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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.
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.
The well-known pumping lemma for regular languages states that, for any regular language L, there is a constant p (depending on L) such that the following holds: If $$w\in L$$ w ∈ L and $$\vert w\vert \ge p$$ | w | ≥ p , then there are words $$x\in V^{*}$$ x ∈ V ∗ , $$y\in V^+$$ y ∈ V + , and $$z\in V^{*}$$ z ∈ V ∗ such that $$w=xyz$$ w = x y z and $$xy^tz\in L$$ x y t z ∈ L for $$t\ge 0$$ t ≥ 0 . The minimal pumping constant $${{{\,\mathrm{mpc}\,}}(L)}$$ mpc ( L ) of L is the minimal number p for which the conditions of the pumping lemma are satisfied. We investigate the behaviour of $${{{\,\mathrm{mpc}\,}}}$$ mpc with respect to operations, i. e., for an n-ary regularity preserving operation $$\circ $$ ∘ , we study the set $${g_{\circ }^{{{\,\mathrm{mpc}\,}}}(k_1,k_2,\ldots ,k_n)}$$ g ∘ mpc ( k 1 , k 2 , … , k n ) of all numbers k such that there are regular languages $$L_1,L_2,\ldots ,L_n$$ L 1 , L 2 , … , L n with $${{{\,\mathrm{mpc}\,}}(L_i)=k_i}$$ mpc ( L i ) = k i for $$1\le i\le n$$ 1 ≤ i ≤ n and $${{{\,\mathrm{mpc}\,}}(\circ (L_1,L_2,\ldots ,L_n)=~k}$$ mpc ( ∘ ( L 1 , L 2 , … , L n ) = k . With respect to Kleene closure, complement, reversal, prefix and suffix-closure, circular shift, union, intersection, set-subtraction, symmetric difference,and concatenation, we determine $${g_{\circ }^{{{\,\mathrm{mpc}\,}}}(k_1,k_2,\ldots ,k_n)}$$ g ∘ mpc ( k 1 , k 2 , … , k n ) completely. Furthermore, we give some results with respect to the minimal pumping length where, in addition, $$\vert xy\vert \le p$$ | x y | ≤ p has to hold.
We investigate the accepting state complexity of deterministic finite automata for regular languages obtained by applying one of the following operations on languages accepted by permutation automata: union, quotient, complement, difference, intersection, Kleene star, Kleene plus, and reversal. The paper thus joins the study of the accepting state complexity of regularity preserving language operations which was initiated in [J. Dassow, J. Autom., Lang. Comb. 21 (2016) 55–67]. We show that for almost all of the above-mentioned 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 numbers 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.
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