Non-commutativity and MELL in the Calculus of Structures

We introduce the calculus of structures: it is more general than the sequent calculus and it allows for cut elimination and the subformula property. We show a simple extension of multiplicative linear logic, by a self-dual noncommutative operator inspired by CCS, that seems not to be expressible in the sequent calculus. Then we show that multiplicative exponential linear logic benefits from its presentation in the calculus of structures, especially because we can replace the ordinary, global promotion rule by a local version. These formal systems, for which we prove cut elimination, outline a range of techniques and properties that were not previously available. Contrarily to what happens in the sequent calculus, the cut elimination proof is modular.

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

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

[3]  Paul Ruet,et al.  Non-Commutative Logic I: The Multiplicative Fragment , 1999, Ann. Pure Appl. Log..

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

[5]  M. Nivat.  Fiftieth volume of theoretical computer science , 1988 .

[6]  Robin Milner,et al.  Communication and concurrency , 1989, PHI Series in computer science.

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

[8]  J. Girard.  Proof Theory and Logical Complexity , 1989 .

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

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

[11]  M. E. Szabo,et al.  The collected papers of Gerhard Gentzen , 1969 .

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

[13]  Radha Jagadeesan,et al.  Games and Full Completeness for Multiplicative Linear Logic , 1994, J. Symb. Log..

[14]  Paul Ruet,et al.  Non-commutative logic II: sequent calculus and phase semantics , 2000, Mathematical Structures in Computer Science.

[15]  Alessio Guglielmi,et al.  A Calculus of Order and Interaction , 1999 .

[16]  Dale Miller.  The pi-Calculus as a Theory in Linear Logic: Preliminary Results , 1992, ELP.

[17]  Lutz Straßburger,et al.  MELL in the calculus of structures , 2003, Theor. Comput. Sci..

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