Operational complexity and pumping lemmas

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∈ L and | w|≥ p , then there are words x∈ V^* , y∈ V^+ , and z∈ V^* such that w=xyz and xy^tz∈ L for t≥ 0 . The minimal pumping constant mpc (L) of L is the minimal number p for which the conditions of the pumping lemma are satisfied. We investigate the behaviour of mpc with respect to operations, i. e., for an n -ary regularity preserving operation ∘ , we study the set g_∘^ mpc (k_1,k_2,… ,k_n) of all numbers k such that there are regular languages L_1,L_2,… ,L_n with mpc (L_i)=k_i for 1≤ i≤ n and 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_∘^ mpc (k_1,k_2,… ,k_n) completely. Furthermore, we give some results with respect to the minimal pumping length where, in addition, | xy|≤ p has to hold.