Dynamic algebras: Examples, constructions, applications

Dynamic algebras combine the classes of Boolean (B ∨ ′ 0) and regular (R ∪ ; ∗) algebras into a single finitely axiomatized variety (B R 3) resembling an R-module with “scalar” multiplication 3. The basic result is that ∗ is reflexive transitive closure, contrary to the intuition that this concept should require quantifiers for its definition. Using this result we give several examples of dynamic algebras arising naturally in connection with additive functions, binary relations, state trajectories, languages, and flowcharts. The main result is that free dynamic algebras are residually finite (i.e. factor as a subdirect product of finite dynamic algebras), important because finite separable dynamic algebras are isomorphic to Kripke structures. Applications include a new completeness proof for the Segerberg axiomatization of propositional dynamic logic, and yet another notion of regular algebra.