Quantified propositional temporal logic with repeating states

Quantified Propositional Temporal Logic (QPTL) is a linear temporal logic that allows quantification over propositional variables. In the usual semantics for QPTL, a model is an infinite discrete linear sequence of states, with each state having some propositional interpretation. The effect of this is that the interpretation of a proposition at one point in time is independent from its interpretation at another point in time. In this paper, we examine the expressivity and decidability of a QPTL, given generalizations of the usual semantics that do not have this restriction. We introduce the repeating semantics (QPTL^R), which allows states to be repeated throughout a model. While semantic interpretation does not affect the unquantified fragment of QPTL it significantly increases the expressive power in the presence of propositional quantification. In the main result of this paper, we show that QPTL^R makes the satisfiability problem highly undecidable through a complicated encoding of a tiling problem. We also investigate two less expressive semantics which still allow states to be repeated. We prove the satisfiability problem for one is undecidable, and decidable for the other.