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

TL;DR

This paper extends the Myhill-Nerode theorem to generalized automata, providing a theoretical foundation and practical applications in pattern matching and data compression.

Contribution

It introduces a full Myhill-Nerode theorem for generalized automata and demonstrates applications to efficient pattern matching and compression.

Findings

Established a full Myhill-Nerode theorem for generalized automata.

Showed Wheeler generalized automata can be stored efficiently.

Pattern matching queries can be performed in logarithmic time.

Abstract

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.