Abstract
Explicit substitutions (ES)were introduced as a bridge between the theory of rewrite systems with binders and substitution,such as the λ -calculus,and their implementation.In a seminal paper P.- A.Melliès observed that the dynamical properties of a rewrite system and its ES-based implementation may not coincide:he showed that a strongly normalising term (i.e. one which does not admit in finite derivations)in the λ -calculus may lose this status in its ES-based implementation.This paper studies normalisation for the latter systems in the general setting of higher-order rewriting:Based on recent work extending the theory of needed strategies to non-orthogonal rewrite systems we show that needed strategies normalise in the ES-based implementation of any orthogonal pattern higher-order rewrite system.
Variables in terms are represented as positive integers.Eg.λx.x is represented as λ 1, λx.λy.x as λλ2,and λx.y as λ2.
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References
M. Abadi, L. Cardelli, P-L. Curien,and J-J. Lévy.Explicit substitutions. Journal of Functional Programming 4(1):375–16,1991.
H.P. Barendregt.The Lambda Calculus: its Syntax and Semantics. Studies in Logic and the Foundations of Mathematics 103.North-Holland, Amsterdam,revised edition,1984.
Z. Benaissa, D. Briaud, P. Lescanne,and J. Rouyer-Degli.λν,a calculus of explicit substitutions which preserves strong normalisation.Journal of Functional Programming 6(5):699–22,1996.
E. Bonelli, D. Kesner,and A. Ríos.A de Bruijn notation for Higher-Order Rewriting.In Proceedings of the 11th RTA number 1833 in LNCS. Springer-Verlag,2000.
E. Bonelli, D. Kesner,and A.Ríos.From Higher-Order to First-Order Rewriting.In Proceedings of the 12th RTA number 2051 in LNCS. Springer-Verlag,2001.
F. Baader and T. Nipkow.Term Rewriting and All That. Cambridge University Press,1998.
E. Bonelli.Term rewriting and explicit substitutions. PhD thesis,Université de Paris Sud,November 2001.
E. Bonelli.A normalisation result for higher-order calculi with explicit substitutions,2003.Full version of this paper. http://www-lifia.info.unlp.edu.ar/~eduardo/.
G. Boudol.Computational semantics of term rewrite systems.In M. Nivat and J.C. Reynolds,editors,Algebraic methods in Semantics Cambridge University Press,1985.
P-L. Curien, T. Hardin,and A. Ríos.Strong normalization of substitutions.In Proceedings of Mathematical Foundations of Computer Science, number 629 in LNCS,pages 209–17.Springer-Verlag,1992.
N.G. de Bruijn.Lambda calculus notation with nameless dummies,a tool for automatic formula manipulation with application to the church-rosser theorem.Indag. Mat., 5(35),1972.
R. David and B. Guillaume.A λ-calculus with explicit weakening and explicit substitutions.Mathematical Structures in Computer Science 11(1), 2001.
N. Dershowitz and J-P. Jouannaud.Rewrite systems.In J. van Leeuwen, editor,Handbook of Theoretical Computer Science volume B,pages 243–309.North-Holland,1990.
J. Glauert, R. Kennaway,and Z. Khasidashvili.Stable results and relative normalization.Journal of Logic and Computation, 10(3),2000.
G. Huet and J-J. Lévy.Computations in orthogonal rewriting systems. In J.L. Lassez and G.D. Plotkin,editors,Computational Logic; Essays in honor of Alan Robinson pages 394–43.MIT Press,1991.
Th. Hardin, L. Maranget, and P. Pagano.Functional back-ends within the lambda-sigma calculus.In Proceedings of the International Conference on Functional Programming LNCS.Springer-Verlag,1996.
J.W. Klop.Combinatory Reduction Systems. PhD thesis,CWI,Amsterdam, 1980.Mathematical Centre Tracts n.127.
J.W. Klop.Term rewriting systems.Handbook of Logic in Computer Science 2:1–16,1992.
J.W. Klop, V. van Oostrom,and F. van Raamsdonk.Combinatory Reduction Systems:introduction and survey.Theoretical Computer Science, 121(1–2):279–08,1993.
F. Kamareddine and A. Ríos. A λ–calculus à la de Bruijn with explicit substitutions.In Proceedings of the International Symposium on Programming Language Implementation and Logic Programming (PLILP), number982 in LNCS.Springer Verlag,1995.
J-J. Lévy.Réductions correctes et optimales dans le lambda-calcul PhD thesis,Université Paris VII,1978.
L. Maranget.La stratégie paresseuse PhD thesis,Université Paris VII, 1992.
P-A. Melliès.Typed λ–calculi with explicit substitutions may not terminate.In Proceedings of Typed Lambda Calculi and Applications, number 902in LNCS.Springer-Verlag,1995.
P-A. Melliès.Description Abstraite des Systèmes de Réécriture. PhD thesis,Université Paris VII,1996.
P-A. Melliès.Axiomatic Rewriting Theory II:The λσ-calculus enjoys finite normalisation cones.Journal of Logic and Computation 10(3):461–487,2000.
D. Miller.A logic programming language with lambda-abstraction,function variables,and simple unification.In P. Schroeder-Heister,editor, Proceedings of the International Workshop on Extensions of Logic Pro-gramming, FRG, 1989 number 475 in Lecture Notes in Artificial Intelligence.Springer-Verlag,December 1991.
T. Nipkow.Higher-order critical pairs.In Proceedings of the Sixth Annual IEEE Symposium on Logic in Computer Science, IEEE Computer Society Press,July 1991.
V.van Oostrom.Confluence for Abstract and Higher-order Rewriting PhD thesis,Vrije University,1994.
V.van Oostrom. Normalization in weakly orthogonal rewriting.In Pro-ceedings of the 10th International Conference on Rewriting Techniques and Applications, number 1631 in LNCS.Springer-Verlag,1999.
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Bonelli, E. (2003). A Normalisation Result for Higher-Order Calculi with Explicit Substitutions. In: Gordon, A.D. (eds) Foundations of Software Science and Computation Structures. FoSSaCS 2003. Lecture Notes in Computer Science, vol 2620. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36576-1_10
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