Game Semantics and Normalization by Evaluation

  • Pierre Clairambault
  • Peter Dybjer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9034)


We show that Hyland and Ong’s game semantics for PCF can be presented using normalization by evaluation (nbe). We use the bijective correspondence between innocent well-bracketed strategies and PCF Böhm trees, and show how operations on PCF Böhm trees, such as composition, can be computed lazily and simply by nbe. The usual equations characteristic of games follow from the nbe construction without reference to low-level game-theoretic machinery. As an illustration, we give a Haskell program computing the application of innocent strategies.


Reduction Rule Semantic Domain Closed Term Game Semantic Recursive Type 
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  1. 1.
    Abel, A.: Normalization by Evaluation: Dependent Types and Impredicativity. Institut für Informatik, Ludwig-Maximilians-Universität München, Habilitation thesis (May 2013)Google Scholar
  2. 2.
    Abel, A., Aehlig, K., Dybjer, P.: Normalization by evaluation for Martin-Löf type theory with one universe. In: 23rd Conference on the Mathematical Foundations of Programming Semantics, MFPS XXIII. Electronic Notes in Theoretical Computer Science, pp. 17–40. Elsevier (2007)Google Scholar
  3. 3.
    Abramsky, S., Jagadeesan, R., Malacaria, P.: Full abstraction for PCF. Inf. Comput. 163(2), 409–470 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Aehlig, K., Joachimski, F.: Operational aspects of untyped normalisation by evaluation. Mathematical Structures in Computer Science 14(4) (2004)Google Scholar
  5. 5.
    Amadio, R., Curien, P.-L.: Domains and lambda-calculi, vol. 46. Cambridge University Press (1998)Google Scholar
  6. 6.
    Berger, U., Schwichtenberg, H.: An inverse to the evaluation functional for typed λ-calculus. In: Proceedings of the 6th Annual IEEE Symposium on Logic in Computer Science, Amsterdam, pp. 203–211 (July 1991)Google Scholar
  7. 7.
    Cartmell, J.: Generalized algebraic theories and contextual categories. Annals of Pure and Applied Logic 32, 209–243 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Clairambault, P., Dybjer, P.: The biequivalence of locally cartesian closed categories and Martin-Löf type theories. Mathematical Structures in Computer Science 24(5) (2013)Google Scholar
  9. 9.
    Clairambault, P., Murawski, A.S.: Böhm trees as higher-order recursive schemes. In: IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2013, Guwahati, India, December 12-14, pp. 91–102 (2013)Google Scholar
  10. 10.
    Curien, P.-L.: Abstract Böhm trees. Mathematical Structures in Computer Science 8(6), 559–591 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Curien, P.-L.: Notes on game semantics. From the authors web page (2006)Google Scholar
  12. 12.
    Dybjer, P.: Internal type theory. In: Berardi, S., Coppo, M. (eds.) TYPES 1995. LNCS, vol. 1158, pp. 120–134. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  13. 13.
    Dybjer, P.: Program testing and the meaning explanations of intuitionistic type theory. In: Epistemology versus Ontology - Essays on the Philosophy and Foundations of Mathematics in Honour of Per Martin-Löf, pp. 215–241 (2012)Google Scholar
  14. 14.
    Dybjer, P., Kuperberg, D.: Formal neighbourhoods, combinatory Böhm trees, and untyped normalization by evaluation. Ann. Pure Appl. Logic 163(2), 122–131 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Filinski, A., Rohde, H.K.: Denotational aspects of untyped normalization by evaluation. Theor. Inf. and App. 39(3), 423–453 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Harmer, R., Hyland, M., Melliès, P.-A.: Categorical combinators for innocent strategies. In: 22nd IEEE Symposium on Logic in Computer Science, Wroclaw, Poland, Proceedings, pp. 379–388. IEEE Computer Society (2007)Google Scholar
  17. 17.
    Hyland, J.M.E., Luke Ong, C.-H.: On full abstraction for PCF: I, II, and III. Inf. Comput. 163(2), 285–408 (2000)CrossRefzbMATHGoogle Scholar
  18. 18.
    McCusker, G.: Games and full abstraction for FPC. Inf. Comput. 160(1-2), 1–61 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Pitts, A.M.: A co-induction principle for recursively defined domains. Theor. Comput. Sci. 124(2), 195–219 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Plotkin, G.D.: LCF considered as a programming language. Theor. Comput. Sci. 5(3), 223–255 (1977)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Plotkin, G.D.: Post-graduate lecture notes in advanced domain theory (incorporating the “Pisa Notes”). Dept. of Computer Science, Univ. of Edinburgh (1981)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Pierre Clairambault
    • 1
  • Peter Dybjer
    • 2
  1. 1.CNRS, ENS Lyon, Inria, UCBLUniversité de LyonLyonFrance
  2. 2.Chalmers Tekniska HögskolaGothenburgSweden

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