## Brzozowski, Janusz and Davies, Sylvie and Liu, Bo Yang Victor - Most Complex Regular Ideal Languages

dmtcs:2167 - Discrete Mathematics & Theoretical Computer Science, October 17, 2016, Vol. 18 no. 3
Most Complex Regular Ideal Languages

Authors: Brzozowski, Janusz and Davies, Sylvie and Liu, Bo Yang Victor

A right ideal (left ideal, two-sided ideal) is a non-empty language $L$ over an alphabet $\Sigma$ such that $L=L\Sigma^*$ ($L=\Sigma^*L$, $L=\Sigma^*L\Sigma^*$). Let $k=3$ for right ideals, 4 for left ideals and 5 for two-sided ideals. We show that there exist sequences ($L_n \mid n \ge k$) of right, left, and two-sided regular ideals, where $L_n$ has quotient complexity (state complexity) $n$, such that $L_n$ is most complex in its class under the following measures of complexity: the size of the syntactic semigroup, the quotient complexities of the left quotients of $L_n$, the number of atoms (intersections of complemented and uncomplemented left quotients), the quotient complexities of the atoms, and the quotient complexities of reversal, star, product (concatenation), and all binary boolean operations. In that sense, these ideals are "most complex" languages in their classes, or "universal witnesses" to the complexity of the various operations.

Source : oai:arXiv.org:1511.00157
Volume: Vol. 18 no. 3
Section: Automata, Logic and Semantics
Published on: October 17, 2016
Submitted on: October 14, 2016
Keywords: Computer Science - Formal Languages and Automata Theory