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Tropical Mathematics and the Lambda Calculus I: Metric and Differential Analysis of Effectful Programs (2311.15704v1)

Published 27 Nov 2023 in cs.LO, cs.PL, and math.LO

Abstract: We study the interpretation of the lambda-calculus in a framework based on tropical mathematics, and we show that it provides a unifying framework for two well-developed quantitative approaches to program semantics: on the one hand program metrics, based on the analysis of program sensitivity via Lipschitz conditions, on the other hand resource analysis, based on linear logic and higher-order program differentiation. To do that we focus on the semantic arising from the relational model weighted over the tropical semiring, and we discuss its application to the study of "best case" program behavior for languages with probabilistic and non-deterministic effects. Finally, we show that a general foundation for this approach is provided by an abstract correspondence between tropical algebra and Lawvere's theory of generalized metric spaces.

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References (78)
  1. Quantales, observational logic and process semantics. Mathematical Structures in Computer Science, 3(2):161–227, 1993. URL: https://www.cambridge.org/core/article/quantales-observational-logic-and-process-semantics/7D1B5D42AEB392EC8DDDA096782AC651, doi:DOI:10.1017/S0960129500000189.
  2. The (in)efficiency of interaction. In Proceedings POPL 2021, volume 5, New York, NY, USA, 2021. Association for Computing Machinery.
  3. Multi types and reasonable space. Proceedings ICFP 2022, 6, 2022.
  4. Tropical polyhedra are equivalent to mean payoff games. International Journal of Algebra and Computation, 22(01):1250001, 2023/01/16 2012. URL: https://doi.org/10.1142/S0218196711006674, doi:10.1142/S0218196711006674.
  5. Best approximation in max-plus semimodules. Linear Algebra and its Applications, 435(12):3261–3296, 2011. URL: https://www.sciencedirect.com/science/article/pii/S0024379511004551, doi:https://doi.org/10.1016/j.laa.2011.06.009.
  6. Cartesian difference categories. In Jean Goubault-Larrecq and Barbara König, editors, Proceedings FoSSaCS 2020, pages 57–76, Cham, 2020. Springer International Publishing.
  7. Change actions: Models of generalised differentiation. In Mikołaj Bojańczyk and Alex Simpson, editors, Proceedings FoSSaCS 2019, pages 45–61, Cham, 2019. Springer International Publishing.
  8. Differential privacy: On the trade-off between utility and information leakage. In Proceedings FAST 2011, FAST–11, pages 39–54, Berlin, Heidelberg, 2011. Springer-Verlag. URL: https://doi.org/10.1007/978-3-642-29420-4_3, doi:10.1007/978-3-642-29420-4_3.
  9. Curry and Howard Meet Borel. In Proceedings LICS 2022, pages 1–13,. IEEE Computer Society, 2022.
  10. A semantic account of metric preservation. In Proceedings POPL 2017, pages 545–556, New York, NY, USA, 2017. Association for Computing Machinery. URL: https://doi.org/10.1145/3009837.3009890, doi:10.1145/3009837.3009890.
  11. Probabilistic relational reasoning via metrics. In Proceedings LICS 2019. IEEE Computer Society, 2019.
  12. Coalgebraic behavioral metrics. Log. Methods Comput. Sci., 14(3), 2018. URL: https://doi.org/10.23638/LMCS-14(3:20)2018, doi:10.23638/LMCS-14(3:20)2018.
  13. Probabilistic relational reasoning for differential privacy. In Proceedings POPL 2012. ACM Press, 2012. URL: http://dx.doi.org/10.1145/2103656.2103670, doi:10.1145/2103656.2103670.
  14. Statistical inference for probabilistic functions of finite state markov chains. The Annals of Mathematical Statistics, 37(6):1554–1563, 2023/01/17/ 1966. URL: http://www.jstor.org/stable/2238772.
  15. Differential categories. Mathematical Structures in Computer Science, 16:1049–1083, 2006.
  16. A convenient differential category. CoRR, abs/1006.3140, 2010. URL: http://arxiv.org/abs/1006.3140, arXiv:1006.3140.
  17. Differential categories revisited. Applied Categorical Structures, 28:171–235, 2020.
  18. Cartesian Differential Categories. Theory and Applications of Categories, 22(23):622–672, 2009.
  19. Gérard Boudol. The lambda-calculus with multiplicities. In Eike Best, editor, Proceedings CONCUR’93, pages 1–6, Berlin, Heidelberg, 1993. Springer Berlin Heidelberg.
