Recall that for linear temporal logic, each Kripke structure may correspond to infinitely many computations.
On-line model-checking for finite linear temporal logic specifications.
In linear temporal logics, each moment in time has a unique possible future, while in branching temporal logics, each moment in time may split into several possible futures.
[1989], is that the assumption in the assume-guarantee pair concerns the interaction of the module with its environment along each computation, and is therefore more naturally expressed in linear temporal logic. Thus, in this approach, an assume-guarantee pair should consist of a linear temporal assumption [Psi] and a branching temporal guarantee [Psi].
The logic LTL is a linear temporal logic. Formulas of LTL are built from a set AP of atomic proposition using the usual Boolean operators and the temporal operators X ("next time"), U ("until"), and U ("duality of until").
In this approach, the assumption in the assume-guarantee pair concerns the interaction of the module with its environment along each computation, and is therefore more naturally expressed in a linear temporal logic. We denote this kind of assertion by [[Psi]]M<[Psi]>.
Since these sequences are labeled with all the state subformulas of [Xi], this causes no difficulty, as we can regard the state subformulas of [Xi] as atomic propositions and regard [Xi] as a linear temporal logic formula.
The complexity of propositional linear temporal logic. Journal ACM 32, 733-749.