A generalization of Deterministic Finite Automata related to discharging

Deterministic Finite Automata (DFAs) are of central importance in automata theory. In view of how state diagrams for DFAs are defined using directed graphs, this leads us to introduce a generalization of DFAs related to a method widely used in graph theory referred to as the discharging method. Given a DFA $(Q, \Sigma, \delta, q_{0}, F)$, the transition function $\delta\colon Q \times \Sigma \to Q$ determines a directed path in the corresponding state diagram based on an input string $a_{1} a_{2} \cdots a_{n}$ consisting of characters in $\Sigma$, and our generalization can be thought of as being based on how each vertex in $D$ ''discharges'' rational values to adjacent vertices (by analogy with the discharging method) depending on the string $a_{1} a_{2} \cdots a_{n}$ and according to a fixed set of rules. We formalize this notion and pursue an exploration of the notion of a Discharging Deterministic Finite Automaton (DDFA) introduced in this paper. Our DDFA construction gives rise to a ring structure consisting of sequences that we refer to as being quasi-$k$-regular, and this ring generalizes the ring of $k$-regular sequences introduced by Allouche and Shallit.