Abstract
Geometric folding processes are ubiquitous in natural systems ranging from protein biochemistry to patterns of insect wings and leaves. In a previous study, a folding operation between strings of formal languages was introduced as a model of such processes. The operation was then used to define a folding system (F-system) as a construct consisting of a core language, containing the strings to be folded, and a folding procedure language, which defines how the folding is done. This paper reviews main definitions associated with F-systems and next it determines necessary conditions for a language to belong to classes generated by such systems. The conditions are stated in the form of pumping lemmas and four classes are considered, in which the core and folding procedure languages are both regular, one of them is regular and the other context-free, or both are context-free. Full demonstrations of the lemmas are provided, and the analysis is illustrated with examples.
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Notes
Sburlan’s (2011) original definition has been modified in order to include the case in which both w and v are empty strings.
Sburlan’s (2011) original version of the lemma states that there exists a positive constant p such that any string \(w\in L\), with \(|w| \ge p\), can be written as \(w =uvxyz\) satisfying \(uv^ixy^iz\in L\) for each \(i\ge 0\), \(|vy| \ge 1\), and \(|vxy|\le p\). In his proof, he set \(p=\max (p_1, p_2)\) where \(p_1\) and \(p_2\) are the pumping lengths for the core and the folding procedure languages. However, it can be shown that such a value of p does not always work. As a simple example, set \(\varPhi =(L_1, L_2)\) with \(L_1=\texttt {aaaab}^*\) and \(L_2=(\texttt {uu})^*\texttt {ddd}\). Then, \(p=5\). However, \(L(\varPhi )= \texttt {aaaab} \cup (\texttt {bb})^*\texttt {aaaabbb}\), and note that string aaaab has length 5 but it can not be pumped as indicated by the lemma. Although the original lemma may still hold for some other value of p (\(p=9\), in case of the example, with \(u=x=y={\epsilon }\), \(v=\texttt {bb}\) and z equal to the rest of the string), a full demonstration was not provided. Other inconsistencies have been detected in the proof. For example, the proof is based on constructing a double stranded structure \(\left[ \frac{r_j}{s_j}\right]\), and the technique is also used here. However, instead of Eqs. (7) and (8), the previous proof sets \(r_j = x_ry_r^{\frac{{{\,\mathrm{lcm}\,}}(|y_r|, |y_s|)}{|y_r|}j}z_r\) and \(s_j = x_sy_s^{\frac{{{\,\mathrm{lcm}\,}}(|y_r|, |y_s|)}{|y_s|}j}z_s\), where \({{\,\mathrm{lcm}\,}}\) denotes the least common multiple. Since the stranded construction requires \(|r_j|=|s_j|\), it follows that \(|x_rz_r|=|x_sz_s|\) and hence \(|y_r|=|y_s|\). However, those conditions do not hold for arbitrary regular languages \(L_1\) and \(L_2\) (note that, in the above example, \(|y_r|=|\texttt {b}|=1\) whereas \(|y_s|=|\texttt {uu}|=2\)).
Similar problems have been found also in the original version of the lemma for the case of \(L\in \mathcal {F}(\textsf {CF}, \textsf {REG})\).
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Acknowledgements
This work was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq, Brazil).
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Lucero, J.C. Pumping lemmas for classes of languages generated by folding systems. Nat Comput 20, 321–327 (2021). https://doi.org/10.1007/s11047-019-09771-5
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DOI: https://doi.org/10.1007/s11047-019-09771-5