Reducing Nondeterminism in the Calculus of Structures

The calculus of structures is a proof theoretical formalism which generalizes the sequent calculus with the feature of deep inference: In contrast to the sequent calculus, inference rules can be applied at any depth inside a formula, bringing shorter proofs than any other formalisms supporting analytical proofs. However, deep applicability of the inference rules causes greater nondeterminism than in the sequent calculus regarding proof search. In this paper, we introduce a new technique which reduces nondeterminism without breaking proof theoretical properties and provides a more immediate access to shorter proofs. We present this technique on system BV, the smallest technically non-trivial system in the calculus of structures, extending multiplicative linear logic with the rules mix, nullary mix, and a self-dual non-commutative logical operator. Because our technique exploits a scheme common to all the systems in the calculus of structures, we argue that it generalizes to these systems for classical logic, linear logic, and modal logics.

[1]  Narciso Martí-Oliet,et al.  The Maude 2.0 System , 2003, RTA.

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

[3]  Alwen Tiu,et al.  A System of Interaction and Structure II: The Need for Deep Inference , 2005, Log. Methods Comput. Sci..

[4]  Student Session , 1993, EACL.

[5]  Ozan Kahramano ˘ gullari.  Reducing Nondeterminism in the Calculus of Structures , 2006 .

[6]  C. Retoré.  Pomset Logic as a Calculus of Directed Cographs , 1999 .

[7]  Steffen Hölldobler,et al.  From the Calculus of Structures to Term Rewriting Systems — Draft — , 2004 .

[8]  Christian Retoré,et al.  Pomset Logic: A Non-commutative Extension of Classical Linear Logic , 1997, TLCA.

[9]  Charles A. Stewart,et al.  A Systematic Proof Theory for Several Modal Logics , 2004, Advances in Modal Logic.

[10]  Pierre-Etienne Moreau,et al.  Implementing Deep Inference in Tom , 2005 .

[11]  H. Reichel,et al.  Deep Inference and Symmetry in Classical Proofs , 2003 .

[12]  Alwen Tiu,et al.  A Local System for Intuitionistic Logic , 2006, LPAR.

[13]  Kai Brünnler,et al.  Atomic Cut Elimination for classical Logic , 2003, CSL.

[14]  Gerard Renardel de Lavalette.  Structures and Deduction - the Quest for the Essence of Proofs , 2005 .

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

[16]  Ozan Kahramanoù.  Implementing System BV of the Calculus of Structures in Maude , 2004 .

[17]  Ozan Kahramanogullari.  System BV without the Equalities for Unit , 2004, ISCIS.

[18]  Lutz Straßburger,et al.  Non-commutativity and MELL in the Calculus of Structures , 2001, CSL.

[19]  J. S. Hodas.  Logic programming in intuitionistic linear logic: theory, design, and implementation , 1995 .

[20]  Alessio Guglielmi,et al.  A system of interaction and structure , 1999, TOCL.

[21]  Paola Bruscoli.  A Purely Logical Account of Sequentiality in Proof Search , 2002, ICLP.

[22]  Ozan Kahramanogullari.  System BV is NP-complete , 2008, Ann. Pure Appl. Log..

[23]  Alessio Guglielmi.  POLYNOMIAL SIZE DEEP-INFERENCE PROOFS INSTEAD OF EXPONENTIAL SIZE SHALLOW-INFERENCE PROOFS , 2008 .

[24]  Steffen Hölldobler,et al.  Properties of a Logical System in the Calculus of Structures , 2001 .

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

[26]  Frank Pfenning,et al.  Efficient Resource Management for Linear Logic Proof Search , 1996, ELP.

[27]  Lev Gordeev,et al.  Basic proof theory , 1998 .

[28]  Narciso Martí-Oliet,et al.  The Maude System , 1999, RTA.

[29]  A. Tiu.  A Local System for Intuitionistic Logic: Preliminary Results ? , 2005 .

[30]  Lutz Straßburger,et al.  A Non-commutative Extension of MELL , 2002, LPAR.

[31]  James Harland,et al.  Resource-distribution via Boolean constraints , 2000, TOCL.

[32]  Christophe Ringeissen,et al.  A Pattern Matching Compiler for Multiple Target Languages , 2003, CC.

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

[34]  Alwen Tiu,et al.  A Local System for Classical Logic , 2001, LPAR.

[35]  Lutz Straßburger,et al.  Linear logic and noncommutativity in the calculus of structures , 2003 .

[36]  Lutz Straßburger.  System NEL is Undecidable , 2003, Electron. Notes Theor. Comput. Sci..

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

[38]  Ozan Kahramano ˘ gullari.  System BV without the Equalities for Unit , 2004 .

[39]  Ozan Kahramano Gullari,et al.  Implementing system BV of the calculus of structures in Maude , 2004 .

[40]  Lutz Straßburger,et al.  A Local System for Linear Logic , 2002, LPAR.