State Complexity of Permutation and Related Decision Problems on Alphabetical Pattern Constraints

TL;DR

This paper analyzes the state complexity of permutation operations on Alphabetical Pattern Constraints, providing sharp bounds and exploring decision problems like inclusion and universality, with implications for automata theory and formal languages.

Contribution

It introduces sharp state complexity bounds for permutation operations on APCs and shows polynomial-time decidability of inclusion and universality problems for fixed alphabets.

Findings

Sharp state complexity bounds for permutation on APCs.

Polynomial-time algorithms for inclusion and universality problems.

Bounds expressed in terms of automaton size and alphabet.

Abstract

We investigate the state complexity of the permutation operation, or the commutative closure, on Alphabetical Pattern Constraints (APC). This class corresponds to level $3/2$ of the Straubing-Th{\'e}rien Hierarchy and includes the finite, the piecewise-testable, or $\mathcal J$-trivial, and the $\mathcal R$-trivial and $\mathcal L$-trivial languages. We give a sharp state complexity bound expressed in terms of the longest strings in the unary projection languages of an associated finite language and which is already sharp for the subclass of finite languages. Additionally, for two subclasses, we give sharp bounds expressed in terms of the size of a recognizing input automaton and the size of the alphabet. Lastly, we investigate the inclusion and universality problem on APCs up to permutational equivalence, two problems known to be PSPACE-complete on APCs even for fixed alphabets in general, and show them to be decidable in polynomial time for fixed alphabets in this case.