The Ranges of State Complexities for Complement, Star, and Reversal of Regular Languages

Jirásková, Galina

doi:10.1142/s0129054114500063

Cited by 6 publications

(9 citation statements)

References 19 publications

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“…3.2, page 67, exercise 3. Many similar results on state complexity are known, e.g., in [7], it is proved that all values until 2 n can be reached as sizes. Proof.…”

Section: The Exponential Gapmentioning

confidence: 67%

Complexity of Automatic Sequences

Zantema

2020

Lecture Notes in Computer Science

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Automatic sequences can be defined by DFAs with output (DFAO) in two natural ways. We propose to consider the minimal size of a corresponding DFAO as the complexity measure of the automatic sequence, for both variants. This paper compares these complexity measures and investigates their properties like the relationships with kernel and morphic sequences. There exist automatic sequences for which the one complexity is exponentially greater than the other one, in both directions. For both complexity measures we investigate the effect of taking basic operations on sequences like removing or adding an element in front, and observe that these operations may increase the complexity by at most a quadratic factor.

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“…3.2, page 67, exercise 3. Many similar results on state complexity are known, e.g., in [7], it is proved that all values until 2 n can be reached as sizes. Proof.…”

Section: The Exponential Gapmentioning

confidence: 67%

Complexity of Automatic Sequences

Zantema

2020

Lecture Notes in Computer Science

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“…We conjecture that if |∆| ≥ 3, the upper bound |∆| n is not reachable over a binary alphabet, despite the fact that it is known to be reachable for |∆| = 2 (the ordinary DFA case). While we could not prove that the upper bound is unreachable in all cases, we have proved it is unreachable when |∆| = n and |∆| ≥ 3, and verified computationally that it is unreachable for (|∆|, n) ∈ { (3,4), (3,5), (3,6), (4, 5)}. We prove a lower bound for the case of a binary input alphabet and 3 ≤ |∆| < n. We provide some preliminary computational evidence showing that this bound may be optimal for n ≥ 7.…”

Section: Introductionmentioning

confidence: 71%

“…Indeed, for (k, n) = (3, 5), the true maximum is 218, and this is achieved by |τ V 1 5 | with τ = [1, 2, 1, 2, 3]. However, for (k, n) = (4, 5): the value 826 is achieved by |τ U 2,3 | with τ = [1,2,3,4,4], while the maximal value of |τ V 1 5 | over all τ and d is 789, despite the fact that |U 2,3 | = 1857 and |V 1 5 | = 2110. Thus maximal monoids M do not necessarily give the maximal values for |τ M |.…”

Section: Resultsmentioning

confidence: 97%

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State Complexity of Reversals of Deterministic Finite Automata with Output

Davies

2018

Implementation and Application of Automata

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We investigate the worst-case state complexity of reversals of deterministic finite automata with output (DFAOs). In these automata, each state is assigned some output value, rather than simply being labelled final or non-final. This directly generalizes the well-studied problem of determining the worst-case state complexity of reversals of ordinary deterministic finite automata. If a DFAO has n states and k possible output values, there is a known upper bound of k n for the state complexity of reversal. We show this bound can be reached with a ternary input alphabet. We conjecture it cannot be reached with a binary input alphabet except when k = 2, and give a lower bound for the case 3 ≤ k < n. We prove that the state complexity of reversal depends solely on the transition monoid of the DFAO and the mapping that assigns output values to states.

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“…. , m + n} for n ≥ 2, m ≥ 2 [ 13] Reversal g nsc R (n) = {n − 1, n, n + 1} for n ≥ 2 [ 14] Kleene-closure g nsc * (n) = {1, 2, . .…”

Section: Introduction and Definitionsmentioning

confidence: 99%

Operational complexity and right linear grammars

Dassow

2021

Acta Informatica

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For a regular language L, let $${{\,\mathrm{Var}\,}}(L)$$ Var ( L ) be the minimal number of nonterminals necessary to generate L by right linear grammars. Moreover, for natural numbers $$k_1,k_2,\ldots ,k_n$$ k 1 , k 2 , … , k n and an n-ary regularity preserving operation f, let $$g_f^{{{\,\mathrm{Var}\,}}}(k_1,k_2,\ldots ,k_n)$$ g f Var ( k 1 , k 2 , … , k n ) be the set of all numbers k such that there are regular languages $$L_1,L_2,\ldots , L_n$$ L 1 , L 2 , … , L n such that $${{\,\mathrm{Var}\,}}(L_i)=k_i$$ Var ( L i ) = k i for $$1\le i\le n$$ 1 ≤ i ≤ n and $${{\,\mathrm{Var}\,}}(f(L_1,L_2,\ldots , L_n))=k$$ Var ( f ( L 1 , L 2 , … , L n ) ) = k . We completely determine the sets $$g_f^{{{\,\mathrm{Var}\,}}}$$ g f Var for the operations reversal, Kleene-closures $$+$$ + and $$*$$ ∗ , and union; and we give partial results for product and intersection.

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The Ranges of State Complexities for Complement, Star, and Reversal of Regular Languages

Cited by 6 publications

References 19 publications

Complexity of Automatic Sequences

Complexity of Automatic Sequences

State Complexity of Reversals of Deterministic Finite Automata with Output

Operational complexity and right linear grammars

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