Compositional Program Verification with Polynomial Functors in Dependent Type Theory
This work addresses program verification for developers and researchers, offering a foundational approach that is incremental in extending categorical methods to verification.
The authors tackled the problem of compositional program verification by introducing a framework based on polynomial functors in dependent type theory, where implementations and verifications compose via wiring diagrams, and they formalized the entire framework in Agda as proof-of-concept.
We present a framework for compositional program verification based on polynomial functors in dependent type theory. In this framework, polynomial functors serve as program interfaces, Kleisli morphisms for the free monad monad serve as implementations, and dependent polynomials encode pre/postcondition specifications. We show that implementations and their verifications compose via wiring diagrams, and that Mealy machines provide a compositional coalgebraic operational semantics. We identify the abstract categorical structure underlying this compositionality as a monoidal functor from specifications to interfaces with a compatible monoidal natural transformation of lax monoidal presheaves; this opens the door to generalizations to other categories, monoidal products, etc., including settings for concurrency and relational verification, which we sketch. As a proof-of-concept, the entire framework has been formalized in Agda.