Multi-focused Proofs with Different Polarity Assignments

In this work, we will reason on how a given focused proof, where atoms are assigned with some polarity, can be transformed into another focused proof, where the polarity assignment to atoms is changed. This will allow, in principle, transforming a proof obtained using one proof system into a proof using another proof system. More specifically, using the intuitionistic focused system LJF restricted to Harrop formulas, we define a procedure, introducing cuts, for transforming a focused proof where an atom is assigned with positive polarity into another focused proof where the same atom is assigned negative polarity and vice-versa. Then we show how to eliminate these cuts, obtaining a very interesting result: while the process of eliminating a cut on a positive atom gives rise to a proof with one smaller cut, in the negative case the number of introduced cuts grows exponentially. We end the paper by showing how to use maximal multi-focusing identify proofs in LJF, giving rise to a 1-1 translation between maximal proofs in LJF and proofs in the natural deduction system for intuitionistic logic NJ, restricted to Harrop formulas.

[1]  Nicolas Guenot,et al.  Computation in focused intuitionistic logic , 2015, PPDP.

[2]  Dale Miller,et al.  A formal framework for specifying sequent calculus proof systems , 2013, Theor. Comput. Sci..

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

[4]  Roy Dyckhoff,et al.  LJQ: A Strongly Focused Calculus for Intuitionistic Logic , 2006, CiE.

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

[6]  Chuck Liang,et al.  Focusing and polarization in linear, intuitionistic, and classical logics , 2009, Theor. Comput. Sci..

[7]  Gopalan Nadathur,et al.  Uniform Proofs as a Foundation for Logic Programming , 1991, Ann. Pure Appl. Log..

[8]  Vincent Danos,et al.  LKQ and LKT: sequent calculi for second order logic based upon dual linear decompositions of classical implication , 1995 .

[9]  Jean-Marc Andreoli.  Focussing and proof construction , 2001, Ann. Pure Appl. Log..

[10]  Hugo Herbelin,et al.  A Lambda-Calculus Structure Isomorphic to Gentzen-Style Sequent Calculus Structure , 1994, CSL.

[11]  Dale Miller,et al.  A Systematic Approach to Canonicity in the Classical Sequent Calculus , 2012, CSL.

[12]  Dale Miller,et al.  Canonical Sequent Proofs via Multi-Focusing , 2008, IFIP TCS.

[13]  Dale Miller,et al.  A Framework for Proof Systems , 2010, Journal of Automated Reasoning.

[14]  Gilles Dowek,et al.  Yet Another Bijection Between Sequent Calculus and Natural Deduction , 2015, LSFA.

[15]  Frank Pfenning,et al.  A Logical Characterization of Forward and Backward Chaining in the Inverse Method , 2007, Journal of Automated Reasoning.

[16]  Jean-Yves Girard,et al.  On the Unity of Logic , 1993, Ann. Pure Appl. Log..