Abstract
We present a new proof of confluence of the untyped lambda calculus by reducing the confluence of β-reduction in the untyped lambda calculus to the confluence of β-reduction in the simply typed lambda calculus. This is achieved by embedding typed lambda terms into simply typed lambda terms. Using this embedding, an auxiliary reduction, and β-reduction on simply typed lambda terms we define a new reduction on all lambda terms. The transitive closure of the reduction defined is β-reduction on all lambda terms. This embedding allows us to use the confluence of β-reduction on simply typed lambda terms and thus prove the confluence of the reduction defined. As a consequence we obtain the confluence of β-reduction in the untyped lambda calculus.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Barendregt, H. P.: The Lambda Calculus-Its Syntax and Semantics. North-Holland, Amsterdam (1984).
Barendregt, H. P.: Lambda calculi with types. In: Abramsky, S., Gabbay, D. M. and T. S. E. Maibaum (eds.): Handbook of Logic in Computer Science, Vol. 2. Oxford University Press, Oxford (1992) 117–309.
Dershowitz, N. and J. P. Jounnaud: Rewrite Systems. In: Leeuwen, J. (ed.): Handbook of Theoretical Computer Science, Elsevier Science Publishers B. V. (1990).
Ghilezan, S.: Application of typed lambda calculi in the untyped lambda calculus. In: Nerode, A. and Yu. Matiyasevich (eds.): Logical Foundations of Computer Science’ 94. Lecture Notes in Computer Science 813, Springer-Verlag, Berlin (1994) 129–139.
Ghilezan, S.: Generalized finiteness of developments in typed lambda calculi. Journal of Automata, Languages and Combinatorics 4 (1996) 247–257.
Klop, J. W., V. van Oostrom, and R. de Vrijer: A geometric proof of confluence by decreasing diagrams. Journal of Logic and Computation 10(3) (2000) 437–460.
Koletsos, G.: Church-Rosser theorem for typed functionals. Journal of Symbolic Logic 50 (1985) 782–790.
Koletsos, G. and G. Stavrinos: Church-Rosser theorem for conjunctive type systems. In: Kakas, A. K. and A. Sinachopoulos (eds.): Proceedings of the First Panhellenic Logic Symposium. Nicosia (1997) 25–37.
Krivine, J. L.: Lambda-calcul types et modèles. Masson, Paris (1990)
Meyer, A. M.: What is a model of lambda calculus? Information and control 122 (1982) 52–87.
Mitchell, J.: Type Systems for Programming Languages. In: Leeuwen, J. (ed.): Handbook of Theoretical Computer Science, Elsevier Science Publishers B. V. (1990) 365–458.
Newman, M. H. A.: On theories with a combinatorial definition of ‘equivalence’. Annals of Mathematics 43 (1942) 223–243.
Pfenning, F.: A Proof of the Church-Rosser theorem and its representation in a Logical Framework, CMU-CS-92-186, (September 1992), forthocoming in Journal of Authomated Reasoning.
Scott, D.: Relating theories of the lambda calculus. In: Seldin, J. P. and J. R. Hindley: To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, Academic Press, London (1980) 403–450.
Statman, R.: Logical relations and the simply typed lambda calculus. Information and Control 65 (1985) 85–97.
Takahashi, M.: Parallel Reductions in λ-Calculus. Journal of Symbolic Computation 7 (1989) 113–123.
van Oostrom, V.: Confluence by decreasing diagrams. Theoretical Computer Science 126 (1994) 259–280.
van Oostrom, V. and F. van Raamsdonk: Weak orthogonality implies confluence: the higher order case. In: Nerode, A. and Yu. Matiyasevich (eds.): Logical Foundations of Computer Science’ 94. Lecture Notes in Computer Science 813, Springer-Verlag, Berlin (1994) 379–392.
Wadsworth, C. P.: The relation between computational and denotational properties for Scott’s D∞-models of the lambda calculus. SIAM Journal of Computing 5(3) (1976) 488–521.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ghilezan, S., Kunčak, V. (2001). Confluence of Untyped Lambda Calculus via Simple Types. In: Theoretical Computer Science. ICTCS 2001. Lecture Notes in Computer Science, vol 2202. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45446-2_3
Download citation
DOI: https://doi.org/10.1007/3-540-45446-2_3
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-42672-1
Online ISBN: 978-3-540-45446-5
eBook Packages: Springer Book Archive