Derivatives and finite automata of expressions in star normal form

Abstract

This paper studies derivatives and automata for expressions in star normal form as defined by Brüggemann-Klein. For an expression in star normal form, the paper shows that the derivatives are either ∅ or unique, while in general Berry and Sethi’s result shows the derivatives are either ∅ or similar. It is known that the partial derivative automaton and the follow automaton are two small automata, each of which is a quotient of the position automaton. For the relation between the partial derivative and follow automata, however, Ilie and Yu stated that a rigorous analysis is necessary but difficult. The paper tackles the issue, and presents several results. Our work shows that there are different conditions under which the relation of the two automata can be different.