Non-strictly positive fixed points for classical natural deduction

Abstract Termination for classical natural deduction is difficult in the presence of commuting/permutative conversions for disjunction. An approach based on reducibility candidates is presented that uses non-strictly positive inductive definitions. It covers second-order universal quantification and also the extension of the logic with fixed points of non-strictly positive operators, which appears to be a new result. Finally, the relation to Parigot’s strictly positive inductive definition of his set of reducibility candidates and to his notion of generalized reducibility candidates is explained.

[1]  W. Tait.  A realizability interpretation of the theory of species , 1975 .

[2]  David J. Pym,et al.  On the intuitionistic force of classical search , 2000, Theor. Comput. Sci..

[3]  M. Sørensen,et al.  The λΔ-calculus , 1994 .

[4]  D. Prawitz.  Ideas and Results in Proof Theory , 1971 .

[5]  Masako Takahashi.  Parallel Reductions in lambda-Calculus , 1995, Inf. Comput..

[6]  D. Prawitz.  Natural Deduction: A Proof-Theoretical Study , 1965 .

[7]  Pawel Urzyczyn,et al.  Type fixpoints: iteration vs. recursion , 1999, ICFP '99.

[8]  Thierry Joly,et al.  Un plongement de la logique classique du 2nd ordre dans AF2 , 1997 .

[9]  Ralph Matthes,et al.  Short proofs of normalization for the simply- typed λ-calculus, permutative conversions and Gödel's T , 2003, Arch. Math. Log..

[10]  Philippe de Groote,et al.  Strong Normalization of Classical Natural Deduction with Disjunction , 2001, TLCA.

[11]  Jean-Louis Krivine,et al.  Lambda-calculus, types and models , 1993, Ellis Horwood series in computers and their applications.

[12]  Ralph Matthes,et al.  Monotone Fixed-Point Types and Strong Normalization , 1998, CSL.

[13]  Y. Andou,et al.  Church-Rosser property of a simple reduction for full first-order classical natural deduction , 2003, Ann. Pure Appl. Log..

[14]  Helmut Schwichtenberg,et al.  Proofs, Lambda Terms and Control Operators , 1997 .

[15]  J. Fenstad.  Proceedings of the Second Scandinavian Logic Symposium , 1971 .

[16]  Michel Parigot,et al.  Lambda-Mu-Calculus: An Algorithmic Interpretation of Classical Natural Deduction , 1992, LPAR.

[17]  Paula Severi,et al.  On normalisation , 1995 .

[18]  Gunnar Stålmarck,et al.  Normalization theorems for full first order classical natural deduction , 1991, Journal of Symbolic Logic.

[19]  N. P. Mendler,et al.  Recursive Types and Type Constraints in Second-Order Lambda Calculus , 1987, LICS.

[20]  J. H. Geuvers,et al.  Inductive and Coinductive Types with Iteration and Recursion , 1992 .

[21]  Ralph Matthes,et al.  Parigot's Second Order lambda-mu-Calculus and Inductive Types , 2001, TLCA.

[22]  Michel Parigot,et al.  Proofs of strong normalisation for second order classical natural deduction , 1997, Journal of Symbolic Logic.

[23]  René David,et al.  A short proof of the strong normalization of classical natural deduction with disjunction , 2003, Journal of Symbolic Logic.

[24]  Paula Severi,et al.  Perpetual Reductions in Lambda-Calculus , 1999, Inf. Comput..

[25]  Takahashi.  Parallel Reduction in calculus , 1989 .

[26]  Helmut Schwichtenberg,et al.  Strict Functionals for Termination Proofs , 1995, TLCA.

[27]  Karim Nour,et al.  A semantical proof of the strong normalization theorem for full propositional classical natural deduction , 2006, Arch. Math. Log..