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.

This work was partially funded by the European Regional Development Fund through the programme COMPETE and by the Portuguese Government through the FCT under project UID/MAT/00144/2013 and project FCOMP-01-0124-FEDER-020486. Eva Maia was also funded by FCT grant SFRH/BD/78392/2011.