On the Computability of Relations on λ-Terms and Rice’s Theorem - The Case of the Expansion Problem for Explicit Substitutions

  • Edward Hermann Haeusler
  • Mauricio Ayala-Rincón
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8392)


Explicit substitutions calculi are versions of the λ-calculus having a concretely defined operation of substitution. An Explicit substitutions calculus, λξ, extends the language Λ, of the λ-calculus including operations and rewriting rules that explicitly implement the implicit substitution involved in β-reduction in Λ. Λξ, that is the language of λξ, might have terms without any computational meaning, i.e., that do not arise from pure lambda terms in Λ. Thus, it is relevant to answer whether for a given t ∈ Λξ, there exists s ∈ Λ such that \(s\rightarrow^*_{\lambda_\xi} t\), i.e., whether there exists a pure λ-term reducing in the extended calculus to the given term. This is known as the expansion problem and was proved to be undecidable for a few explicit substitutions calculi by using Scott’s theorem. In this note we prove the undecidability of the expansion problem for the λσ calculus by using a version of Rice’s theorem. This method is more straightforward and general than the one based on Scott’s theorem.


Explicit Substitution Lambda-Calculus Rice’s and Scott’s Theorem 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abadi, M., Cardelli, L., Curien, P.-L., Lévy, J.-J.: Explicit Substitutions. J. of Functional Programming 1(4), 375–416 (1991)CrossRefMATHGoogle Scholar
  2. 2.
    Arbiser, A.: The Expansion Problem in Lambda Calculi with Explicit Substitution. J. Log. Comput. 18(6), 849–883 (2008)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Benaissa, Z.-E.-A., Briaud, D., Lescanne, P., Rouyer-Degli, J.: λυ, a Calculus of Explicit Substitutions which Preserves Strong Normalization. J. of Functional Programming 6(5), 699–722 (1996)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Bloo, R., Rose, K.H.: Preservation of Strong Normalisation in Named Lambda Calculi with Explicit Substitution and Garbage Collection. In: CSN-95: Computer Science in the Netherlands, pp. 62–72 (1995)Google Scholar
  5. 5.
    da Silva, F.H.: Expansibilidade em Cálculos de Substituições Explícitas. Master’s thesis, Graduate Program in Informatics, Universidade de Brasília (December 2012) (in Portuguese)Google Scholar
  6. 6.
    de Bruijn, N.G.: Lambda-Calculus Notation with Nameless Dummies, a Tool for Automatic Formula Manipulation, with Application to the Church-Rosser Theorem. Indag. Mat. 34(5), 381–392 (1972)CrossRefGoogle Scholar
  7. 7.
    de Moura, F.L.C., Ayala-Rincón, M., Kamareddine, F.: Higher-Order Unification: A structural relation between Huet’s method and the one based on explicit substitutions. J. Applied Logic 6(1), 72–108 (2008)CrossRefMATHGoogle Scholar
  8. 8.
    Dowek, G., Hardin, T., Kirchner, C.: Higher-order Unification via Explicit Substitutions. Information and Computation 157(1/2), 183–235 (2000)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Kamareddine, F., Ríos, A.: A λ-calculus à la de Bruijn with Explicit Substitutions. In: Swierstra, S.D. (ed.) PLILP 1995. LNCS, vol. 982, pp. 45–62. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  10. 10.
    Kesner, D.: The Theory of Calculi with Explicit Substitutions Revisited. In: Duparc, J., Henzinger, T.A. (eds.) CSL 2007. LNCS, vol. 4646, pp. 238–252. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  11. 11.
    Machtey, M., Young, P.: An introduction to the general theory of algorithms. Theory of computation series. Elsevier North-Holland, New York (1978)Google Scholar
  12. 12.
    Rice, H.G.: Classes of Recursively Enumerable Sets and Their Decision Problems. Trans. Amer. Math. Soc. 74, 358–366 (1953)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Rogers Jr., H.: Theory of recursive functions and effective computability. MIT Press, Cambridge (1987)Google Scholar
  14. 14.
    Rose, K.H.: Explicit Cyclic Substitutions. In: Rusinowitch, M., Remy, J.-L. (eds.) CTRS 1992. LNCS, vol. 656, pp. 36–50. Springer, Heidelberg (1993)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Edward Hermann Haeusler
    • 1
  • Mauricio Ayala-Rincón
    • 2
  1. 1.Departamento de InformáticaPUC-RioBrasil
  2. 2.Departamentos de Computação e MatemáticaUniversidade de BrasíliaBrasil

Personalised recommendations