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Diamonds Are Forever

Theoretical and Empirical Support for a Dependency-Enhanced Type Logic

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Logic and Algorithms in Computational Linguistics 2021 (LACompLing2021)

Part of the book series: Studies in Computational Intelligence ((SCI,volume 1081))

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Abstract

Extended Lambek calculi enlarge the type language with adjoint pairs of unary modalities. In previous work, modalities have been used as licensors for controlled forms of restructuring, reordering and copying. Here, we study a complementary use of the modalities as dependency features coding for grammatical roles. The result is a multidimensional type logic simultaneously inducing dependency and function argument structure on the linguistic material. We discuss the new perspective on constituent structure suggested by the dependency-enhanced type logic, and we experimentally evaluate how well a neural language model like BERT can deal with the subtle interplay between logical and structural reasoning that this type logic gives rise to.

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Notes

  1. 1.

    The notation \(\Gamma [\ \ ]\) is for a context: a structure with a hole.

  2. 2.

    The application \((\lambda x.M) N\) \(\beta \)-reduces to M[x/N] which replaces the occurrences of x in M by N.

  3. 3.

    In practice, one will work with indexed families \(\{\diamondsuit _{i},\Box _{i}\}_{i\in I}\) so as to parameterize type assignment to particular structural choices. To avoid clutter, we avoid indexing in this section, specifying the intended structural packages in the text.

  4. 4.

    The analysis of non-peripheral extraction presented here is quite similar in spirit to the ‘subexponentials’ \(!^i\) used by [3] for controlled structural rules in a non-associative, non-commutative setting. As a matter of fact, one can see the combination \(\diamondsuit \Box (-)\) as the decomposition of an exponential \(\mathop {!}(-)\) into an adjoint pair, with the \(\diamondsuit \) part licensing structural reasoning by means of (15) and the \(\mathop {!}\) Dereliction rule turned into a theorem \(\diamondsuit \Box A\vdash A\). We leave a detailed comparison for another occasion.

  5. 5.

    Infix ‘+’, defined as \(\lambda x\lambda y\lambda i.(x\ (y\ i))\), represents concatenation.

  6. 6.

    This homomorphism can be defined using standard tools from \(\lambda \) calculus for modelling tuples and lists.

References

  1. Baillot, P., Mogbil, V.: Soft lambda-calculus: a language for polynomial time computation. In: Walukiewicz, I. (ed.) Foundations of Software Science and Computation Structures. Lecture Notes in Computer Science, vol. 2987, pp. 27–41. Springer (2004). https://doi.org/10.1007/978-3-540-24727-2_4

  2. van Benthem, J.: Language in Action: Categories, Lambdas and Dynamic Logic. MIT Press (1995)

    Google Scholar 

  3. Blaisdell, E., Kanovich, M.I., Kuznetsov, S.L., Pimentel, E., Scedrov, A.: Non-associative, non-commutative multi-modal linear logic. In: Blanchette, J., Kovács, L., Pattinson, D. (eds.) Automated Reasoning - 11th International Joint Conference, IJCAR 2022, Haifa, Israel, August 8-10, 2022, Proceedings. Lecture Notes in Computer Science, vol. 13385, pp. 449–467. Springer (2022). https://doi.org/10.1007/978-3-031-10769-6_27

  4. Boullier, P.: Range concatenation grammars. In: Proceedings of the Sixth International Workshop on Parsing Technologies, IWPT, pp. 53–64 (2000). https://aclanthology.org/2000.iwpt-1.8/

  5. Devlin, J., Chang, M.W., Lee, K., Toutanova, K.: BERT: Pre-training of deep bidirectional transformers for language understanding. In: Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies. Long and Short Papers, vol. 1, pp. 4171–4186. Association for Computational Linguistics, Minneapolis, Minnesota (2019). https://doi.org/10.18653/v1/N19-1423, https://aclanthology.org/N19-1423

  6. Girard, J.: Linear logic. Theor. Comput. Sci. 50, 1–102 (1987). https://doi.org/10.1016/0304-3975(87)90045-4

    Article  MathSciNet  MATH  Google Scholar 

  7. de Groote, P., Pogodalla, S.: On the expressive power of abstract categorial grammars: representing context-free formalisms. J. Log. Lang. Inf. 13(4), 421–438 (2004). https://doi.org/10.1007/s10849-004-2114-x

