Abstract
We consider two approaches to generating formal string languages: context-free grammars and Lambek grammars, which are based on the Lambek calculus. They are equivalent in the sense that they generate the same set of languages (disregarding the empty word). It is well known that context-free grammars can be generalized to hyperedge replacement grammars (HRGs) preserving their main principles and properties. In this paper, we study a generalization of the Lambek grammars to hypergraphs and investigate the recognizing power of the new formalism. We show how to define the hypergraph Lambek calculus (\(\mathrm {HL}\)), and then introduce hypergraph Lambek grammars based on \(\mathrm {HL}\). It turns out that such grammars recognize all isolated-node bounded languages generated by HRGs. However, they are more powerful than HRGs: they recognize at least finite intersections of such languages. Thus the Pentus theorem along with the pumping lemma and the Parikh theorem have no place for hypergraph Lambek grammars. Besides, it can be shown that hypergraph Lambek grammars are NP-complete, so they constitute an attractive alternative to HRGs, which are also NP-complete.
The study was funded by RFBR, project number 20-01-00670 and by the Interdisciplinary Scientific and Educational School of Moscow University “Brain, Cognitive Systems, Artificial Intelligence”. The author is a Scholarship holder of “BASIS” Foundation.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
To be consistent with the definition of a hypergraph one may assume that there are different symbols \(\$_n,n\ge 0\) instead such that \(type(\$_n)=n\).
References
Bar-Hillel, Y., Gaifman, C., Shamir, E.: On categorial and phrase-structure grammars. Bulletin of the Research Council of Israel F(9), 1–16 (1963)
Drewes, F., Kreowski, H.-J., Habel, A.: Hyperedge replacement graph grammars (1997)
Kanazawa, M.: Second-order abstract categorial grammars as hyperedge replacement grammars. J. Log. Lang. Inf. 19, 137–161 (2010)
Lambek, J.: The mathematics of sentence structure. Am. Math. Monthly 65(3), 154–170 (1958)
Moortgat, M.: Multimodal linguistic inference. J. Log. Lang. Inf. 5(3/4), 349–385 (1996)
Pentus, M.: Lambek calculus is NP-complete. Theor. Comput. Sci. 357(1–3), 186–201 (2006)
Pentus, M.: Lambek grammars are context free. In: Proceedings of the Eighth Annual Symposium on Logic in Computer Science (LICS 1993), Montreal, Canada. IEEE Computer Society (1993)
Gadducci, F., Kehrer, T. (eds.): ICGT 2020. LNCS, vol. 12150. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-51372-6
Pshenitsyn, T.: Introduction to a Hypergraph Logic Unifying Different Variants of the Lambek Calculus. Preprint at arXiv.org. https://arxiv.org/abs/2103.01199
Pshenitsyn, T.: Weak Greibach Normal Form for Hyperedge Replacement Grammars. In: Electronic Proceedings in Theoretical Computer Science, vol. 330, pp. 108–125. Open Publishing Association (2020)
Acknowledgments
I thank my scientific advisor prof. Mati Pentus for fruitful discussions.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Pshenitsyn, T. (2021). Powerful and NP-Complete: Hypergraph Lambek Grammars. In: Gadducci, F., Kehrer, T. (eds) Graph Transformation. ICGT 2021. Lecture Notes in Computer Science(), vol 12741. Springer, Cham. https://doi.org/10.1007/978-3-030-78946-6_6
Download citation
DOI: https://doi.org/10.1007/978-3-030-78946-6_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-78945-9
Online ISBN: 978-3-030-78946-6
eBook Packages: Computer ScienceComputer Science (R0)