On the coverage of Partial Validations

The validation of implementations is an essential part of the design of both hardware and software systems in order to establish the correctness of such systems. As such it has been an important application area for all kinds of formal methods to support this activity. Many of such methods, however, aim at a complete proof of correctness, which become unmanageable in the case of larger, realistic designs. In practice, therefore, attention is limited to such methods that can be applied partially or in an approximative manner. Albeit more pragmatic, these approaches usually lack a good measure for the extent to which correctness is established. Such coverage measures are needed to compare and assess different strategies for partial validation in the context of a given specification. In this article we propose to follow a measure-theoretic approach in which an exogenous cost function (quantifying the effect of certain properties in an implementation) is integrated over a measure that is induced by the probability of error occurrences in implementations. In this way, in fact, we do not only obtain a notion of coverage, but a general way of assigning measures to specification theories in the context of a given class of implementation structures.