Quotient complexities of atoms of regular languages

• Authors: Janusz Brzozowski;Hellis Tamm
• Venue: DLT'12 Proceedings of the 16th international conference on Developments in Language Theory
• Year: 2012

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 2n−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.