On the Descriptional Complexity of Operations on Semilinear Sets

TL;DR

This paper studies the complexity of operations on semilinear sets, providing upper bounds for union, intersection, complementation, and inverse homomorphism, and resolving an open problem in the field.

Contribution

It establishes new upper bounds on the descriptional complexity of key operations on semilinear sets, including a solution to an open problem on complementation.

Findings

Upper bounds for union, intersection, and inverse homomorphism operations.

A new upper bound for the complementation of semilinear sets.

Resolution of an open problem from prior research on semilinear set complementation.

Abstract

We investigate the descriptional complexity of operations on semilinear sets. Roughly speaking, a semilinear set is the finite union of linear sets, which are built by constant and period vectors. The interesting parameters of a semilinear set are: (i) the maximal value that appears in the vectors of periods and constants and (ii) the number of such sets of periods and constants necessary to describe the semilinear set under consideration. More precisely, we prove upper bounds on the union, intersection, complementation, and inverse homomorphism. In particular, our result on the complementation upper bound answers an open problem from [G. J. LAVADO, G. PIGHIZZINI, S. SEKI: Operational State Complexity of Parikh Equivalence, 2014].