Calculating upward and downward simulations of state-based specifications

This paper concerns calculational methods of refinement in state-based specification languages. Data refinement is a well-established technique for transforming specifications of abstract data types into ones, which are closer to an eventual implementation. The conditions under which a transformation is a correct refinement are encapsulated into two simulation rules: downward and upward simulations. One approach for refining an abstract system is to specify the concrete data type, and then attempt to verify that it is a valid refinement of the abstract type. An alternative approach is to calculate the concrete specification based upon the abstract specification and a retrieve relation, which links the abstract and concrete states. In this paper we generalise existing calculational methods for downward simulations and derive similar results for upward simulations; we also document their use and application in a particular specification language, namely Z.