Abstract

Inspired by recent work on normalisation by evaluation for sums, we propose a normalising and confluent extensional rewriting theory for the simply-typed λ-calculus extended with sum types. As a corollary of confluence we obtain decidability for the extensional equational theory of simply-typed λ-calculus extended with sum types. Unlike previous decidability results, which rely on advanced rewriting techniques or advanced category theory, we only use standard techniques.

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References

  1. Altenkirch, T., Dybjer, P., Hofmann, M., Scott, P.: Normalization by evaluation for typed lambda calculus with coproducts. In: 16th Annual IEEE Symposium on Logic in Computer Science, Boston, MA, June 2001, pp. 303–310. IEEE Computer Society Press, Los Alamitos (2001)CrossRefGoogle Scholar
  2. Balat, V., Cosmo, R.D., Fiore, M.: Extensional normalisation and type-directed partial evaluation for typed lambda calculus with sums. In: POPL 2004. 31st Symposium on Principles of Programming Languages, January 2004, pp. 64–76. ACM Press, New York (2004)CrossRefGoogle Scholar
  3. Barendregt, H.P.: The Lambda Calculus: Its Syntax and Semantics. Studies in Logics and the Foundations of Mathmatics, vol. 103. North Holland, Amsterdam (1984)MATHGoogle Scholar
  4. Berger, U., Eberl, M., Schwichtenberg, H.: Normalization by evaluation. In: Möller, B., Tucker, J.V. (eds.) Prospects for Hardware Foundations. LNCS, vol. 1546, pp. 117–137. Springer, Heidelberg (1998)Google Scholar
  5. Ghani, N.: Beta-eta equality for coproducts. In: Dezani-Ciancaglini, M., Plotkin, G. (eds.) TLCA 1995. LNCS, vol. 902, pp. 171–185. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  6. Girard, J.-Y., Lafont, Y., Taylor, P.: Proofs and Types. Cambridge University Press, Cambridge (1989)MATHGoogle Scholar
  7. Huet, G.P.: Confluent reductions: Abstract properties and applications to term rewriting systems. J. ACM 27(4), 797–821 (1980)MATHCrossRefGoogle Scholar
  8. Lindley, S.: Normalisation by Evaluation in the Compilation of Typed Functional Programming Languages. PhD thesis, University of Edinburgh (2005)Google Scholar
  9. Lindley, S., Stark, I.: Reducibility and \({\top\top}\)-lifting for computation types. In: Urzyczyn, P. (ed.) TLCA 2005. LNCS, vol. 3461, pp. 262–277. Springer, Heidelberg (2005)Google Scholar
  10. Ohta, Y., Hasegawa, M.: A terminating and confluent linear lambda calculus. RTA, pp. 166–180 (2006)Google Scholar
  11. Prawitz, D.: Ideas and results in proof theory. In: Proceedings of the 2nd Scandinavian Logic Symposium. Studies in Logics and the Foundations of Mathmatics, vol. 63, pp. 235–307. North Holland, Amsterdam (1971)Google Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Sam Lindley
    • 1
  1. 1.Laboratory for Foundations of Computer Science, School of Informatics, The University of Edinburgh 

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