### Technical Report: DCC-2010-04

###
On the Average Size of PD automata: an Analytic Combinatorics
Approach

### Sabine Broda

DCC-FC & LIACC, Universidade do Porto

E-mail: sbb@ncc.up.pt

### António Machiavelo

CMUP, Universidade do Porto

E-mail: ajmachia@fc.up.pt

### Nelma Moreira

### Rogério Reis

DCC-FC & LIACC, Universidade do Porto

E-mail: \{nam,rvr\}@ncc.up.pt}

Rua do Campo Alegre, 4169-007 Porto, Portugal

July 2010

## Abstract

The partial derivative automaton (NFA_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 (NFA_POS). By estimating the number of regular
expressions that have epsilon as a partial derivative, we
compute a lower bound of the average number of mergings of states in
NFA_POS.
and describe its asymptotic behaviour. This depends on the alphabet
size, k, and its limit, as k goes to infinity, is 1\2.
The lower bound corresponds exactly to consider the NFA_PD
automaton for the marked version of the RE, i.e. where all its
letters are made different. Experimental results suggest that the average
number of states of this automaton, and of the NFA_PD automaton for the
unmarked RE, are very close to each other.