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Logical Foundations for Hybrid Type-Logical Grammars

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Abstract

This paper explores proof-theoretic aspects of hybrid type-logical grammars, a logic combining Lambek grammars with lambda grammars. We prove some basic properties of the calculus, such as normalisation and the subformula property and also present both a sequent and a proof net calculus for hybrid type-logical grammars. In addition to clarifying the logical foundations of hybrid type-logical grammars, the current study opens the way to variants and extensions of the original system, including but not limited to a non-associative version and a multimodal version incorporating structural rules and unary modes.

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Notes

  1. Linear Categorial Grammar (Mihaliček and Pollard 2012; Pollard 2015; Smith 2010; Mihaličik 2012; Martin 2013; Worth 2014) is the restriction of HTLG to the linear implication rules. LCG has been criticised (Muskens 2001; Kubota and Levine 2015) for licensing proofs parallel to the one in Fig. 3. Note that given the types (with \(X=np\multimap s\)) all lexical term assignments for this type overgenerate (Moot 2014), so this problem is not easily fixed.

  2. The rule for the \(/\) directly parallels that for \(\backslash \), modulo directionality.

  3. In case \(((\lambda x. M)\, N)\) is an \(\eta \) redex and we perform an \(\eta \) reduction on it, its trace must be of the form \(((\lambda x. (M'\, x))\, N')\), and the \(\eta \) redex of this term is identical to its \(\beta \) redex.

  4. As is usual in the lambda calculus, we do not distinguish alpha-equivalent lambda terms.

  5. For natural deduction, rule permutations are a problem only for the \(\bullet E\) and the \(\Diamond E\) rules.

  6. To ensure confluence of ‘/’ and ‘\(\backslash \)’ in the presence of \(\epsilon \) we can add the side condition to the [/I] and \([\backslash I]\) contractions that the component to which the par link is attached has at least one hypothesis other than the auxiliary conclusion of the par link. This forbids empty antecedent derivations and restores confluence.

  7. To avoid making the structure overly complicated, neither “studies” nor the tv hypothesis with which it is matched have been unfolded.

  8. We need to prove cut elimination for the multiplicative-additive logic for this to hold, which is rather simple extension of the cut elimination proof in Sect. 3.2.

  9. Although [Kubota and Levine (2020), Sect. 7.2] give a treatment of parasitic gapping assigning non-linear lambda terms to the empty string.

  10. An alternative would be to forbid Lambek calculus connectives in assignments to the empty string. In this case, the question of decidability would be open.

  11. A possible exception would be the treatment of discontinuous gapping of Kubota and Levine (2012, Sect. 3.2). However, we would need to carefully weigh the added complexity at the level of prosodic terms (and types) that type polymorphism would introduce at this level.

References

  • Bondy, J. A., & Murty, U. S. R. (2008). Graph theory, Graduate Texts in Mathematics, vol. 244. Springer.

  • Buszkowski, W. (1982). Some decision problems in the theory of syntactic categories. Mathematical Logic Quarterly, 28, 539–548.

    Article  Google Scholar 

  • Danos, V. (1990). La logique linéaire appliquée à l’étude de divers processus de normalisation (principalement du \(\lambda \)-calcul). Ph.D. thesis, University of Paris VII.

  • de Groote, P., & Retoré, C. (1996). On the semantic readings of proof nets. In G. J. Kruijff, G. Morrill, R. T. Oehrle (Eds.) Formal grammar, pp. 57–70.

  • Di Cosmo, R., & Kesner, D. (1996). Combining algebraic rewriting, extensional lambda calculi, and fixpoints. Theoretical Computer Science, 169, 201–220.

    Article  Google Scholar 

  • Emms, M. (1993). Parsing with polymorphism. In Proceedings of the sixth conference of the European association of computational linguistics, pp. 120–129.

  • Emms, M. (1995). An undecidability result for polymorphic Lambek calculus. In Proceedings of the 10th Amsterdam colloquium.

  • Girard, J. Y. (1987). Linear logic. Theoretical Computer Science, 50, 1–102.

    Article  Google Scholar 

  • Girard, J. Y., Lafont, Y., & Taylor, P. (1988). Proofs and types. Cambridge Tracts in Theoretical Computer Science 7. Cambridge University Press.

  • Kanovich, M. I., Kuznetsov, S., & Scedrov, A. (2019). The complexity of multiplicative-additive lambek calculus: 25 years later. In R. Iemhoff, M. Moortgat, R. J. G. B. de Queiroz (Eds.) Logic, language, information, and computation - 26th international workshop, WoLLIC 2019, Utrecht, The Netherlands, July 2–5, 2019, Proceedings, Lecture Notes in Computer Science, vol. 11541, pp. 356–372. Springer.

  • Klop, J., van Oostrom, V., & van Raamsdonk, F. (1993). Combinatory reduction systems: Introduction and survey. Theoretical Computer Science, 121, 279–308.

    Article  Google Scholar 

  • Kubota, Y. (2010). (in)flexibility of constituency in japanese in multi-modal categorial grammar with structured phonology. Ph.D. thesis, The Ohio State University.

  • Kubota, Y., & Levine, R. (2012). Gapping as like-category coordination. Lecture Notes in Computer Science. In D. Béchet & A. Dikovsky (Eds.), Logical aspects of computational linguistics (Vol. 7351, pp. 135–150). Nantes: Springer.

  • Kubota, Y., & Levine, R. (2013a). Coordination in hybrid type-logical grammar. In Ohio State University working papers in linguistics, vol. 60. Columbus, Ohio.

