skip to main content
research-article

On coinductive equivalences for higher-order probabilistic functional programs

Published:08 January 2014Publication History
Skip Abstract Section

Abstract

We study bisimulation and context equivalence in a probabilistic lambda-calculus. The contributions of this paper are threefold. Firstly we show a technique for proving congruence of probabilistic applicative bisimilarity. While the technique follows Howe's method, some of the technicalities are quite different, relying on non-trivial "disentangling" properties for sets of real numbers. Secondly we show that, while bisimilarity is in general strictly finer than context equivalence, coincidence between the two relations is attained on pure lambda-terms. The resulting equality is that induced by Levy-Longo trees, generally accepted as the finest extensional equivalence on pure lambda-terms under a lazy regime. Finally, we derive a coinductive characterisation of context equivalence on the whole probabilistic language, via an extension in which terms akin to distributions may appear in redex position. Another motivation for the extension is that its operational semantics allows us to experiment with a different congruence technique, namely that of logical bisimilarity.

Skip Supplemental Material Section

Supplemental Material

d2_left_t4.mp4

References

  1. S. Abramsky. The Lazy λ-Calculus. In D. Turner, editor, Research Topics in Functional Programming, pages 65--117. Addison Wesley, 1990. Google ScholarGoogle ScholarDigital LibraryDigital Library
  2. S. Abramsky and C.-H. L. Ong. Full abstraction in the lazy lambda calculus. Inf. Comput., 105(2):159--267, 1993. Google ScholarGoogle ScholarDigital LibraryDigital Library
  3. E. Astesiano and G. Costa. Distributive semantics for nondeterministic typed lambda-calculi. Theor. Comput. Sci., 32:121--156, 1984.Google ScholarGoogle ScholarCross RefCross Ref
  4. H. P. Barendregt. The Lambda Calculus -- Its Syntax and Semantics, volume 103 of Studies in Logic and the Foundations of Mathematics. North-Holland, 1984.Google ScholarGoogle Scholar
  5. M. Bernardo, R. De Nicola, and M. Loreti. A uniform framework for modeling nondeterministic, probabilistic, stochastic, or mixed processes and their behavioral equivalences. Inf. Comput., 225:29--82, 2013. Google ScholarGoogle ScholarDigital LibraryDigital Library
  6. G. Boudol. Lambda-calculi for (strict) parallel functions. Inf. Comput., 108(1):51--127, 1994. Google ScholarGoogle ScholarDigital LibraryDigital Library
  7. G. Boudol and C. Laneve. The discriminating power of the calculus with multiplicities. Inf. Comput., 126(1):83--102, 1996. Google ScholarGoogle ScholarDigital LibraryDigital Library
  8. D. Comaniciu, V. Ramesh, and P. Meer. Kernel-based object tracking. IEEE Trans. on Pattern Analysis and Machine Intelligence,, 25(5): 564--577, 2003. Google ScholarGoogle ScholarDigital LibraryDigital Library
  9. U. Dal Lago and M. Zorzi. Probabilistic operational semantics for the lambda calculus. RAIRO - Theor. Inf. and Applic., 46(3):413--450, 2012.Google ScholarGoogle Scholar
  10. U. Dal Lago, D. Sangiorgi, and M. Alberti. On coinductive equivalences for probabilistic higher-order functional programs (long version). Available at http://arxiv.org/abs/1311.1722, 2013.Google ScholarGoogle Scholar
  11. V. Danos and R. Harmer. Probabilistic game semantics. ACM Trans. Comput. Log., 3(3):359--382, 2002. Google ScholarGoogle ScholarDigital LibraryDigital Library
  12. R. De Nicola and M. Hennessy. Testing equivalences for processes. Theor. Comput. Sci., 34:83--133, 1984.Google ScholarGoogle ScholarCross RefCross Ref
