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Observed Communication Semantics for Classical Processes

  • Robert Atkey
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10201)

Abstract

Classical Linear Logic (CLL) has long inspired readings of its proofs as communicating processes. Wadler’s CP calculus is one of these readings. Wadler gave CP an operational semantics by selecting a subset of the cut-elimination rules of CLL to use as reduction rules. This semantics has an appealing close connection to the logic, but does not resolve the status of the other cut-elimination rules, and does not admit an obvious notion of observational equivalence. We propose a new operational semantics for CP based on the idea of observing communication. We use this semantics to define an intuitively reasonable notion of observational equivalence. To reason about observational equivalence, we use the standard relational denotational semantics of CLL. We show that this denotational semantics is adequate for our operational semantics. This allows us to deduce that, for instance, all the cut-elimination rules of CLL are observational equivalences.

Keywords

Relational Semantic Operational Semantic Label Transition System Reduction Rule Denotational Semantic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

Thanks to Sam Lindley, J. Garrett Morris, Conor McBride and Phil Wadler for helpful discussions and comments on this paper. This work was partly funded by a Science Faculty Starter Grant from the University of Strathclyde.

References

  1. 1.
    Abramsky, S.: Computational interpretations of linear logic. Theor. Comput. Sci. 111, 3–57 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Abramsky, S.: Proofs as processes. Theor. Comput. Sci. 135(1), 5–9 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Accattoli, B.: Linear logic and strong normalization. In: 24th International Conference on Rewriting Techniques and Applications, RTA 2013, 24–26 June 2013, Eindhoven, The Netherlands, pp. 39–54 (2013)Google Scholar
  4. 4.
    Atkey, R., Lindley, S., Morris, J.G.: Conflation confers concurrency. In: Lindley, S., McBride, C., Trinder, P., Sannella, D. (eds.) A List of Successes That Can Change the World. LNCS, vol. 9600, pp. 32–55. Springer, Heidelberg (2016). doi: 10.1007/978-3-319-30936-1_2 CrossRefGoogle Scholar
  5. 5.
    Baelde, D.: Least, greatest fixed points in linear logic. ACM Trans. Comput. Logic 13(1), 2:1–2:44 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Barr, M.: *-Autonomous categories and linear logic. Math. Struct. Comput. Sci. 1(2), 159–178 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bellin, G., Scott, P.J.: On the \(\uppi \)-calculus and linear logic. Theoret. Comput. Sci. 135(1), 11–65 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Berger, M., Honda, K., Yoshida, N.: Genericity and the \(\uppi \)-calculus. In: Gordon, A.D. (ed.) FoSSaCS 2003. LNCS, vol. 2620, pp. 103–119. Springer, Heidelberg (2003). doi: 10.1007/3-540-36576-1_7 CrossRefGoogle Scholar
  9. 9.
    Caires, L., Pfenning, F.: Session types as intuitionistic linear propositions. In: Gastin, P., Laroussinie, F. (eds.) CONCUR 2010. LNCS, vol. 6269, pp. 222–236. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-15375-4_16 CrossRefGoogle Scholar
  10. 10.
    Curry, H.B.: Functionality in combinatory logic. Proc. Natl. Acad. Sci. 20, 584–590 (1934)CrossRefzbMATHGoogle Scholar
  11. 11.
    Danos, V., Ehrhard, T.: Probabilistic coherence spaces as a model of higher-order probabilistic computation. Inf. Comput. 209(6), 966–991 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order, 2nd edn. Cambridge University Press, Cambridge (2002)CrossRefzbMATHGoogle Scholar
  13. 13.
    Ehrhard, T.: Finiteness spaces. Math. Struct. Comput. Sci. 15(4), 615–646 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ehrhard, T., Laurent, O.: Interpreting a finitary \(\uppi \)-calculus in differential interaction nets. Inf. Comput. 208(6), 606–633 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gay, S.J., Vasconcelos, V.T.: Linear type theory for asynchronous session types. J. Funct. Program. 20(01), 19–50 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Girard, J.-Y.: Linear logic. Theor. Comput. Sci. 50, 1–101 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Honda, K.: Types for dyadic interaction. In: Best, E. (ed.) CONCUR 1993. LNCS, vol. 715, pp. 509–523. Springer, Heidelberg (1993). doi: 10.1007/3-540-57208-2_35 Google Scholar
