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10.5555/2168279.2168293guidebooksArticle/Chapter ViewAbstractPublication PagesBookacm-pubtype
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Higher-Order rewriting: framework, confluence and termination

Published: 01 January 2005 Publication History

Abstract

Equations are ubiquitous in mathematics and in computer science as well. This first sentence of a survey on first-order rewriting borrowed again and again characterizes best the fundamental reason why rewriting, as a technology for processing equations, is so important in our discipline [10]. Here, we consider higher-order rewriting, that is, rewriting higher-order functional expressions at higher-types. Higher-order rewriting is a useful generalization of first-order rewriting: by rewriting higher-order functional expressions, one can process abstract syntax as done for example in program verification with the prover Isabelle [27]; by rewriting expressions at higher-types, one can implement complex recursion schemas in proof assistants like Coq [12].

References

[1]
Franco Barbanera, Maribel Fernández, and Herman Geuvers. Modularity of strong normalization and confluence in the algebraic-?-cube. In Proc. of the 9th Symp. on Logic in Computer Science, IEEE Computer Society, 1994.
[2]
Henk Barendregt. Handbook of Logic in Computer Science, chapter Typed lambda calculi. Oxford Univ. Press, 1993. eds. Abramsky et al.
[3]
Terese. Term Rewriting Systems. Cambridge Tracts in Theoretical Computer Science 55, Marc Bezem, Jan Willem Klop and Roel de Vrijer eds., Cambridge University Press, 2003.
[4]
Frédéric Blanqui, Jean-Pierre Jouannaud, and Mitsuhiro Okada. The Calculus of Algebraic Constructions. In Narendran and Rusinowitch, Proc. RTA, LNCS 1631, 1999.
[5]
Frédéric Blanqui. Inductive Types Revisited. Available from the web.
[6]
Frédéric Blanqui. Definitions by rewriting in the Calculus of Constructions, 2003. To appear in Mathematical Structures in Computer Science.
[7]
Frédéric Blanqui, Jean-Pierre Jouannaud, and Mitsuhiro Okada. Inductive Data Types. Theoretical Computer Science 277, 2001.
[8]
Jacek Chrzaczsz and DariaWalukiewicz-Chrzaczsz. Consistency and Completeness of Rewriting in the Calculus of Constructions. Draft.
[9]
Nachum Dershowitz. Orderings for term rewriting systems. Theoretical Computer Science, 17(3):279-301, March 1982.
[10]
Nachum Dershowitz and Jean-Pierre Jouannaud. Rewrite systems. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, volume B, pages 243- 309. North-Holland, 1990.
[11]
Gilles Dowek. Higher-Order Unification and Matching. Handbook of Automated Reasonning, A. Voronkov ed., vol 2, pages 1009-1062.
[12]
Gilles Dowek, Amy Felty, Hugo Herbelin, Gérard Huet, Christine Paulin-Mohring, and Benjamin Werner. The Coq proof assistant user's guide version 5.6. INRIA Rocquencourt and ENS Lyon.
[13]
Gilles Dowek, Thérèse Hardin, Claude Kichner and Franck Pfenning. Unification via explicit substitutions: The case of Higher-Order Patterns. In JICSLP:259-273, 1996.
[14]
Jean-Yves Girard, Yves Lafont, and Patrick Taylor. Proofs and Types. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989.
[15]
Gérard Huet. Confluent reductions: abstract properties and applications to term rewriting systems. In Journal of the ACM 27:4(797-821), 1980.
[16]
Jean-Pierre Jouannaud and Hélène Kirchner. Completion of a Set of Rules Modulo a Set of Equations. In Siam Journal of Computing 15:4(1155-1194), 1984.
[17]
Jean-Pierre Jouannaud and Mitsuhiro Okada. Abstract data type systems. Theoretical Computer Science, 173(2):349-391, February 1997.
[18]
