Intuitionistic Linear Temporal Logics

We consider intuitionistic variants of linear temporal logic with “next,” “until,” and “release” based on expanding posets: partial orders equipped with an order-preserving transition function. This class of structures gives rise to a logic that we denote ITLe, and by imposing additional constraints, we obtain the logics ITLp of persistent posets and ITLht of here-and-there temporal logic, both of which have been considered in the literature. We prove that ITLe has the effective finite model property and hence is decidable, while ITLp does not have the finite model property. We also introduce notions of bounded bisimulations for these logics and use them to show that the “until” and “release” operators are not definable in terms of each other, even over the class of persistent posets.