General Concepts of Reduction, Minimization and Realization

Abstract

In this chapter we want to find a general setting for notions like “behavior”, “minimal” and “reduced” in order to formulate some general concepts of reduction, minimization and realization and especially their relationships. All these notions were introduced in the last chapter, concerning the input-output behavior as well as the transition monoid of automata. Moreover in the literature these notions have also been used for several other kinds of automata, machines and systems unfortunately mostly with different meaning. Motivated by the idea of minimal realization given in [45], which will also be extended to cover the nondeterministic cases, we are going to study the concepts of reduction, minimization and realization in the general setting of categories and functors. The explicit constructions for automata of deterministic and nondeterministic type listed in 1.12 will be given in the following chapters. In fact this chapter is a short version of [30] where in addition several other examples (cf. [3,5,8,35,36,45,53,56,72]) were studied and classified with respect to this general concept.