The Ranges of Accepting State Complexities of Languages Resulting From Some Operations

Hospodár, Michal; Holzer, Markus

doi:10.1007/978-3-319-94812-6_17

Cited by 4 publications

(9 citation statements)

References 13 publications

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“…Here the restriction for the input automaton to be a PFA provides magic numbers which are not magic if the input automaton is not restricted. For deterministic finite automata with no restrictions the following result was proven in [6]:…”

Section: Reversalmentioning

confidence: 99%

“…Since for unary languages the left and right quotient coincide no distinction is made at this point and we use the right quotient unless otherwise stated. For regular languages in general the following was shown in [6]:…”

Section: Quotientmentioning

confidence: 99%

“…Thus, the conversion from nondeterministic to equivalent deterministic finite automata can produce unbounded deterministic accepting state complexity for a regular language. Moreover, the operational accepting state complexity was studied in [6]. The obtained results on the accepting state complexity prove that this measure is significantly different to the original state complexity.…”

Section: Introductionmentioning

confidence: 99%

“…Following the terminology of [8] we call values α "magic" if there are no such automata A and B. The following results were shown in [2] and [6] for the operational accepting state complexity on languages accepted by DFAs-for accepting state complexities m and n one can obtain all values from the given number set for α:…”

Section: Introductionmentioning

confidence: 99%

“…This is entirely different compared to the general case. Finally, the unary case for the accepting state complexity of the intersection operation for DFAs in general was left open in [6]. We close this gap by considering this problem in detail.…”

Section: Introductionmentioning

confidence: 99%

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On the Accepting State Complexity of Operations on Permutation Automata

Rauch

Holzer

2022

Electron. Proc. Theor. Comput. Sci.

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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.

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Section: Reversalmentioning

confidence: 99%

Section: Quotientmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On the Accepting State Complexity of Operations on Permutation Automata

Rauch

Holzer

2022

Electron. Proc. Theor. Comput. Sci.

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Operational complexity and pumping lemmas

Dassow

Jecker

2022

Acta Informatica

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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.

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On the accepting state complexity of operations on permutation automata

Rauch,

Holzer

2023

RAIRO-Theor. Inf. Appl.

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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 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.

show abstract

The Ranges of Accepting State Complexities of Languages Resulting From Some Operations

Cited by 4 publications

References 13 publications

On the Accepting State Complexity of Operations on Permutation Automata

On the Accepting State Complexity of Operations on Permutation Automata

Operational complexity and pumping lemmas

On the accepting state complexity of operations on permutation automata

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