When Discrete Met Continuous: on the integration of discrete- and continuous-time metric temporal logics

Real-time systems usually encompass parts that are best described by a continuous-time model, such as physical processes under control, together with other components that are more naturally formalized by a discrete-time model, such as digital computing modules. Describing such systems in a unified framework based on metric temporal logic requires to integrate formulas which are interpreted over discrete and continuous time. In this paper, we tackle this problem with reference to the metric temporal logic TRIO, that admits both a discrete-time and a continuous-time semantics. We identify sufficient conditions for a TRIO specification to be invariant under change of time model from discrete to continuous and vice versa. These conditions basically involve the restriction to a proper subset of the TRIO language (which we call TRIOsi) and a requirement on the finite variability over time of the values of the basic items that constitute the formulas of the specification. A specification which is invariant can then be verified entirely under the simpler discrete-time model, with the results of the verification holding for the continuous-time model as well. We believe that this approach is general enough to be easily extendible to other temporal logics of comparable expressive power.