Liveness Properties in Geometric Logic for Domain-Theoretic Streams

We devise a version of Linear Temporal Logic (LTL) on a denotational domain of streams. We investigate this logic in terms of domain theory, (point-free) topology and geometric logic. This yields the first steps toward an extension of the "Domain Theory in Logical Form" paradigm to temporal liveness properties. We show that the negation-free formulae of LTL induce sober subspaces of streams, but that this is in general not the case in presence of negation. We propose a direct, inductive, translation of negation-free LTL to geometric logic. This translation reflects the approximations used to compute the usual fixpoint representations of LTL modalities. As a motivating example, we handle a natural input-output specification for the usual filter function on streams.