On the Transition Reduction Problem for Finite Automata

In this paper we apply the concept of common follow sets CFS of a regular expression to homogeneous finite state automaton. Based on this concept and using particular binary trees, we devise an efficient algorithm to reduce minimize the number of transitions of the automaton recognizing the language LEn denoted by the regular expression $E_n = 1 + \epsiv \cdot 2 + \epsiv \cdot 3 + \epsiv \dots n+\epsiv$. Experiments reveal that for small values of n, all e-free NFAs with n+1 states and with a minimum number of transitions for LEn are exactly those obtained by our construction. Also, the produced e-free NFA is asymptotically minimal, in the sense that the number of transitions is equivalent to nlog2 n2 which corresponds in the same time to the upper and the lower bound. We conjecture that our construction is not only a reduction but a minimization for LEn.