Abstract
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 \(\textsf{co}\textsf{NP}\)-complete, even for deterministic finite automata (DFAs), and it becomes \(\textsf{PSPACE}\)-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 \(o(n^{1-\delta })\) with \(0\le \delta \le 1/2\), for a given n-state NFA, unless the Exponential Time Hypothesis (ETH) fails.
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Notes
- 1.
Observe that the words \(w=xyz\) and \(xy^tz\), for all \(t\ge 0\), belong to the same Myhill-Nerode equivalence class of the language L. Thus, one can say that the pumping of the word y in w respects equivalence classes.
- 2.
Here a path is called simple if it does not have repeated states/vertices.
- 3.
A language \(L\subseteq \varSigma ^*\) is homogeneous if all words in L are of same length.
- 4.
Instead of a Hamiltonian cycle we ask for a Hamiltonian s-t-path.
- 5.
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Gruber, H., Holzer, M., Rauch, C. (2023). The Pumping Lemma for Regular Languages is Hard. In: Nagy, B. (eds) Implementation and Application of Automata. CIAA 2023. Lecture Notes in Computer Science, vol 14151. Springer, Cham. https://doi.org/10.1007/978-3-031-40247-0_9
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