On the Average Number of States of Partial Derivative Automata

Abstract

The partial derivative automaton ($\mathcal{A}_{\rm pd}$) is usually smaller than other non-deterministic finite automata constructed from a regular expression, and it can be seen as a quotient of the Glushkov automaton ($\mathcal{A}_{\rm pos}$). By estimating the number of regular expressions that have ε as a partial derivative, we compute a lower bound of the average number of mergings of states in $\mathcal{A}_{\rm pos}$ and describe its asymptotic behaviour. This depends on the alphabet size, k, and its limit, as k goes to infinity, is $\frac12$. The lower bound corresponds exactly to consider the $\mathcal{A}_{\rm pd}$ automaton for the marked version of the regular expression, i.e. where all its letters are made different. Experimental results suggest that the average number of states of this automaton, and of the $\mathcal{A}_{\rm pd}$ automaton for the unmarked regular expression, are very close to each other.

This work was partially funded by Fundação para a Ciência e Tecnologia (FCT) and Program POSI, and by projects RESCUE (PTDC/EIA/65862/2006) and CANTE (PTDC/EIA-CCO/101904/2008).