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Confluence of extensional and non-extensional \(\lambda\)-calculi with explicit substitutions. (English) Zbl 0944.68033
Summary: This paper studies confluence of extensional and non-extensional \(\lambda\)-calculi with explicit substitutions, where extensionality is interpreted by \(\eta\)-expansion. For that, we propose a scheme for explicit substitutions which describes those abstract properties that are sufficient to guarantee confluence. Our method makes it possible to treat at the same time many well-known calculi such as \(\lambda_{\sigma},\lambda_{\sigma \Uparrow},\lambda_{\varphi},\lambda_{s},\lambda_{v},\lambda_{f},\lambda_{d}\) and \(\lambda_{d}n\).

MSC:
68N18 Functional programming and lambda calculus
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[1] Abadi, M.; Cardelli, L.; Curien, P.L.; Lévy, J.-J., Explicit substitutions, J. funct. programming, 4, 1, 375-416, (1991) · Zbl 0941.68542
[2] Y. Akama, On Mints’ reductions for ccc-Calculus, in: Proc. Int. Conf. on Typed Lambda Calculi and Applications (TLCA), Lecture Notes in Computer Science, vol. 664, Springer, Berlin, 1993. · Zbl 0786.03006
[3] Benaissa, Z.-E.-A.; Briaud, D.; Lescanne, P.; Rouyer-Degli, J., \(λυ\), a calculus of explicit substitutions which preserves strong normalisation, J. funct. programming, 6, 5, 699-722, (1996) · Zbl 0873.68108
[4] Bloo, R.; Rose, K., Combinatory reduction systems with explicit substitution that preserve strong normalisation, ()
[5] Briaud, D., An explicit eta rewrite rule, () · Zbl 1063.68552
[6] P.-L. Curien, Combinateurs Catégoriques, algorithmes séquentiels et programmation applicative, Thèse d’etat, Université Paris VII, 1983.
[7] P.-L. Curien, Categorical combinators, sequential algorithms and functional programming, Progress in Theoretical Computer Science, 1st ed., Birkhaüser, Basel, 1986. · Zbl 0643.68004
[8] P.-L. Curien, R. Di Cosmo, A confluent reduction system for the \(λ\)-calculus with surjective pairing and terminal object, in: Leach, Monien, Artalejo (Eds.), Internat. Conf. on Automata, Languages and Programming (ICALP), Lecture Notes in Computer Science, vol. 510, Springer, Berlin, 1991, pp. 291-302. · Zbl 0769.68056
[9] P.-L. Curien, T. Hardin, J.-J. Lévy, Confluence Properties of Weak and Strong Calculi of Explicit substitutions, Technical Report 1617, INRIA-Rocquencourt, 1992.
[10] P.-L. Curien, T. Hardin, A. Rı́os, Strong normalisation of Substitutions, MFCS’92, Lecture Notes in Computer Science, vol. 629, Springer, Berlin, 1992, pp. 209-218.
[11] de Bruijn, N., Lambda-calculus notation with nameless dummies, a tool for automatic formula manipulation, with application to the church-rosser theorem, Indag. mat., 5, 35, 381-392, (1972) · Zbl 0253.68007
[12] de Bruijn, N., Lambda-calculus notation with namefree formulas involving symbols that represent reference transforming mappings, Indag. mat., 40, 356-384, (1978) · Zbl 0393.03009
[13] Di Cosmo, R.; Kesner, D., A confluent reduction for the extensional typed \(λ\)-calculus with pairs, sums, recursion and terminal object, () · Zbl 1422.03022
[14] R. Di Cosmo, D. Kesner, Combining first order algebraic rewriting systems, recursion and extensional typed lambda calculi, Internat Conf. on Automata, Languages and Programming (ICALP), Lecture Notes in Computer Science, vol. 820, Springer, Berlin, 1994. · Zbl 0874.68158
[15] Di Cosmo, R.; Kesner, D., Rewriting with extensional polymorphic \(λ\)-calculus (extended abstract), ()
[16] D. Dougherty, Some lambda calculi with categorical sums and products, Proc. 5th Int. Conf. on Rewriting Techniques and Applications (RTA), Lecture Notes in Computer Science, vol. 690, Springer, Berlin, 1993.
[17] G. Dowek, T. Hardin, C. Kirchner, Higher-Order Unification via explicit substitutions, Proc. Symp. on Logic in Computer Science (LICS), 1995. · Zbl 1005.03016
[18] T. Ehrhard, Une sémantique catégorique des types dépendants. Application au calcul des constructions, Thèse de doctorat, Université de Paris VII, 1988.
[19] Ferreira, M.C.F.; Kesner, D.; Puel, L., \(λ\)-calculi with explicit substitutions and composition which preserve \(β\)-strong normalization (extended abstract), (), 284-298 · Zbl 1355.68039
[20] 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.