  20. On intersection types and probabilistic lambda calculi. In Proceedings PPDP 2018, PPDP ’18, New York, NY, USA, 2018. Association for Computing Machinery. URL: https://doi.org/10.1145/3236950.3236968, doi:10.1145/3236950.3236968.
  21. Categorical models for simply typed resource calculi. Electronic Notes in Theoretical Computer Science, 265:213 – 230, 2010. Proceedings of the 26th Conference on the Mathematical Foundations of Programming Semantics (MFPS 2010). URL: http://www.sciencedirect.com/science/article/pii/S1571066110000927, doi:https://doi.org/10.1016/j.entcs.2010.08.013.
  22. Exponentiation in V-categories. Topology and its Applications, 153(16):3113 – 3128, 2006. doi:10.1016/j.topol.2005.01.038.
  23. Stefan Cobzaş̧. Lipschitz properties of convex functions. Advances in Operator Theory, 2(1):21 – 49, 2017. URL: https://doi.org/10.22034/aot.1610.1022, doi:10.22034/aot.1610.1022.
  24. Raphaëlle Crubillé. Probabilistic stable functions on discrete cones are power series. In Anuj Dawar and Erich Grädel, editors, Proceedings LICS 2018, pages 275–284. ACM, 2018. URL: https://doi.org/10.1145/3209108.3209198, doi:10.1145/3209108.3209198.
  25. Differential logical relations, part I: the simply-typed case. In Proceedings ICALP 2019, pages 111:1–111:14, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.111, doi:10.4230/LIPIcs.ICALP.2019.111.
  26. On quantitative algebraic higher-order theories. In Proceedings FSCD 2022, volume 228 of LIPIcs, pages 4:1–4:18, 2022.
  27. Computation by interaction for space-bounded functional programming. Information and Computation, 248:150–194, June 2016. doi:10.1016/j.ic.2015.04.006.
  28. Probabilistic coherence spaces as a model of higher-order probabilistic computation. Inf. Comput., 209(6):966–991, 2011. URL: https://doi.org/10.1016/j.ic.2011.02.001, doi:10.1016/j.ic.2011.02.001.
  29. Daniel de Carvalho. Execution time of λ𝜆\lambdaitalic_λ-terms via denotational semantics and intersection types. Mathematical Structures in Computer Science, 28(7):1169–1203, 2018.
  30. Thomas Ehrhard. Finiteness spaces. Mathematical Structures in Computer Science, 15(4):615–646, 2005. URL: https://www.cambridge.org/core/article/finiteness-spaces/E5E9CE1FA4050A56EF25CFB6F6A5754F.
  31. Thomas Ehrhard. An introduction to differential linear logic: proof-nets, models and antiderivatives. Mathematical Structures in Computer Science, pages 1–66, February 2017. doi:10.1017/S0960129516000372.
  32. Thomas Ehrhard. Cones as a model of intuitionistic linear logic. In Proceedings LICS 2020, pages 370–383. IEEE Computer Society, 2020. URL: https://doi.org/10.1145/3373718.3394758, doi:10.1145/3373718.3394758.
  33. Thomas Ehrhard. Differentials and distances in probabilistic coherence spaces. Logical Methods in Computer Science, 18(3):2:1–2:33, 2022.
  34. The computational meaning of probabilistic coherence spaces. In Proceedings LICS 2011, pages 87–96. IEEE Computer Society, 2011. doi:10.1109/LICS.2011.29.
  35. Full Abstraction for Probabilistic PCF. Journal of the ACM, 65(4), 2018.
  36. Measurable cones and stable, measurable functions: a model for probabilistic higher-order programming. In Proceedings POPL 2018, volume 2, pages 59:1–59:28, 2018. URL: https://doi.org/10.1145/3158147, doi:10.1145/3158147.
  37. The differential lambda-calculus. Theoretical Computer Science, 309(1):1–41, December 2003. doi:10.1016/S0304-3975(03)00392-X.
