The number of DFAs for a given spanning tree

Abstract

In the last few decades, several techniques to randomly generate a deterministic finite automaton have been developed. These techniques have implications in the enumeration and random generation of automata of size n. One of the ways to generate a finite automaton is to generate a random tree and to complete it to a deterministic finite automaton, assuming that the tree will be the automaton's breadth-first spanning tree. In this paper we explore further this method, and the string representation of a tree, and use it to counting the number of automata having a tree as a breadth-first spanning subtrees. We introduce the notions of characteristic and difference sequence of a tree, and define the weight of a tree. We also present a recursive formula for this quantity in terms of the "derivative" of a tree. Finally, we analyze the implications of this formula in terms of exploring trees with the largest and smallest number of automata in the span of the tree and present simple applications for finding densities of subsets of DFAs.