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Why Are Proofs Relevant in Proof-Relevant Models?

Published:11 January 2023Publication History
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Abstract

Relational models of λ-calculus can be presented as type systems, the relational interpretation of a λ-term being given by the set of its typings. Within a distributors-induced bicategorical semantics generalizing the relational one, we identify the class of ‘categorified’ graph models and show that they can be presented as type systems as well. We prove that all the models living in this class satisfy an Approximation Theorem stating that the interpretation of a program corresponds to the filtered colimit of the denotations of its approximants. As in the relational case, the quantitative nature of our models allows to prove this property via a simple induction, rather than using impredicative techniques. Unlike relational models, our 2-dimensional graph models are also proof-relevant in the sense that the interpretation of a λ-term does not contain only its typings, but the whole type derivations. The additional information carried by a type derivation permits to reconstruct an approximant having the same type in the same environment. From this, we obtain the characterization of the theory induced by the categorified graph models as a simple corollary of the Approximation Theorem: two λ-terms have isomorphic interpretations exactly when their B'ohm trees coincide.

References

  1. Samson Abramsky. 1991. Domain theory in logical form. Annals of Pure and Applied Logic, 51, 1 (1991), 1–77. issn:0168-0072 https://doi.org/10.1016/0168-0072(91)90065-T Google ScholarGoogle ScholarCross RefCross Ref
  2. Roberto M. Amadio and Pierre-Louis Curien. 1998. Domains and Lambda-calculi. Cambridge University Press, New York, NY, USA. isbn:0-521-62277-8 Google ScholarGoogle Scholar
  3. Henk Barendregt, Mario Coppo, and Mariangiola Dezani-Ciancaglini. 1983. A filter lambda model and the completeness of type assignment. Journal of Symbolic Logic, 48, 4 (1983), 931–940. https://doi.org/10.2307/2273659 Google ScholarGoogle ScholarCross RefCross Ref
  4. Henk P. Barendregt. 1977. The type free lambda calculus. In Handbook of Mathematical Logic, J. Barwise (Ed.) (Studies in Logic and the Foundations of Mathematics, Vol. 90). North-Holland, Amsterdam, 1091–1132. Google ScholarGoogle Scholar
  5. Henk P. Barendregt. 1984. The lambda-calculus, its syntax and semantics (revised ed.) (Studies in Logic and the Foundations of Mathematics). North-Holland. Google ScholarGoogle Scholar
  6. Hendrik Pieter Barendregt, Wil Dekkers, and Richard Statman. 2013. Lambda Calculus with Types. Cambridge University Press. isbn:978-0-521-76614-2 http://www.cambridge.org/de/academic/subjects/mathematics/logic-categories-and-sets/lambda-calculus-types Google ScholarGoogle ScholarDigital LibraryDigital Library
  7. J. Benabou. 1973. Les distributeurs: d’après le cours de Questions spéciales de mathématique. Institut de mathématique pure et appliquée, Université catholique de Louvain. https://books.google.fr/books?id=XiauHAAACAAJ Google ScholarGoogle Scholar
  8. Chantal Berline. 2000. From computation to foundations via functions and application: The λ -calculus and its webbed models. Theor. Comput. Sci., 249, 1 (2000), 81–161. https://doi.org/10.1016/S0304-3975(00)00057-8 Google ScholarGoogle ScholarDigital LibraryDigital Library
  9. Francis Borceux. 1994. Handbook of Categorical Algebra (Encyclopedia of Mathematics and its Applications, Vol. 1). Cambridge University Press. https://doi.org/10.1017/CBO9780511525858 Google ScholarGoogle ScholarCross RefCross Ref
