Decidable restrictions of the Post Correspondence Problem

The Post Correspondence Problem (PCP) is undecidable.

The bounded version of the PCP is $\mathrm{NP}$-complete and the marked version of the PCP (the words of one of the two lists are required to differ in the first letter) is in $\mathrm{PSPACE}$ [1].

• Are these restricted versions used to prove some complexity results of other problems (through reduction)?

• Are there other restricted versions of the PCP that make it decidable (and in particular $\mathrm{PSPACE}$-complete)?

[1] "Marked PCP is decidable" by V. Halava, M. Hirvensalo, R. De Wolf (1999)

Ehrenfeucht, Karhumäki and Rozenberg have shown that binary PCP (where the domain is binary) is decidable. Halava and Holub later showed that it's actually in P.

StackExchange Computer Science Q#701, answer score: 11