# On the Dependencies of Logical Rules

• Marc Bagnol
• Amina Doumane
• Alexis Saurin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9034)

## Abstract

Many correctness criteria have been proposed since linear logic was introduced and it is not clear how they relate to each other. In this paper, we study proof-nets and their correctness criteria from the perspective of dependency, as introduced by Mogbil and Jacobé de Naurois. We introduce a new correctness criterion, called DepGraph, and show that together with Danos’ contractibility criterion and Mogbil and Naurois criterion, they form the three faces of a notion of dependency which is crucial for correctness of proof-structures. Finally, we study the logical meaning of the dependency relation and show that it allows to recover and characterize some constraints on the ordering of inferences which are implicit in the proof-net.

## Keywords

Linear logic MLL Proof nets Correctness criterion Contractibility Mogbil-Naurois Criterion Permutability of inferences

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