Advertisement

The Omega Rule is \(\mathbf{\Pi_{1}^{1}}\)-Complete in the λβ-Calculus

  • Benedetto Intrigila
  • Richard Statman
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4583)

Abstract

We give a many-one reduction of the set of true \(\mathbf{\Pi_{1}^{1}}\) sentences to the set of consequences of the λ-calculus with the ω-rule. This solves in the affirmative a long-standing problem of H. Barendregt (1975).

Keywords

Induction Hypothesis Head Part Lambda Calculus Recursive Tree Closed Term 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Barendregt, H.P.: The Lambda Calculus. Its Syntax and Semantics. North-Holland, Amsterdam (1984)zbMATHGoogle Scholar
  2. Böhm, C. (ed.): Lambda-Calculus and Computer Science Theory. LNCS, vol. 37. Springer, Heidelberg (1975)zbMATHGoogle Scholar
  3. Flagg, R.C., Myhill, J.: Implication and Analysis in Classical Frege Structure. Annals of Pure and Applied Logic 34, 33–85 (1987)zbMATHCrossRefGoogle Scholar
  4. Rogers Jr., H.: Theory of Recursive Functions and Effective Computability. MacGraw Hill, New York (1967)zbMATHGoogle Scholar
  5. Intrigila, B., Statman, R.: The Omega Rule is \({\Pi}^{0}_{2}\)-Hard in the λβ-Calculus. LICS 2004, pp. 202–210. IEEE Computer Society Press, Los Alamitos (2004)Google Scholar
  6. Intrigila, B., Statman, R.: Some Results on Extensionality in Lambda Calculus. Annals of Pure and Applied Logic 132(2-3), 109–125 (2005)zbMATHCrossRefGoogle Scholar
  7. Intrigila, B., Statman, R.: Solution of a Problem of Barendregt on Sensible λ-Theories. Logical Methods in Computer Science, 2(4) (2006)Google Scholar
  8. Plotkin, G.: The λ-Calculus is ω-incomplete. J. Symbolic Logic, 39, 313–317.Google Scholar
  9. Schütte, K.: Proof Theory. Springer, Heidelberg (1977)zbMATHGoogle Scholar
  10. Statman, R.: Gentzen’s Notion of a Direct Proof. In: Barwise, K.J. (ed.) Handbook of Mathematical Logic, North-Holland, Amsterdam (1978)Google Scholar
  11. Statman, R.: Normal Varieties of Combinators. In: Moschovakis, Y.N. (ed.) Logic from Computer Science, pp. 585–596. Springer, Heidelberg (1992)Google Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Benedetto Intrigila
    • 1
  • Richard Statman
    • 2
  1. 1.Università di Roma “Tor Vergata”, RomeItaly
  2. 2.Carnegie-Mellon University, Pittsburgh, PAUSA

Personalised recommendations