Advertisement

The Free Exponential Modality of Probabilistic Coherence Spaces

  • Raphaëlle Crubillé
  • Thomas Ehrhard
  • Michele PaganiEmail author
  • Christine Tasson
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10203)

Abstract

Probabilistic coherence spaces yield a model of linear logic and lambda-calculus with a linear algebra flavor. Formulas/types are associated with convex sets of \({\mathbb R^+}^{}\)-valued vectors, linear logic proofs with linear functions and \(\lambda \)-terms with entire functions, both mapping the convex set of their domain into the one of their codomain.

Previous results show that this model is particularly precise in describing the observational equivalences between probabilistic functional programs. We prove here that the exponential modality is the free commutative comonad, giving a further mark of canonicity to the model.

Notes

Acknowledgments

We thank Sam Staton, Hugh Steele, Lionel Vaux and the anonymous reviewers for useful comments and discussions. This work has been partly funded by the French project ANR-14-CE25-0005 Elica and by the French-Chinese project ANR-11-IS02-0002 and NSFC 61161130530 Locali.

References

  1. 1.
    Breuvart, F., Pagani, M.: Modelling coeffects in the relational semantics of linear logic. In: Kreutzer, S. (ed.) Proceedings of the 24th EACSL Annual Conference on Computer Science Logic, CSL15, Berlin, Germany. LIPICS (2015)Google Scholar
  2. 2.
    Carraro, A., Ehrhard, T., Salibra, A.: Exponentials with infinite multiplicities. In: Dawar, A., Veith, H. (eds.) CSL 2010. LNCS, vol. 6247, pp. 170–184. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-15205-4_16 CrossRefGoogle Scholar
  3. 3.
    Danos, V., Ehrhard, T.: Probabilistic coherence spaces as a model of higher-order probabilistic computation. Inf. Comput. 209(6), 966–991 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ehrhard, T., Pagani, M., Tasson, C.: Probabilistic coherence spaces are fully abstract for probabilistic PCF. In: Sewell, P. (ed.) The 41th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL14, San Diego, USA. ACM (2014)Google Scholar
  5. 5.
    Ehrhard, T., Tasson, C.: Probabilistic call by push value (2016). preprint http://arxiv.org/abs/1607.04690
  6. 6.
    Girard, J.-Y.: Linear logic. Theor. Comput. Sci. 50, 1–102 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Girard, J.-Y.: Linear logic: its syntax and semantics. In: Girard, J.-Y., Lafont, Y., Regnier, L. (eds.) Advances in Linear Logic, London Math. Soc. Lect. Notes Ser, vol. 222, pp. 1–42. (1995)Google Scholar
  8. 8.
    Girard, J.-Y.: Between logic and quantic: a tract. In: Ehrhard, T., Girard, J.-Y., Ruet, P., Scott, P. (eds.) Linear Logic in Computer Science, London Math. Soc. Lect. Notes Ser., vol. 316. CUP (2004)Google Scholar
  9. 9.
    Lafont, Y.: Logiques, catégories et machines. Ph.D. thesis, Université Paris 7 (1988)Google Scholar
  10. 10.
    Melliès, P.-A.: Categorical semantics of linear logic. Panoramas et Synthèses, 27 (2009)Google Scholar
  11. 11.
    Melliès, P.-A., Tabareau, N., Tasson, C.: An explicit formula for the free exponential modality of linear logic. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009. LNCS, vol. 5556, pp. 247–260. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-02930-1_21 CrossRefGoogle Scholar
  12. 12.
    van de Wiele, J.: Manuscript (1987)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Raphaëlle Crubillé
    • 1
  • Thomas Ehrhard
    • 1
  • Michele Pagani
    • 1
    Email author
  • Christine Tasson
    • 1
  1. 1.IRIF, UMR 8243, Université Paris Diderot, Sorbonne Paris CitéParisFrance

Personalised recommendations