A Myhill-Nerode Theorem for Generalized Automata, with Applications to Pattern Matching and Compression

The model of generalized automata, introduced by Eilenberg in 1974, allows representing a regular language more concisely than conventional automata by allowing edges to be labeled not only with characters, but also strings. Giammarresi and Montalbano introduced a notion of determinism for generalized automata [STACS 1995]. While generalized deterministic automata retain many properties of conventional deterministic automata, the uniqueness of a minimal generalized deterministic automaton is lost. In the first part of the paper, we show that the lack of uniqueness can be explained by introducing a set $\mathcal{W(A)}$ associated with a generalized automaton $\mathcal{A}$. In this way, we derive for the first time a full Myhill-Nerode theorem for generalized automata, which contains the textbook Myhill-Nerode theorem for conventional automata as a degenerate case. In the second part of the paper, we show that the set $\mathcal{W(A)}$ leads to applications for pattern matching and data compression. We show that a Wheeler generalized automata can be stored using $\mathfrak{e} \log \sigma (1 + o(1)) + O(e)$ bits so that pattern matching queries can be solved in $O(m \log \log \sigma)$ time, where $\mathfrak{e}$ is the total length of all edge labels, $e$ is the number of edges, $\sigma$ is the size of the alphabet and $m$ is the length of the pattern.