Abstract
We propose a calculus of explicit substitutions with de Bruijn indices for implementing objects and functions which is confluent and preserves strong normalization. We start from Abadi and Cardelli’s ς-calculus [1] for the object calculus and from the λν-calculus [20] for the functional calculus. The de Bruijn setting poses problems when encoding the λν-calculus within the ς-calculus following the style proposed in [1]. We introduce fields as a primitive construct in the target calculus in order to deal with these difficulties. The solution obtained greatly simplifies the one proposed in [17] in a named variable setting. We also eliminate the conditional rules present in the latter calculus obtaining in this way a full non-conditional first order system.
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
M. Abadi and L. Cardelli. A Theory of Objects. Springer Verlag, 1996.
M. Abadi, L. Cardelli, P.-L. Curien, and J.-J. Lévy. Explicit Substitutions. Journal of Functional Programming, 4(1):375–416, 1991.
R. Bloo. Preservation of Termination for Explicit Substitution. PhD. Thesis, Eindhoven University, 1997.
R. Bloo and H. Geuvers. Explicit Substitution: on the edge of strong normalization. Theoretical Computer Science, 204, 1998.
R. Bloo, K. Rose. Combinatory Reduction Systems with explicit substitution that preserve strong normalization. In RTA’96, LNCS 1103, 1996.
E. Bonelli. Using fields and explicit substitutions to implement objects and functions in a de Bruijn setting. Full version obtainable by ftp at ftp://ftp.lri.fr/LRI/articles/bonelli/objectdbfull.ps.gz.
D. Briaud. An explicit eta rewrite rule. In M. Dezani Ed., Int. Conference on Typed Lambda Calculus and Applications, LNCS vol. 902, 1995.
N.G. de Bruijn. Lambda calculus notation with nameless dummies, a tool for automatic formual manipulation with application to the Church-Rosser theorem. Koninklijke Nederlandse Akademie van Wetenschappen, Series A, Mathematical Sciences, 75:381–392, 1972.
P.-L. Curien, T. Hardin, and J.-J. Lévy. Confluence properties of weak and strong calculi of explicit substitutions. Technical Report, Centre d’Etudes et de Recherche en Informatique, CNAM, 1991.
N. Dershowitz. Orderings for term rewriting systems. Theoretical Computer Science, 17(3):279–301, 1982.
M. Ferreira, D. Kesner. and L. Puel. Lambda-calculi with explicit substitutions and composition which preserve beta-strong normalization. Proceedings of the 5th International Conference on Algebraic and Logic Programming (ALP’96), LNCS 1139, 1996.
T. Hardin. Résultats de confluence pour les règles fortes de la logique combinatoire catégorique et liens avec les lambda-calculs. Thèse de doctorat, Université de Paris VII, 1987.
T. Hardin and J-J. Lévy. A confluent calculus of substitutions. In France-Japan Artificial Intelligence and Computer Science Symposium, 1989.
F. Kamareddine and A. Ríos. A lambda calculus a la de Bruijn with Explicit Substitutions. Proceedings of the 7th International Symposium on Programming Languages: Implementations, Logics and Programs (PLILP’95), LNCS 982, 1995.
F. Kamareddine and A. Ríos. Extending a λ-calculus with Explicit Substitutions which Preserves Strong Normalization into a Confluent Calculus on Open Terms. In Journal of Functional Programming, Vol.7 No.4, 1997.
D. Kesner. Confluence properties of extensional and non-extensional lambda-calculi with explicit substitutions. Proceedings of the 7th International Conference on Rewriting Techniques and Applications (RTA’96), LNCS 1103, 1996.
D. Kesner, P.E. Martínez López. Explicit Substitutions for Objects and Functions. Proceedings of the Joint International Symposiums: Programming Languages, Implementations, Logics and Program (PLILP’98) and Algebraic and Logic Programming (ALP), LNCS 1490,pp. 195–212, Sept. 1998.
J.W. Klop. Combinatory Reduction Systems. PhD Thesis, University of Utrecht, 1980.
J.W. Klop. Term Rewriting Systems. In S. Abramsky, D. Gabbay, and T. Maibaum, editors, Handbook of Logic in Computer Science, Volume II, Oxford University Press, 1992.
P. Lescanne. From λ σto λ ν, a journey through calculi of explicit substitutions. In Ann. ACM Symp. on Principles of Programming Languages (POPL), pp. 60–69. ACM, 1994.
P. Lescanne and J. Rouyer Degli. Explicit substitutions with de Bruijn’s levels. In P. Lescanne editor, RTA’95, LNCS 914, 1995.
P.A. Melliès. Typed λ-calculi with explicit substitutions may not terminate. In TLCA’95, LNCS 902, 1995.
C.A. Muñoz. Confluence and Preservation of Strong Normalisation in an Explicit Substitutions Calculus. Proceedings of the Eleven Annual IEEE Symposium on Logic in Computer Science, 1996.
B. Pagano. Des calculs de substitution explicit et leur application à la compilation des langages fonctionnels. Thèse de doctorat, Université de Paris VII, 1998.
A. Ríos. Contribution à l’étude des λ-calculs avec substitutions explicites. Ph.D Thesis, Université de Paris VII, 1993.
K. Rose. Explicit Cyclic Substitutions. In CTRS’92, LNCS 656, 1992.
M. Takahashi. Parallel reduction in the λ-calculus. Journal of Symbolic Computation, 7:113–123, 1989.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bonelli, E. (1999). Using Fields and Explicit Substitutions to Implement Objects and Functions in a de Bruijn Setting. In: Flum, J., Rodriguez-Artalejo, M. (eds) Computer Science Logic. CSL 1999. Lecture Notes in Computer Science, vol 1683. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48168-0_15
Download citation
DOI: https://doi.org/10.1007/3-540-48168-0_15
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-66536-6
Online ISBN: 978-3-540-48168-3
eBook Packages: Springer Book Archive