Amalgams of inverse semigroups and reversible two-counter machines

We show that the word problem for an amalgam [S1,S2;U,ω1,ω2] of inverse semigroups may be undecidable even if we assume S1 and S2 (and therefore U) to have finite R-classes and ω1,ω2 to be computable functions, interrupting a series of positive decidability results on the subject. This is achieved by encoding into an appropriate amalgam of inverse semigroups 2-counter machines with sufficient universality, and relating the nature of certain Schützenberger graphs to sequences of computations in the machine.