Remarks on Semantic Completeness for Proof-Terms with Laird's Dual Affine/Intuitionistic lambda -Calculus

The purpose of this note is to give a demonstration of the completeness theorem of type assignment system for λ-terms of [Hindley 83] and [Coquand 05] with two directions of slight extensions. Firstly, using the idea of [Okada 96], [Okada-Terui 99] and [Hermant-Okada 07], we extend their completeness theorem to a stronger form which implies a normal form theorem. Secondly, we extend the simple type (the implicational fragment of intuitionistic logic) framework of [Hindley 83] and [Coquand 05] to a linear (affine) types (the {-,&,→}-fragment of affine logic) framework of [Laird 03, 05].

[1]  Mitsuhiro Okada,et al.  A uniform semantic proof for cut-elimination and completeness of various first and higher order logics , 2002, Theor. Comput. Sci..

[2]  John C. Mitchell,et al.  Type Systems for Programming Languages , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[3]  James Laird.  A Game Semantics of Linearly Used Continuations , 2003, FoSSaCS.

[4]  Michael J. Fischer,et al.  Lambda-calculus schemata , 1993 .

[5]  J. Roger Hindley,et al.  Introduction to combinators and λ-calculus , 1986, Acta Applicandae Mathematicae.

[6]  V. Michele Abrusci.  Sequent Calculus for Intuitionistic Linear Propositional Logic , 1990 .

[7]  Masahito Hasegawa.  Linearly Used Effects: Monadic and CPS Transformations into the Linear Lambda Calculus , 2001, APLAS.

[8]  Thierry Coquand.  Completeness Theorems and lambda-Calculus , 2005, TLCA.

[9]  Andrew Barber,et al.  Dual Intuitionistic Linear Logic , 1996 .

[10]  Alex K. Simpson,et al.  Computational Adequacy in an Elementary Topos , 1998, CSL.

[11]  Jean-Yves Girard,et al.  Linear Logic , 1987, Theor. Comput. Sci..

[12]  C.-H. Luke Ong,et al.  On Full Abstraction for PCF: I, II, and III , 2000, Inf. Comput..

[13]  J. Roger Hindley,et al.  The Completeness Theorem for Typing lambda-Terms , 1983, Theor. Comput. Sci..

[14]  James Laird.  Game semantics and linear CPS interpretation , 2005, Theor. Comput. Sci..

[15]  Masahito Hasegawa.  Classical Linear Logic of Implications , 2002, CSL.

[16]  P. Martin-Löf.  Hauptsatz for the Intuitionistic Theory of Iterated Inductive Definitions , 1971 .

[17]  Mitsuhiro Okada.  Phase Semantic Cut-Elimination and Normalization Proofs of First- and Higher-Order Linear Logic , 1999, Theor. Comput. Sci..

[18]  Gordon D. Plotkin,et al.  Call-by-Name, Call-by-Value and the lambda-Calculus , 1975, Theor. Comput. Sci..

[19]  Peter W. O'Hearn,et al.  Kripke Logical Relations and PCF , 1995, Inf. Comput..

[20]  Mitsuhiro Okada.  Phase Semantics for Higher Order Completeness, Cut-Elimination and Normalization Proofs , 1996, Electron. Notes Theor. Comput. Sci..

[21]  Ian Stark,et al.  Free-Algebra Models for the pi-Calculus , 2005, FoSSaCS.

[22]  J. Roger Hindley,et al.  Introduction to Combinators and Lambda-Calculus , 1986 .

[23]  Kazushige Terui,et al.  The finite model property for various fragments of intuitionistic linear logic , 1999, Journal of Symbolic Logic.

[24]  Sam Lindley,et al.  Extensional Rewriting with Sums , 2007, TLCA.