Continuous Additive Algebras and Injective Simulations of Synchronization Trees

Abstract

The (in)equational properties of the least fixed point operation on
(omega-)continuous functions on (omega-)complete partially ordered sets are
captured by the axioms of (ordered) iteration algebras, or iteration
theories. We show that the inequational laws of the sum operation in
conjunction with the least fixed point operation in continuous additive
algebras have a finite axiomatization over the inequations of ordered
iteration algebras. As a byproduct of this relative axiomatizability
result, we obtain complete infinite inequational and finite implicational
axiomatizations. Along the way of proving these results, we give a
concrete description of the free algebras in the corresponding variety of
ordered iteration algebras. This description uses injective simulations of
regular synchronization trees. Thus, our axioms are also sound and
complete for the injective simulation (resource bounded simulation) of
(regular) processes.

Keywords: equational logic, fixed points, synchronization trees, simulation.