Modal Kleene Algebra and Partial Correctness

We enrich Kleene algebra by domain and codomain operators. These abstractions of relational notions give rise to four modal operators. The boxes and diamonds enjoy various symmetries via Galois connections and dualities. Lifting modal statements to modal operator semirings yields a further abstraction and thus a more elegant and concise “statefree” reasoning about modalities. We use this modal Kleene algebra for calculating soundness and completeness proofs for propositional Hoare logic. While our soundness proof is more direct than related ones, our algebraic completeness proof seems entirely novel. It uses a modal symmetry that relates the wlp predicate transformer with partial correctness assertions and that is beyond the expressibility of formalisms like propositional dynamic logic.