Nondeterministic operational complexity in subregular languages

Abstract

We study the nondeterministic state complexity of basic regular operations on subregular language families. In particular, we focus on the classes of combinational, singleton, finite, finitely generated left ideal, symmetric definite, group, star, comet, two-sided comet, ordered, and power-separating languages, and consider the operations of intersection, union, concatenation, power, Kleene star, reversal, and complementation. We get the exact complexity in all cases, except for complementation of singleton languages where we only have a lower bound n and an upper bound n. The complexity of all operations on combinational languages is given by a constant function, except for the k-th power where it is k+1. For all considered operations, the known upper bounds for left ideals are met by finitely generated left ideal languages, and the known lower bound 2n−1 for complementation on left ideals is tight also for symmetric definite languages. The complexity of intersection on singleton languages is min⁡{m,n}, and for all other operations, except for complementation, the known upper bounds for finite languages are met by singleton languages. The nondeterministic state complexity of the k-th power, star, and reversal on star languages is n, and the nondeterministic state complexity of complementation on group languages is max⁡{n,(n−1⌊n/2⌋)}. In all the remaining cases, the nondeterministic state complexity of all considered operations is the same as in the regular case, although sometimes we need to use a larger alphabet to describe the corresponding witnesses.