Extending PPTL for Verifying Heap Evolution Properties

In this paper, we integrate separation logic with Propositional Projection Temporal Logic (PPTL) to obtain a two-dimensional logic, namely PPTL$^{{\tiny\mbox{SL}}$.} The spatial dimension is realized by a decidable fragment of separation logic which can be used to describe linked lists, and the temporal dimension is expressed by PPTL. We show that PPTL and PPTL$^{{\tiny\mbox{SL}}$} are closely related in their syntax structures. That is, for any PPTL$^{{\tiny\mbox{SL}}$} formula in a restricted form, there exists an "isomorphic" PPTL formula. The "isomorphic" PPTL formulas can be obtained by first an equisatisfiable translation and then an isomorphic mapping. As a result, existing theory of PPTL, such as decision procedure for satisfiability and model checking algorithm, can be reused for PPTL$^{{\tiny\mbox{SL}}$.}