  10. Flavien Breuvart, Giulio Manzonetto, and Domenico Ruoppolo. 2018. Relational Graph Models at Work. Log. Methods Comput. Sci., 14, 3 (2018), https://doi.org/10.23638/LMCS-14(3:2)2018 Google ScholarGoogle ScholarCross RefCross Ref
  11. Antonio Bucciarelli, Thomas Ehrhard, and Giulio Manzonetto. 2007. Not Enough Points Is Enough. In Computer Science Logic, 21st International Workshop, CSL 2007, 16th Annual Conference of the EACSL, Lausanne, Switzerland, September 11-15, 2007, Proceedings, Jacques Duparc and Thomas A. Henzinger (Eds.) (Lecture Notes in Computer Science, Vol. 4646). Springer, 298–312. https://doi.org/10.1007/978-3-540-74915-8_24 Google ScholarGoogle ScholarCross RefCross Ref
  12. Antonio Bucciarelli, Delia Kesner, and Simona Ronchi Della Rocca. 2014. The Inhabitation Problem for Non-idempotent Intersection Types. In IFIP TCS (Lecture Notes in Computer Science, Vol. 8705). Springer, 341–354. Google ScholarGoogle ScholarCross RefCross Ref
  13. Antonio Bucciarelli, Delia Kesner, and Daniel Ventura. 2017. Non-idempotent intersection types for the Lambda-Calculus. Log. J. IGPL, 25, 4 (2017), 431–464. https://doi.org/10.1093/jigpal/jzx018 Google ScholarGoogle ScholarCross RefCross Ref
  14. Mario Coppo, Mariangiola Dezani-Ciancaglini, Furio Honsell, and Giuseppe Longo. 1984. Extended Type Structures and Filter Lambda Models. In Logic Colloquium ’82, G. Lolli, G. Longo, and A. Marcja (Eds.) (Studies in Logic and the Foundations of Mathematics, Vol. 112). Elsevier, 241–262. https://doi.org/10.1016/S0049-237X(08)71819-6 Google ScholarGoogle ScholarCross RefCross Ref
  15. Mario Coppo, Mariangiola Dezani-Ciancaglini, and Maddalena Zacchi. 1987. Type Theories, Normal Forms and D_∞ Lambda-Models. Inf. Comput., 72, 2 (1987), 85–116. https://doi.org/10.1016/0890-5401(87)90042-3 Google ScholarGoogle ScholarDigital LibraryDigital Library
  16. Daniel de Carvalho. 2007. Sémantiques de la logique linéaire. Aix-Marseille Université. Google ScholarGoogle Scholar
  17. Daniel de Carvalho. 2009. Execution Time of λ -Terms via Denotational Semantics and Intersection Types. CoRR, abs/0905.4251 (2009), arXiv:0905.4251. arxiv:0905.4251 Google ScholarGoogle Scholar
  18. Daniel de Carvalho. 2018. Execution time of λ -terms via denotational semantics and intersection types. Math. Struct. Comput. Sci., 28, 7 (2018), 1169–1203. https://doi.org/10.1017/S0960129516000396 First submitted in 2009, see abs-0905-4251 Google ScholarGoogle ScholarCross RefCross Ref
  19. Thomas Ehrhard. 2012. Collapsing non-idempotent intersection types. In Computer Science Logic (CSL’12) - 26th International Workshop/21st Annual Conference of the EACSL, CSL 2012 (LIPIcs, Vol. 16). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 259–273. https://doi.org/10.4230/LIPIcs.CSL.2012.259 Google ScholarGoogle ScholarCross RefCross Ref
  20. Thomas Ehrhard. 2016. Call-By-Push-Value from a Linear Logic Point of View. In Programming Languages and Systems, Peter Thiemann (Ed.). Springer Berlin Heidelberg, Berlin, Heidelberg. 202–228. isbn:978-3-662-49498-1 Google ScholarGoogle Scholar
  21. Thomas Ehrhard and Laurent Regnier. 2003. The differential lambda-calculus. Theor. Comput. Sci., 309, 1-3 (2003), 1–41. https://doi.org/10.1016/S0304-3975(03)00392-X Google ScholarGoogle ScholarDigital LibraryDigital Library
  22. Thomas Ehrhard and Laurent Regnier. 2006. Böhm Trees, Krivine’s Machine and the Taylor Expansion of Lambda-Terms. In CiE (Lecture Notes in Computer Science, Vol. 3988). Springer, 186–197. Google ScholarGoogle Scholar
  23. Thomas Ehrhard and Laurent Regnier. 2008. Uniformity and the Taylor expansion of ordinary λ -terms. Theor. Comput. Sci., 403, 2-3 (2008), 347–372. https://doi.org/10.1016/j.tcs.2008.06.001 Google ScholarGoogle ScholarDigital LibraryDigital Library
