post Correspondence Problem (PCP) are studied. Specifically, we describe our efforts to find difficult instances of the PCP, where a "difficult" instance is defined to mean an instance whose shortest solution is long. As a result, we attempt to quantify the difficulty of the PCP in the same way the Busy Beaver Problem does for the Turing Halting Problem. We find instances of the PCP that have quite long solutions even when the number of pairs and the length of the strings is small, e.g., four and three, respectively. We discuss algorithms for solving the PCP and for generating difficult PCP instances. This problem poses unique difficulties because the size of the search space is unbounded.