10.46298/dmtcs.1343
Brzozowski, Janusz
Janusz
Brzozowski
Davies, Sylvie
Sylvie
Davies
Liu, Bo Yang Victor
Bo Yang Victor
Liu
Most Complex Regular Ideal Languages
episciences.org
2016
Computer Science - Formal Languages and Automata Theory
contact@episciences.org
episciences.org
2016-10-14T21:51:30+02:00
2021-08-23T23:09:16+02:00
2016-10-17
eng
Journal article
https://dmtcs.episciences.org/1343
arXiv:1511.00157
1365-8050
PDF
1
Discrete Mathematics & Theoretical Computer Science ; Vol. 18 no. 3 ; Automata, Logic and Semantics ; 1365-8050
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.
Comment: 25 pages, 11 figures. To appear in Discrete Mathematics and
Theoretical Computer Science. arXiv admin note: text overlap with
arXiv:1311.4448