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)


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).


Induction Hypothesis Head Part Lambda Calculus Recursive Tree Closed Term 
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  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

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