Graded Hoare Logic and its Categorical Semantics

Deductive verification techniques based on program logics (i.e., the family of Floyd-Hoare logics) are a powerful approach for program reasoning. Recently, there has been a trend of increasing the expressive power of such logics by augmenting their rules with additional information to reason about program side-effects. For example, general program logics have been augmented with cost analyses, logics for probabilistic computations have been augmented with estimate measures, and logics for differential privacy with indistinguishability bounds. In this work, we unify these various approaches via the paradigm of grading,\ud adapted from the world of functional calculi and semantics. We propose Graded Hoare Logic (GHL), a parameterisable framework for augmenting program logics with a preordered monoidal analysis. We develop a\ud semantic framework for modelling GHL such that grading, logical assertions (pre- and post-conditions) and the underlying effectful semantics of an imperative language can be integrated together. Central to our\ud framework is the notion of a graded category which we extend here, introducing graded Freyd categories which provide a semantics that can interpret many examples of augmented program logics from the literature.\ud We leverage coherent fibrations to model the base assertion language, and thus the overall setting is also fibrational.