Skip to main content
SpringerLink
Log in

Effective longest and infinite reduction paths in untyped λ-calculi

  • Conference paper
  • First Online:
Book cover Trees in Algebra and Programming — CAAP '96 (CAAP 1996)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1059))

Included in the following conference series:

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. H.P. Barendregt. The Lambda Calculus: Its Syntax and Semantics. N.-H., 1984.

    Google Scholar 

  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. J.A. Bergstra and J.W. Klop. Church-Rosser strategies in the lambda calculus. Th. Comp. Sci., 9:27–38, 1979.

    Google Scholar 

  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. M. Bezem and J.F. Groote, editors. Typed Lambda Calculi and Applications, volume 664 of LNCS. S.-V., 1993.

    Google Scholar 

  6. A. Church and J.B. Rosser. Some properties of conversion. Transactions of the American Mathematical Society, 39:11–21, 1936.

    Google Scholar 

  7. H.B. Curry and R. Feys. Combinatory Logic. North-Holland, Amsterdam, 1958.

    Google Scholar 

  8. R.C. de Vrijer. A direct proof of the finite developments theorem. J. of Symbolic Logic, 50:339–343, 1985.

    Google Scholar 

  9. R.C. de Vrijer. Surjective Pairing and Strong Normalization: Two Themes in Lambda Calculus. PhD thesis, University of Amsterdam, 1987.

    Google Scholar 

  10. R.O. Gandy. Proofs of strong normalization. In Seldin and Hindley [20], pp. 457–477.

    Google Scholar 

  11. J. Hudelmaier. Bounds for cut elimmination in intuitionistic propositional logic. Archive for Mathematical Logic, 31:331–353, 1992.

    Google Scholar 

  12. B. Jacobs. Semantics of lambda-I and of other substructure calculi. In Bezem and Groote [5], pp. 195–208.

    Google Scholar 

  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. 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. J.W. Klop. Combinatory Reduction Systems. PhD thesis, Utrecht University, 1980. CWI Tract, Amsterdam.

    Google Scholar 

  16. J.-J. Lévy. Optimal reductions in the lambda-calculus. In Seldin and Hindley [20], pp. 159–191.

    Google Scholar 

  17. L. Regnier. Une équivalence sur le lambda-termes. Th. Comp. Sci., 126:281–292, 1994.

    Google Scholar 

  18. H. Rogers. Theory of Recursive Functions and Effective Computability. McGraw Hill, New York, 1967.

    Google Scholar 

  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. 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. 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. 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. F. van Raamsdonk and P. Severi. On normalisation. TR CS-R9545, CWI, 1995.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics & Informatics, Catholic University of Nijmegen (KUN), The Netherlands

    Morten Heine SØrensen

  2. Department of Computer Science, University of Copenhagen (DIKU), The Netherlands

    Morten Heine SØrensen

Authors
  1. Morten Heine SØrensen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Morten Heine SØrensen .

Editor information

Hélène Kirchner

Rights and permissions

Reprints and permissions

Copyright information

© 1996 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

SØrensen, M.H. (1996). Effective longest and infinite reduction paths in untyped λ-calculi. In: Kirchner, H. (eds) Trees in Algebra and Programming — CAAP '96. CAAP 1996. Lecture Notes in Computer Science, vol 1059. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61064-2_44

Download citation

  • DOI: https://doi.org/10.1007/3-540-61064-2_44

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-61064-9

  • Online ISBN: 978-3-540-49944-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation