Asymptotic Approximation for the Quotient Complexities of Atoms

In a series of papers, Brzozowski together with Tamm, Davies, and Szyku la studied the quotient complexities of atoms of regular languages [6, 7, 3, 4]. The authors obtained precise bounds in terms of binomial sums for the most complex situations in the following five cases: (G): general, (R): right ideals, (L): left ideals, (T ): two-sided ideals and (S): suffix-free languages. In each case let κC(n) be the maximal complexity of an atom of a regular language L, where L has complexity n ≥ 2 and belongs to the class C ∈ {G,R,L, T ,S}. It is known that κT (n) ≤ κL(n) = κR(n) ≤ κG(n) < 3 and κS(n) = κL(n − 1). We show that the ratio κC(n) κC(n−1) tends exponentially fast to 3 in all five cases but it remains different from 3. This behaviour was suggested by experimental results of Brzozowski and Tamm; and the result for G was shown independently by Luke Schaeffer and the first author soon after the paper of Brzozowski and Tamm appeared in 2012. However, proofs for the asymptotic behavior of κG(n) κG(n−1) were never published; and the results here are valid for all five classes above. Moreover, there is an interesting oscillation for all C: for almost all n we have κC(n) κC(n−1) > 3 if and only if κC(n+1) κC(n) < 3.