Focusing in linear meta-logic: extended report

It is well known how to use an intuitionistic meta-logic to specify natural deduction systems. It is also possible to use linear logic as a meta-logic for the specification of a variety of sequent calculus proof systems. Here, we show that if we adopt different {\em focusing} annotations for such linear logic specifications, a range of other proof systems can also be specified. In particular, we show that natural deduction (normal and non-normal), sequent proofs (with and without cut), tableaux, and proof systems using general elimination and general introduction rules can all be derived from essentially the same linear logic specification by altering focusing annotations. By using elementary linear logic equivalences and the completeness of focused proofs, we are able to derive new and modular proofs of the soundness and completeness of these various proofs systems for intuitionistic and classical logics.

[1]  Furio Honsell,et al.  A framework for defining logics , 1993, JACM.

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

[3]  Dale Miller,et al.  Forum: A Multiple-Conclusion Specification Logic , 1996, Theor. Comput. Sci..

[4]  Dale Miller,et al.  Logic Programming in a Fragment of Intuitionistic Linear Logic , 1994, Inf. Comput..

[5]  Michel Parigot,et al.  Free Deduction: An Analysis of "Computations" in Classical Logic , 1990, RCLP.

[6]  Frank Pfenning.  Structural Cut Elimination: I. Intuitionistic and Classical Logic , 2000, Inf. Comput..

[7]  Elaine Pimentel.  LINEAR LOGIC AS A FRAMEWORK FOR SPECIFYING SEQUENT CALCULUS , 2002 .

[8]  Dale Miller,et al.  Focusing and Polarization in Intuitionistic Logic , 2007, CSL.

[9]  Jan von Plato,et al.  Natural deduction with general elimination rules , 2001, Arch. Math. Log..

[10]  Dale Miller,et al.  Focusing in Linear Meta-logic , 2008, IJCAR.

[11]  JEAN-MARC ANDREOLI,et al.  Logic Programming with Focusing Proofs in Linear Logic , 1992, J. Log. Comput..

[12]  Frank Pfenning,et al.  Elf: a language for logic definition and verified metaprogramming , 1989, [1989] Proceedings. Fourth Annual Symposium on Logic in Computer Science.

[13]  Peter Schroeder-Heister,et al.  A natural extension of natural deduction , 1984, Journal of Symbolic Logic.

[14]  Wilfried Sieg,et al.  Normal Natural Deduction Proofs (in classical logic) , 1998, Stud Logica.

[15]  Dale Miller,et al.  On the Specification of Sequent Systems , 2005, LPAR.

[16]  Dale Miller,et al.  Incorporating Tables into Proofs , 2007, CSL.

[17]  Amy P. Felty,et al.  Specifying Theorem Provers in a Higher-Order Logic Programming Language , 1988, CADE.

[18]  Elaine Pimentel,et al.  Using Linear Logic to Reason about Sequent Systems , 2002, TABLEAUX.

[19]  Gerhard Gentzen,et al.  Investigations into Logical Deduction , 1970 .

[20]  Marcello D'Agostino,et al.  The Taming of the Cut. Classical Refutations with Analytic Cut , 1994, J. Log. Comput..

[21]  Helmut Schwichtenberg,et al.  Basic proof theory , 1996, Cambridge tracts in theoretical computer science.

[22]  Sara Negri,et al.  Structural proof theory , 2001 .