  13. U. de'Liguoro and A. Piperno. Non deterministic extensions of untyped lambda-calculus. Inf. Comput., 122(2):149--177, 1995. Google ScholarGoogle ScholarDigital LibraryDigital Library
  14. M. Dezani-Ciancaglini and E. Giovannetti. From bohm's theorem to observational equivalences: an informal account. Electr. Notes Theor. Comput. Sci., 50(2):83--116, 2001.Google ScholarGoogle ScholarCross RefCross Ref
  15. M. Dezani-Ciancaglini, J. Tiuryn, and P. Urzyczyn. Discrimination by parallel observers: The algorithm. Inf. Comput., 150(2):153--186, 1999. Google ScholarGoogle ScholarDigital LibraryDigital Library
  16. T. Ehrhard, M. Pagani, and C. Tasson. The computational meaning of probabilistic coherence spaces. In LICS, pages 87--96, 2011. Google ScholarGoogle ScholarDigital LibraryDigital Library
  17. S. Goldwasser and S. Micali. Probabilistic encryption. J. Comput. Syst. Sci., 28(2):270--299, 1984.Google ScholarGoogle ScholarCross RefCross Ref
  18. N. D. Goodman. The principles and practice of probabilistic programming. In POPL, pages 399--402, 2013. Google ScholarGoogle ScholarDigital LibraryDigital Library
  19. A. D. Gordon. Bisimilarity as a theory of functional programming. Electr. Notes Theor. Comput. Sci., 1:232--252, 1995.Google ScholarGoogle ScholarCross RefCross Ref
  20. A. D. Gordon, M. Aizatulin, J. Borgström, G. Claret, T. Graepel, A. V. Nori, S. K. Rajamani, and C. V. Russo. A model-learner pattern for bayesian reasoning. In POPL, pages 403--416, 2013. Google ScholarGoogle ScholarDigital LibraryDigital Library
  21. M. Hennessy. Exploring probabilistic bisimulations, part I. Formal Asp. Comput., 24(4--6):749--768, 2012. Google ScholarGoogle ScholarCross RefCross Ref
  22. D. J. Howe. Proving congruence of bisimulation in functional programming languages. Inf. Comput., 124(2):103--112, 1996. Google ScholarGoogle ScholarDigital LibraryDigital Library
  23. R. Jagadeesan and P. Panangaden. A domain-theoretic model for a higher-order process calculus. In ICALP, pages 181--194, 1990. Google ScholarGoogle ScholarDigital LibraryDigital Library
  24. C. Jones and G. D. Plotkin. A probabilistic powerdomain of evaluations. In LICS, pages 186--195, 1989. Google ScholarGoogle ScholarDigital LibraryDigital Library
  25. V. Koutavas, P. B. Levy, and E. Sumii. From applicative to environmental bisimulation. Electr. Notes Theor. Comput. Sci., 276: 215?235, 2011. Google ScholarGoogle ScholarDigital LibraryDigital Library
  26. K. G. Larsen and A. Skou. Bisimulation through probabilistic testing. Inf. Comput., 94(1):1?28, 1991. Google ScholarGoogle ScholarDigital LibraryDigital Library
  27. S. B. Lassen. Relational Reasoning about Functions and Nondeterminism. PhD thesis, University of Aarhus, 1998.Google ScholarGoogle Scholar
  28. S. Lenglet, A. Schmitt, and J.-B. Stefani. Howe's method for calculi with passivation. In CONCUR, pages 448--462, 2009. Google ScholarGoogle ScholarDigital LibraryDigital Library
  29. C. D. Manning and H. Schütze. Foundations of statistical natural language processing, volume 999. MIT Press, 1999. Google ScholarGoogle ScholarDigital LibraryDigital Library
  30. C.-H. L. Ong. Non-determinism in a functional setting. In LICS, pages 275--286, 1993.Google ScholarGoogle ScholarCross RefCross Ref
  31. P. Panangaden. Labelled Markov Processes. Imperial College Press, 2009. Google ScholarGoogle ScholarDigital LibraryDigital Library
  32. S. Park, F. Pfenning, and S. Thrun. A probabilistic language based on sampling functions. ACM Trans. Program. Lang. Syst., 31(1), 2008. Google ScholarGoogle ScholarDigital LibraryDigital Library
