ON THE AVERAGE STATE COMPLEXITY OF PARTIAL DERIVATIVE AUTOMATA: AN ANALYTIC COMBINATORICS APPROACH
The partial derivative automaton (A(pd)) is usually smaller than other nondeterministic finite automata constructed from a regular expression, and it can be seen as a quotient of the Glushkov automaton (A(pos)). By estimating the number of regular expressions that have E as a partial derivative, we compute a lower bound of the average number of mergings of states in A(pos) and describe its asymptotic behaviour. This depends on the alphabet size, k, and for growing k's its limit approaches half the number of states in A(pos). The lower bound corresponds to consider the A(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 A(pd) automaton for the unmarked regular expression, are very close to each other.
partial derivatives analytic combinatorics regular expressions regular languages conversion between regular expressions and nondeterministic finite automata average case analysis