Orthogonal Higher-Order Rewrite Systems are Confluent

The results about higher-order critical pairs and the confluence of OHRSs provide a firm foundation for the further study of higher-order rewrite systems. It should now be interesting to lift more results and techniques both from term-rewriting and λ-calculus to the level of HRSs. For example termination proof techniques are much studied for TRSs and are urgently needed for HRSs; similarly the extension of our result to weakly orthogonal HRSs or even to Huet's “parallel closed” systems is highly desirable. Conversely, a large body of λ-calculus reduction theory has been lifted to CRSs [10] already and should be easy to carry over to HRSs.

[1]  Sören Stenlund.  Combinators, λ-Terms and Proof Theory , 2011 .

[2]  Daniel J. Dougherty.  Adding Algebraic Rewriting to the Untyped Lambda Calculus , 1992, Inf. Comput..

[3]  Femke Raamsdonk.  A simple proof of confluence for weakly orthogonal combinatory reduction systems , 1992 .

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

[5]  Lawrence C. Paulson,et al.  Isabelle: The Next 700 Theorem Provers , 2000, ArXiv.

[6]  Jan Willem Klop,et al.  Term Rewriting Systems: From Church-Rosser to Knuth-Bendix and Beyond , 1990, ICALP.

[7]  Fritz Müller,et al.  Confluence of the Lambda Calculus with Left-Linear Algebraic Rewriting , 1992, Inf. Process. Lett..

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

[9]  T. Nipkom.  Functional unification of higher-order patterns , 1993, LICS 1993.

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

[11]  V. Breazu-Tannen,et al.  Combining algebra and higher-order types , 1988, [1988] Proceedings. Third Annual Information Symposium on Logic in Computer Science.

[12]  Daniel J. Dougherty.  Adding Algebraic Rewriting to the Untyped Lambda Calculus (Extended Abstract) , 1991, RTA.

[13]  Jan Willem Klop,et al.  Combinatory reduction systems , 1980 .

[14]  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).

[15]  Amy P. Felty,et al.  A Logic Programming Approach to Implementing Higher-Order Term Rewriting , 1991, ELP.

[16]  Henk Barendregt,et al.  The Lambda Calculus: Its Syntax and Semantics , 1985 .

[17]  Jean-Pierre Jouannaud,et al.  Rewrite Systems , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.