  18. 18.
    Howard, W.A.: The formulae-as-types notion of construction. In: Seldin, J.P., Hindley, J.R. (eds.) To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism. Academic Press, Boston (1980)Google Scholar
  19. 19.
    Laird, J., Manzonetto, G., McCusker, G., Pagani, M.: Weighted relational models of typed \(\lambda \)-calculi. In: 28th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2013, 25–28 June 2013, New Orleans, LA, USA, pp. 301–310 (2013)Google Scholar
  20. 20.
    Lindley, S., Morris, J.G.: A semantics for propositions as sessions. In: Vitek, J. (ed.) ESOP 2015. LNCS, vol. 9032, pp. 560–584. Springer, Heidelberg (2015). doi: 10.1007/978-3-662-46669-8_23 CrossRefGoogle Scholar
  21. 21.
    Lindley, S., Morris, J.G.: Talking bananas: structural recursion for session types. In: ICFP (2016, to appear)Google Scholar
  22. 22.
    Loader, R.: Linear logic, totality and full completeness. In: Proceedings of the Ninth Annual Symposium on Logic in Computer Science (LICS 1994), 4–7 July 1994, Paris, France, pp. 292–298 (1994)Google Scholar
  23. 23.
    Mazza, D.: The true concurrency of differential interaction nets. Math. Struct. Comput. Sci. (2015, to appear)Google Scholar
  24. 24.
    McBride, C.: I got plenty o’ nuttin’. In: Lindley, S., McBride, C., Trinder, P., Sannella, D. (eds.) A List of Successes That Can Change the World. LNCS, vol. 9600, pp. 207–233. Springer, Heidelberg (2016). doi: 10.1007/978-3-319-30936-1_12 CrossRefGoogle Scholar
  25. 25.
    Melliès, P.-A.: Categorical semantics of linear logic. In: Curien, P.-L., Herbelin, H., Krivine, J.-L., Melliès, P.-A. (eds.) Interactive Models of Computation and Program Behavior, Number 27 in Panoramas et Synthèses. Société Mathématique de France (2009)Google Scholar
  26. 26.
    Milner, R., Sangiorgi, D.: Barbed bisimulation. In: Kuich, W. (ed.) ICALP 1992. LNCS, vol. 623, pp. 685–695. Springer, Heidelberg (1992). doi: 10.1007/3-540-55719-9_114 CrossRefGoogle Scholar
  27. 27.
    Pérez, J.A., Caires, L., Pfenning, F., Toninho, B.: Linear logical relations and observational equivalences for session-based concurrency. Inf. Comput. 239, 254–302 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Plotkin, G.D.: LCF considered as a programming language. Theor. Comput. Sci. 5(3), 223–255 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Retoré, C.: Pomset logic: a non-commutative extension of classical linear logic. In: Groote, P., Roger Hindley, J. (eds.) TLCA 1997. LNCS, vol. 1210, pp. 300–318. Springer, Heidelberg (1997). doi: 10.1007/3-540-62688-3_43 CrossRefGoogle Scholar
  30. 30.
    Roscoe, A.W.: The Theory and Practice of Concurrency. Prentice Hall, Upper Saddle River (1998)Google Scholar
  31. 31.
    Sangiorgi, D., Walker, D.: The \(\uppi \)-Calculus - A Theory of Mobile Processes. Cambridge University Press, Cambridge (2001)zbMATHGoogle Scholar
  32. 32.
    Stark, I.: A fully abstract domain model for the \(\uppi \)-calculus. In: Proceedings of the 11th Annual IEEE Symposium on Logic in Computer Science, 27–30 July 1996, New Brunswick, New Jersey, USA, pp. 36–42 (1996)Google Scholar
  33. 33.
    Toninho, B., Caires, L., Pfenning, F.: Dependent session types via intuitionistic linear type theory. In: Proceedings of the 13th International ACM SIGPLAN Conference on Principles and Practice of Declarative Programming, 20–22 July 2011, Odense, Denmark, pp. 161–172 (2011)Google Scholar
  34. 34.
    Toninho, B., Yoshida, N.: Certifying data in multiparty session types. In: A List of Successes That Can Change the World - Essays Dedicated to Philip Wadler on the Occasion of His 60th Birthday, pp. 433–458 (2016)Google Scholar
  35. 35.
    Wadler, P.: Propositions as sessions. In: Proceedings of the 17th ACM SIGPLAN International Conference on Functional Programming, ICFP 2012. ACM (2012)Google Scholar
  36. 36.
    Wadler, P.: Propositions as sessions. J. Funct. Program. 24(2–3), 384–418 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.MSP GroupUniversity of StrathclydeGlasgowUK

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