  38. Martín Hötzen Escardó. A metric model of PCF. Unpublished note presented at the Workshop on Realizability Semantics and Applications, June 1999. Available at the author’s webpage., 1999.
  39. Soichiro Fuji. Enriched categories and tropical mathematics. https://arxiv.org/abs/1909.07620, 2019.
  40. Linear dependent types for differential privacy. SIGPLAN Not., 48(1):357–370, jan 2013. URL: https://doi.org/10.1145/2480359.2429113, doi:10.1145/2480359.2429113.
  41. Zeinab Galal. A bicategorical model for finite nondeterminism. In Proceedings FSCD 2021, volume 195 of LIPIcs, pages 10:1–10:17, 2021.
  42. A partial metric semantics of higher-order types and approximate program transformations. In Proceedings CSL 2021, volume 183 of LIPIcs, pages 35:1–35:18, 2021.
  43. Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50(1):1–101, 1987. doi:10.1016/0304-3975(87)90045-4.
  44. Bounded linear logic: A modular approach to polynomial time computability. Theoretical Computer Science, 97:1–66, 1992. Extended abstract in Feasible Mathematics, S. R. Buss and P. J. Scott editors, Proceedings of the MCI Workshop, Ithaca, NY, June 1989, Birkhauser, Boston, pp. 195–209.
  45. Dima Grigoriev. Tropical differential equations. Advances in Applied Mathematics, 82:120–128, 2017. URL: https://www.sciencedirect.com/science/article/pii/S0196885816300719, doi:https://doi.org/10.1016/j.aam.2016.08.002.
  46. A convenient category for higher-order probability theory. In Proceedings LICS 2017. IEEE Computer Society, 2017.
  47. Monoidal Topology: a Categorical Approach to Order, Metric and Topology. Cambridge University Press, New York, 2014.
  48. Supertropical semirings and supervaluations. Journal of Pure and Applied Algebra, 215(10):2431–2463, 2011. URL: https://www.sciencedirect.com/science/article/pii/S0022404911000193, doi:https://doi.org/10.1016/j.jpaa.2011.01.002.
  49. C. Kahlert and L.O. Chua. The complete canonical piecewise-linear representation. i. the geometry of the domain space. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 39(3):222–236, 1992. doi:10.1109/81.128016.
  50. Shin-ya Katsumata. A double category-theoretic analysis of graded linear exponential comonads. In Proceedings FoSSaCS 2018, pages 110–127. Springer International Publishing, 2018.
  51. Weighted relational models of typed lambda-calculi. In Proceedings LICS 2013, pages 301–310. IEEE Computer Society, 2013. doi:10.1109/LICS.2013.36.
  52. F. William Lawvere. Metric spaces, generalized logic, and closed categories. Rendiconti del Seminario Matematico e Fisico di Milano, 43(1):135–166, Dec 1973. URL: https://doi.org/10.1007/BF02924844, doi:10.1007/BF02924844.
  53. Jean-Simon Pacaud Lemay. Convenient antiderivatives for differential linear categories, 2020. arXiv:1808.08513.
  54. Jean-Simon Pacaud Lemay. Coderelictions for Free Exponential Modalities. In Fabio Gadducci and Alexandra Silva, editors, 9th Conference on Algebra and Coalgebra in Computer Science (CALCO 2021), volume 211 of Leibniz International Proceedings in Informatics (LIPIcs), pages 19:1–19:21, Dagstuhl, Germany, 2021. Schloss Dagstuhl – Leibniz-Zentrum für Informatik. URL: https://drops.dagstuhl.de/opus/volltexte/2021/15374, doi:10.4230/LIPIcs.CALCO.2021.19.
  55. Yves Lucet. What shape is your conjugate? a survey of computational convex analysis and its applications. SIAM Journal on Optimization, 20(1):216–250, 2009. URL: https://doi.org/10.1137/080719613, arXiv:https://doi.org/10.1137/080719613, doi:10.1137/080719613.
  56. Introduction to tropical geometry, volume 161 of Graduate Studies in Mathematics. American Mathematical Society, 2015.
  57. Giulio Manzonetto. What is a categorical model of the differential and the resource λ𝜆\lambdaitalic_λ-calculi? Mathematical Structures in Computer Science, 22(3):451–520, 2012. URL: https://www.cambridge.org/core/article/what-is-a-categorical-model-of-the-differential-and-the-resource-calculi/3DE9773BD8769205EEBE588114F8E7C3, doi:DOI:10.1017/S0960129511000594.
