CSL 2005: Computer Science Logic pp 184-199

# Permutative Logic

• Jean-Marc Andreoli
• Gabriele Pulcini
• Paul Ruet
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3634)

## Abstract

Recent work establishes a direct link between the complexity of a linear logic proof in terms of the exchange rule and the topological complexity of its corresponding proof net, expressed as the minimal rank of the surfaces on which the proof net can be drawn without crossing edges. That surface is essentially computed by sequentialising the proof net into a sequent calculus which is derived from that of linear logic by attaching an appropriate structure to the sequents. We show here that this topological calculus can be given a better-behaved logical status, when viewed in the variety-presentation framework introduced by the first author. This change of viewpoint gives rise to permutative logic, which enjoys cut elimination and focussing properties and comes equipped with new modalities for the management of the exchange rule. Moreover, both cyclic and linear logic are shown to be embedded into permutative logic. It provides the natural logical framework in which to study and constrain the topological complexity of proofs, and hence the use of the exchange rule.

## Keywords

Linear Logic Structural Rule Sequent Calculus Minimal Rank Categorial Grammar
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Authors and Affiliations

• Jean-Marc Andreoli
• 1
• 3
• Gabriele Pulcini
• 2
• Paul Ruet
• 3
1. 1.Xerox Research Centre EuropeMeylanFrance
2. 2.Facoltà di Lettere e FilosofiaUniversità Roma TreRomaItaly
3. 3.CNRS – Institut de Mathématiques de LuminyMarseille Cedex 9France