Logic as a distributive law

We present an algorithm for deriving a spatial-behavioral type system from a formal presentation of a computational calculus. Given a 2-monad Calc : Cat → Cat for the free calculus on a category of terms and rewrites and a 2-monad BoolAlg for the free Boolean algebra on a category, we get a 2-monad Form = BoolAlg + Calc for the free category of formulae and proofs. We also get the 2-monad BoolAlg◦Calc for subsets of terms. The interpretation of formulae is a natural transformation J−K : Form ⇒ BoolAlg◦Calc defined by the units and multiplications of the monads and a distributive law transformation δ : Calc ◦ BoolAlg ⇒ BoolAlg ◦ Calc. This interpretation is consistent both with the Curry-Howard isomorphism and with realizability. We give an implementation of the “possibly” modal operator parametrized by a two-hole term context and show that, surprisingly, the arrow type constructor in the λ-calculus is a specific case. We also exhibit nontrivial formulae encoding confinement and liveness properties for a reflective higher-order variant of the π-calculus.