Most Complex Regular Right-Ideal Languages

Abstract

A right ideal is a language L over an alphabet Sigma that satisfies the equation L = L Sigma*. We show that there exists a sequence (Rn vertical bar n >= 3) of regular right-ideal languages, where R-n has n left quotients and is most complex among regular right ideals under the following measures of complexity: the state complexities of the left quotients, the number of atoms (intersections of complemented and uncomplemented left quotients), the state complexities of the atoms, the size of the syntactic semigroup, the state complexities of reversal, star, product, and all binary boolean operations that depend on both arguments. Thus (Rn vertical bar n >= 3) is a universal witness reaching the upper bounds for these measures.