Operational complexity and pumping lemmas

Abstract The well-known pumping lemma for regular languages states that, any language L , there is a constant p (depending on ) such that the following holds: If $$w\in L$$ w ∈ L and $$\vert w\vert \ge p$$ | ≥ p then are words $$x\in V^{*}$$ x V ∗ $$y\in V^+$$ y + $$z\in z $$w=xyz$$ = $$xy^tz\in t $$t\ge 0$$ 0 . minimal $${{{\,\mathrm{mpc}\,}}(L)}$$ mpc ( ) of number which conditions satisfied. We investigate behaviour $${{{\,\mathrm{mpc}\,}}}$$ with respect to operations, i. e., an n -ary regularity preserving operation $$\circ $$ ∘ we study set $${g_{\circ }^{{{\,\mathrm{mpc}\,}}}(k_1,k_2,\ldots ,k_n)}$$ g k 1 , 2 … n all numbers k $$L_1,L_2,\ldots ,L_n$$ $${{{\,\mathrm{mpc}\,}}(L_i)=k_i}$$ i $$1\le i\le n$$ ≤ $${{{\,\mathrm{mpc}\,}}(\circ (L_1,L_2,\ldots ,L_n)=~k}$$ With Kleene closure, complement, reversal, prefix suffix-closure, circular shift, union, intersection, set-subtraction, symmetric difference,and concatenation, determine completely. Furthermore, give some results length where, in addition, xy\vert \le has hold.