Specifications, Models, and Implementations of Data Abstractions

Abstract We consider the specification and verification of modules in hierarchically structured programs, as proposed by Parnas and Hoare. We argue that a specification for such a module is a set of sentences in some logical language in which the names to be exported by the module appear as nonlogical symbols. We further argue that an implementation of one module in terms of another module is a translation of the nonlogical symbols of the first specification into the language of the second. Equality must also be interpreted. We proposed necessary conditions which any such notion of ‘correct implementation’ ought to satisfy. These criteria provide a basis for judging the logical adequacy of any proposed specification language and definition of implementation. We then study DLP, a specification language obtained by adding uninterpreted procedure symbols to Pratt's first order dynamic logic. We present a definition of ‘implementation’ for DLP, and we show it satisfies these conditions. The main theorem, called the implementation Theorem, extends the Interpretation Theorem from first-order logic to DLP. The proof of this theorem is complicated by the necessity of dealing with modalities, parameters to procedures, interpretations of equality, and interpretations of sorts as tuples.