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Taylor subsumes Scott, Berry, Kahn and Plotkin

Published:20 December 2019Publication History
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Abstract

The speculative ambition of replacing the old theory of program approximation based on syntactic continuity with the theory of resource consumption based on Taylor expansion and originating from the differential λ-calculus is nowadays at hand. Using this resource sensitive theory, we provide simple proofs of important results in λ-calculus that are usually demonstrated by exploiting Scott’s continuity, Berry’s stability or Kahn and Plotkin’s sequentiality theory. A paradigmatic example is given by the Perpendicular Lines Lemma for the Böhm tree semantics, which is proved here simply by induction, but relying on the main properties of resource approximants: strong normalization, confluence and linearity.

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References

  1. Beniamino Accattoli. 2018. (In)Efficiency and Reasonable Cost Models. Electr. Notes Theor. Comput. Sci. 338 (2018), 23–43.Google ScholarGoogle ScholarCross RefCross Ref
  2. Beniamino Accattoli, Stéphane Graham-Lengrand, and Delia Kesner. 2018. Tight typings and split bounds. PACMPL 2, ICFP (2018), 94:1–94:30. Google ScholarGoogle ScholarDigital LibraryDigital Library
  3. Davide Barbarossa and Giulio Manzonetto. 2019. About the power of Taylor expansion. 3rd International Workshop on Trends in Linear Logic and Applications.Google ScholarGoogle Scholar
  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.). Number 103 in Studies in Logic and the Foundations of Mathematics. North-Holland.Google ScholarGoogle Scholar
  6. Atilim Gunes Baydin, Barak A. Pearlmutter, Alexey Andreyevich Radul, and Jeffrey Mark Siskind. 2017. Automatic Differentiation in Machine Learning: a Survey. Journal of Machine Learning Research 18 (2017), 153:1–153:43. http: //jmlr.org/papers/v18/17- 468.htmlGoogle ScholarGoogle Scholar
  7. Gérard Berry. 1978. Stable Models of Typed lambda-Calculi. In ICALP (Lecture Notes in Computer Science), Vol. 62. Springer, 72–89.Google ScholarGoogle Scholar
  8. Richard Blute, J. Robin B. Cockett, and Robert A. G. Seely. 2006. Differential categories. Mathematical Structures in Computer Science 16, 6 (2006), 1049–1083.Google ScholarGoogle ScholarDigital LibraryDigital Library
  9. Richard Blute, J. Robin B. Cockett, and Robert A. G. Seely. 2009. Cartesian differential categories. Theory and Applications of Categories 22, 23 (2009), 622–672.Google ScholarGoogle Scholar
  10. Gérard Boudol. 1993. The Lambda-Calculus with Multiplicities (Abstract). In CONCUR ’93, 4th International Conference on Concurrency Theory, Hildesheim, Germany, August 23-26, 1993, Proceedings (Lecture Notes in Computer Science), Eike Best (Ed.), Vol. 715. Springer, 1–6. Google ScholarGoogle ScholarCross RefCross Ref
  11. Antonio Bucciarelli, Thomas Ehrhard, and Giulio Manzonetto. 2012. A relational semantics for parallelism and nondeterminism in a functional setting. Ann. Pure Appl. Logic 163, 7 (2012), 918–934. Google ScholarGoogle ScholarCross RefCross Ref
  12. J. Robin B. Cockett and Geoff S. H. Cruttwell. 2014. Differential Structure, Tangent Structure, and SDG. Applied Categorical Structures 22, 2 (2014), 331–417. Google ScholarGoogle ScholarCross RefCross Ref
  13. J. Robin B. Cockett and Jean-Simon Lemay. 2019. Integral categories and calculus categories. Mathematical Structures in Computer Science 29, 2 (2019), 243–308. Google ScholarGoogle ScholarCross RefCross Ref
