Church-Rosser Properties of Normal Rewriting

We prove a general purpose abstract Church-Rosser result that captures most existing such results that rely on termination of computations. This is achieved by studying abstract normal rewriting in a way that allows to incorporate positions at the abstract level. New concrete Church-Rosser results are obtained, in particular for higher-order rewriting at higher types.

[1]  Hélène Kirchner,et al.  Completion of a Set of Rules Modulo a Set of Equations , 1986, SIAM J. Comput..

[2]  Nachum Dershowitz,et al.  Orderings for Equational Proofs , 1986, LICS.

[3]  Terese.  Term rewriting systems , 2003, Cambridge tracts in theoretical computer science.

[4]  Vincent van Oostrom.  Confluence by Decreasing Diagrams , 2008, RTA.

[5]  Tobias Nipkow,et al.  Higher-Order Rewrite Systems and Their Confluence , 1998, Theor. Comput. Sci..

[6]  Jean-Pierre Jouannaud,et al.  Termination and Completion Modulo Associativity, Commutativity and Identity , 1992, Theor. Comput. Sci..

[7]  Mark E. Stickel,et al.  Complete Sets of Reductions for Some Equational Theories , 1981, JACM.

[8]  Claude Marché,et al.  Normalised rewriting and normalised completion , 1994, Proceedings Ninth Annual IEEE Symposium on Logic in Computer Science.

[9]  Hendrik Pieter Barendregt,et al.  Functional Programming and Lambda Calculus , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[10]  Femke van Raamsdonk,et al.  Higher-Order Rewriting , 1999, RTA.

[11]  Gérard P. Huet,et al.  Confluent Reductions: Abstract Properties and Applications to Term Rewriting Systems , 1980, J. ACM.

[12]  Tobias Nipkow,et al.  Higher-order critical pairs , 1991, [1991] Proceedings Sixth Annual IEEE Symposium on Logic in Computer Science.

[13]  D. Knuth,et al.  Simple Word Problems in Universal Algebras , 1983 .

[14]  Evelyne Contejean,et al.  AC-Unification of Higher-Order Patterns , 1997, CP.

[15]  Gerard Huet,et al.  Conflunt reductions: Abstract properties and applications to term rewriting systems , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

[16]  L. Bachmair.  Canonical Equational Proofs , 1991, Progress in Theoretical Computer Science.

[17]  Dale Miller,et al.  A Logic Programming Language with Lambda-Abstraction, Function Variables, and Simple Unification , 1991, J. Log. Comput..

[18]  J. Van Leeuwen,et al.  Handbook of theoretical computer science - Part A: Algorithms and complexity; Part B: Formal models and semantics , 1990 .

[19]  Vincent van Oostrom,et al.  Confluence by Decreasing Diagrams Converted Vincent van Oostrom , 2008 .

[20]  Ralph W. Wilkerson,et al.  Complete Sets of Reductions Modulo Associativity, Commutativity and Identity , 1989, RTA.

[21]  Albert Rubio,et al.  Higher-Order Orderings for Normal Rewriting , 2006, RTA.

[22]  Jean-Jacques Lévy,et al.  An abstract standardisation theorem , 1992, [1992] Proceedings of the Seventh Annual IEEE Symposium on Logic in Computer Science.

[23]  Volker Sorge,et al.  A Term Rewriting System for Kuratowski's Closure-Complement Problem , 2012, RTA.