Equivalences within Non-Finite State Systems

In this paper, we discuss the issues associated with proving complete equivalences within Non-Finite State Systems. The various forms of equivalence that can exist between two algebraically defined systems can be intractable to demonstrate if the systems have infinite state order. This problem leads us to outline the following approach for enabling us to talk about, albeit weaker, approximations of equivalence for infinite systems: The definition of successive finite approximations to equivalences, allows for the construction of simulation tools, which can provide meaningful analysis for Non-Finite State Systems (NFSS) as well as the more usual Finite State Systems. The introduction of this extra variant form of equivalence leads to a discussion of their strictness in comparison with the traditional strict equivalence definitions. The intention being, to show that it is permissible to talk about successive finite approximations approaching strict equivalence in the limit. And finally w..