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 \(w\in L\) and \(\vert w\vert \ge p\), then there are words \(x\in V^{*}\), \(y\in V^+\), and \(z\in V^{*}\) such that \(w=xyz\) and \(xy^tz\in L\) for \(t\ge 0\). 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 \(1\le i\le n\) 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, \(\vert xy\vert \le p\) has to hold.