    Article  MathSciNet  MATH  Google Scholar 

  8. Hendriks, H.: The logic of tune – a proof-theoretic analysis of intonation. In: Lecomte, A., Lamarche, F., Perrier, G. (eds.) Logical Aspects of Computational Linguistics, Selected Papers. Lecture Notes in Computer Science, vol. 1582, pp. 132–159. Springer (1997). https://doi.org/10.1007/3-540-48975-4_7

  9. Huybregts, R.: The weak inadequacy of context-free phrase structure grammars. In: de Haan, G., Trommelen, M., Zonneveld, W. (eds.) Van Periferie Naar Kern, pp. 81–99. Foris Publications (1984)

    Google Scholar 

  10. Kallmeyer, L.: Parsing Beyond Context-Free Grammars. Springer (2010)

    Google Scholar 

  11. Kanovich, M.I., Kuznetsov, S.L., Scedrov, A.: The multiplicative-additive Lambek Calculus with subexponential and bracket modalities. J. Log. Lang. Inf. 30(1), 31–88 (2021). https://doi.org/10.1007/s10849-020-09320-9

    Article  MathSciNet  MATH  Google Scholar 

  12. Kogkalidis, K., Moortgat, M.: Geometry-aware supertagging with heterogeneous dynamic convolutions. CoRR abs/2203.12235 (2022). https://doi.org/10.48550/arXiv.2203.12235

  13. Kogkalidis, K., Moortgat, M., Moot, R.: Æthel: automatically extracted typelogical derivations for Dutch. In: Proceedings of The 12th Language Resources and Evaluation Conference, LREC 2020, Marseille, pp. 5257–5266. European Language Resources Association (2020)

    Google Scholar 

  14. Kogkalidis, K., Moortgat, M., Moot, R.: Neural proof nets. In: CoNLL2020, Proceedings of the 24th Conference on Computational Natural Language Learning, pp. 26–40. Association for Computational Linguistics (2020)

    Google Scholar 

  15. Kogkalidis, K., Moortgat, M., Moot, R.: Neural proof nets. In: Proceedings of the 24th Conference on Computational Natural Language Learning, pp. 26–40. Association for Computational Linguistics (2020). https://doi.org/10.18653/v1/2020.conll-1.3, https://aclanthology.org/2020.conll-1.3

  16. Kogkalidis, K., Wijnholds, G.: Discontinuous constituency and BERT: a case study of Dutch. In: Findings of the Association for Computational Linguistics: ACL 2022, pp. 3776–3785. Association for Computational Linguistics, Dublin, Ireland (2022). https://doi.org/10.18653/v1/2022.findings-acl.298, https://aclanthology.org/2022.findings-acl.298

  17. Kurtonina, N., Moortgat, M.: Structural control. In: Blackburn, P., de Rijke, M. (eds.) Specifying Syntactic Structures, pp. 75–113. CSLI, Stanford (1997)

    Google Scholar 

  18. Lafont, Y.: Soft linear logic and polynomial time. Theor. Comput. Sci. 318(1–2), 163–180 (2004). https://doi.org/10.1016/j.tcs.2003.10.018

    Article  MathSciNet  MATH  Google Scholar 

  19. Lambek, J.: The mathematics of sentence structure. Am. Math. Mon. 65(3), 154–170 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lambek, J.: On the calculus of syntactic types. Struct. Lang. Math. Asp. 12, 166–178 (1961)

    Google Scholar 

  21. de Marneffe, M.C., Manning, C.D., Nivre, J., Zeman, D.: Universal dependencies. Comput. Linguist. 47(2), 255–308 (2021). https://doi.org/10.1162/coli_a_00402

    Article  Google Scholar 

  22. Montague, R.: Universal grammar. In: Thomason, R.H. (ed.) Formal Philosophy: Selected Papers of Richard Montague, pp. 222–247. Yale University Press, New Haven, London (1974)

    Google Scholar 

  23. Moortgat, M.: Constants of grammatical reasoning. In: Bouma, G., Hinrichs, E., Kruijff, G.J.M., Oehrle, R.T. (eds.) Constraints and Resources in Natural Language Syntax and Semantics, pp. 195–219. CSLI Publications (1999)

    Google Scholar 

  24. Moortgat, M.: Meaningful patterns. In: Gerbrandy, J., Marx, M., de Rijke, M., Venema, Y. (eds.) JFAK. Assays dedicated to Johan van Benthem on the occasion of his 50th birthday. ILLC, Amsterdam (1999)