  14. Daniel de Carvalho. 2018. Execution time of λ-terms via denotational semantics and intersection types. Mathematical Structures in Computer Science 28, 7 (2018), 1169–1203. Google ScholarGoogle ScholarCross RefCross Ref
  15. Thomas Ehrhard. 2002. On Köthe Sequence Spaces and Linear Logic. Mathematical Structures in Computer Science 12, 5 (2002), 579–623. Google ScholarGoogle ScholarDigital LibraryDigital Library
  16. 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, September 3-6, 2012, Fontainebleau, France (LIPIcs), Patrick Cégielski and Arnaud Durand (Eds.), Vol. 16. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 259–273. Google ScholarGoogle ScholarCross RefCross Ref
  17. Thomas Ehrhard. 2018. An introduction to differential linear logic: proof-nets, models and antiderivatives. Mathematical Structures in Computer Science 28, 7 (2018), 995–1060.Google ScholarGoogle ScholarCross RefCross Ref
  18. Thomas Ehrhard and Giulio Guerrieri. 2016. The Bang Calculus: an untyped lambda-calculus generalizing call-by-name and call-by-value. In Proceedings of the 18th International Symposium on Principles and Practice of Declarative Programming, Edinburgh, United Kingdom, September 5-7, 2016, James Cheney and Germán Vidal (Eds.). ACM, 174–187. Google ScholarGoogle ScholarDigital LibraryDigital Library
  19. Thomas Ehrhard and Laurent Regnier. 2003. The differential lambda-calculus. Theor. Comput. Sci. 309, 1-3 (2003), 1–41.Google ScholarGoogle ScholarDigital LibraryDigital Library
  20. Thomas Ehrhard and Laurent Regnier. 2006a. 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
  21. Thomas Ehrhard and Laurent Regnier. 2006b. Differential interaction nets. Theor. Comput. Sci. 364, 2 (2006), 166–195.Google ScholarGoogle ScholarDigital LibraryDigital Library
  22. Thomas Ehrhard and Laurent Regnier. 2008. Uniformity and the Taylor expansion of ordinary lambda-terms. Theor. Comput. Sci. 403, 2-3 (2008), 347–372.Google ScholarGoogle ScholarDigital LibraryDigital Library
  23. Thomas Ehrhard and Christine Tasson. 2019. Probabilistic call by push value. Logical Methods in Computer Science 15, 1 (2019).Google ScholarGoogle Scholar
  24. Jörg Endrullis and Roel C. de Vrijer. 2008. Reduction Under Substitution. In Rewriting Techniques and Applications, 19th International Conference, RTA 2008 (Lecture Notes in Computer Science), Andrei Voronkov (Ed.), Vol. 5117. Springer, 425–440.Google ScholarGoogle Scholar
  25. Sabrina Fiege, Andrea Walther, Kshitij Kulshreshtha, and Andreas Griewank. 2018. Algorithmic differentiation for piecewise smooth functions: a case study for robust optimization. Optimization Methods and Software 33, 4-6 (2018), 1073–1088. Google ScholarGoogle ScholarCross RefCross Ref
  26. Marcelo P. Fiore, Nicola Gambino, J. Martin E. Hyland, and Glynn Winskel. 2007. The cartesian closed bicategory of generalised species of structures. J. London Maths. Soc 77 (2007), 203–220.Google ScholarGoogle ScholarCross RefCross Ref
  27. Jean Goubault-Larrecq. 2019. A Probabilistic and Non-Deterministic Call-by-Push-Value Language. In 34th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2019, Vancouver, BC, Canada, June 24-27, 2019. IEEE, 1–13. Google ScholarGoogle ScholarCross RefCross Ref
  28. Andreas Griewank, Richard Hasenfelder, Manuel Radons, Lutz Lehmann, and Tom Streubel. 2018. Integrating Lipschitzian dynamical systems using piecewise algorithmic differentiation. Optimization Methods and Software 33, 4-6 (2018), 1089–1107. Google ScholarGoogle ScholarCross RefCross Ref
