Abstract
We define a notion of stable and measurable map between cones endowed with measurability tests and show that it forms a cpo-enriched cartesian closed category. This category gives a denotational model of an extension of PCF supporting the main primitives of probabilistic functional programming, like continuous and discrete probabilistic distributions, sampling, conditioning and full recursion. We prove the soundness and adequacy of this model with respect to a call-by-name operational semantics and give some examples of its denotations.
Supplemental Material
- Tsuyoshi Andô. 1962. On fundamental properties of a Banach space with a cone. Pacific J. Math. 12, 4 (1962), 1163–1169. Google Scholar
Cross Ref
- Robert J. Aumann. 1961. Borel structures for function spaces. Illinois J. Math. 5, 4 (12 1961), 614–630. https://projecteuclid.org: 443/euclid.ijm/1255631584Google Scholar
Cross Ref
- Tyler Barker. 2016. A monad for randomized algorithms. Ph.D. Dissertation. Tulane University.Google Scholar
- Gérard Berry. 1978. Stable Models of Typed Lambda-calculi. In Automata, Languages and Programming, Fifth Colloquium, Udine, Italy, July 17-21, 1978, Proceedings (Lecture Notes in Computer Science), Vol. 62. Springer, 72–89.Google Scholar
Cross Ref
- Johannes Borgström, Ugo Dal Lago, Andrew D. Gordon, and Marcin Szymczak. 2016. A lambda-calculus foundation for universal probabilistic programming. In Proceedings of the 21st ACM SIGPLAN International Conference on Functional Programming, ICFP 2016, Nara, Japan, September 18-22, 2016, Jacques Garrigue, Gabriele Keller, and Eijiro Sumii (Eds.). ACM, 33–46. Google Scholar
Digital Library
- Vincent Danos and Thomas Ehrhard. 2011. Probabilistic coherence spaces as a model of higher-order probabilistic computation. Information and Computation 209, 6 (2011), 966–991. Google Scholar
Digital Library
- Vincent Danos and Russel Harmer. 2002. Probabilistic game semantics. ACM Transactions on Computational Logic 3, 3 (July 2002), 359–382. Google Scholar
Digital Library
- Thomas Ehrhard, Michele Pagani, and Christine Tasson. 2011. The Computational Meaning of Probabilistic Coherence Spaces. In Proceedings of the 26th Annual IEEE Symposium on Logic in Computer Science (LICS 2011), Martin Grohe (Ed.). IEEE Computer Society Press, 87–96. Google Scholar
Digital Library
- Thomas Ehrhard, Michele Pagani, and Christine Tasson. 2014. Probabilistic Coherence Spaces are Fully Abstract for Probabilistic PCF. In The 41th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL14, San Diego, USA, P. Sewell (Ed.). ACM. Google Scholar
Digital Library
- Thomas Ehrhard, Michele Pagani, and Christine Tasson. 2017. The cartesian closed category of measurable cones and stable, measurable functions A model for probabilistic higher-order programming. (Oct. 2017). https://hal.archives- ouvertes.fr/ hal- 01622046 working paper or preprint.Google Scholar
- Thomas Ehrhard and Christine Tasson. 2016. Probabilistic call by push value. (2016). Submitted, preprint available at http://arxiv.org/abs/1607.04690.Google Scholar
- Martin Escardó, Martin Hofmann, and Thomas Streicher. 2004. On the non-sequential nature of the interval-domain model of real-number computation. Mathematical Structures in Computer Science 14, 6 (2004), 803–814. Google Scholar
Digital Library
- Martin Escardó. 1996. PCF extended with real numbers. Theoretical Computer Science 162, 1 (1996), 79 – 115. Google Scholar
Digital Library
- Florian Faissole and Bas Spitters. 2017. Synthetic topology in Homotopy Type Theory for probabilistic programming. (2017). available at: https://pps2017.soic.indiana.edu/files/2016/12/ProbProg.pdf .Google Scholar
- Michèle Giry. 1982. A categorical approach to probability theory. Springer Berlin Heidelberg, Berlin, Heidelberg, 68–85. Google Scholar
Cross Ref
- Noah D. Goodman and Joshua B. Tenenbaum. 2014. Probabilistic models of cognition. (2014). http://probmods.org.Google Scholar
- Jean Goubault-Larrecq and Daniele Varacca. 2011. Continuous Random Variables. In LICS. IEEE Computer Society, 97–106. Google Scholar
Digital Library
- Chris Heunen, Ohad Kammar, Sam Staton, and Hongseok Yang. 2017. A Convenient Category for Higher-Order Probability Theory. In Proceedings of the 32st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’17, Reykjavik, June 20-23, 2017, Joel Ouaknine (Ed.). ACM. Google Scholar
Cross Ref
- Daniel Huang and Greg Morrisett. 2017. An application of computable distributions to the semantics of probabilistic programs: part 2. (2017). available at: https://pps2017.soic.indiana.edu/files/2016/12/comp- dist- sem.pdf .Google Scholar
- Claire Jones and Gordon D. Plotkin. 1989. A Probabilistic Powerdomain of Evaluations. In Proceedings of the Fourth Annual Symposium on Logic in Computer Science (LICS ’89), Pacific Grove, California, USA, June 5-8, 1989. IEEE Computer Society, 186–195. Google Scholar
