Least and Greatest Fixed Points in Linear Logic

The first-order theory of MALL (multiplicative, additive linear logic) over only equalities is a well-structured but weak logic since it cannot capture unbounded (infinite) behavior. Instead of accounting for unbounded behavior via the addition of the exponentials (! and ?), we add least and greatest fixed point operators. The resulting logic, which we call μMALL, satisfies two fundamental proof theoretic properties: we establish weak normalization for it, and we design a focused proof system that we prove complete with respect to the initial system. That second result provides a strong normal form for cut-free proof structures that can be used, for example, to help automate proof search. We show how these foundations can be applied to intuitionistic logic.

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

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

[3]  J. Hannan,et al.  A logical framework for reasoning about logical specifications , 2004 .

[4]  Jean-Marc Andreoli,et al.  Linear objects: Logical processes with built-in inheritance , 1990, New Generation Computing.

[5]  N. P. Mendler,et al.  Inductive Types and Type Constraints in the Second-Order lambda Calculus , 1991, Ann. Pure Appl. Log..

[6]  Dale Miller,et al.  Proof and refutation in MALL as a game , 2010, Ann. Pure Appl. Log..

[7]  David Baelde.  On the Proof Theory of Regular Fixed Points , 2009, TABLEAUX.

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

[9]  Olivier Laurent,et al.  Étude de la polarisation en logique , 2001 .

[10]  Olivier Laurent,et al.  Polarized and focalized linear and classical proofs , 2005, Ann. Pure Appl. Log..

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

[12]  Luigi Santocanale.  A Calculus of Circular Proofs and Its Categorical Semantics , 2002, FoSSaCS.

[13]  Dale Miller,et al.  Focused Inductive Theorem Proving , 2010, IJCAR.

[14]  Dale Miller,et al.  A Neutral Approach to Proof and Refutation in MALL , 2008, 2008 23rd Annual IEEE Symposium on Logic in Computer Science.

[15]  Jean-Yves Girard.  Locus Solum: From the Rules of Logic to the Logic of Rules , 2001, CSL.

[16]  Vivek Nigam,et al.  Exploiting non-canonicity in the sequent calculus , 2009 .

[17]  Michael Rathjen,et al.  Lambda Calculus with Types , 2014 .

[18]  Alberto Momigliano,et al.  Cut elimination for a logic with induction and co-induction , 2012, J. Appl. Log..

[19]  A. Burroni.  Récursivité graphique (1e partie) : catégorie des fonctions récursives primitives formelles , 1986 .

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

[21]  Yves Lafont,et al.  Soft linear logic and polynomial time , 2004, Theor. Comput. Sci..

[22]  Dale Miller,et al.  Least and Greatest Fixed Points in Linear Logic , 2007, LPAR.

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

[24]  Gopalan Nadathur,et al.  Mixing Finite Success and Finite Failure in an Automated Prover , 2005 .

[25]  Frank Pfenning,et al.  Focusing the Inverse Method for Linear Logic , 2005, CSL.

[26]  Ian Mackie,et al.  The Power of Linear Functions , 2006, CSL.

[27]  Dale Miller,et al.  Unification Under a Mixed Prefix , 1992, J. Symb. Comput..

[28]  Krzysztof R. Apt,et al.  Contributions to the Theory of Logic Programming , 1982, JACM.

[29]  Patrick Lincoln,et al.  Linear logic , 1992, SIGA.

[30]  A. Tiu.  Model Checking for π-Calculus Using Proof Search , 2005 .

[31]  Gopalan Nadathur,et al.  A User Guide to Bedwyr ∗ , 2006 .

[32]  Dale Miller,et al.  From Proofs to Focused Proofs: A Modular Proof of Focalization in Linear Logic , 2007, CSL.

[33]  James Brotherston,et al.  Cyclic Proofs for First-Order Logic with Inductive Definitions , 2005, TABLEAUX.

[34]  Alberto Momigliano,et al.  Induction and Co-induction in Sequent Calculus , 2003, TYPES.

[35]  Vincent Danos,et al.  The Structure of Exponentials: Uncovering the Dynamics of Linear Logic Proofs , 1993, Kurt Gödel Colloquium.

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

[37]  David Baelde.  On the Expressivity of Minimal Generic Quantification , 2009, Electron. Notes Theor. Comput. Sci..

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

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

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

[41]  Pierre Clairambault,et al.  Least and Greatest Fixpoints in Game Semantics , 2009, FICS.

[42]  Dale Miller,et al.  A Game Semantics for Proof Search: Preliminary Results , 2005, MFPS.

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

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

[45]  Dale Miller,et al.  A proof theory for generic judgments , 2005, TOCL.

[46]  Jean-Yves Girard,et al.  Light Linear Logic , 1998, Inf. Comput..

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

[48]  Gopalan Nadathur,et al.  The Bedwyr System for Model Checking over Syntactic Expressions , 2007, CADE.

[49]  Peter Schroeder-Heister,et al.  Rules of definitional reflection , 1993, [1993] Proceedings Eighth Annual IEEE Symposium on Logic in Computer Science.

[50]  Dale Miller,et al.  Cut-elimination for a logic with definitions and induction , 2000, Theor. Comput. Sci..