Quotient Complexities of Atoms of Regular Languages

Abstract

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.

This work was supported by the Natural Sciences and Engineering Research Council of Canada under grant No. OGP0000871, the ERDF funded Estonian Center of Excellence in Computer Science, EXCS, the Estonian Science Foundation grant 7520, and the Estonian Ministry of Education and Research target-financed research theme no. 0140007s12.