Jean-Pierre Jouannaud and Mitsuhiro Okada. Higher-Order Algebraic Specifications. In Annual IEEE Symposium on Logic in Computer Science, Amsterdam, The Netherlands, 1991. IEEE Comp. Soc. Press.
[19]
Jean-Pierre Jouannaud and Albert Rubio. The higher-order recursive path ordering. In Giuseppe Longo, editor, Fourteenth Annual IEEE Symposium on Logic in Computer Science, Trento, Italy, July 1999. IEEE Comp. Soc. Press.
[20]
Jean-Pierre Jouannaud and Albert Rubio. Higher-order recursive path orderings. Available from the web.
[21]
Jean-Pierre Jouannaud and Albert Rubio. Higher-order orderings for normal rewriting. Available from the web.
[22]
Jean-Pierre Jouannaud and Albert Rubio and Femke van Raamsdonk. Higherorder Rewriting with Types and Arities. Available from the web.
[23]
Jan Willem Klop. Combinatory Reduction Systems. Mathematical Centre Tracts 127. Mathematisch Centrum, Amsterdam, 1980.
[24]
Richard Mayr and Tobias Nipkow. Higher-order rewrite systems and their confluence. Theoretical Computer Science, 192(1):3-29, February 1998.
[25]
Dale Miller. A Logic Programming Language with Lambda-Abstraction, Function Variables, and Simple Unification. In Journal and Logic and Computation 1(4):497- 536, 1991.
[26]
Tobias Nipkow. Higher-order critical pairs. In 6th IEEE Symp. on Logic in Computer Science, pages 342-349. IEEE Computer Society Press, 1991.
[27]
Tobias Nipkow, Laurence C. Paulson and Markus Wenzel. Isabelle/HOL -- A Proof Assistant for Higher-Order Logic. LNCS 2283, Springer Verlag, 2002.
[28]
Tobias Nipkow and Christian Prehofer. Higher-Order Rewriting and Equational Reasonning. In Automated deduction -- A basis for Applications. Volume I: Foundations, Bibel and Schmitt editors. Applied Logic Series 8:399-430, Kluwer, 1998.
[29]
Femke van Raamsdonk. Higher-order rewriting. In {3}.
[30]
Gerald E. Peterson and Mark E. Stickel. Complete sets of reductions for some equational theories. In JACM 28(2):233-264, 1981.
[31]
Franck Pfenning. Logic Programming in the LF Logical Framework. In Logical Frameworks, Gérard Huet and Gordon D. Plotkin eds., Cambridge University Press, 1991.
[32]
Femke van Raamsdonk. Confluence and Normalization for Higher-Order Rewrite Systems. phd thesis, Vrije Universiteit, Amsterdam, The Netherlands, 1996.
[33]
Daria Walukiewicz-Chrzaszcz. Termination of rewriting in the Calculus of Constructions. In Proceedings of the Workshop on Logical Frameworks and Metalanguages, Santa Barbara, California, 2000.

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cover image Guide books
Processes, Terms and Cycles: steps on the Road to Infinity
January 2005
638 pages
ISBN:354030911X
  • Editors:
  • Aart Middeldorp,
  • Vincent Oostrom,
  • Femke Raamsdonk,
  • Roel Vrijer

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Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 01 January 2005

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  • (2018)Typed Nominal RewritingACM Transactions on Computational Logic (TOCL)10.1145/316155819:1(1-46)Online publication date: 8-Feb-2018
  • (2015)Normal Higher-Order TerminationACM Transactions on Computational Logic10.1145/269991316:2(1-38)Online publication date: 9-Mar-2015
  • (2007)Superdeduction at workRewriting Computation and Proof10.5555/2391276.2391285(132-166)Online publication date: 1-Jan-2007
  • (2007)Polymorphic higher-order recursive path orderingsJournal of the ACM (JACM)10.1145/1206035.120603754:1(1-48)Online publication date: 1-Mar-2007
  • (2006)Higher-order orderings for normal rewritingProceedings of the 17th international conference on Term Rewriting and Applications10.1007/11805618_29(387-399)Online publication date: 12-Aug-2006

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