  29. Giulio Guerrieri and Giulio Manzonetto. 2018. The Bang Calculus and the Two Girard’s Translations. In Proceedings Joint International Workshop on Linearity & Trends in Linear Logic and Applications, [email protected] 2018, Oxford, UK, 7-8 July 2018. (EPTCS), Thomas Ehrhard, Maribel Fernández, Valeria de Paiva, and Lorenzo Tortora de Falco (Eds.), Vol. 292. 15–30. Google ScholarGoogle ScholarCross RefCross Ref
  30. Martin Hyland. 1975. A syntactic characterization of the equality in some models for the λ-calculus. Journal London Mathematical Society (2) 12(3) (1975), 361–370.Google ScholarGoogle Scholar
  31. Bart Jacobs and Jan Rutten. 1997. A Tutorial on (Co)Algebras and (Co)Induction. EATCS Bulletin 62 (1997), 62–222.Google ScholarGoogle Scholar
  32. Gilles Kahn and Gordon D. Plotkin. 1978. Domaines Concrets. Technical Report 333. Rapport INRIA-LABORIA.Google ScholarGoogle Scholar
  33. Emma Kerinec, Giulio Manzonetto, and Michele Pagani. 2018. Revisiting Call-by-value Böhm trees in light of their Taylor expansion. CoRR abs/1809.02659 (2018). Accepted in Logical Methods in Computer Science.Google ScholarGoogle Scholar
  34. Dexter Kozen and Alexandra Silva. 2017. Practical coinduction. Mathematical Structures in Computer Science 27, 7 (2017), 1132–1152.Google ScholarGoogle Scholar
  35. Jan Kuper. 1995. Proving the Genericity Lemma by Leftmost Reduction is Simple. In RTA (Lecture Notes in Computer Science), Vol. 914. Springer, 271–278.Google ScholarGoogle Scholar
  36. Ugo Dal Lago and Thomas Leventis. 2019. On the Taylor Expansion of Probabilistic lambda-terms. In 4th International Conference on Formal Structures for Computation and Deduction, FSCD 2019, June 24-30, 2019, Dortmund, Germany. (LIPIcs), Herman Geuvers (Ed.), Vol. 131. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 13:1–13:16. Google ScholarGoogle ScholarCross RefCross Ref
  37. Ugo Dal Lago and Margherita Zorzi. 2012. Probabilistic operational semantics for the lambda calculus. RAIRO - Theor. Inf. and Applic. 46, 3 (2012), 413–450.Google ScholarGoogle ScholarCross RefCross Ref
  38. Jim Laird, Giulio Manzonetto, Guy McCusker, and Michele Pagani. 2013. Weighted Relational Models of Typed LambdaCalculi. In 28th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2013, New Orleans, LA, USA, June 25-28, 2013. IEEE Computer Society, 301–310. Google ScholarGoogle ScholarDigital LibraryDigital Library
  39. Soren B. Lassen. 1999. Bisimulation in Untyped Lambda Calculus: Böhm Trees and Bisimulation up to Context. Electr. Notes Theor. Comput. Sci. 20 (1999), 346–374.Google ScholarGoogle ScholarCross RefCross Ref
  40. Thomas Leventis. 2018. Probabilistic Böhm Trees and Probabilistic Separation. In Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2018, Oxford, UK, July 09-12, 2018, Anuj Dawar and Erich Grädel (Eds.). ACM, 649–658. Google ScholarGoogle ScholarDigital LibraryDigital Library
  41. Paul Blain Levy. 2006. Call-by-push-value: Decomposing call-by-value and call-by-name. Higher-Order and Symbolic Computation 19, 4 (2006), 377–414. Google ScholarGoogle ScholarDigital LibraryDigital Library
  42. Stefania Lusin and Antonino Salibra. 2004. The Lattice of Lambda Theories. J. Log. Comput. 14, 3 (2004), 373–394.Google ScholarGoogle ScholarDigital LibraryDigital Library
  43. Giulio Manzonetto. 2009. A General Class of Models of H ∗ . In Mathematical Foundations of Computer Science 2009, 34th International Symposium, MFCS 2009, Novy Smokovec, High Tatras, Slovakia, August 24-28, 2009. Proceedings (Lecture Notes in Computer Science), Rastislav Královic and Damian Niwinski (Eds.), Vol. 5734. Springer, 574–586. Google ScholarGoogle ScholarDigital LibraryDigital Library
