Prefix and Right-Partial Derivative Automata

Abstract

Recently, Yamamoto presented a new method for the conversion from regular expressions (REs) to non-deterministic finite automata (NFA) based on the Thompson \(\varepsilon \)-NFA (\(\mathcal {A}_\mathsf {T}\)). The \(\mathcal {A}_\mathsf {T}\) automaton has two quotients discussed: the suffix automaton \(\mathcal {A}_\mathsf {suf}\) and the prefix automaton, \(\mathcal {A}_\mathsf {pre}\). Eliminating \(\varepsilon \)-transitions in \(\mathcal {A}_\mathsf {T}\), the Glushkov automaton (\(\mathcal {A}_{\mathsf {pos}}\)) is obtained. Thus, it is easy to see that \(\mathcal {A}_\mathsf {suf}\) and the partial derivative automaton (\(\mathcal {A}_\mathsf {pd})\) are the same. In this paper, we characterise the \(\mathcal {A}_\mathsf {pre}\) automaton as a solution of a system of left RE equations and express it as a quotient of \(\mathcal {A}_{\mathsf {pos}}\) by a specific left-invariant equivalence relation. We define and characterise the right-partial derivative automaton (\(\overleftarrow{\mathcal {A}}_\mathsf {pd}\)). Finally, we study the average size of all these constructions both experimentally and from an analytic combinatorics point of view.