Operational complexity and pumping lemmas

Abstract

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 win L and vert wvert ge p, then there are words xin V^{*}, yin V^+, and zin V^{*} such that w=xyz and xy^tzin L for tge 0. The minimal pumping constant {{{,mathrm{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},}}} 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)} of all numbers k such that there are regular languages L_1,L_2,ldots ,L_n with {{{,mathrm{mpc},}}(L_i)=k_i} for 1le ile n and {{{,mathrm{mpc},}}(circ (L_1,L_2,ldots ,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)} completely. Furthermore, we give some results with respect to the minimal pumping length where, in addition, vert xyvert le p has to hold.