  44. Giulio Manzonetto and Domenico Ruoppolo. 2014. Relational Graph Models, Taylor Expansion and Extensionality. Electr. Notes Theor. Comput. Sci. 308 (2014), 245–272.Google ScholarGoogle ScholarDigital LibraryDigital Library
  45. Rob P. Nederpelt, J. Herman. Geuvers, and Roel C. de Vrijer (Eds.). 1994. Selected Papers on Automath. Studies in Logic and the Foundations of Mathematics, Vol. 133. North-Holland, Amsterdam.Google ScholarGoogle Scholar
  46. Michele Pagani, Peter Selinger, and Benoît Valiron. 2014. Applying quantitative semantics to higher-order quantum computing. In The 41st Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL ’14, San Diego, CA, USA, January 20-21, 2014, Suresh Jagannathan and Peter Sewell (Eds.). ACM, 647–658. Google ScholarGoogle ScholarDigital LibraryDigital Library
  47. Luca Paolini, Mauro Piccolo, and Simona Ronchi Della Rocca. 2017. Essential and relational models. Mathematical Structures in Computer Science 27, 5 (2017), 626–650. Google ScholarGoogle ScholarCross RefCross Ref
  48. Gordon D. Plotkin. 1974. The lambda-Calculus is ω-Incomplete. Journal of Symbolic Logic 39, 2 (1974), 313–317.Google ScholarGoogle ScholarCross RefCross Ref
  49. Gordon D. Plotkin. 1975. Call-by-Name, Call-by-Value and the lambda-Calculus. Theor. Comput. Sci. 1, 2 (1975), 125–159. Google ScholarGoogle ScholarCross RefCross Ref
  50. Dana S. Scott. 1972. Continuous lattices. In Toposes, Algebraic Geometry and Logic (Lecture Notes in Mathematics), Lawvere (Ed.), Vol. 274. Springer, 97–136.Google ScholarGoogle Scholar
  51. Richard Statman and Henk Barendregt. 1999. Applications of Plotkin-Terms: Partitions and Morphisms for Closed Terms. J. Funct. Program. 9, 5 (1999), 565–575.Google ScholarGoogle ScholarDigital LibraryDigital Library
  52. Masako Takahashi. 1994. A Simple Proof of the Genericity Lemma. In Logic, Language and Computation (Lecture Notes in Computer Science), Vol. 792. Springer, 117–118.Google ScholarGoogle Scholar
  53. 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 LICS. ACM, 889–898.Google ScholarGoogle Scholar
  54. Dirk van Daalen. 1980. The Language Theory of Automath. Ph.D. thesis. Technical University Eindhoven. Large parts of the thesis, including the treatment of reduction under substitution, have been reproduced in [ Nederpelt et al. 1994 ].Google ScholarGoogle Scholar
  55. Lionel Vaux. 2009. The algebraic lambda calculus. Mathematical Structures in Computer Science 19, 5 (2009), 1029–1059.Google ScholarGoogle ScholarDigital LibraryDigital Library
  56. Lionel Vaux. 2019. Normalizing the Taylor expansion of non-deterministic λ-terms, via parallel reduction of resource vectors. Logical Methods in Computer Science Volume 15, Issue 3 (2019).Google ScholarGoogle Scholar
  57. 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.Google ScholarGoogle ScholarCross RefCross Ref
  58. Sebastian F. Walter and Lutz Lehmann. 2013. Algorithmic differentiation in Python with AlgoPy. J. Comput. Science 4, 5 (2013), 334–344. Google ScholarGoogle ScholarCross RefCross Ref

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      • Published in

        cover image Proceedings of the ACM on Programming Languages
        Proceedings of the ACM on Programming Languages  Volume 4, Issue POPL
        January 2020
        1984 pages
        EISSN:2475-1421
        DOI:10.1145/3377388
        Issue’s Table of Contents

        Copyright © 2019 Owner/Author

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        Association for Computing Machinery

        New York, NY, United States

        Publication History

        • Published: 20 December 2019
        Published in pacmpl Volume 4, Issue POPL

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