Timed Denotational Semantics for Causal Functions over Timed Streams

We consider timed system behaviors seen as functions acting over streams of timed events (timed streams) that satisfy a temporal causality property: at every instant, current outputs only depends on inputs that have already been received. We aim at defining a timed model for describing their denotational semantics in the style of Strachey’s and Scott’s domain approach. For such a purpose, we extend timed streams with a notion of partial timed streams that form together a directed complete partial order (DCPO). Continuous functions over these DCPOs are then natural candidates for modeling timed denotational semantics of causal functions. Indeed, restricting to some notion of pre-synchronous continuous functions, we show that every causal function admits a non empty lattice of possible semantics which least element corresponds to a latest or laziest semantics and which greatest element corresponds a earliest or eagerest semantics. These operational aspects are then made explicit by defining a operational semantics model for causal functions that is inspired by timed-IO automata theory.