Monodic temporal logic with quantified propositional variables
2014-06-13T07:59:33Z (GMT) by
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.