  24. Erwin Engeler. 1981. Algebras and combinators. Algebra Universalis, 13, 3 (1981), 389–392. Google ScholarGoogle Scholar
  25. Marcelo Fiore, Nicola Gambino, Martin Hyland, and Glynn Winskel. 2008. The cartesian closed bicategory of generalised species of structures. Journal of the London Mathematical Society, 77, 1 (2008), 203––220. https://doi.org/10.1112/jlms/jdm096 Google ScholarGoogle ScholarCross RefCross Ref
  26. Marcelo Fiore, Nicola Gambino, Martin Hyland, and Glynn Winskel. 2017. Relative pseudomonads, Kleisli bicategories, and substitution monoidal structures. Selecta Mathematica, 24, 3 (2017), 11, 2791–2830. issn:1420-9020 https://doi.org/10.1007/s00029-017-0361-3 Google ScholarGoogle ScholarCross RefCross Ref
  27. Marcelo Fiore and Philip Saville. 2019. A type theory for cartesian closed bicategories (Extended Abstract). In 34th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2019, Vancouver, BC, Canada, June 24-27, 2019. 1–13. https://doi.org/10.1109/LICS.2019.8785708 Google ScholarGoogle ScholarCross RefCross Ref
  28. Marcelo Fiore and Philip Saville. 2020. Coherence and Normalisation-by-Evaluation for Bicategorical Cartesian Closed Structure. In Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS ’20). Association for Computing Machinery, New York, NY, USA. 425–439. https://doi.org/10.1145/3373718.3394769 Google ScholarGoogle ScholarDigital LibraryDigital Library
  29. Zeinab Galal. 2020. A Profunctorial Scott Semantics. In 5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020), Zena M. Ariola (Ed.) (Leibniz International Proceedings in Informatics (LIPIcs), Vol. 167). Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Dagstuhl, Germany. 16:1–16:18. isbn:978-3-95977-155-9 issn:1868-8969 https://doi.org/10.4230/LIPIcs.FSCD.2020.16 Google ScholarGoogle ScholarCross RefCross Ref
  30. Nicola Gambino and André Joyal. 2017. On operads, bimodules and analytic functors. Memoirs of the American Mathematical Society, 249, 1184 (2017), 9, issn:1947-6221 https://doi.org/10.1090/memo/1184 Google ScholarGoogle ScholarCross RefCross Ref
  31. Jean-Yves Girard. 1987. Linear Logic. Theor. Comput. Sci., 50 (1987), 1–102. Google ScholarGoogle ScholarDigital LibraryDigital Library
  32. Jean-Yves Girard. 1988. Normal Functors, Power Series and Lambda-Calculus. Annals of Pure and Applied Logic, 37, 2 (1988), 129. Google ScholarGoogle ScholarCross RefCross Ref
  33. Giulio Guerrieri and Federico Olimpieri. 2021. Categorifying Non-Idempotent Intersection Types. In 29th EACSL Annual Conference on Computer Science Logic (CSL 2021), Christel Baier and Jean Goubault-Larrecq (Eds.) (Leibniz International Proceedings in Informatics (LIPIcs), Vol. 183). Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Dagstuhl, Germany. 25:1–25:24. isbn:978-3-95977-175-7 issn:1868-8969 https://doi.org/10.4230/LIPIcs.CSL.2021.25 Google ScholarGoogle ScholarCross RefCross Ref
  34. Barney P. Hilken. 1996. Towards a proof theory of rewriting: the simply typed 2λ -calculus. Theor. Comput. Sci., 170, 1-2 (1996), 407–444. https://doi.org/10.1016/S0304-3975(96)80713-4 Google ScholarGoogle ScholarCross RefCross Ref
  35. J. Martin E. Hyland. 1976. A syntactic characterization of the equality in some models for the λ -calculus. Journal London Mathematical Society (2), 12(3) (1976), 361–370. Google ScholarGoogle ScholarCross RefCross Ref
  36. J. Martin E. Hyland. 2014. Towards a Notion of Lambda Monoid. Electron. Notes Theor. Comput. Sci., 303 (2014), 59–77. https://doi.org/10.1016/j.entcs.2014.02.004 Google ScholarGoogle ScholarDigital LibraryDigital Library
