Cascade products and Wheeler automata

The Krohn-Rhodes Decomposition Theorem (KRDT) holds a central position in automata and semigroup theories. It asserts that any finite-state automaton can be broken down into a collection, a cascade, of automata of two simple types (reset and permutation) that, combined, simulate the original automaton. In this paper we show how the cascade product operation and the related decomposition are particularly well-suited for the class of Wheeler automata. In these automata, recently introduced in the context of data compression, states are ordered and transitions map state-intervals to state-intervals. First, we prove that Wheeler DFAs are closed under cascade products in an efficient way: the cascade product of two Wheeler automata is still a Wheeler automaton and has always a number of states which is at most the sum (after removing unreachable states) of the number of states of the two input automata, a result that cannot be achieved for general (even counter-free) automata. Second, we prove that each Wheeler automaton can be decomposed into a cascade of a linear number of reset blocks. Crucially, our line of reasoning avoids the necessity of using full KRDT and proves our results directly by an inductive argument.