Effective longest and infinite reduction paths in untyped λ-calculi

  • Morten Heine SØrensen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1059)


A maximal reduction strategy in untyped λ-calculus computes for a term a longest (finite or infinite) reduction path. Some types of reduction strategies in untyped λ-calculus have been studied, but maximal strategies have received less attention. We give a systematic study of maximal strategies, recasting the few known results in our framework and giving a number of new results, the most important of which is an effective maximal strategy in λΒη. We also present a number of applications illustrating the relevance and usefulness of maximal strategies.


λ-calculus reduction strategies effectiveness 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    H.P. Barendregt. The Lambda Calculus: Its Syntax and Semantics. N.-H., 1984.Google Scholar
  2. 2.
    H.P. Barendregt, J. Bergstra, J.W. Klop, and H. Volken. Degrees, reductions and representability in the lambda calculus. TR Preprint 22, University of Utrecht, Department of Mathematics, 1976.Google Scholar
  3. 3.
    J.A. Bergstra and J.W. Klop. Church-Rosser strategies in the lambda calculus. Th. Comp. Sci., 9:27–38, 1979.Google Scholar
  4. 4.
    J.A. Bergstra and J.W. Klop. Strong normalization and perpetual reductions in the lambda calculus. J. of Information Processing and Cybernetics, 18:403–417, 1982.Google Scholar
  5. 5.
    M. Bezem and J.F. Groote, editors. Typed Lambda Calculi and Applications, volume 664 of LNCS. S.-V., 1993.Google Scholar
  6. 6.
    A. Church and J.B. Rosser. Some properties of conversion. Transactions of the American Mathematical Society, 39:11–21, 1936.Google Scholar
  7. 7.
    H.B. Curry and R. Feys. Combinatory Logic. North-Holland, Amsterdam, 1958.Google Scholar
  8. 8.
    R.C. de Vrijer. A direct proof of the finite developments theorem. J. of Symbolic Logic, 50:339–343, 1985.Google Scholar
  9. 9.
    R.C. de Vrijer. Surjective Pairing and Strong Normalization: Two Themes in Lambda Calculus. PhD thesis, University of Amsterdam, 1987.Google Scholar
  10. 10.
    R.O. Gandy. Proofs of strong normalization. In Seldin and Hindley [20], pp. 457–477.Google Scholar
  11. 11.
    J. Hudelmaier. Bounds for cut elimmination in intuitionistic propositional logic. Archive for Mathematical Logic, 31:331–353, 1992.Google Scholar
  12. 12.
    B. Jacobs. Semantics of lambda-I and of other substructure calculi. In Bezem and Groote [5], pp. 195–208.Google Scholar
  13. 13.
    Z. Khasidashvili. The longest perpetual reductions in orthogonal expression reduction systems. In A. Nerode and Yu. V. Matiyasevich, editors, Logical Foundations of Computer Science, volume 813 of LNCS, pp. 191–203. S.-V., 1994.Google Scholar
  14. 14.
    Z. Khasidashvili. Perpetuality and strong normalization in orthogonal term rewriting systems. In P. Enjalbert, et al., editors, 11th Annual Symposium on Theoretical Aspects of Computer Science, volume 775 of LNCS, pp. 163–174. S.-V., 1994.Google Scholar
  15. 15.
    J.W. Klop. Combinatory Reduction Systems. PhD thesis, Utrecht University, 1980. CWI Tract, Amsterdam.Google Scholar
  16. 16.
    J.-J. Lévy. Optimal reductions in the lambda-calculus. In Seldin and Hindley [20], pp. 159–191.Google Scholar
  17. 17.
    L. Regnier. Une équivalence sur le lambda-termes. Th. Comp. Sci., 126:281–292, 1994.Google Scholar
  18. 18.
    H. Rogers. Theory of Recursive Functions and Effective Computability. McGraw Hill, New York, 1967.Google Scholar
  19. 19.
    H. Schwichtenberg. An upper bound for reduction sequences in the typed lambda-calculus. Archive for Mathematical Logic, 30:405–408, 1991.Google Scholar
  20. 20.
    J.P. Seldin and J.R. Hindley, editors. To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism. Academic Press Limited, 1980.Google Scholar
  21. 21.
    M.H. SØrensen. Properties of infinite reduction paths in untyped λ-calculus. In Proc. of the Tbilisi Symp. on Logic, Language, and Computation, 1995. To appear.Google Scholar
  22. 22.
    J. Springintveld. Lower and upper bounds for reductions of types in 301-01 and λP. In Bezem and Groote [5], pp. 391–405.Google Scholar
  23. 23.
    F. van Raamsdonk and P. Severi. On normalisation. TR CS-R9545, CWI, 1995.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  1. 1.Faculty of Mathematics & InformaticsCatholic University of Nijmegen (KUN)The Netherlands
  2. 2.Department of Computer ScienceUniversity of Copenhagen (DIKU)The Netherlands

Personalised recommendations