Improving QED-Tutrix by Automating the Generation of Proofs

The idea of assisting teachers with technological tools is not new. Mathematics in general, and geometry in particular, provide interesting challenges when developing educative softwares, both in the education and computer science aspects. QED-Tutrix is an intelligent tutor for geometry offering an interface to help high school students in the resolution of demonstration problems. It focuses on specific goals: 1) to allow the student to freely explore the problem and its figure, 2) to accept proofs elements in any order, 3) to handle a variety of proofs, which can be customized by the teacher, and 4) to be able to help the student at any step of the resolution of the problem, if the need arises. The software is also independent from the intervention of the teacher. QED-Tutrix offers an interesting approach to geometry education, but is currently crippled by the lengthiness of the process of implementing new problems, a task that must still be done manually. Therefore, one of the main focuses of the QED-Tutrix' research team is to ease the implementation of new problems, by automating the tedious step of finding all possible proofs for a given problem. This automation must follow fundamental constraints in order to create problems compatible with QED-Tutrix: 1) readability of the proofs, 2) accessibility at a high school level, and 3) possibility for the teacher to modify the parameters defining the "acceptability" of a proof. We present in this paper the result of our preliminary exploration of possible avenues for this task. Automated theorem proving in geometry is a widely studied subject, and various provers exist. However, our constraints are quite specific and some adaptation would be required to use an existing prover. We have therefore implemented a prototype of automated prover to suit our needs. The future goal is to compare performances and usability in our specific use-case between the existing provers and our implementation.Comment: In Proceedings ThEdu'17, arXiv:1803.0072

Évaluation et amélioration de la qualité de DBpedia pour la représentation de la connaissance du domaine

