Abstract
We make a formal analogy between random sampling and fresh name generation. We show that quasi-Borel spaces, a model for probabilistic programming, can soundly interpret the ν-calculus, a calculus for name generation. Moreover, we prove that this semantics is fully abstract up to first-order types. This is surprising for an ‘off-the-shelf’ model, and requires a novel analysis of probability distributions on function spaces. Our tools are diverse and include descriptive set theory and normal forms for the ν-calculus.
- S. Abramsky, D. R. Ghica, A. S. Murawski, C.-H. L. Ong, and I. D. B. Stark. 2004. Nominal games and full abstraction for the nu-Calculus. In Proc. LICS 2004. 150-159.Google Scholar
Cross Ref
- Robert J. Aumann. 1961. Borel structures for function spaces. Illinois Journal of Mathematics 5 ( 1961 ).Google Scholar
- Giorgio Bacci, Robert Furber, Dexter Kozen, Radu Mardare, Prakash Panangaden, and Dana Scott. 2018. Boolean-valued semantics for stochastic lambda-calculus. In Proc. LICS 2018.Google Scholar
- Nick Benton and Vasileios Koutavas. 2008. A Mechanized Bisimulation for the Nu-Calculus. Technical Report MSR-TR-2008-129. Microsoft Research.Google Scholar
- Chung chieh Shan and Norman Ramsey. 2017. Exact Bayesian inference by symbolic disintegration. In Proc. POPL 2017.Google Scholar
Digital Library
- Fredrik Dahlqvist and Dexter Kozen. 2020. Semantics of higher-order probabilistic programs with conditioning. Proc. ACM Program. Lang. 4, POPL, Article 19 ( Dec. 2020 ).Google Scholar
Digital Library
- Ugo Dal Lago and Naohiko Hoshino. 2019. The geometry of Bayesian programming. In Proc. LICS 2019.Google Scholar
Cross Ref
- Thomas Ehrhard, Michele Pagani, and Christine Tasson. 2018. Measurable cones and stable, measurable functions. In Proc. POPL 2018.Google Scholar
- Thomas Ehrhard, Charistine Tasson, and Michele Pagani. 2014. Probabilistic coherence spaces are fully abstract for probabilistic PCF. In Proc. POPL 2014. 309-320.Google Scholar
- M.H. Escardo. 2009. Semi-decidability of may, must and probabilistic testing in a higher-type setting. In Proc. MFPS 2009.Google Scholar
Digital Library
- Tobias Fritz. 2020. A synthetic approach to Markov kernels, conditional independence and theorems on suficient statistics. Adv. Math. 370, 107239 (Aug. 2020 ).Google Scholar
Cross Ref
- T. Gehr, S. Stefen, and M. T. Vechev. 2020. PSI: exact inference for higher-order probabilistic programs. In Proc. PLDI 2020.Google Scholar
Digital Library
- Michèle Giry. 1982. A categorical approach to probability theory. In Categorical Aspects of Topology and Analysis. Lecture Notes in Mathematics, Vol. 915. Springer, 68-85.Google Scholar
- Chris Heunen, Ohad Kammar, Sam Staton, and Hongseok Yang. 2017. A Convenient Category for Higher-Order Probability Theory. In Proceedings of the 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (Reykjavík, Iceland) (LICS '17). IEEE Press, Article 77, 12 pages.Google Scholar
Digital Library
- Daniel Huang, Greg Morrisett, and Bas Spitters. 2018. An application of computable distributions to the semantics of probabilistic programs. arxiv:1806.07966.Google Scholar
- A. Jefrey and J. Rathke. 1999. Towards a theory of bisimulation for local names. In Proc. LICS 1999.Google Scholar
- Paul Jung, Jiho Lee, Sam Staton, and Hongseok Yang. 2020. A generalization of hierarchical exchangeability on trees to directed acyclic graphs. Annales Henri Lebesgue ( 2020 ). to appear.Google Scholar
- Olav Kallenberg. 2002. Foundations of Modern Probability. Springer, New York.Google Scholar
