Brzozowski, JanuszTamm, Hellis2017-09-292017-09-292013-11-01http://dx.doi.org/10.1142/S0129054113400285http://hdl.handle.net/10012/12513Electronic version of an article published as International Journal of Foundations of Computer Science, 24(07), 2013, 1009–1027. http://dx.doi.org/10.1142/S0129054113400285 © World Scientific Publishing Company http://www.worldscientific.com/The quotient complexity of a regular language L, which is the same as its state complexity the number of left quotients of L. An atom of a non-empty regular language L with n quotients is a non-empty intersection of the n quotients, which can be uncomplemented or complemented. An NFA is atomic if the right language of every state is a union of atoms. We characterize all reduced atomic NFAs of a given language, i.e., those NFAs that have no equivalent states, 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; for 1 <= r <= n - 1 the bound is 1+ (k=1)Sigma(r) (h=k+1)Sigma(k+n-r) ((n)(h)) ((h)(k)). For each n >= 2 we exhibit a language with 2(n) atoms which meet these bounds.enAtomsfinite automatonatomic NFAquotient complexityregular languagestate complexitysyntactic semigroupwitnessArticle