[21] T. Hardin, \(η\)-reduction for explicit substitutions, Algebraic and Logic Programming’92, Lecture Notes in Computer Science, vol. 632, Springer, Berlin, 1992.
[22] Hardin, T., Eta-conversion for the languages of explicit substitutions, Applicable alg. eng. commun. comput., 5, 317-341, (1994) · Zbl 0810.03010
[23] T. Hardin, A. Laville, Proof of termination of the rewriting system subst on c.c.l., Theoret. Comput. Sci., 1986. · Zbl 0618.68031
[24] T. Hardin, J.-J. Lévy, A confluent calculus of substitutions, France-Japan Artificial Intelligence and Computer Science Symp., 1989.
[25] T. Hardin, L. Maranget, B. Pagano, Functional back-ends and compilers within the lambda-sigma calculus, 1995, Draft. · Zbl 1345.68056
[26] G. Huet, Résolution d’équations dans les langages d’ordre \(1,2,…,ω\), Thèse de Doctorat d’état, Université Paris VII, 1976.
[27] Jay, C.B.; Ghani, N., The virtues of eta-expansion, J. funct. programming, 5, 2, 135-154, (1995) · Zbl 0833.68072
[28] F. Kamareddine, A. Rı́os, A \(λ\)-calculus à la de Bruijn with explicit substitutions, Proc. Int. Symp. on Programming Language Implementation and Logic Programming, Lecture Notes in Computer Science, vol. 982, Springer, Berlin, 1995.
[29] Kesner, D., Confluence properties of extensional and non-extensional \(λ\)-calculi with explicit substitutions, (), 184-199 · Zbl 0944.68033
[30] D. Kesner, Confluence of extensional and non-extensional \(λ\)-calculi with explicit substitutions, Technical Report 1103, LRI, Université Paris-Sud, 1997. · Zbl 0944.68033
[31] D. Kesner, Reasoning about redundant patterns, J. Funct. Logic Programming, 1997, to appear. · Zbl 0924.68047
[32] J.W. Klop, Combinatory Reduction Systems, Mathematical Centre Tracts, vol. 127, CWI, Amsterdam, 1980, Ph.D. Thesis. · Zbl 0466.03006
[33] Klop, J.W.; van Oostrom, V.; van Raamsdonk, F., Combinatory reduction systems: introduction and survey, Theoret. comput. sci., 121, 279-308, (1993) · Zbl 0796.03024
[34] P. Lescanne, From \(λσ\) to \(λv\), a journey through calculi of explicit substitutions, in: Ann. ACM Symp. on Principles of Programming Languages (POPL), ACM, 1994, pp. 60-69.
[35] P. Lescanne, Personal Communication, 1996.
[36] P. Lescanne, J. Rouyer-Degli, The calculus of explicit substitutions \(λν\), Technical Report, INRIA, Lorraine, 1994. · Zbl 0873.68108
[37] M. Mauny, Compilation des langages fonctionnels dans les combinateurs catégoriques. Applications au langage ML, Thèse 3em̀e cycle, Université Paris VII, 1985.
[38] P.-A. Mellies, Four typed-lambda calculi with explicit substitutions may not terminate: the first examples, 1994, Draft.
[39] Mellies, P.-A., Typed \(λ\)-calculi with explicit substitutions may not terminate, () · Zbl 1063.03522
[40] Mints, G., Closed categories and the theory of proofs, Zapiski nauchnykh seminarov leningradskogo otdeleniya matematicheskogo instituta im. V.A. steklova AN SSSR, 68, 83-114, (1977)
[41] G. Mints, Teorija categorii i teoria dokazatelstv.I, Aktualnye problemy logiki i metodologii nauky, 1979, pp. 252-278.
[42] Muñoz, C., A left-linear variant of \(λσ\), () · Zbl 0884.03012
[43] B. Pagano, Des calculs de substitution explicite et de leur application à la compilation des langages fonctionnels, Thèse de doctorat, Université Pierre et Marie Curie, 1998.
[44] G. Pottinger, The Church Rosser theorem for the typed lambda-calculus with Surjective pairing, Notre Dame J. Formal Logic 22 (3) (1981) pp. 264-268. · Zbl 0476.03026
[45] D. Prawitz, Ideas and results in proof theory, Proc. 2nd Scandinavian Logic Symp., 1971, pp. 235-307. · Zbl 0226.02031
[46] A. Rı́os, Contribution à l’étude des \(λ\)-calculus avec substitutions explicites, Thèse de doctorat, Université de Paris VII, 1993.
[47] V. van Oostrom, F. van Raamsdonk, Weak orthogonality implies confluence: the higher-order case, Proc. 3rd Internat. Symp. on Logical Foundations of Computer Science, 1994, pp. 379-392. · Zbl 0964.68523
[48] Zantema, H., Termination on term rewriting: interpretation and type elimination, J. symbolic comput., 1, 17, 23-50, (1994) · Zbl 0810.68087
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