Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Unrestricted State Complexity of Binary Operations on Regular and Ideal
  Languages   We study the state complexity of binary operations on regular languages over
different alphabets. It is known that if $L'_m$ and $L_n$ are languages of
state complexities $m$ and $n$, respectively, and restricted to the same
alphabet, the state complexity of any binary boolean operation on $L'_m$ and
$L_n$ is $mn$, and that of product (concatenation) is $m 2^n - 2^{n-1}$. In
contrast to this, we show that if $L'_m$ and $L_n$ 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 $m2^n+2^{n-1}$. 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 $m+2^{n-2}+2^{n-1}$ ($m+2^{n-2}$); 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.

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The categories of this paper are: mtrl-th
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The categories of this paper are: funct-an
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The categories of this paper are: solv-int
9
The categories of this paper are: nlin
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The categories of this paper are: chao-dyn
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The categories of this paper are: cond-mat
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The categories of this paper are: q-alg
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-th
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The categories of this paper are: supr-con
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The categories of this paper are: nucl-th
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The categories of this paper are: patt-sol
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The categories of this paper are: atom-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-lat
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The categories of this paper are: econ