Kleene Algebra Modulo Theories

Kleene algebras with tests (KATs) o er sound, complete, and decidable equational reasoning about regularly structured programs. Since NetKAT demonstrated how well various extensions of KATs apply to computer networks, interest in KATs has increased greatly. Unfortunately, extending a KAT to a particular domain by adding custom primitives, proving its equational theory sound and complete, and coming up with e cient automata-theoretic implementations is still an expert’s task. We present a general framework for deriving KATs we call Kleene algebra modulo theories: given primitives and notions of state, we can automatically derive a corresponding KAT’s semantics, prove its equational theory sound and complete, and generate an automata-based implementation of equivalence checking. Our framework is based on pushback, a way of specifying how predicates and actions interact, rst used in Temporal NetKAT. We o er several case studies, including theories for bitvectors, increasing natural numbers, unbounded sets and maps, temporal logic, and network protocols. Finally, we provide an OCaml implementation that closely matches the theory: with only a few declarations, users can automatically derive an automata-theoretic decision procedure for a KAT.