Post Correspondence Problem with Partially Commutative Alphabets

We introduce a version of Post Correspondence Problem (PCP, in short) generalized to words over partially commutative alphabets. Several observations are presented about the algorithmic status of the introduced problem. In particular solvability is shown for the partially commutative PCP for two special cases: the binary case of PCP (denoted by PCP(2) ), and the case with one periodic morphism. This extends solvability results for the classical PCP for these cases. Also a weaker version of PCP, named here Weak-PCP, is discussed. This version distinguishes (in the sense of solvability) the case of noncommutative from the case of partially commutative alphabets. We consider also a solvable (though NP-hard) simple version of Weak-PCP. Our solvability results demonstrate the power of Ibarra’s algorithms for reversal bounded multi-counter machines.