What is the complexity of model checking Process Logic (LTL fragment)? Process Logic is a modal logic allowing to reason about temporal properties of programs. Its formulae take the form similar to (Propositional) Dynamic Logic $[P]\phi$, with $P$ being a program (think regular expression) and $\phi$ being another process logic formula, however unlike PDL, it can contain temporal LTL-style modalities. Unlike LTL, the formulae are read as follows "along traces resulting from execution of $P$, $\phi$ holds", or plainly "during execution of $P$, $\phi$ holds". Temporal modalities correspond to an extension of LTL with a chop and slice operators.

I am interested in a fragment of Process Logic where $\phi$ wouldn't be a full PL formula, but rather a plain LTL formula without modalities of the form $[P]$.

Question: Are there complexity analysis results relevant to model-checking formulae of Process Logic, or related formalisms? I am interested in both, the the complexity analysis, as well as some ideas for algorithms for doing it.
More along the "related formalisms" direction.


PDL with intersection and converse: satisfiability and infinite-state model checking, Stefan Göller, Markus Lohrey, and Carsten Lutz, 2009
The Effects of Bounding Syntactic Resources on Presburger LTL, Stéphane Demri and Régis Gascon


The introduction section of the first paper has an extensive discussion of complexity results for PDL fragments and extensions. I include the second mainly because you may be be able to encode some parts of your problem as a Presburger extension and transfer some results.