  Specification theories as a tool in model-driven development processes of
component-based software systems have recently attracted a considerable
attention. Current specification theories are however qualitative in nature,
and therefore fragile in the sense that the inevitable approximation of systems
by models, combined with the fundamental unpredictability of hardware
platforms, makes it difficult to transfer conclusions about the behavior, based
on models, to the actual system. Hence this approach is arguably unsuited for
modern software systems. We propose here the first specification theory which
allows to capture quantitative aspects during the refinement and implementation
process, thus leveraging the problems of the qualitative setting.
  Our proposed quantitative specification framework uses weighted modal
transition systems as a formal model of specifications. These are labeled
transition systems with the additional feature that they can model optional
behavior which may or may not be implemented by the system. Satisfaction and
refinement is lifted from the well-known qualitative to our quantitative
setting, by introducing a notion of distances between weighted modal transition
systems. We show that quantitative versions of parallel composition as well as
quotient (the dual to parallel composition) inherit the properties from the
Boolean setting.
