Lynch-Morawska Systems on Strings

We investigate properties of convergent and forward-closed string rewriting systems in the context of the syntactic criteria introduced in \cite{LynchMorawska} by Christopher Lynch and Barbara Morawska (we call these $LM$-Systems). Since a string rewriting system can be viewed as a term-rewriting system over a signature of purely monadic function symbols, we adapt their definition to the string rewriting case. We prove that the subterm-collapse problem for convergent and forward-closed string rewriting systems is effectively solvable. Therefore, there exists a decision procedure that verifies if such a system is an $LM$-System. We use the same construction to prove that the \emph{cap problem} from the field of cryptographic protocol analysis, which is undecidable for general $LM$-systems, is decidable when restricted to the string rewriting case.
Comment: Revised based on reviewers' feedback