Operational complexity and right linear grammars

Dassow, Jürgen

doi:10.1007/s00236-020-00386-3

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“…Proof We prove the theorem by determining which pairs (n, k) ∈ N × N can appear as (mpc(L), mpc(Circ(L))) for some language L. Lemma 3 immediately implies that n = 0 if and only if k = 0. We partition the remaining cases in three parts: First we look at 1 ≤ n ≤ 2, 1 ≤ k ≤ 3, second we deal with k ≥ 4 and 1 ≤ n ≤ k, and finally we consider n ≥ 3 and (2), we show how to adapt the pumpings of the words of size 1 and 2 from L to Circ(L).…”

Section: {0}mentioning

confidence: 99%

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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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Section: {0}mentioning

confidence: 99%

“…In the papers [2] and [3], the measure Var(L) which is defined as the minimal number of nonterminals in a right linear grammar which generate L was considered. It was shown that often the behaviour of Var is nearly related to the behaviour of nsc, however, the ranges of Var and circular closure or right and left quotient are infinite.…”

Section: Introductionmentioning

confidence: 99%