On the Accepting State Complexity of Operations on Permutation Automata

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.

Sweeping Permutation Automata

“…A suggested question for future research is the state complexity of operations on 2PerFA. Indeed, state complexity of operations on 1PerFA has recently been investigated [3,12], state complexity of operations on 2DFA of the general form was studied as well [4], and it would be interesting to know how the case of 2PerFA compares to these related models.…”

“…The language family recognized by permutation automata is known as the group languages, because their syntactic monoid is a group, and it has received some attention in the literature on algebraic automata theory [9]. Recently, Hospodár and Mlynárčik [3] determined the state complexity of operations on these automata, while Rauch and Holzer [12] investigated the effect of operations on permutation automata on the number of accepting states. Permutation automata are reversible, in the sense that, indeed, knowing the current state and the last read symbol one can always reconstruct the state at the previous step.…”