Cross Ref
- Achim Jung and Regina Tix. 1998. The troublesome probabilistic powerdomain. Electr. Notes Theor. Comput. Sci. 13 (1998), 70–91. Google Scholar
Cross Ref
- Klaus Keimel and Gordon D. Plotkin. 2017. Mixed powerdomains for probability and nondeterminism. Logical Methods in Computer Science 13, 1 (2017). Google Scholar
Cross Ref
- Dexter Kozen. 1981. Semantics of Probabilistic Programs. J. Comput. System Sci. 22, 3 (1981), 328–350. Google Scholar
Cross Ref
- Jim Laird, Giulio Manzonetto, Guy McCusker, and Michele Pagani. 2013. Weighted relational models of typed lambda-calculi. In 28th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2013, New Orleans, LA, USA, June 25-28, 2013. IEEE Computer Society. Google Scholar
Digital Library
- Brockway McMillan. 1954. Absolutely Monotone Functions. Annals of Mathematics 60, 3 (1954), 467–501. Google Scholar
Cross Ref
- Michael W. Mislove. 2016. Domains and Random Variables. CoRR abs/1607.07698 (2016). http://arxiv.org/abs/1607.07698Google Scholar
- Prakash Panangaden. 1999. The Category of Markov Kernels. Electronic Notes in Theoretical Computer Science 22 (1999), 171 – 187. Google Scholar
Cross Ref
- Sungwoo Park, Frank Pfenning, and Sebastian Thrun. 2008. A Probabilistic Language Based on Sampling Functions. ACM Trans. Program. Lang. Syst. 31, 1, Article 4 (Dec. 2008), 46 pages. Google Scholar
Digital Library
- Gordon D. Plotkin. 1977. LCF Considered as a Programming Language. Theor. Comput. Sci. 5, 3 (1977), 225–255. Google Scholar
Cross Ref
- Mathys Rennela. 2016. Convexity and Order in Probabilistic Call-by-Name FPC. CoRR abs/1607.04332 (2016). http: //arxiv.org/abs/1607.04332Google Scholar
- N. Saheb-Djahromi. 1980. CPO’S of Measures for Nondeterminism. Theor. Comput. Sci. 12 (1980), 19–37. Google Scholar
- Peter Selinger. 2004. Toward a semantics for higher-order quantum computation. In Proceedings of the 2nd International Workshop on Quantum Programming Languages, Peter Selinger (Ed.), Vol. 33. TUCS General Publication, 127–143.Google Scholar
- Sam Staton. 2017. Commutative Semantics for Probabilistic Programming. In Programming Languages and Systems - 26th European Symposium on Programming, ESOP 2017, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2017, Uppsala, Sweden, April 22-29, 2017, Proceedings (Lecture Notes in Computer Science), Hongseok Yang (Ed.), Vol. 10201. Springer, 855–879. Google Scholar
Digital Library
- Sam Staton, Hongseok Yang, Frank Wood, Chris Heunen, and Ohad Kammar. 2016. Semantics for probabilistic programming: higher-order functions, continuous distributions, and soft constraints. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, New York, NY, USA, July 5-8, 2016. ACM, 525–534. Google Scholar
Digital Library
- Regina Tix, Klaus Keimel, and Gordon D. Plotkin. 2009. Semantic Domains for Combining Probability and Non-Determinism. Electr. Notes Theor. Comput. Sci. 222 (2009), 3–99. Google Scholar
Digital Library
- Jean Vuillemin. 1988. Exact Real Computer Arithmetic with Continued Fractions. In Proceedings of the 1988 ACM Conference on LISP and Functional Programming (LFP ’88). ACM, New York, NY, USA, 14–27. Google Scholar
Digital Library
- Glynn Winskel. 2014. Probabilistic and Quantum Event Structures. In Horizons of the Mind. A Tribute to Prakash Panangaden - Essays Dedicated to Prakash Panangaden on the Occasion of His 60th Birthday (Lecture Notes in Computer Science), Franck van Breugel, Elham Kashefi, Catuscia Palamidessi, and Jan Rutten (Eds.), Vol. 8464. Springer, 476–497. Google Scholar
Cross Ref
Index Terms
Measurable cones and stable, measurable functions: a model for probabilistic higher-order programming
Recommendations
Linear lambda calculus with non-linear first-class continuations
ICSCA '17: Proceedings of the 6th International Conference on Software and Computer ApplicationsThe Curry-Howard isomorphism is the correspondence between propositions and types, proofs and lambda-terms, and proof normalization and evaluation. In Curry-Howard isomorphism, we find a duality between values and continuations in pure functional ...
Higher-order representation of substructural logics
ICFP '10We present a technique for higher-order representation of substructural logics such as linear or modal logic. We show that such logics can be encoded in the (ordinary) Logical Framework, without any linear or modal extensions. Using this encoding, ...
Higher-order representation of substructural logics
ICFP '10: Proceedings of the 15th ACM SIGPLAN international conference on Functional programmingWe present a technique for higher-order representation of substructural logics such as linear or modal logic. We show that such logics can be encoded in the (ordinary) Logical Framework, without any linear or modal extensions. Using this encoding, ...






Comments