On the size of the universal automaton of a regular language

Abstract

The universal automaton of a regular language is the maximal NFA without merging states that recognizes this language. This automaton is directly inspired by the factor matrix defined by Conway thirty years ago. We prove in this paper that a tight bound on its size with respect to the size of the smallest equivalent NFA is given by Dedekind's numbers. At the end of the paper, we deal with the unary case. Chrobak has proved that the size of the minimal deterministic automaton with respect to the smallest NFA is tightly bounded by the Landau's function; we show that the size of the universal automaton is in this case an exponential of the Landau's function. © Springer-Verlag Berlin Heidelberg 2007.