Abstract
We demonstrate that for all practical purposes, Lambek Grammars (LG) are strongly equivalent to Context-Free Grammars (CFG) and hence to second-order Abstract Categorial Grammars (ACG). To be precise, for any Lambek Grammar LG there exists a second-order ACG with a second-order lexicon such that: the set of LG derivations (with a bound on the ‘nesting’ of introduction rules) is the abstract language of the ACG, and the set of yields of those derivations is its object language. Furthermore, the LG lexicon is represented in the abstract ACG signature with no duplications. The fixed, and small, bound on the nesting of introduction rules seems adequate for natural languages. One may therefore say that ACGs are not merely just as expressive as LG, but strongly equivalent.
The key is the algebraic description of Lambek Grammar derivations, and the avoidance of the Curry-Howard correspondence with lambda calculus.
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Acknowledgments
We thank Yusuke Kubota for his inspiring challenge. The comments by Richard Moot and the anonymous reviewers are gratefully acknowledged.
This work was partially supported by JSPS KAKENHI Grant Number 17K00091.
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Kiselyov, O., Hoshino, Y. (2020). Lambek Grammars as Second-Order Abstract Categorial Grammars. In: Sakamoto, M., Okazaki, N., Mineshima, K., Satoh, K. (eds) New Frontiers in Artificial Intelligence. JSAI-isAI 2019. Lecture Notes in Computer Science(), vol 12331. Springer, Cham. https://doi.org/10.1007/978-3-030-58790-1_15
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