Relative Completeness for Logics of Functional Programs

We prove a relative completeness result for a logic of functional programs extending D. Scott’s LCF. For such a logic, contrary to results for Hoare logic, it does not make sense to ask whether it is complete relative to the full theory of its first-order part, since the first order part does not determine uniquely the model at higher-order types. Therefore, one has to fix a model and choose an appropriate data theory w.r.t. which the logic is relatively complete. We establish relative completeness for two models: for the Scott model we use the theory of Baire Space as data theory, and for the effective Scott model we take first-order arithmetic. In both cases we need to extend traditional LCF in order to capture a sufficient amount of domain theory. 1998 ACM Subject Classification D.2.4 Program Verification; D.3.1 Formal Definitions and Theory; F.3.1 Specifying and Verifying and Reasoning about Programs; F.3.2 Semantics of Programming Languages