Compositional Program Verification with Polynomial Functors in Dependent Type Theory

TL;DR

This paper introduces a categorical framework for compositional program verification using polynomial functors in dependent type theory, enabling modular reasoning about programs.

Contribution

It develops a novel categorical approach with polynomial functors and monoidal structures, formalized in Agda, for compositional verification and potential extensions.

Findings

Framework formalized in Agda demonstrates practical implementation.

Wiring diagrams enable compositionality of implementations and verifications.

Categorical structures support generalizations to concurrency and relational verification.

Abstract

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.