A System of Interaction and Structure

This paper introduces a logical system, called BV, which extends multiplicative linear logic by a non-commutative self-dual logical operator. This extension is particularly challenging for the sequent calculus, and so far it is not achieved therein. It becomes very natural in a new formalism, called the calculus of structures, which is the main contribution of this work. Structures are formulae submitted to certain equational laws typical of sequents. The calculus of structures is obtained by generalising the sequent calculus in such a way that a new top-down symmetry of derivations is observed, and it employs inference rules that rewrite inside structures at any depth. These properties, in addition to allow the design of BV, yield a modular proof of cut elimination.Comment: This is the authoritative version of the article, with readable pictures, in colour, also available at . (The published version contains errors introduced by the editorial processing.) Web site for Deep Inference and the Calculus of Structures at <http://alessio.guglielmi.name/res/cos

Towards a Combinatorial Proof Theory

International audienceThe main part of a classical combinatorial proof is a skew fi-bration, which precisely captures the behavior of weakening and contraction. Relaxing the presence of these two rules leads to certain substruc-tural logics and substructural proof theory. In this paper we investigate what happens if we replace the skew fibration by other kinds of graph homomorphism. This leads us to new logics and proof systems that we call combinatorial

On the Length of Medial-Switch-Mix Derivations

International audienceSwitch and medial are two inference rules that play a central role in many deep inference proof systems. In specific proof systems, the mix rule may also be present. In this paper we show that the maximal length of a derivation using only the inference rules for switch, medial, and mix, modulo associativity and commutativity of the two binary con-nectives involved, is quadratic in the size of the formula at the conclusion of the derivation. This shows, at the same time, the termination of the rewrite system

Eleven strategies for making reproducible research and open science training the norm at research institutions

Reproducible research and open science practices have the potential to accelerate scientific progress by allowing others to reuse research outputs, and by promoting rigorous research that is more likely to yield trustworthy results. However, these practices are uncommon in many fields, so there is a clear need for training that helps and encourages researchers to integrate reproducible research and open science practices into their daily work. Here, we outline eleven strategies for making training in these practices the norm at research institutions. The strategies, which emerged from a virtual brainstorming event organized in collaboration with the German Reproducibility Network, are concentrated in three areas: (i) adapting research assessment criteria and program requirements; (ii) training; (iii) building communities. We provide a brief overview of each strategy, offer tips for implementation, and provide links to resources. We also highlight the importance of allocating resources and monitoring impact. Our goal is to encourage researchers - in their roles as scientists, supervisors, mentors, instructors, and members of curriculum, hiring or evaluation committees - to think creatively about the many ways they can promote reproducible research and open science practices in their institutions

A Logical Basis for Quantum Evolution and Entanglement

International audienceWe reconsider discrete quantum causal dynamics where quan-tum systems are viewed as discrete structures, namely directed acyclic graphs. In such a graph, events are considered as vertices and edges de-pict propagation between events. Evolution is described as happening between a special family of spacelike slices, which were referred to as locative slices. Such slices are not so large as to result in acausal influ-ences, but large enough to capture nonlocal correlations. In our logical interpretation, edges are assigned logical formulas in a spe-cial logical system, called BV, an instance of a deep inference system. We demonstrate that BV, with its mix of commutative and noncommutative connectives, is precisely the right logic for such analysis. We show that the commutative tensor encodes (possible) entanglement, and the non-commutative seq encodes causal precedence. With this interpretation, the locative slices are precisely the derivable strings of formulas. Several new technical results about BV are developed as part of this analysis

Structures for multiplicative cyclic linear logic: Deepness vs cyclicity

Abstract. The aim of this work is to give an alternative presentation for the multiplicative fragment of Yetter’s cyclic linear logic. The new presentation is inspired by the calculus of structures, and has the interesting feature of avoiding the cyclic rule. The main point in this work is to show how cyclicity can be substituted by deepness, i.e. the possibility of applying an inference rule at any point of a formula. We finally derive, through a new proof technique, the cut elimination property of the calculus