Reduction and Minimization of Automata in Pseudoclosed Categories

Abstract

Automata in pseudoclosed categories, including nondeterministic, relational, stochastic and relational topological automata, have been introduced in chapter 6. In 2.7 it is shown that,in contrast to the deterministic case,reduction and minimization do not coincide for nondeterministic automata. Thus we have to study reduction, minimization and realization problems for automata in pseudoclosed categories separately. The construction of an observable realization, already given in (6.7.2), turns out to be a weak minimal realization functor (cf. 7.8). Considering reduction and minimization we will construct for each automaton A in a pseudoclosed category an equivalent observable automaton A′ and an equivalent reduced automaton R(A) together with a reduction u(A):A → R(A) (cf.7.3, 7.4). Uniqueness and other properties of these constructions are studied in 7.7 and 7.8 using the theory and classification of systematics which are introduced in chapter 3. Unfortunately, observable automata are not minimal in the sense of systematics in general but only with respect to “weak morphisms”. This is the reason for the fact that equivalent observable automata are not necessarily isomorphic, but that only their state objects are. On the other hand minimality can be obtained regarding “strong observable” automata meaning that not only the states but also different subsets of states are inequivalent.