  33. J. Pearl. Probabilistic reasoning in intelligent systems: networks of plausible inference. Morgan Kaufmann, 1988. Google ScholarGoogle ScholarDigital LibraryDigital Library
  34. A. Pfeffer. IBAL: A probabilistic rational programming language. In IJCAI, pages 733--740. Morgan Kaufmann, 2001. Google ScholarGoogle ScholarDigital LibraryDigital Library
  35. A. M. Pitts. Operationally-based theories of program equivalence. In Semantics and Logics of Computation, pages 241--298. Cambridge University Press, 1997.Google ScholarGoogle ScholarCross RefCross Ref
  36. A. M. Pitts. Howe's method for higher-order languages. In D. Sangiorgi and J. Rutten, editors, Advanced Topics in Bisimulation and Coinduction, pages 197--232. Cambridge University Press, 2011.Google ScholarGoogle ScholarCross RefCross Ref
  37. N. Ramsey and A. Pfeffer. Stochastic lambda calculus and monads of probability distributions. In POPL, pages 154--165, 2002. Google ScholarGoogle ScholarDigital LibraryDigital Library
  38. N. Saheb-Djahromi. Probabilistic LCF. In MFCS, volume 64 of LNCS, pages 442--451, 1978.Google ScholarGoogle Scholar
  39. D. Sands. From SOS rules to proof principles: An operational metatheory for functional languages. In POPL, pages 428--441, 1997. Google ScholarGoogle ScholarDigital LibraryDigital Library
  40. D. Sangiorgi and D. Walker. The pi-Calculus -- a theory of mobile processes. Cambridge University Press, 2001. Google ScholarGoogle ScholarDigital LibraryDigital Library
  41. D. Sangiorgi, N. Kobayashi, and E. Sumii. Logical bisimulations and functional languages. In FSEN, volume 4767 of LNCS, pages 364--379, 2007. Google ScholarGoogle ScholarDigital LibraryDigital Library
  42. D. Sangiorgi, N. Kobayashi, and E. Sumii. Environmental bisimulations for higher-order languages. ACM Trans. Program. Lang. Syst., 33(1):5, 2011. Google ScholarGoogle ScholarDigital LibraryDigital Library
  43. K. Sieber. Call-by-value and nondeterminism. In TLCA, volume 664 of LNCS, pages 376--390, 1993. Google ScholarGoogle ScholarDigital LibraryDigital Library
  44. S. Thrun. Robotic mapping: A survey. Exploring artificial intelligence in the new millennium, pages 1--35, 2002. Google ScholarGoogle ScholarDigital LibraryDigital Library

Index Terms

  1. On coinductive equivalences for higher-order probabilistic functional programs

        Recommendations

        Comments

        Login options

        Check if you have access through your login credentials or your institution to get full access on this article.

        Sign in

        Full Access

        • Published in

          cover image ACM SIGPLAN Notices
          ACM SIGPLAN Notices  Volume 49, Issue 1
          POPL '14
          January 2014
          661 pages
          ISSN:0362-1340
          EISSN:1558-1160
          DOI:10.1145/2578855
          Issue’s Table of Contents
          • cover image ACM Conferences
            POPL '14: Proceedings of the 41st ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages
            January 2014
            702 pages
            ISBN:9781450325448
            DOI:10.1145/2535838

          Copyright © 2014 ACM

          Publisher

          Association for Computing Machinery

          New York, NY, United States

          Publication History

          • Published: 8 January 2014

          Check for updates

          Qualifiers

          • research-article

        PDF Format

        View or Download as a PDF file.

        PDF

        eReader

        View online with eReader.

        eReader
        About Cookies On This Site

        We use cookies to ensure that we give you the best experience on our website.

        Learn more

        Got it!