RÉSUMÉ L’évolution récente du Web sémantique, tant par la quantité d’information offerte que par la multiplicité des usages possibles, rend indispensable l’évaluation de la qualité des divers ensembles de données (datasets) disponibles. Le Web sémantique étant basé sur la syntaxe RDF, i.e. des triplets (par exemple ), on peut le voir comme un immense graphe, où un triplet relie un nœud « sujet » et un nœud « objet » par une arête « relation ». Chaque dataset représente ainsi un sous-graphe. Dans cette représentation, DBpedia, un des datasets majeurs du Web sémantique, en est souvent considéré comme le nœud central. En effet, DBpedia a pour vocation, à terme, de pouvoir représenter toute l’information présente dans Wikipedia, et couvre donc une très grande variété de sujets, permettant de faire le lien avec tous les autres datasets, incluant les plus spécialisés. C’est de cette multiplicité des sujets couverts qu’apparait un point fondamental de ce projet : la notion de « domaine ». Informellement, nous considérons un domaine comme étant un ensemble de sujets reliés par une thématique commune. Par exemple, le domaine Mathématiques contient plusieurs sujets, comme algèbre, fonction ou addition. Formellement, nous considérons un domaine comme un sous-graphe de DBpedia, où l’on ne conserve que les nœuds représentant des concepts liés à ce domaine. En l’état actuel, les méthodes d’extraction de données de DBpedia sont généralement beaucoup moins efficaces lorsque le sujet est abstrait, conceptuel, que lorsqu’il s’agit d’une entité nommée, par exemple une personne, ville ou compagnie. Par conséquent, notre première hypothèse est que l’information disponible sur DBpedia liée à un domaine est souvent pauvre, car nos domaines sont essentiellement constitués de concepts abstraits. La première étape de ce travail de recherche fournit une évaluation de la qualité de l’information conceptuelle d’un ensemble de 17 domaines choisis semi-aléatoirement, et confirme cette hypothèse. Pour cela, nous identifions plusieurs axes permettant de chiffrer la « qualité » d’un domaine : 1 - nombre de liens entrants et sortants pour chaque concept, 2 - nombre de liens reliant deux concepts du domaine par rapport aux liens reliant le domaine au reste de DBpedia, 3 - nombre de concepts typés (i.e. représentant l’instance d’une classe, par exemple Addition est une instance de la classe Opération mathématique : le concept Addition est donc typé si la relation apparait dans DBpedia). Nous arrivons à la conclusion que l’information conceptuelle contenue dans DBpedia est effectivement incomplète, et ce selon les trois axes. La seconde partie de ce travail de recherche est de tenter de répondre au problème posé dans la première partie. Pour cela, nous proposons deux approches possibles. La première permet de fournir des classes potentielles, répondant en partie à la problématique de la quantité de concepts typés. La seconde utilise des systèmes d’extraction de relations à partir de texte (ORE – Open Relation Extraction) sur l’abstract (i.e. premier paragraphe de la page Wikipedia) de chaque concept. En classifiant les relations extraites, cela nous permet 1) de proposer des relations inédites entre concepts d’un domaine, 2) de proposer des classes potentielles, comme dans la première approche. Ces deux approches ne sont, en l’état, qu’un début de solution, mais nos résultats préliminaires sont très encourageants, et indiquent qu’il s’agit sans aucun doute de solutions pertinentes pour aider à corriger les problèmes démontrés dans la première partie.----------ABSTRACT In the current state of the semantic web, the quantity of available data and the multiplicity of its uses impose the continuous evaluation of the quality of this data, on the various Linked Open Data (LOD) datasets. These datasets are based on the RDF syntax, i.e. triples, such as . As a consequence, the LOD cloud can be represented as a huge graph, where every triple links the two nodes “subject” and “object”, by an edge “relation”. In this representation, each dataset is a sub-graph. DBpedia, one of the major datasets, is colloquially considered to be the central hub of this cloud. Indeed, the ultimate purpose of DBpedia is to provide all the information present in Wikipedia, “translated” into RDF, and therefore covers a wide range of domains, allowing a linkage with every other LOD dataset, including the most specialized. From this wide coverage arises one of the fundamental concepts of this project: the notion of “domain”. Informally, a domain is a set of subjects with a common thematic. For instance, the domain Mathematics contains several subjects such as algebra, function or addition. More formally, a domain is a sub-graph of DBpedia, where the nodes represent domain-related concepts. Currently, the automatic extraction methods for DBpedia are usually far less efficient when the target subject is conceptual than when it is a named entity (such as a person, city or company). Hence our first hypothesis: the domain-related information available on DBpedia is often poor, since domains are constituted of concepts. In the first part of this research project, we confirm this hypothesis by evaluating the quality of domain-related knowledge in DBpedia for 17 domains chosen semi-randomly. This evaluation is based on three numerical aspects of the “quality” of a domain: 1 – number of inbound and outbound links for each concepts, 2 – number of links between two domain concepts compared to the number of links between the domain and the rest of DBpedia, 3- number of typed concepts (i.e. representing the instance of a class : for example, Addition is an instance of the class Mathematical operation : the concept Addition is typed if the relation appears in DBpedia). We reach the conclusion that the domain-related, conceptual information present in DBpedia is indeed poor on the three axis. In the second half of this work, we give two solutions to the quality problem highlighted in the first half. The first one allows to propose potential classes that could be added in DBpedia, addressing the 3rd quality aspect: number of typed concepts. The second one uses an Open Relation Extraction (ORE) system that allows to detect relations in a text. By using this system on the abstract (i.e. the first paragraph of the Wikipedia page) of each concept, and classifying the extracted relation depending on their semantic meaning, we can 1) propose novel relations between domain concepts, and 2) propose additional potential classes. These two methods currently only represent the first step, but the preliminary results we obtain are very encouraging, and seem to indicate that they are absolutely relevant to help correcting the issues highlighted in the first part

Assessing and Improving Domain Knowledge Representation in DBpedia

With the development of knowledge graphs and the billions of triples generated on the Linked Data cloud, it is paramount to ensure the quality of data. In this work, we focus on one of the central hubs of the Linked Data cloud, DBpedia. In particular, we assess the quality of DBpedia for domain knowledge representation. Our results show that DBpedia has still much room for improvement in this regard, especially for the description of concepts and their linkage with the DBpedia ontology. Based on this analysis, we leverage open relation extraction and the information already available on DBpedia to partly correct the issue, by providing novel relations extracted from Wikipedia abstracts and discovering entity types using the dbo:type predicate. Our results show that open relation extraction can indeed help enrich domain knowledge representation in DBpedia

Rapport d’expertise de la procédure de qualification des équipements ILS. Limites et Probabilités de confiance.

