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2021 | 50 | 1 | 55-80
Tytuł artykułu

One-Sided Sequent Systems for Nonassociative Bilinear Logic: Cut Elimination and Complexity

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Języki publikacji
EN
Abstrakty
EN
Bilinear Logic of Lambek amounts to Noncommutative MALL of Abrusci. Lambek proves the cut–elimination theorem for a one-sided (in fact, left-sided) sequent system for this logic. Here we prove an analogous result for the nonassociative version of this logic. Like Lambek, we consider a left-sided system, but the result also holds for its right-sided version, by a natural symmetry. The treatment of nonassociative sequent systems involves some subtleties, not appearing in associative logics. We also prove the PTime complexity of the multiplicative fragment of NBL.
Rocznik
Tom
50
Numer
1
Strony
55-80
Opis fizyczny
Daty
wydano
2021-03-30
Twórcy
  • Adam Mickiewicz University, Faculty of Mathematics and Computer Science Uniwersytetu Poznanskiego 4, 61-614 Poznan, Poland
Bibliografia
  • [1] V. M. Abrusci, Phase semantics and sequent calculus for pure noncommutative classical linear propositional logic, The Journal of Symbolic Logic, vol. 56(4) (1991), pp. 1403–1451, DOI: http://dx.doi.org/10.2307/2275485
  • [2] A. Bastenhof, Categorial Symmetry, Ph.D. thesis, Utrecht University (2013).
  • [3] W. Buszkowski, On classical nonassociative Lambek calculus, [in:] M. Amblard, P. de Groote, S. Pogodalla, C. Retoré (eds.), Logical Aspects of Computational Linguistics, vol. 10054 of Lecture Notes in Computer Science, Springer (2016), pp. 68–84, DOI: http://dx.doi.org/10.1007/978-3-662-53826-5_5
  • [4] W. Buszkowski, Involutive nonassociative Lambek calculus: Sequent systems and complexity, Bulletin of the Section of Logic, vol. 46(1/2) (2017), DOI: http://dx.doi.org/10.18778/0138-0680.46.1.2.07
  • [5] P. De Groote, F. Lamarche, Classical non-associative Lambek calculus, Studia Logica, vol. 71(3) (2002), pp. 355–388, DOI: http://dx.doi.org/10.1023/A:1020520915016
  • [6] N. Galatos, P. Jipsen, Residuated frames with applications to decidability, Transactions of the American Mathematical Society, vol. 365(3) (2013), pp. 1219–1249, URL: https://www.jstor.org/stable/23513444
  • [7] N. Galatos, H. Ono, Cut elimination and strong separation for substructural logics: an algebraic approach, Annals of Pure and Applied Logic, vol. 161(9) (2010), pp. 1097–1133, DOI: http://dx.doi.org/0.1016/j.apal.2010.01.003
  • [8] J.-Y. Girard, Linear logic, Theoretical Computer Science, vol. 50(1) (1987), pp. 1–101, DOI: http://dx.doi.org/10.1016/0304-3975(87)90045-4
  • [9] J. Lambek, On the calculus of syntactic types, [in:] R. Jakobson (ed.), Structure of language and its mathematical aspects, vol. 12, Providence, RI: American Mathematical Society (1961), pp. 166–178, DOI: http://dx.doi.org/10.1090/psapm/012/9972
  • [10] J. Lambek, Cut elimination for classical bilinear logic, Fundamenta Informaticae, vol. 22(1, 2) (1995), pp. 53–67, DOI: http://dx.doi.org/10.3233/FI-1995-22123
  • [11] M. Pentus, Lambek calculus is NP-complete, Theoretical Computer Science, vol. 357(1-3) (2006), pp. 186–201, DOI: http://dx.doi.org/10.1016/j.tcs.2006.03.018Get.
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Bibliografia
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