Abstract
We show how higher-order rewriting may be encoded into first-order rewriting modulo an equational theory ε. We obtain a characterization of the class of higher-order rewriting systems which can be encoded by first-order rewriting modulo an empty theory (that is, ε = 0). This class includes of course the λ-calculus. Our technique does not rely on a particular substitution calculus but on a set of abstract properties to be verified by the substitution calculus used in the translation.
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References
M. Abadi, L. Cardelli, P.-L. Curien, and J.-J. Lévy. Explicit substitutions. Journal of Functional Programming, 4(1):375–416, 1991.
T. Arts and J. Giesl. Termination of term rewriting using dependency pairs. Theoretical Computer Science, 236:133–178, 2000.
L. Bachmair and N. Dershowitz. Critical pair criteria for completion. Journal of Symbolic Computation, 6(1):1–18, 1988.
Z. Benaissa, D. Briaud, P. Lescanne, and J. Rouyer-Degli. λv, a calculus of explicit substitutions which preserves strong normalisation. Journal of Functional Programming, 6(5):699–722, 1996.
E. Bonelli, D. Kesner, and A. Ríos. A de Bruijn notation for higher-order rewriting. In Proc. of the Eleventh Int. Conference on Rewriting Techniques and Applications, Norwich, UK, July 2000.
R. David and B. Guillaume. A λ-calculus with explicit weakening and explicit substitutions. Journal of Mathematical Structures in Computer Science, 2000. To appear.
N. Dershowitz. Orderings for term rewriting systems. Theoretical Computer Science, 17(3):279–301, 1982.
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.
G. Dowek, T. Hardin and C. Kirchner. Theorem proving modulo. Technical Report RR 3400, INRIA, 1998.
G. Dowek, T. Hardin, C. Kirchner and F. Pfenning. Unification via Explicit Substitutions: The Case of Higher-Order Patterns. Technical Report RR3591. INRIA. 1998.
G. Dowek, T. Hardin and C. Kirchner. Higher-order unification via explicit substitutions. Information and Computation, 157:183–235, 2000.
G. Dowek, T. Hardin and C. Kirchner. Hol-lambda-sigma: an intentional first-order expression of higher-order logic. Mathematical Structures in Computer Science, 11:1–25, 2001.
T. Hardin and J-J. Lévy. A confluent calculus of substitutions. In France-Japan Artificial Intelligence and Computer Science Symposium, 1989.
J-P. Jouannaud and A. Rubio. The higher-order recursive path ordering. In Fourteenth Annual IEEE Symposium on Logic in Computer Science, Trento, Italy, 1999.
F. Kamareddine and A. Ríos. A λ-calculus à la de Bruijn with explicit substitutions. In Proc. of the Int. Symposium on Programming Language Implementation and Logic Programming (PLILP), LNCS 982. 1995.
S. Kamin and J. J. Lévy. Attempts for generalizing the recursive path orderings. University of Illinois, 1980.
D. Kesner. Confluence of extensional and non-extensional lambda-calculi with explicit substitutions. Theoretical Computer Science, 238(1-2):183–220, 2000.
Z. Khasidashvili. Expression Reduction Systems. In Proc. of I. Vekua Institute of Applied Mathematics, volume 36, Tbilisi, 1990.
Z. Khasidashvili and M. Ogawa and V. van Oostrom. Uniform Normalization beyond Orthogonality. Submitted for publication, 2000.
P. Lescanne and J. Rouyer-Degli. Explicit substitutions with de Bruijn levels. In Proc. of the Sixth Int. Conference on Rewriting Techniques and Applications, LNCS 914, 1995.
P-A. Melliès. Description Abstraite des Systèmes de Réécriture. Thèse de l’Université Paris VII, December 1996.
C. Muñoz. A left-linear variant of λσ. In Michael Hanus, J. Heering, and K. (Karl) Meinke, editors, Proc. of the 6th Int. Conference on Algebraic and Logic Programming (ALP’97), LNCS 1298. 1997.
T. Nipkow. Higher-order critical pairs. in Proc. of the Sixth Annual IEEE Symposium on Logic in Computer Science, 1991.
B. Pagano. Des Calculs de Susbtitution Explicite et leur application à la compilation des langages fonctionnels. PhD thesis, Université Paris VI, 1998.
R. Pollack. Closure under alpha-conversion. In Henk Barendregt and Tobias Nipkow, editors, Types for Proofs and Programs (TYPES), LNCS 806. 1993.
H. Zantema. Termination of Term Rewriting by Semantic Labelling. Fundamenta Informaticae, volume 24, pages 89–105, 1995.
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Bonelli, E., Kesner, D., Ríos, A. (2001). From Higher-Order to First-Order Rewriting. In: Middeldorp, A. (eds) Rewriting Techniques and Applications. RTA 2001. Lecture Notes in Computer Science, vol 2051. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45127-7_6
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DOI: https://doi.org/10.1007/3-540-45127-7_6
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