We present a compositional theory of system verification, where specifications assign real-numbered costs to systems. These costs can express a wide variety of quantitative system properties, such as resource consumption, price, or a measure of how well a system satisfies its specification. The theory supports the composition of systems and specifications, and the hiding of variables. Boolean refinement relations are replaced by real-numbered distances between descriptions of a system at different levels of detail. We show that the classical boolean rules for compositional reasoning have quantitative counterparts in our setting.

While our general theory allows costs to be specified by arbitrary cost functions, we also consider a class of linear cost functions, which give rise to an instance of our framework where all operations are computable in polynomial time.