Quotient Complexity of Closed Languages

Abstract

Abstract. A language L is prefix-closed if, whenever a word w is in L, then every prefix of w is also in L. We define suffix-, factor-, and subwordclosed languages in the same way, where by subword we mean subsequence. We study the quotient complexity (usually called state complexity) of operations on prefix-, suffix-, factor-, and subword-closed languages. We find tight upper bounds on the complexity of the prefix-, suffix-, factor-, and subword-closure of arbitrary languages, and on the complexity of boolean operations, concatenation, star and reversal in each of the four classes of closed languages. We show that repeated application of positive closure and complement to a closed language results in at most four distinct languages, while Kleene closure and complement gives at most eight languages.

A New Technique for Reachability of States in Concatenation Automata

“…Theorem 11 (Prefix-Closed Witness. Brzozowski, Jirásková and Zou, 2014 [6]). Define A and B as follows:…”

“…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.…”

Complexity of Left-Ideal, Suffix-Closed and Suffix-Free Regular Languages

“…The complexity of suffix-closed languages was studied in [10] in the restricted case, and the syntactic semigroup of these languages, in [14,17,20]; however, most complex suffix-closed languages have not been examined. 1.…”

“…5. Star The upper bound n was proved in [10]. To construct an NFA recognizing (L n (a, −, −, d, e)) * we add an ε-transition from the final state of D n (a, −, −, d, e) to the initial state 0; however in this case the ε-transition is a loop at 0, which does not affect the language recognized by the automaton.…”

“…Thus (L n (a, −, −, d, e)) * = L n (a, −, −, d, e) and its complexity is n. 6. Product (a) Restricted complexity: The upper bound mn − n + 1 was derived in [10]. The NFA for the product L ′ m (a, b, −, d, e)L n (a, e, −, d, b) is shown in Figure 7 for m = n = 4.…”