Most Complex Regular Right-Ideal Languages

Abstract

Abstract.A right ideal is a language L over an alphabet Σ that satisfies the equation L = LΣ * . We show that there exists a sequence (Rn | n 3) of regular right-ideal languages, where Rn has n left quotients and is most complex among regular right ideals under the following measures of complexity: the state complexities of the left quotients, the number of atoms (intersections of complemented and uncomplemented left quotients), the state complexities of the atoms, the size of the syntactic semigroup, the state complexities of reversal, star, product, and all binary boolean operations that depend on both arguments. Thus (Rn | n 3) is a universal witness reaching the upper bounds for these measures.

Asymptotic Approximation for the Quotient Complexities of Atoms

“…We show that the ratio κ C (n) κ C (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.…”

A New Technique for Reachability of States in Concatenation Automata

“…Subclass Complexity Regular [1,9,15,19] (m − 1)2 n + 2 n−1 Prefix-closed [6,9] (m + 1)2 n−2 Unary [16,17,19] ∼mn (asymptotically) Prefix-free [9,13,14] m + n − 2 Finite unary [10,18] m + n − 2 Suffix-closed [6,8] mn − n + 1 Finite binary [10] (m − n + 3)2 n−2 − 1 Suffix-free [8,12] (m − 1)2 n−2 + 1 Star-free [7] (m − 1)2 n + 2 n−1 Right ideal [4,5,9] m + 2 n−2 Non-returning [3,11] (m − 1)2 n−1 + 1 Left ideal [4,5,8] m + n − 1 This suggests that our technique is widely applicable and should be considered as an viable alternative to the traditional induction argument when attempting reachability proofs in concatenation automata. The rest of the paper is structured as follows.…”

“…Additionally, we have (m ′ , {1, n − 1}) c −→ (m ′ , {1, n}), giving m + 2 n−2 . For distinguishability of the reached states, see [4]. Hence {ε, a, ab, ab 2 , .…”

Complexity of Right-Ideal, Prefix-Closed, and Prefix-Free Regular Languages

“…The results in this section are based on [8,9,18]; however, the stream below is different from that of [18], where c : (n − 2 → 0) and d : (n − 2 → n − 1). 1.…”

“…Since every prefix-closed language has an empty quotient, it is sufficient to consider boolean operations on languages over the same alphabet. The problems are the same as those in [9], except that there the transformation induced by d is d : (n − 2 → n − 1).…”