Dynamic algebras and the nature of induction

Abstract

Dynamic algebras constitute the variety (equationally defined class) of models of the Segerberg axioms for propositional dynamic logic. We obtain the following results (to within inseparability). (i) In any dynamic algebra * is reflexive transitive closure. (ii) Every free dynamic algebra can be factored into finite dynamic algebras. (iii) Every finite dynamic algebra is isomorphic to a Kripke structure. (ii) and (iii) imply Parikh's completeness theorem for the Segerberg axioms. We also present an approach to treating the inductive aspect of recursion within dynamic algebras.