  58. Tropical geometry and machine learning. Proceedings of the IEEE, 109(5):728–755, 2021. doi:10.1109/JPROC.2021.3065238.
  59. Quantitative algebraic reasoning. In Proceedings LICS 2016. IEEE Computer Society, 2016.
  60. Polyadic approximations, fibrations and intersection types. In Proceedings POPL 2018. ACM, 2018.
  61. An explicit formula for the free exponential modality of linear logic. Mathematical Structures in Computer Science, 28(7):1253–1286, 2018. URL: https://www.cambridge.org/core/article/an-explicit-formula-for-the-free-exponential-modality-of-linear-logic/A73D77D7862E94F2F7CBDCA33113BFAC, doi:DOI:10.1017/S0960129516000426.
  62. Tropical laurent series, their tropical roots, and localization results for the eigenvalues of nonlinear matrix functions. https://arxiv.org/abs/2107.07982, 2021.
  63. Tropical roots as approximations to eigenvalues of matrix polynomials. SIAM J. Matrix Anal. Appl., 36(1):138–157, jan 2015. URL: https://doi.org/10.1137/14096637X, doi:10.1137/14096637X.
  64. Federico Olimpieri. Intersection type distributors. In Proceedings LICS 2021. IEEE Computer Society, 2021. URL: https://doi.org/10.1109/LICS52264.2021.9470617, doi:10.1109/LICS52264.2021.9470617.
  65. Tropical geometry of statistical models. Proceedings of the National Academy of Sciences, 101(46):16132–16137, 2023/01/16 2004. URL: https://doi.org/10.1073/pnas.0406010101, doi:10.1073/pnas.0406010101.
  66. Parallel reduction in resource λ𝜆\lambdaitalic_λ-calculus. aplav09, pages 226–242, 2009.
  67. Paolo Pistone. On generalized metric spaces for the simply typed λ𝜆\lambdaitalic_λ-calculus. In Proceedings LICS 2021, pages 1–14. IEEE Computer Society, 2021.
  68. Gordon Plotkin. LCF considered as a programming language. Theoretical Computer Science, 5(3):223–255, 1977.
  69. Distance makes the types grow stronger. Proceedings ICFP 2010, pages 157–168, 2010.
  70. Ciro Russo. Quantale Modules, with Applications to Logic and Image Processing. PhD thesis, Università degli Studi di Salerno, available at https://arxiv.org/pdf/0909.4493.pdf, 2007.
  71. Ulrich Schöpp. Stratified Bounded Affine Logic for Logarithmic Space. In 22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007), pages 411–420, July 2007. doi:10.1109/LICS.2007.45.
  72. Peter Selinger. Towards a semantics for higher-order quantum computation. In Proceedings QPL 2004, TUCS General Publication No 33, pages 127–143, 2004.
  73. Imre Simon. On semigroups of matrices over the tropical semiring. Informatique Théorique et Applications, 28:277–294, 1994.
  74. Isar Stubbe. Categorical structures enriched in a quantaloid: Tensored and cotensored categories. Theory and Applications of Categories, 16(14):283–306, 2006.
  75. Isar Stubbe. An introduction to quantaloid-enriched categories. Fuzzy Sets and Systems, 256:95 – 116, 2014. Special Issue on Enriched Category Theory and Related Topics (Selected papers from the 33rd Linz Seminar on Fuzzy Set Theory, 2012). doi:https://doi.org/10.1016/j.fss.2013.08.009.
  76. Franck van Breugel. An introduction to metric semantics: operational and denotational models for programming and specification languages. Theoretical Computer Science, 258(1):1 – 98, 2001. doi:10.1016/S0304-3975(00)00403-5.
  77. Simon Willerton. Tight spans, Isbell completions and semi-tropical modules. Theory and Applications of Categories, 28(22):696–732, 2013.
  78. Tropical geometry of deep neural networks. In Proceedings ICML 2018, volume 80 of Proceedings of Machine Learning Research, pages 5819–5827. PMLR, 2018. URL: http://proceedings.mlr.press/v80/zhang18i.html.

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