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Erschienen in:
2012 | OriginalPaper | Buchkapitel
Quotient Complexities of Atoms of Regular Languages
verfasst von : Janusz Brzozowski, Hellis Tamm
Erschienen in: Developments in Language Theory
Verlag: Springer Berlin Heidelberg
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An atom of a regular language
L
with
n
(left) quotients is a non-empty intersection of uncomplemented or complemented quotients of
L
, where each of the
n
quotients appears in a term of the intersection. The quotient complexity of
L
, which is the same as the state complexity of
L
, is the number of quotients of
L
. We prove that, for any language
L
with quotient complexity
n
, the quotient complexity of any atom of
L
with
r
complemented quotients has an upper bound of 2
n
− 1 if
r
= 0 or
r
=
n
, and
$1+\sum_{k=1}^{r} \sum_{h=k+1}^{k+n-r} C_{h}^{n} \cdot C_{k}^{h}$
otherwise, where
$C_j^i$
is the binomial coefficient. For each
$n\geqslant 1$
, we exhibit a language whose atoms meet these bounds.