🤖 AI Summary
Traditional deterministic finite automata (DFA) employ uniform, unweighted state transitions, limiting their capacity to model quantitative or graded behaviors.
Method: This paper introduces the *discharging deterministic finite automaton* (DDFA), the first automaton model incorporating graph-theoretic discharging principles: states “discharge” rational-valued weights to adjacent states upon reading input symbols, enabling dynamic, weighted transitions. We formally define DDFA and analyze its algebraic structure.
Contribution/Results: We prove that the set of sequences generated by a DDFA forms a ring—the *quasi-k-regular sequence ring*—which strictly generalizes the classical k-regular sequence ring of Allouche and Shallit. Crucially, quasi-k-regularity properly contains k-regularity and exhibits superior algebraic completeness. By integrating formal language theory, directed graph modeling, ring theory, and discrete dynamical systems, this work significantly extends the algebraic expressiveness and applicability of automata models.
📝 Abstract
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 $deltacolon Q imes Sigma o Q$ 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.