Complexity Of Atoms Of Regular Languages
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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.
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Janusz Brzozowski, Hellis Tamm (2013). Complexity Of Atoms Of Regular Languages. UWSpace. http://hdl.handle.net/10012/12513