  37. J. Martin E. Hyland. 2017. Classical lambda calculus in modern dress. Math. Struct. Comput. Sci., 27, 5 (2017), 762–781. https://doi.org/10.1017/S0960129515000377 Google ScholarGoogle ScholarCross RefCross Ref
  38. J. Martin E. Hyland, Misao Nagayama, John Power, and Giuseppe Rosolini. 2006. A Category Theoretic Formulation for Engeler-style Models of the Untyped Lambda Calculus. Electron. Notes Theor. Comput. Sci., 161 (2006), 43–57. https://doi.org/10.1016/j.entcs.2006.04.024 Google ScholarGoogle ScholarCross RefCross Ref
  39. Benedetto Intrigila, Giulio Manzonetto, and Andrew Polonsky. 2019. Degrees of extensionality in the theory of Böhm trees and Sallé’s conjecture. Log. Methods Comput. Sci., 15, 1 (2019), https://doi.org/10.23638/LMCS-15(1:6)2019 Google ScholarGoogle ScholarCross RefCross Ref
  40. Bart Jacobs and Jan Rutten. 1997. A Tutorial on (Co)Algebras and (Co)Induction. EATCS Bulletin, 62 (1997), 62–222. Google ScholarGoogle Scholar
  41. Niles Johnson and Donald Yau. 2021. 2-Dimensional Categories. Oxford University Press. https://doi.org/10.1093/oso/9780198871378.001.0001 Google ScholarGoogle ScholarCross RefCross Ref
  42. André Joyal. 1986. Foncteurs analytiques et espèces de structures. In Combinatoire énumérative. Springer Berlin Heidelberg, Berlin, Heidelberg. 126–159. Google ScholarGoogle Scholar
  43. Panagis Karazeris. 2001. Categorical domain theory: Scott topology, powercategories, coherent categories.. Theory and Applications of Categories [electronic only], 9 (2001), 106–120. http://eudml.org/doc/122250 Google ScholarGoogle Scholar
  44. G. Maxwell Kelly. 1980. A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on. Bulletin of the Australian Mathematical Society, 22, 1 (1980), 1–83. https://doi.org/10.1017/S0004972700006353 Google ScholarGoogle ScholarCross RefCross Ref
  45. Max Kelly. 1982. Basic Concepts of Enriched Category Theory (Lecture Notes in Mathematics, Vol. 64). Cambridge University Press, Cambridge. Republished as: Reprints in Theory and Applications of Categories, No. 10 (2005) pp. 1–136 Google ScholarGoogle Scholar
  46. James Laird. 2017. From Qualitative to Quantitative Semantics - By Change of Base. In Foundations of Software Science and Computation Structures - 20th International Conference, FOSSACS 2017, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2017, Uppsala, Sweden, April 22-29, 2017, Proceedings, Javier Esparza and Andrzej S. Murawski (Eds.) (Lecture Notes in Computer Science, Vol. 10203). 36–52. https://doi.org/10.1007/978-3-662-54458-7_3 Google ScholarGoogle ScholarDigital LibraryDigital Library
  47. Jim Laird, Giulio Manzonetto, Guy McCusker, and Michele Pagani. 2013. Weighted Relational Models of Typed Lambda-Calculi. In Proceedings of the 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS ’13). IEEE Computer Society, USA. https://doi.org/10.1109/LICS.2013.36 Google ScholarGoogle ScholarDigital LibraryDigital Library
  48. Fosco Loregian. 2021. (Co)end Calculus. Cambridge University Press, Cambridge. https://doi.org/10.1017/9781108778657 Google ScholarGoogle ScholarCross RefCross Ref
  49. Stefania Lusin and Antonino Salibra. 2004. The Lattice of Lambda Theories. J. Log. Comput., 14, 3 (2004), 373–394. Google ScholarGoogle ScholarDigital LibraryDigital Library
  50. Giulio Manzonetto and Domenico Ruoppolo. 2014. Relational Graph Models, Taylor Expansion and Extensionality. In Proceedings of the 30th Conference on the Mathematical Foundations of Programming Semantics, MFPS 2014, Ithaca, NY, USA, June 12-15, 2014, Bart Jacobs, Alexandra Silva, and Sam Staton (Eds.) (Electronic Notes in Theoretical Computer Science, Vol. 308). Elsevier, 245–272. https://doi.org/10.1016/j.entcs.2014.10.014 Google ScholarGoogle ScholarDigital LibraryDigital Library
