The Pumping Lemma for Regular Languages is Hard.

We investigate the computational complexity of the Pumping-Problem, that is, for a given finite automaton A and a value p , to decide whether the language L ( A ) satisfies a previously fixed pumping lemma w.r.t. the value p . Here we concentrate on two different pumping lemmata from the literature. It turns out that this problem is intractable, namely, it is already co NP -complete, even for deterministic finite automata (DFAs), and it becomes PSPACE -complete for nondeterministic finite state devices (NFAs), for at least one of the considered pumping lemmata. In addition we show that the minimal pumping constant for the considered particular pumping lemmata cannot be approximated within a factor of o ( n 1 - δ ) with 0 ≤ δ ≤ 1 / 2 , for a given n -state NFA, unless the Exponential Time Hypothesis (ETH) fails.