Monodic temporal logic with quantified propositional variables

We extend the monodic fragment of first-order linear temporal logic to include right-linear grammar operators and quantification of propositional variables. Unlike propositional temporal logic, the use of grammar operators in first-order temporal logic is not equivalent to general propositional quantification, as the latter admit satisfiable formulae without countable models. We consider the decision problem for fragments where propositional quantification occurs outside of quantification of individual variables and temporal (grammar) operators. We show that if externally quantified propositions inside temporal operators occur within positive occurrences of universal quantifiers for individual variables then validity for all propositional prefix classes is recursively enumerable, and decidable in the two-variable case. Without this condition we show that, even with very severe restrictions on the first-order part of the logic, no non-trivial prefix class is recursively enumerable.