Power, positive closure, and quotients on convex languages

Abstract

We study the state complexity and nondeterministic state complexity of the k-th power, positive closure, right quotient, and left quotient on the classes of prefix-, suffix-, factor-, and subword-free, -closed, and -convex regular languages, and on the classes of right, left, two-sided, and all-sided ideal languages. We show that the nondeterministic state complexity of the k-th power is kn for closed and convex languages, and k(n−1)+1 in the remaining classes, while its state complexity is n+(k−1)2n−2 for right ideals, k(n−1)+1 for other ideals and factor- and subword-closed languages, and k(n−2)+2 for prefix-, factor-, and subword-free languages. We next prove that the nondeterministic state complexity of positive closure is 1 for factor- and subword-closed languages and n for all other classes, while its state complexity is 2 for factor- and subword-closed languages, 2n−2+1 for prefix-closed and suffix-free languages, and n for all other considered classes. We also study left and right quotients by a language from the considered class or by a regular language, and get tight upper bounds in all cases except for state complexity of left quotient of all-sided ideals and subword-closed languages by a regular language, and nondeterministic state complexity of left quotient on subword-convex languages. Assuming that both dividend K and divisor L belong to a given class, the upper bound m on the complexity of the right quotient is tight for free, closed, convex, and right ideal languages, while the complexity of the right quotient of two left ideals is one. The nondeterministic state complexity of the left quotient is m+1 for left ideal, prefix-free, prefix-closed, and factor convex languages, it is m for suffix-free and suffix-closed languages and it is one for right ideals. If the divisor is an arbitrary regular language, then the upper bound m on the complexity of right quotient is always tight, and the upper bound m+1 on the complexity of left quotient is tight unless the dividend is an all-sided ideal, suffix-free, or subword-closed language. In these cases, the complexity of the left quotient is at most m. In four cases, our witnesses are defined over a growing alphabet. All the remaining witnesses are described over a small fixed alphabet, and whenever the binary alphabet is used, it is optimal.