- Ohad Kammar and Gordon D. Plotkin. 2012. Algebraic foundations for efect-dependent optimisations. In Proc. POPL 2012. 349-360.Google Scholar
- Alexander Kechris. 1987. Classical Descriptive Set Theory. Springer.Google Scholar
- Anders Kock. 2011. Commutative monads as a theory of distributions. Theory and Applications of Categories 26 (Aug. 2011 ).Google Scholar
- Dexter Kozen. 1981. Semantics of probabilistic programs. J. Comput. Syst. Sci. 22, 3 ( 1981 ), 328-350.Google Scholar
Cross Ref
- Ugo Dal Lago and Francesco Gavazzo. 2019. On bisimilarity in lambda calculi with continuous probabilistic choice. Electron. Notes Theoret. Comput. Sci. 347 ( 2019 ), 121-141. Proc. MFPS 2019.Google Scholar
- James Laird. 2004. A game semantics of local names and good variables. In Proc. FOSSACS 2004. 289-303.Google Scholar
Cross Ref
- J Lambek and P J Scott. 1988. Introduction to higher order categorical logic. CUP.Google Scholar
- Alexander K. Lew, Marco F. Cusumano-Towner, Benjamin Sherman, Michael Carbin, and Vikash K. Mansinghka. 2019. Trace types and denotational semantics for sound programmable inference in probabilistic languages. Proc. ACM Program. Lang. 4, POPL, Article 19 ( Dec. 2019 ).Google Scholar
- Robin Milner. 1999. Communicating and mobile systems-the Pi-calculus. CUP.Google Scholar
- Eugenio Moggi. 1991. Notions of computation and monads. Inform. Comput. 93, 1 ( 1991 ), 55-92.Google Scholar
- Andrzej S. Murawski and Nikos Tzevelekos. 2016. Nominal game semantics. Found. Trends Program. Lang. ( 2016 ).Google Scholar
- Lawrence M. Murray and Thomas B. Schön. 2018. Automated learning with a probabilistic programming language: Birch. Annual Reviews in Control 46 ( 2018 ), 29-43.Google Scholar
- Aditya Nori, Chung-Kil Hur, Sriram Rajamani, and Selva Samuel. 2014. R2: An eficient MCMC sampler for probabilistic programs. In Proc. AAAI 2014.Google Scholar
- Martin Odersky. 1994. A Functional Theory of Local Names. In Proc. POPL 1994. 48-59.Google Scholar
- Peter Orbanz and Daniel M. Roy. 2015. Bayesian models of graphs, arrays and other exchangeable random structures. IEEE Trans. Pattern Anal. Mach. Intell. 2 ( 2015 ), 437-461.Google Scholar
- Hugo Paquet and Glynn Winskel. 2018. Continuous probability distributions in concurrent games. In Proc. MFPS 2018. 321-344.Google Scholar
Cross Ref
- Arthur J. Parzygnat. 2020. Inverses, disintegrations, and Bayesian inversion in quantum Markov categories. arXiv: 2001.08375.Google Scholar
- Evan Patterson. 2020. The algebra and machine representation of statistical models. Ph.D. Dissertation. Stanford University Department of Statistics.Google Scholar
- Andrew M. Pitts. 2013. Nominal Sets: Names and Symmetry in Computer Science. Cambridge University Press.Google Scholar
Digital Library
- Andrew M. Pitts and Ian Stark. 1993. Observable properties of higher order functions that dynamically create local names, or: What's new?. In Proc. MFCS 1993 (Lecture Notes in Computer Science). 122-141.Google Scholar
- G. D. Plotkin. 1973. Lambda-definability and logical relations. Technical Report SAI-RM-4. School of A.I., Univ.of Edinburgh.Google Scholar
- David Pollard. 2001. A users' guide to measure-theoretic probability. CUP.Google Scholar
- Daniel Roy, Vikash Mansinghka, Noah Goodman, and Josh Tenenbaum. 2008. A stochastic programming perspective on nonparametric Bayes. In Proc. ICML Workshop on Nonparametric Bayes.Google Scholar
- Tetsuya Sato, Alejandro Aguirre, Gilles Barthe, Marco Gaboardi, Deepak Garg, and Justin Hsu. 2019. Formal verification of higher-order probabilistic programs: reasoning about approximation, convergence, bayesian inference, and optimization. Proc. ACM Program. Lang. 3, POPL, Article 38 ( Jan. 2019 ), 30 pages.Google Scholar