  51. Damiano Mazza. 2017. Polyadic Approximations in Logic and Computation. Université Sorbonne Paris Nord. Google ScholarGoogle Scholar
  52. Damiano Mazza, Luc Pellissier, and Pierre Vial. 2018. Polyadic approximations, fibrations and intersection types. Proc. ACM Program. Lang., 2, POPL, 6:1–6:28. https://doi.org/10.1145/3158094 Google ScholarGoogle ScholarDigital LibraryDigital Library
  53. Federico Olimpieri. 2020. Intersection Types and Resource Calculi in the Denotational Semantics of Lambda-Calculus. Ph. D. Dissertation. Aix-Marseille Université. Google ScholarGoogle Scholar
  54. Federico Olimpieri. 2021. Intersection Type Distributors. In 36th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2021, Rome, Italy, June 29 - July 2, 2021. IEEE, 1–15. https://doi.org/10.1109/LICS52264.2021.9470617 Google ScholarGoogle ScholarDigital LibraryDigital Library
  55. C.-H. Luke Ong. 2017. Quantitative semantics of the lambda calculus: Some generalisations of the relational model. In 32nd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2017, Reykjavik, Iceland, June 20-23, 2017. IEEE Computer Society, 1–12. https://doi.org/10.1109/LICS.2017.8005064 Google ScholarGoogle ScholarCross RefCross Ref
  56. Luca Paolini, Mauro Piccolo, and Simona Ronchi Della Rocca. 2017. Essential and relational models. Math. Struct. Comput. Sci., 27, 5 (2017), 626–650. https://doi.org/10.1017/S0960129515000316 Google ScholarGoogle ScholarCross RefCross Ref
  57. Simona Ronchi Della Rocca. 1982. Characterization Theorems for a Filter Lambda Model. Inf. Control., 54, 3 (1982), 201–216. https://doi.org/10.1016/S0019-9958(82)80022-3 Google ScholarGoogle ScholarCross RefCross Ref
  58. Philip Saville. 2020. Cartesian closed bicategories: type theory and coherence. Ph. D. Dissertation. University of Cambridge. arxiv:2007.00624. Google ScholarGoogle Scholar
  59. Dana S. Scott. 1976. Data Types as Lattices. SIAM J. Comput., 5, 3 (1976), 522–587. https://doi.org/10.1137/0205037 Google ScholarGoogle ScholarDigital LibraryDigital Library
  60. Robert A. G. Seely. 1987. Modelling Computations: A 2-Categorical Framework. In Proceedings of the Symposium on Logic in Computer Science (LICS ’87), Ithaca, New York, USA, June 22-25, 1987. IEEE Computer Society, 65–71. Google ScholarGoogle Scholar
  61. William W. Tait. 1966. A nonconstructive proof of Gentzen’s Hauptsatz for second order predicate logic. Bull. Amer. Math. Soc., 72 (1966), 980–983. Google ScholarGoogle ScholarCross RefCross Ref
  62. Takeshi Tsukada, Kazuyuki Asada, and C.-H. Luke Ong. 2017. Generalised Species of Rigid Resource Terms. In Proceedings of the 32rd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2017). https://doi.org/10.1109/LICS.2017.8005093 Google ScholarGoogle ScholarCross RefCross Ref
  63. Takeshi Tsukada, Kazuyuki Asada, and C.-H. Luke Ong. 2018. Species, Profunctors and Taylor Expansion Weighted by SMCC: A Unified Framework for Modelling Nondeterministic, Probabilistic and Quantum Programs. In Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS ’18). 889–898. isbn:978-1-4503-5583-4 https://doi.org/10.1145/3209108.3209157 Google ScholarGoogle ScholarDigital LibraryDigital Library
  64. Steffen van Bakel. 2011. Strict intersection types for the Lambda Calculus. ACM Comput. Surv., 43, 3 (2011), 20:1–20:49. https://doi.org/10.1145/1922649.1922657 Google ScholarGoogle ScholarDigital LibraryDigital Library
  65. Christopher P. Wadsworth. 1976. The Relation Between Computational and Denotational Properties for Scott’s D_∞ -Models of the Lambda-Calculus. SIAM J. Comput., 5, 3 (1976), 488–521. https://doi.org/10.1137/0205036 Google ScholarGoogle ScholarDigital LibraryDigital Library

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