Completely Reachable Automata: An Interplay Between Automata, Graphs, and Trees

Abstract

A deterministic finite automaton in which every non-empty set of states occurs as the image of the whole state set under the action of a suitable input word is called completely reachable. We characterize such automata in terms of graphs and trees.

Binary Completely Reachable Automata

“…The two results collected in this section are basically known as their versions have been scattered over the literature; see, e.g., [4,6,1]. We believe it is more convenient for the reader to see direct arguments rather than follow references to various sources where similar ideas might have appeared under different terminology and notation.…”

“…It readily follows from the definition of the greatest common divisor that g s is the greatest common divisor of g s−1 and d s . Hence g s−1 = mg s and d s = kg s for some coprime m and k. The subgroup ⟨g s ⟩ is equal to the union of the m cosets of the subgroup ⟨g s−1 ⟩ that are contained in ⟨g s ⟩; these m cosets are (1) ⟨g s−1 ⟩, g s ⊕ ⟨g s−1 ⟩, . .…”

“…cyclically connect all m cosets in (1). By the induction assumption, each of these cosets is the vertex set of a strongly connected component of the digraph Γ (s−1) .…”

“…The newly added edges 0 → 18, 18 → 36, 6 → 24, 12 → 30 cyclically connect these four cosets, producing a strongly connected component of Γ (1) as shown in Figure 2. The three other strongly connected components of the digraph Γ (1) are constructed in the same way.…”

“…0 24 6 18 30 12 36 42 FIGURE 2. One of the strongly connected components of the digraph Γ (1) constructed for the DFA E 48 from Figure 1. The solid edges are inherited from Γ (0) ; the dashed edges are newly added.…”