Digital Library
- Adam Ścibior, Ohad Kammar, Matthijs Vákár, Sam Staton, Hongseok Yang, Yufei Cai, Klaus Ostermann, Sean Moss, Chris Heunen, and Zoubin Ghahramani. 2017. Denotational validation of higher-order Bayesian inference. Proceedings of the ACM on Programming Languages 2 ( Nov. 2017 ).Google Scholar
- Dan Shiebler. 2020. Categorical stochastic processes and likelihood. arXiv: 2005.04735.Google Scholar
- Alex Simpson. 2017. Probability Sheaves and the Giry Monad. In Proc. CALCO 2017.Google Scholar
- Shashi M. Srivastava. 1998. A Course on Borel Sets. Springer, New York.Google Scholar
- Ian Stark. 1994. Names and Higher-Order Functions. Ph.D. Dissertation. University of Cambridge. Also available as Technical Report 363, University of Cambridge Computer Laboratory.Google Scholar
- Ian Stark. 1996. Categorical models for local names. LISP and Symbolic Computation 9, 1 (Feb. 1996 ), 77-107.Google Scholar
Cross Ref
- Sam Staton. 2010. Completeness for algebraic theories of local state. In Proc. FOSSACS 2010. 48-63.Google Scholar
Digital Library
- Sam Staton. 2017. Commutative semantics for probabilistic programming. In Proc. ESOP 2017.Google Scholar
Digital Library
- Sam Staton, Dario Stein, Hongseok Yang, Nathanael L. Ackerman, Cameron E. Freer, and Daniel M. Roy. 2018. The Beta-Bernoulli process and algebraic efects. Proc. ICALP 2018.Google Scholar
- S. Staton, H. Yang, N. L.. Ackerman, C. Freer, and D. Roy. 2017. Exchangeable random process and data abstraction. In Proc. PPS 2017.Google Scholar
- 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 Proc. LICS 2016. 525-534.Google Scholar
Digital Library
- Eijiro Sumii and Benjamin C. Pierce. 2003. Logical relations for encryption. J. Comput. Secur. 11, 4 ( 2003 ), 521-554.Google Scholar
- Nikos Tzevelekos. 2008. Nominal game semantics. Ph.D. Dissertation. Oxford University Computing Laboratory.Google Scholar
- Jan-Willem van de Meent, Brooks Paige, Hongseok Yang, and Frank Wood. 2018. An introduction to probabilistic programming. arxiv:1809.10756.Google Scholar
- Alexander Vandenbroucke and Tom Schrijvers. 2020. P NK: functional probabilistic NetKAT. In Proc. POPL 2020.Google Scholar
- Yu Zhang and David Nowak. 2003. Logical relations for dynamic name creation. In Proc. CSL 2003. 575-588.Google Scholar
Cross Ref
Index Terms
Probabilistic programming semantics for name generation
Recommendations
Higher-order probabilistic adversarial computations: categorical semantics and program logics
Adversarial computations are a widely studied class of computations where resource-bounded probabilistic adversaries have access to oracles, i.e., probabilistic procedures with private state. These computations arise routinely in several domains, ...
A program logic for fresh name generation
Highlights- A program logic for fresh name generation.
- A logic to reason about the nu-...
AbstractWe present a program logic for Pitts and Stark's ν-calculus, an extension of the call-by-value simply-typed λ-calculus with a mechanism for the generation of fresh names. Names can be compared for equality and inequality, producing ...
Probability monads with submonads of deterministic states
LICS '22: Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer ScienceProbability theory can be studied synthetically as the computational effect embodied by a commutative monad. In the recently proposed Markov categories, one works with an abstraction of the Kleisli category and then defines deterministic morphisms ...






Comments