On some equivalence notions of synchronous systems

An important optimization tool in the design of synchronous systems is retiming, which in many cases allows a significant reduction in the length of the systems’ clock period. Even though the internal structure of systems changes upon retiming, their input-output behavior remains essentially the same. The original system and the one after the retiming can simulate each other in a suitable way. The equivalence notion arising from this kind of mutual simulation is called simulation equivalence, and the aim of this paper is to characterize simulation equivalence in an algebraic setting. It is shown that simulation equivalence is a congruence relation of the algebra of synchronous schemes, and that this congruence is the smallest one containing retiming equivalence and finitary strong equivalence. An axiomatization of these equivalences is presented in the general framework of strictly monoidal categories with feedback.