Combining the Temporal and Epistemic Dimensions for MTL Monitoring

We define a new notion of satisfaction of a temporal logic formula phi by a behavior w. This notion, denoted by (w, t, t') |= phi, is characterized by two time parameters: the position t from which satisfaction is considered, and the end of the (finite) behavior t' which indicates how much do we know about the behavior. We define this notion in dense time where phi is a formula in the future fragment of metric temporal logic (MTL) and w is a Boolean signal of bounded variability. We show that the set of all pairs (t, t') such that (w, t, t') |= phi can be expressed as a finite union of two-dimensional zones and give an effective procedure to compute it.