Classifying Regular Languages via Cascade Products of Automata

A significant result in the structure theory of regular languages is the Krohn-Rhodes Theorem, which states that any finite automaton can be decomposed into simple “prime factors”. This theorem can be stated in different versions using different products on different structures, namely the cascade product of automata, the wreath product of transformation semigroups and the block product of semigroups. We explore the connection between these three products and use these results to state how the three versions mentioned relate. We then use the Krohn-Rhodes Theorem to characterize families of regular languages in terms of the decompositions of the corresponding minimal automata. We study the case of piecewise testable, R-trivial and commutative languages. We introduce the concept of a biased reset and a locally i-triggered cascade product in order to characterize piecewise testable languages. Dropping the requirement of a locally itriggered product, we then show that a language is R-trivial iff its minimal automaton is covered by a cascade product of biased resets. In order to characterize commutative languages, we introduce the notion of a one letter automaton (OLA) and a one letter cascade product, in which acceptance of a word is determined solely by the number of occurrences of a single alphabet letter. We show that a language is commutative iff its minimal automaton is covered by a direct product of a one letter cascade products of biased resets and one letter simple cyclic grouplike automata, i.e. grouplike automata, the transition monoid of which is a simple cyclic group. Finally we introduce the scope of resets within a cascade product in order to further refine our analysis of the Krohn-Rhodes decomposition. The scope measures a notion of locality in the product. As initial results we show that the scope of cascade products recognizing R-trivial languages is bounded by a constant and that for a certain family of languages ˆLnnC1, such that Ln is of dot-depth n, there exists a product of scope precisely n, which recognizes Ln.