    Google Scholar 

  25. Moortgat, M., Morrill, G.: Heads and phrases. Type calculus for dependency and constituent structure. Ms Utrecht University (1991)

    Google Scholar 

  26. Moortgat, M., Wijnholds, G.: Lexical and derivational meaning in vector-based models of relativisation. CoRR abs/1711.11513 (2017)

    Google Scholar 

  27. Moot, R., Retoré, C.: Natural language semantics and computability. J. Log. Lang. Inf. 28(2), 287–307 (2019). https://doi.org/10.1007/s10849-019-09290-7

    Article  MathSciNet  MATH  Google Scholar 

  28. Morrill, G.: Type Logical Grammar. Springer (1994). https://doi.org/10.1007/978-94-011-1042-6

  29. Morrill, G.: Parsing/theorem-proving for logical grammar CatLog3. J. Log. Lang. Inf. 28(2), 183–216 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  30. van Noord, G., Bouma, G., Van Eynde, F., de Kok, D., van der Linde, J., Schuurman, I., Sang, E.T.K., Vandeghinste, V.: Large Scale Syntactic Annotation of Written Dutch: Lassy, pp. 147–164. Springer, Berlin, Heidelberg (2013). https://doi.org/10.1007/978-3-642-30910-6_9

  31. Pollard, C.: Head grammars, generalized phrase structure grammars, and natural language. Ph.D. thesis, Stanford (1984)

    Google Scholar 

  32. Sadrzadeh, M., Muskens, R.: Static and dynamic vector semantics for lambda calculus models of natural language. J. Lang. Model. 6(2), 319–351 (2018). https://doi.org/10.15398/jlm.v6i2.228

    Article  Google Scholar 

  33. Seki, H., Matsumura, T., Fujii, M., Kasami, T.: On multiple context-free grammars. Theor. Comput. Sci. 88(2), 191–229 (1991). https://doi.org/10.1016/0304-3975(91)90374-B

    Article  MathSciNet  MATH  Google Scholar 

  34. Shieber, S.M.: Evidence against the context-freeness of natural language. Linguist. Philos. 8, 333–343 (1985). https://doi.org/10.1007/BF00630917

    Article  Google Scholar 

  35. de Vries, W., van Cranenburgh, A., Bisazza, A., Caselli, T., van Noord, G., Nissim, M.: BERTje: a Dutch BERT model (2019). arXiv:1912.09582

  36. Wansing, H.: Formulas-as-types for a hierarchy of sublogics of intuitionistic propositional logic. In: Pearce, D., Wansing, H. (eds.) Nonclassical Logics and Information Processing, International Workshop, Berlin, Germany, 9–10 November 1990, Proceedings. Lecture Notes in Computer Science, vol. 619, pp. 125–145. Springer (1990). https://doi.org/10.1007/BFb0031928

  37. Wijnholds, G., Moortgat, M.: SICK-NL: a dataset for Dutch natural language inference. In: Proceedings of the 16th Conference of the European Chapter of the Association for Computational Linguistics: Main Volume, pp. 1474–1479. Association for Computational Linguistics, Online (2021). https://aclanthology.org/2021.eacl-main.126

  38. Wijnholds, G., Sadrzadeh, M.: A type-driven vector semantics for ellipsis with anaphora using Lambek calculus with limited contraction. J. Log. Lang. Inf. 28(2), 331–358 (2019). https://doi.org/10.1007/s10849-019-09293-4

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. UiL-OTS Utrecht University, Utrecht, The Netherlands

    Michael Moortgat, Konstantinos Kogkalidis & Gijs Wijnholds

Authors
  1. Michael Moortgat
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  2. Konstantinos Kogkalidis
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  3. Gijs Wijnholds
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Correspondence to Michael Moortgat .

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Editors and Affiliations

  1. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Roussanka Loukanova

  2. Department of Mathematics, Faculty of Science, Stockholm University, Stockholm, Sweden

    Peter LeFanu Lumsdaine

  3. Faculty of Science, ILLC—Institute for Logic, Language and Computation, University of Amsterdam, Amsterdam, The Netherlands

    Reinhard Muskens

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Moortgat, M., Kogkalidis, K., Wijnholds, G. (2023). Diamonds Are Forever. In: Loukanova, R., Lumsdaine, P.L., Muskens, R. (eds) Logic and Algorithms in Computational Linguistics 2021 (LACompLing2021). Studies in Computational Intelligence, vol 1081. Springer, Cham. https://doi.org/10.1007/978-3-031-21780-7_3

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