We study the state complexity of binary operations on regular languages over
different alphabets. It is known that if Lm′ and Ln are languages of
state complexities m and n, respectively, and restricted to the same
alphabet, the state complexity of any binary boolean operation on Lm′ and
Ln is mn, and that of product (concatenation) is m2n−2n−1. In
contrast to this, we show that if Lm′ and Ln are over different
alphabets, the state complexity of union and symmetric difference is
(m+1)(n+1), that of difference is mn+m, that of intersection is mn, and
that of product is m2n+2n−1. We also study unrestricted complexity of
binary operations in the classes of regular right, left, and two-sided ideals,
and derive tight upper bounds. The bounds for product of the unrestricted cases
(with the bounds for the restricted cases in parentheses) are as follows: right
ideals m+2n−2+2n−1 (m+2n−2); left ideals mn+m+n (m+n−1);
two-sided ideals m+2n (m+n−1). The state complexities of boolean operations
on all three types of ideals are the same as those of arbitrary regular
languages, whereas that is not the case if the alphabets of the arguments are
the same. Finally, we update the known results about most complex regular,
right-ideal, left-ideal, and two-sided-ideal languages to include the
unrestricted cases.Comment: 30 pages, 15 figures. This paper is a revised and expanded version of
the DCFS 2016 conference paper, also posted previously as arXiv:1602.01387v3.
The expanded version has appeared in J. Autom. Lang. Comb. 22 (1-3), 29-59,
2017, the issue of selected papers from DCFS 2016. This version corrects the
proof of distinguishability of states in the difference operation on p. 12 in
arXiv:1609.04439v