On the Power of Simple Diagrams

In this paper we focus on a set of abstract lemmas that are easy to apply and turn out to be quite valuable in order to establish confluence and/or normalization modularly, especially when adding rewriting rules for extensional equalities to various calculi. We show the usefulness of the lemmas by applying them to various systems, ranging from simply typed lambda calculus to higher order lambda calculi, for which we can establish systematically confluence and/or normalization (or decidability of equality) in a simple way. Many result are new, but we also discuss systems for which our technique allows to provide a much simpler proof than what can be found in the literature.

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

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

[3]  Mitsuhiro Okada,et al.  Strong normalizability for the combined system of the typed lmbda calculus and an arbitrary convergent term rewrite system , 1989, ISSAC '89.

[4]  J. Girard,et al.  Proofs and types , 1989 .

[5]  J. Gallier.  On Girard's "Candidats de Reductibilité" , 1989 .

[6]  Franco Barbanera.  Combining Term Rewriting and Type Assignment Systems , 1990, Int. J. Found. Comput. Sci..

[7]  Piergiorgio Odifreddi,et al.  Logic and computer science , 1990 .

[8]  Alfons Geser,et al.  Relative Termination , 1990 .

[9]  Roberto Di Cosmo,et al.  A Concluent Reduction for the Lambda-Calculus with Surjective Pairing and Terminal Object , 1991, ICALP.

[10]  Val Tannen,et al.  Polymorphic Rewriting Conserves Algebraic Strong Normalization , 1991, Theor. Comput. Sci..

[11]  Limsoon Wong,et al.  Naturally Embedded Query Languages , 1992, ICDT.

[12]  Daniel J. Dougherty.  Some Lambda Calculi with Categorial Sums and Products , 1993, RTA.

[13]  Giuseppe Longo,et al.  The Genericity Theorem and Parametricity in the Polymorphic lambda-Calculus , 1993, Theor. Comput. Sci..

[14]  Yohji Akama.  On Mints' Reduction for ccc-Calculus , 1993, TLCA.

[15]  Val Tannen,et al.  Polymorphic Rewriting Conserves Algebraic Confluence , 1994, Inf. Comput..

[16]  Herman Geuvers,et al.  Modularity of strong normalization and confluence in the algebraic-/spl lambda/-cube , 1994, Proceedings Ninth Annual IEEE Symposium on Logic in Computer Science.

[17]  Limsoon Wong,et al.  Querying Nested Collections , 1994 .

[18]  Bruno Courcelle,et al.  Recognizable sets of graphs: equivalent definitions and closure properties , 1994, Mathematical Structures in Computer Science.

[19]  Roberto Di Cosmo,et al.  Simulating expansions without expansions , 1994, Mathematical Structures in Computer Science.

[20]  Neil Ghani,et al.  ßn-Equality for Coproducts , 1995, TLCA.

[21]  Neil Ghani,et al.  The virtues of eta-expansion , 1995, Journal of Functional Programming.

[22]  Roberto Di Cosmo,et al.  Expanding Extensional Polymorphism , 1995, TLCA.

[23]  Gianna Bellè.  Syntactical properties of an extension of Girard's System F where types can be taken as "generic" in , 1995 .

[24]  Roberto Di Cosmo,et al.  Combining Algebraic Rewriting, Extensional Lambda Calculi, and Fixpoints , 1996, Theor. Comput. Sci..

[25]  Vincent van Oostrom,et al.  Developing Developments , 1997, Theor. Comput. Sci..