An Alpha-Corecursion Principle for the Infinitary Lambda Calculus

  • Alexander Kurz
  • Daniela Petrişan
  • Paula Severi
  • Fer-Jan de Vries
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7399)

Abstract

Gabbay and Pitts proved that lambda-terms up to alpha-equivalence constitute an initial algebra for a certain endofunctor on the category of nominal sets. We show that the terms of the infinitary lambda-calculus form the final coalgebra for the same functor. This allows us to give a corecursion principle for alpha-equivalence classes of finite and infinite terms. As an application, we give corecursive definitions of substitution and of infinite normal forms (Böhm, Lévy-Longo and Berarducci trees).

References

  1. 1.
    Abramsky, S.: The lazy lambda calculus. In: Research Topics in Functional Programming, pp. 65–116. Addison-Wesley (1990)Google Scholar
  2. 2.
    Abramsky, S., Luke Ong, C.-H.: Full abstraction in the lazy lambda calculus. Inform. and Comput. 105(2), 159–267 (1993)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Adámek, J.: On final coalgebras of continuous functors. Theor. Comput. Sci. 294(1/2), 3–29 (2003)MATHCrossRefGoogle Scholar
  4. 4.
    Arnold, A., Nivat, M.: The metric space of infinite trees. algebraic and topological properties. Fundamenta Informaticae 4, 445–476 (1980)MathSciNetGoogle Scholar
  5. 5.
    Barendregt, H.P.: The Lambda Calculus: Its Syntax and Semantics, Revised edition. North-Holland, Amsterdam (1984)Google Scholar
  6. 6.
    Barr, M.: Terminal coalgebras for endofunctors on sets. Theor. Comp. Sci. 114(2), 299–315 (1999)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Berarducci, A.: Infinite λ-calculus and non-sensible models. In: Logic and Algebra (Pontignano, 1994), pp. 339–377. Dekker, New York (1996)Google Scholar
  8. 8.
    Cheney, J.: Completeness and Herbrand theorems for nominal logic. J. Symb. Log. 71(1), 299–320 (2006)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Duppen, Y.D.: A coalgebraic approach to lambda calculus. Master’s thesis, Vrije Universiteit Amsterdam (2000)Google Scholar
  10. 10.
    Fernández, M., Gabbay, M.: Nominal rewriting. Inf. Comput. 205(6), 917–965 (2007)MATHCrossRefGoogle Scholar
  11. 11.
    Gabbay, M., Pitts, A.M.: A new approach to abstract syntax involving binders. In: LICS, pp. 214–224 (1999)Google Scholar
  12. 12.
    Gabbay, M.J.: A general mathematics of names. Inf. Comput. 205(7), 982–1011 (2007)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Gabbay, M.J., Mathijssen, A.: Nominal (universal) algebra: Equational logic with names and binding. J. Log. Comput. 19(6), 1455–1508 (2009)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Gabbay, M.J., Pitts, A.M.: A new approach to abstract syntax with variable binding. Formal Aspects of Computing 13(3–5), 341–363 (2001)Google Scholar
  15. 15.
    Kennaway, J.R., de Vries, F.J.: Infinitary rewriting. In: Terese (ed.) Term Rewriting Systems. Cambridge Tracts in Theor. Comp. Sci, vol. 55, pp. 668–711. Cambridge University Press (2003)Google Scholar
  16. 16.
    Kennaway, J.R., Klop, J.W., Sleep, M.R., de Vries, F.J.: Infinite Lambda Calculus and Böhm Models. In: Hsiang, J. (ed.) RTA 1995. LNCS, vol. 914, pp. 257–270. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  17. 17.
    Kennaway, J.R., Klop, J.W., Sleep, M.R., de Vries, F.J.: Infinitary lambda calculus. Theor. Comp. Sci. 175(1), 93–125 (1997)MATHCrossRefGoogle Scholar
  18. 18.
    Lévy, J.-J.: An algebraic interpretation of the λβK-calculus, and an application of a labelled λ-calculus. Theor. Comp. Sci. 2(1), 97–114 (1976)MATHCrossRefGoogle Scholar
  19. 19.
    Longo, G.: Set-theoretical models of λ-calculus: theories, expansions, isomorphisms. Ann. Pure Appl. Logic 24(2), 153–188 (1983)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Matthes, R., Uustalu, T.: Substitution in non-wellfounded syntax with variable binding. Theor. Comput. Sci. 327, 155–174 (2004)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Moss, L.S.: Parametric corecursion. Theor. Comp. Sci. 260, 139–163 (2001)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Pitts, A.M.: Nominal logic, a first order theory of names and binding. Information and Computation 186, 165–193 (2003)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Pitts, A.M.: Alpha-Structural Recursion and Induction. In: Hurd, J., Melham, T. (eds.) TPHOLs 2005. LNCS, vol. 3603, pp. 17–34. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  24. 24.
    Salibra, A.: Nonmodularity results for lambda calculus. Fundamenta Informaticae 45, 379–392 (2001)MathSciNetMATHGoogle Scholar
  25. 25.
    Severi, P.G., de Vries, F.J.: Weakening the axiom of overlap in the infinitary lambda calculus. In: RTA. LIPIcs, vol. 10, pp. 313–328. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik (2011)Google Scholar

Copyright information

© IFIP International Federation for Information Processing 2012

Authors and Affiliations

  • Alexander Kurz
    • 1
  • Daniela Petrişan
    • 1
  • Paula Severi
    • 1
  • Fer-Jan de Vries
    • 1
  1. 1.Department of Computer ScienceUniversity of LeicesterUK

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