A Sequent Calculus for a Semi-Associative Law

We introduce a sequent calculus with a simple restriction of Lambek\u27s product rules that precisely captures the classical Tamari order, i.e., the partial order on fully-bracketed words (equivalently, binary trees) induced by a semi-associative law (equivalently, tree rotation). We establish a focusing property for this sequent calculus (a strengthening of cut-elimination), which yields the following coherence theorem: every valid entailment in the Tamari order has exactly one focused derivation. One combinatorial application of this coherence theorem is a new proof of the Tutte-Chapoton formula for the number of intervals in the Tamari lattice Y_n. Elsewhere, we have also used the sequent calculus and the coherence theorem to build a surprising bijection between intervals of the Tamari order and a natural fragment of lambda calculus, consisting of the beta-normal planar lambda terms with no closed proper subterms

The Sequent Calculus of Skew Monoidal Categories

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Combinatorics of the Permutahedra, Associahedra, and Friends

I present an overview of the research I have conducted for the past ten years in algebraic, bijective, enumerative, and geometric combinatorics. The two main objects I have studied are the permutahedron and the associahedron as well as the two partial orders they are related to: the weak order on permutations and the Tamari lattice. This document contains a general introduction (Chapters 1 and 2) on those objects which requires very little previous knowledge and should be accessible to non-specialist such as master students. Chapters 3 to 8 present the research I have conducted and its general context. You will find: * a presentation of the current knowledge on Tamari interval and a precise description of the family of Tamari interval-posets which I have introduced along with the rise-contact involution to prove the symmetry of the rises and the contacts in Tamari intervals; * my most recent results concerning q, t-enumeration of Catalan objects and Tamari intervals in relation with triangular partitions; * the descriptions of the integer poset lattice and integer poset Hopf algebra and their relations to well known structures in algebraic combinatorics; * the construction of the permutree lattice, the permutree Hopf algebra and permutreehedron; * the construction of the s-weak order and s-permutahedron along with the s-Tamari lattice and s-associahedron. Chapter 9 is dedicated to the experimental method in combinatorics research especially related to the SageMath software. Chapter 10 describes the outreach efforts I have participated in and some of my approach towards mathematical knowledge and inclusion.Comment: 163 pages, m\'emoire d'Habilitation \`a diriger des Recherche

On Differentiable Interpreters

Neural networks have transformed the fields of Machine Learning and Artificial Intelligence with the ability to model complex features and behaviours from raw data. They quickly became instrumental models, achieving numerous state-of-the-art performances across many tasks and domains. Yet the successes of these models often rely on large amounts of data. When data is scarce, resourceful ways of using background knowledge often help. However, though different types of background knowledge can be used to bias the model, it is not clear how one can use algorithmic knowledge to that extent. In this thesis, we present differentiable interpreters as an effective framework for utilising algorithmic background knowledge as architectural inductive biases of neural networks. By continuously approximating discrete elements of traditional program interpreters, we create differentiable interpreters that, due to the continuous nature of their execution, are amenable to optimisation with gradient descent methods. This enables us to write code mixed with parametric functions, where the code strongly biases the behaviour of the model while enabling the training of parameters and/or input representations from data. We investigate two such differentiable interpreters and their use cases in this thesis. First, we present a detailed construction of ∂4, a differentiable interpreter for the programming language FORTH. We demonstrate the ability of ∂4 to strongly bias neural models with incomplete programs of variable complexity while learning missing pieces of the program with parametrised neural networks. Such models can learn to solve tasks and strongly generalise to out-of-distribution data from small datasets. Second, we present greedy Neural Theorem Provers (gNTPs), a significant improvement of a differentiable Datalog interpreter NTP. gNTPs ameliorate the large computational cost of recursive differentiable interpretation, achieving drastic time and memory speedups while introducing soft reasoning over logic knowledge and natural language

Generating referring expressions in a domain of objects and processes

This thesis presents a collection of algorithms and data structures for the generation of pronouns, anaphoric definite noun phrases, and one-anaphoric phrases. After a close analysis of the particular kinds of referring expressions that appear in a particular domain -that of cookery recipes -the thesis presents an appropriate ontology and a corresponding representation language. This ontology is then integrated into a wider framework for language generation as a whole, whereupon we show how the representation language can be successfully used to produce appropriate referring expressions for a range of complex object types.Amongst the more important ideas explored in the thesis are the following:• We introduce the notion of a generalized physical object as a way of representing singular entities, mass entities, and entities which are sets.• We adopt the view that planning operators are essentially underspecified events, and use this, in conjunction with a simple model of the hearer, to allow us to determine the appropriate level of detail at which a given plan should be described.• We make use of a discourse model that distinguishes local and global focus, and is closely tied to a notion of discourse structure; and we introduce a notion of DISCRIMINATORY POWER as a means to choosing the content of a referring expression.• We present a model of the generation of referring expressions that makes use of two levels of intermediate representation, and integrate this model with the use of a linguistically- founded grammar for noun phrases.The thesis ends by making some suggestions for further extensions to the work reported here

Epidemiology of Injury in English Women's Super league Football: A Cohort Study

INTRODUCTION: The epidemiology of injury in male professional football has been well documented (Ekstrand, Hägglund, & Waldén, 2011) and used as a basis to understand injury trends for a number of years. The prevalence and incidence of injuries occurring in womens super league football is unknown. The aim of this study is to estimate the prevalence and incidence of injury in an English Super League Women’s Football squad. METHODS: Following ethical approval from Leeds Beckett University, players (n = 25) signed to a Women’s Super League Football club provided written informed consent to complete a self-administered injury survey. Measures of exposure, injury and performance over a 12-month period was gathered. Participants were classified as injured if they reported a football injury that required medical attention or withdrawal from participation for one day or more. Injuries were categorised as either traumatic or overuse and whether the injury was a new injury and/or re-injury of the same anatomical site RESULTS: 43 injuries, including re-injury were reported by the 25 participants providing a clinical incidence of 1.72 injuries per player. Total incidence of injury was 10.8/1000 h (95% CI: 7.5 to 14.03). Participants were at higher risk of injury during a match compared with training (32.4 (95% CI: 15.6 to 48.4) vs 8.0 (95% CI: 5.0 to 10.85)/1000 hours, p 28 days) of which there were three non-contact anterior cruciate ligament (ACL) injuries. The epidemiological incidence proportion was 0.80 (95% CI: 0.64 to 0.95) and the average probability that any player on this team will sustain at least one injury was 80.0% (95% CI: 64.3% to 95.6%) CONCLUSION: This is the first report capturing exposure and injury incidence by anatomical site from a cohort of English players and is comparable to that found in Europe (6.3/1000 h (95% CI 5.4 to 7.36) Larruskain et al 2017). The number of ACL injuries highlights a potential injury burden for a squad of this size. Multi-site prospective investigations into the incidence and prevalence of injury in women’s football are require

A sequent calculus for a semi-associative law

This article is an extended version of a paper presented at FSCD 2017.International audienceWe introduce a sequent calculus with a simple restriction of Lambek's product rules that precisely captures the classical Tamari order, i.e., the partial order on fully-bracketed words (equivalently, binary trees) induced by a semi-associative law (equivalently, right rotation). We establish a focusing property for this sequent calculus (a strengthening of cut-elimination), which yields the following coherence theorem: every valid entailment in the Tamari order has exactly one focused derivation. We then describe two main applications of the coherence theorem, including: 1. A new proof of the lattice property for the Tamari order, and 2. A new proof of the Tutte-Chapoton formula for the number of intervals in the Tamari lattice $Y_n$