On the Accepting State Complexity of Operations on Permutation Automata

TL;DR

This paper studies the accepting state complexity of regular languages obtained through various operations on permutation automata, revealing that most complexities match general DFA cases except for reversal and quotient, and resolving an open problem for unary language intersection.

Contribution

It demonstrates that for most operations, permutation automata have similar accepting state complexities to general DFA, and it solves an open problem regarding unary language intersection complexities.

Findings

Most operations have similar complexity for permutation automata and general DFA.

Reversal and quotient operations have inherent complexity limitations ('magic' numbers).

The accepting state complexity problem for unary language intersection is resolved.

Abstract

We investigate the accepting state complexity of deterministic finite automata for regular languages obtained by applying one of the following operations to languages accepted by permutation automata: union, quotient, complement, difference, intersection, Kleene star, Kleene plus, and reversal. The paper thus joins the study of accepting state complexity of regularity preserving language operations which was initiated by the work [J. Dassow: On the number of accepting states of finite automata, J. Autom., Lang. Comb., 21, 2016]. We show that for almost all of the operations, except for reversal and quotient, there is no difference in the accepting state complexity for permutation automata compared to deterministic finite automata in general. For both reversal and quotient we prove that certain accepting state complexities cannot be obtained; these number are called "magic" in the literature. Moreover, we solve the left open accepting state complexity problem for the intersection of unary languages accepted by permutation automata and deterministic finite automata in general.