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From Lambda Calculus to Cybersecurity Through Program Analysis pp 86–102Cite as

A Type Theory for Probabilistic \(\lambda \)–calculus

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12065))

Abstract

We present a theory of types where formulas may contain a choice constructor. This constructor allows for the selection of a particular type among a finite set of options, each corresponding to a given probabilistic term. We show that this theory induces a type assignment system for the probabilistic \(\lambda \)–calculus introduced in an earlier work by Chris Hankin, Herbert Wiklicky and the author, where probabilistic terms represent probability distributions on classical terms of the simply typed \(\lambda \)–calculus. We prove the soundness of the type assignment with respect to a probabilistic term reduction and a normalization property of the latter.

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Notes

  1. 1.

    Having in mind a model in which a type is a vector space, it is natural to consider its subspaces as its subtypes.

  2. 2.

    In fact, probabilities are used at the semantical level only to estimate the likelihood that a certain value is actually otained after the reduction process.

References

  1. Abramsky, S., Hankin, C. (eds.): Abstract Interpretation of Declarative Languages. Halsted Press, Sydney (1987)

    Google Scholar 

  2. Abramsky, S., Jensen, T.P.: A relational approach to strictness analysis for higher-order polymorphic functions. In: Proceedings of ACM Symposium on Principles of Programming Languages, pp. 49–54 (1991)

    Google Scholar 

  3. Barbanera, F., Dezani-Ciancaglini, M., de’Liguoro, U.: Intersection and union types: syntax and semantics. Inf. Comput. 119(2), 202–230 (1995). https://doi.org/10.1006/inco.1995.1086. https://dx.doi.org/10.1006/inco.1995.1086

    Article  MathSciNet  MATH  Google Scholar 

  4. Barendregt, H.P.: The Lambda Calculus, Studies in Logic and the Foundations of Mathematics, vol. 103, revised edn. North-Holland (1991)

    Google Scholar 

  5. Barendregt, H.P.: Lambda calculi with types. In: Abramsky, S., Gabbay, D.M., Maibaum, S.E. (eds.) Handbook of Logic in Computer Science, vol. 2, pp. 117–309. Oxford University Press Inc., New York (1992), http://dl.acm.org/citation.cfm?id=162552.162561

  6. Barendregt, H.: The Lambda Calculus, Its Syntax and Semantics, Studies in Logic and the Foundations of Mathematics, vol. 103. Elsevier (1984)

    Google Scholar 

  7. Cardelli, L., Wegner, P.: On understanding types, data abstraction, and polymorphism. ACM Comput. Surv. 17(4), 471–523 (1985). https://doi.org/10.1145/6041.6042

    Article  Google Scholar 

  8. Castagna, G., Ghelli, G., Longo, G.: A calculus for overloaded functions with subtyping. Inform. Comput. 117(1), 115–135 (1995). https://doi.org/10.1006/inco.1995.1033

    Article  MathSciNet  MATH  Google Scholar 

  9. Dal Lago, U., Zorzi, M.: Probabilistic operational semantics for the lambda calculus. RAIRO - Theor. Inform. Appl. 46(3), 413–450 (2012)

    Article  MathSciNet  Google Scholar 

  10. Dezani, M.: Logical semantics for concurrent lambda calculus, ph.D. Thesis, Radboud University Nijmegen (1996)

    Google Scholar 

  11. Di Pierro, A., Hankin, C., Wiklicky, H.: Probabilistic lambda-calculus and quantitative program analysis. J. Logic Comput. 15(2), 159–179 (2005)

    Article  MathSciNet  Google Scholar 

  12. Di Pierro, A., Wiklicky, H.: Measuring the precision of abstract interpretations. LOPSTR 2000. LNCS, vol. 2042, pp. 147–164. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-45142-0_9

    Chapter  Google Scholar 

  13. Draheim, D.: Semantics of the Probabilistic Typed Lambda Calculus - Markov Chain Semantics, Termination Behavior, and Denotational Semantics. Springer, Heidelberg (2017). https://doi.org/10.1007/978-3-642-55198-7

    Book  MATH  Google Scholar 

  14. Girard, J.Y.: Une extension de l’interprétation de Gödel à l’analyse, et son application à l’élimination des coupures dans l’analyse et la théorie des types. In: Proceedings of the 2nd Scandinavian Logic Symposium. Studies in Logic and the Foundations of Mathematics, vol. 63, pp. 63–92. North-Holland (1971)

    Google Scholar 

  15. Gunter, C.A.: Foundations of Computing. MIT Press, Cambridge (1992)

    Google Scholar 

  16. Coppo, M., Dezani-Ciancaglini, M., Venneri, B.: Functional characters of solvable terms. Math. Logic Q. 27(2–6), 45–58 (1981)

    Article  MathSciNet  Google Scholar 

  17. Park, S.: A calculus for probabilistic languages. In: Proceedings of the ACM SIGPLAN International Workshop on Types in Languages Design and Implementation, pp. 38–49. ACM (2003)

    Google Scholar 

  18. Park, S., Pfenning, F., Thrun, S.: A monadic probabilistic language (2003). manuscript. http://www.cs.cmu.edu/~fp/papers/prob03.pdf

  19. Park, S., Pfenning, F., Thrun, S.: A probabilistic language based on sampling functions. ACM Trans. Program. Lang. Syst. 31(1), 4:1–4:46 (2008). https://doi.org/10.1145/1452044.1452048. https://doi.acm.org/10.1145/1452044.1452048

    Article  MATH  Google Scholar 

  20. Pierce, B.C.: Types and Programming Languages. MIT Press, Cambridge (2002)

    MATH  Google Scholar 

  21. Plotkin, G.D.: Call-by-name, call-by-value and the \(\lambda \)-calculus. Theor. Comput. Sci. 1(2), 125–159 (1975)

    Article  MathSciNet  Google Scholar 

  22. Plotkin, G.D.: LCF considered as a programming language. Theor. Comput. Sci. 5(3), 223–255 (1977)

    Article  MathSciNet  Google Scholar 

  23. Ramsey, N., Pfeffer, A.: Stochastic lambda calculus and monads of probability distributions. In: Proceedings of ACM Symposium on Principles of Programming Languages, pp. 154–165. ACM Press (2002)

    Google Scholar 

  24. Reynolds, J.C.: Towards a theory of type structure. In: Robinet, B. (ed.) Programming Symposium. LNCS, vol. 19, pp. 408–425. Springer, Heidelberg (1974). https://doi.org/10.1007/3-540-06859-7_148

    Chapter  Google Scholar 

  25. Roman, S.: Advanced Linear Algebra. Graduate Texts in Mathematics 135. Springer, New York (2008). https://doi.org/10.1007/978-0-387-72831-5

    Book  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Informatica, Università di Verona, Strada le Grazie, 15, 37134, Verona, Italy

    Alessandra Di Pierro

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  1. Alessandra Di Pierro
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Correspondence to Alessandra Di Pierro .

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Editors and Affiliations

  1. University of Verona, Verona, Italy

    Alessandra Di Pierro

  2. School of Electronic Engineering and Computer Science, Queen Mary University of London, London, UK

    Pasquale Malacaria

  3. Department of Computer Science, Middlesex University, London, UK

    Rajagopal Nagarajan

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Di Pierro, A. (2020). A Type Theory for Probabilistic \(\lambda \)–calculus. In: Di Pierro, A., Malacaria, P., Nagarajan, R. (eds) From Lambda Calculus to Cybersecurity Through Program Analysis. Lecture Notes in Computer Science(), vol 12065. Springer, Cham. https://doi.org/10.1007/978-3-030-41103-9_3

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