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

Abstract

A language L over an alphabet Σ is suffix-convex if, for any words x, y, z ∈ Σ * , whenever z and xyz are in L, then so is yz. Suffixconvex languages include three special cases: left-ideal, suffix-closed, and suffix-free languages. We examine complexity properties of these three special classes of suffix-convex regular languages. In particular, we study the quotient/state complexity of boolean operations, product (concatenation), star, and reversal on these languages, as well as the size of their syntactic semigroups, and the quotient complexity of their atoms.

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.…”

“…For distinguishability of the reached states, see [8]. Note that the authors of [8] use a different concatenation DFA from our C: they first delete the sink states m ′ from A and n from B, and then form the concatenation of these modified DFAs. However, the same words used for distinguishing states in [8] can be used to distinguish states of C.…”

Towards a Theory of Complexity of Regular Languages

“…The complexities of common operations using various witnesses were studied in [17], and most complex suffix-closed languages in [25].…”

“…The complexities of common operations using various witnesses were studied in [25,30,36,44,48], semigroup size lower bound in [20], and upper bound in [31]. Suffix-free languages were the first example found of a class in which a most complex stream does not exist [30].…”

Complexity of Proper Suffix-Convex Regular Languages

“…Similarly, if • is a binary operation on languages, then the quotient/state complexity of • is the maximal value of κ(L ′ m • L n ), expressed as a function of m and n, as L ′ m and L n range over all regular languages of complexity m and n, respectively. We assume in this paper that L ′ m and L n are over a common alphabet Σ, however the unrestricted complexity of binary operations, where the two languages may use different alphabets, has recently been studied as well [5,10]. The complexity of an operation gives a worst-case bound on the time and space complexity of the operation, and it has been studied extensively (see [3,4,11,12,18]).…”