State complexity bounds for projection, shuffle, up- and downward closure and interior on commutative regular languages

We consider the state complexity of projection, shuffle, up- and downward closure and interior on commutative regular languages. We deduce the state complexity bound n |E| for upward closure and downward interior, and ( 2 nm ) || , ( nm ) || , ( n + m - 1 ) || and ( n + m - 2 ) | E | for the shuffle on commutative regular, group, aperiodic and finite languages, respectively, with state complexities n and m over the alphabet E . We do not know whether these bounds are sharp. For projection, downward closure and upward interior, we give the sharp bound n . Our results are obtained by using the index and period vectors of a regular language, which we introduce in the present work and investigate w.r.t. the above operations and also union and intersection. Furthermore, we characterize the commutative aperiodic and commutative group languages in terms of these parameters and prove that a commutative regular language equals a finite union of shuffle products of commutative finite and commutative group languages. (c) 2024 Published by Elsevier Inc.