Completeness of Park induction

AbstractThe (in)equational properties of iteration, i.e., least (pre-)fixed point solutions over cpo's, are captured by the axioms of iteration theories. All known axiomatizations of iteration theories consist of the Conway identities and a complicated equation scheme, the commutative identity. The results of this paper show that the commutative identity is implied by the Conway identities and a weak form of the Park induction principle. Hence, we obtain a simple first order axiomatization of the (in)equational theory of iteration. It follows that a few simple identities and a weak form of the Scott induction principle, formulated to involve only inequations, are also complete. We also show that the Conway identities and the Park induction principle are not complete for the universal Horn theory of iteration