Abstract
Lambek categorial grammars is a class of formal grammars based on the Lambek calculus. Pentus proved in 1993 that they generate exactly the class of context-free languages without the empty word. In this paper, we study categorial grammars based on the Lambek calculus with the permutation rule LP. Of particular interest is the product-free fragment of LP called the Lambek-van Benthem calculus LBC. Buszkowski in his 1984 paper conjectured that grammars based on the Lambek-van Benthem calculus (LBC-grammars for short) generate exactly permutation closures of context-free languages. In this paper, we disprove this conjecture by presenting a language generated by an LBC-grammar that is not a permutation closure of any context-free language. Firstly, we introduce an ad-hoc modification of vector addition systems called linearly-restricted branching vector addition systems with states and additional memory (LRBVASSAMs for short) and prove that the latter are equivalent to LBC-grammars. Then we construct an LRBVASSAM that generates a non-semilinear set and thus disprove Buszkowski’s conjecture.
Since Buszkowski’s conjecture is false, not so much is known about the languages generated by LBC-grammars or by LP-grammars. The equivalence of LRBVASSAMs and LBC-grammars allows us to establish a number of their properties. We show that LP-grammars generate the same class of languages as LBC-grammars; that is, removing product from LP does not decrease expressive power of corresponding categorial grammars. We also prove that this class of languages is closed under union, intersection, concatenation, and Kleene plus.
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Notes
Here, however, we use a slightly different notation.
An \(\textrm{LBC}\)-grammar \(\langle S, \triangleright \rangle \) is of order less than or equal to 1 if and only if \(a \triangleright T\) implies that T is of the form \(p_1 \backslash \left( \dotsc \backslash \left( p_{k-1} \backslash \left( p_k \backslash q\right) \right) \dotsc \right) \) for some \(k \in \mathbb {N}\) and some primitive formulas \(p_i,q\).
We apologize to the reader for such long abbreviations.
Actually, it suffices to require that \(\Pi \) is non-empty in the rule \((\backslash _R)\); then, one can prove that all antecedents of derivable sequents are non-empty.
QL is an abbreviation for “quadratic language”.
As an exercise, the reader might check that LRBVASSAMs generate the same class of languages as LRBVASAMs, which are LRBVASSAMs with only one state.
DPO grammars extend unrestricted Chomsky grammars to hypergraphs.
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Acknowledgements
I thank Stepan L. Kuznetsov for bringing my attention to this problem and for suggesting valuable ideas to explore. I also thank the anonymous reviewer for his careful attention and remarks that helped me to improve my work.
Funding
The study was supported by the Theoretical Physics and Mathematics Advancement Foundation “BASIS”; by RFBR, project number 20-01-00670; and by the Interdisciplinary Scientific and Educational School of Moscow University “Brain, Cognitive Systems, Artificial Intelligence”.
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Pshenitsyn, T. Commutative Lambek Grammars. J of Log Lang and Inf 32, 887–936 (2023). https://doi.org/10.1007/s10849-023-09407-z
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DOI: https://doi.org/10.1007/s10849-023-09407-z