  • Kubota, Y., & Levine, R. (2013b). Determiner gapping as higher-order discontinuous constituency. In G. Morrill, M. J. Nederhof (Eds.) Formal grammar, lecture notes in computer science, vol. 8036, pp. 225–241. Springer.

  • Kubota, Y., & Levine, R. (2015). Against ellipsis: Arguments for the direct licensing of ‘noncanonical’ coordinations. Linguistics and Philosophy, 38, 521–576. https://doi.org/10.1007/s10988-015-9179-7.

  • Kubota, Y., & Levine, R. (2020). Type-logical syntax. MIT Press.

  • Lambek, J. (1958). The mathematics of sentence structure. American Mathematical Monthly, 65, 154–170.

    Article  Google Scholar 

  • Lincoln, P., Mitchell, J., Scedrov, A., & Shankar, N. (1992). Decision problems for propositional linear logic. Annals of Pure and Applied Logic, 56(1–3), 239–311.

    Article  Google Scholar 

  • Martin, S. (2013). The dynamics of sense and implicature. Ph.D. thesis, The Ohio State University.

  • Mihaliček, V., & Pollard, C. (2012). Distinguishing phenogrammar from tectogrammar simplifies the analysis of interrogatives. In Formal grammar: FG2010, FG2011, Lecture Notes in Computer Science, vol. 7395, pp. 130–145. Springer. https://doi.org/10.1007/978-3-642-32024-8_9

  • Mihaličik, V. (2012). Serbo-croatian word order: A logical approach. Ph.D. thesis, The Ohio State University.

  • Moortgat, M. (1988). Categorial investigations: Logical and linguistic aspects of the Lambek calculus. Dordrecht: Foris.

    Book  Google Scholar 

  • Moortgat, M. (1997). Categorial type logics. In J. van Benthem, A. ter Meulen (Eds.) Handbook of logic and language, chap. 2, pp. 93–177. Elsevier/MIT Press.

  • Moortgat, M., & Moot, R. (2013). Proof nets for the Lambek-Grishin calculus. In E. Grefenstette, C. Heunen, M. Sadrzadeh (Eds.) Quantum physics and linguistics: A compositional, diagrammatic discourse, pp. 283–320. Oxford University Press.

  • Moot, R. (2002). Proof nets for linguistic analysis. Ph.D. thesis, Utrecht Institute of Linguistics OTS, Utrecht University.

  • Moot, R. (2014). Hybrid type-logical grammars, first-order linear logic and the descriptive inadequacy of lambda grammars. Tech. rep.: LaBRI (CNRS), Bordeaux University.

  • Moot, R., & Puite, Q. (2002). Proof nets for the multimodal Lambek calculus. Studia Logica, 71(3), 415–442.

    Article  Google Scholar 

  • Moot, R., & Stevens-Guille, S. J. (2019). Proof-theoretic aspects of hybrid type-logical grammars. In Proceedings of formal grammar 2019, Lecture Notes in Computer Science, vol. 11668, pp. 84–100. Springer.

  • Morrill, G. (1999). Geometry of lexico-syntactic interaction. In Proceedings of the ninth conference of the European chapter of the association for computational linguistics, pp. 61–70. Bergen, Norway.

  • Muskens, R. (2001). Categorial grammar and lexical-functional grammar. In Proceedings of the LFG01 conference, pp. 259–279. University of Hong Kong.

  • Newman, M. H. A. (1942). On theories with a combinatorial definition of “equivalence.” Annals of mathematics, 43(2), 223–243.

  • Pentus, M. (2006). Lambek calculus is NP-complete. Theoretical Computer Science, 357(1), 186–201.

    Article  Google Scholar 

  • Perrier, G. (1999). A PSPACE-complete fragment of second order linear logic. Theoretical Computer Science, 222(1–2), 267–289.

    Article  Google Scholar 

  • Pollard, C. (2015). Agnostic hyperintensional semantics. Synthese, 192, 535–562. https://doi.org/10.1007/s11229-013-0373-2

    Article  Google Scholar 

  • Savateev, Y. (2009). Product-free Lambek calculus is NP-complete. In Symposium on logical foundations of computer science (LFCS), pp. 380–394.

  • Smith, E. A. (2010). Correlational comparison in english. Ph.D. thesis, The Ohio State University.

  • Worth, C. (2014). The phenogrammar of coordination. In Proceedings of the EACL 2014 workshop on type theory and natural language semantics (TTNLS), pp. 28–36.

  • Yoshinaka, R., & Kanazawa, M. (2005). The complexity and generative capacity of lexicalized abstract categorial grammars. In Proceedings of logical aspects of computational linguistics 2005, lecture notes in artificial intelligence, vol. 3492, pp. 330–346. Springer, Bordeaux, France.

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  1. LIRMM, Université de Montpellier, CNRS, Montpellier, France

    Richard Moot

  2. The Ohio State University, Columbus, OH, USA

    Symon Jory Stevens-Guille

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Correspondence to Richard Moot.

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This is a significantly extended version of a paper presented at Formal Grammar 2019 (Moot and Stevens-Guille 2019). We thank the referees of Formal Grammar as well as the audience of the conference for their invaluable feedback. We also thank the referees of the Journal of Logic, Language and Information for their comments on this revised paper.

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Moot, R., Stevens-Guille, S.J. Logical Foundations for Hybrid Type-Logical Grammars. J of Log Lang and Inf 31, 35–76 (2022). https://doi.org/10.1007/s10849-021-09348-5

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