L'objet de cette étude est de clarifier certains passages du rapport "DERA/WSS/WX1/CR 980799/2.3 ILS Certification Requirements". Nous étudions en particulier le paragraphe 3.3 "Confidence Limits for Sequential Tests" p.28-34 et le paragraphe 2 de l'appendice D, p.69-72

Automating the Generation of High School Geometry Proofs using Prolog in an Educational Context

When working on intelligent tutor systems designed for mathematics education and its specificities, an interesting objective is to provide relevant help to the students by anticipating their next steps. This can only be done by knowing, beforehand, the possible ways to solve a problem. Hence the need for an automated theorem prover that provide proofs as they would be written by a student. To achieve this objective, logic programming is a natural tool due to the similarity of its reasoning with a mathematical proof by inference. In this paper, we present the core ideas we used to implement such a prover, from its encoding in Prolog to the generation of the complete set of proofs. However, when dealing with educational aspects, there are many challenges to overcome. We also present the main issues we encountered, as well as the chosen solutions.Comment: In Proceedings ThEdu'19, arXiv:2002.1189

Study of the ILS certification process. Confidence limits and probabilities.

The purpose of this study is to explain some parts of the DERA report DERA/WSS/WX1/CR 980799/2.3. "ILS Certification Requirements" (Ref. 2). We mainly study section 3.3 "Confidence Limits for Sequential Tests" p.28-34 and section 2 of the appendix D, p.69-72

Interleukin-13 Activates Distinct Cellular Pathways Leading to Ductular Reaction, Steatosis, and Fibrosis

Fibroproliferative diseases are driven by dysregulated tissue repair responses and are major cause of morbidity and mortality as they affect nearly every organ system. Type-2 cytokine responses are critically involved in tissue repair; however, the mechanisms that regulate beneficial regeneration versus pathological fibrosis are not well understood. Here, we have shown that the type-2 effector cytokine interleukin-13 simultaneously, yet independently, directed hepatic fibrosis and the compensatory proliferation of hepatocytes and biliary cells in progressive models of liver disease induced by interleukin-13 over-expression or following infection with Schistosoma mansoni. Using transgenic mice with interleukin-13 signaling genetically disrupted in hepatocytes, cholangiocytes, or resident tissue fibroblasts, we have revealed direct and distinct roles for interleukin-13 in fibrosis, steatosis, cholestasis, and ductular reaction. Together, these studies show that these mechanisms are simultaneously controlled but distinctly regulated by interleukin-13 signaling. Thus, it may be possible to promote interleukin-13-dependent hepatobiliary expansion without generating pathological fibrosis

Génération automatique de preuves pour un logiciel tuteur en géométrie

RÉSUMÉ : Le logiciel QED-Tutrix est un tuteur intelligent qui offre un cadre aidant les élèves de secondaire à résoudre des problèmes de géométrie. Une de ses caractéristiques fondamentales est l’aspect tuteur, qui accompagne l’élève dans sa résolution du problème, et l’aide à avancer en cas de blocage par le biais de messages, personnalisés selon son avancement dans la preuve. Une autre de ses caractéristiques est de rester proche de la façon dont un élève résoudrait le problème avec un papier et un crayon, et, par conséquent, lui permet d’explorer la construction de sa preuve dans l’ordre de son choix. En d’autres termes, si le premier instinct de l’élève est de remarquer une caractéristique de la situation géométrique n’étant ni un conséquent direct des hypothèses, ni un élément directement nécessaire à la conclusion, QED-Tutrix lui permet de commencer à résoudre le problème à partir de ce résultat intermédiare. Ces deux caractéristiques créent une difficulté importante. En effet, le logiciel doit fournir une aide personnalisée selon l’avancement de l’élève, qui peut contenir des éléments de preuve très variés et sans nécessairement de rapport direct entre eux, pouvant par exemple appartenir à plusieurs chemins de preuve distincts. Par conséquent, le logiciel doit connaître à l’avance les différents chemins de résolution possibles pour chaque problème, et être capable d’analyser ces chemins pour déterminer dans quelle direction l’élève se dirige. Pour résoudre cet enjeu, le logiciel associe à chaque problème une structure de données, nommée le graphe HPDIC et présentée plus en détail dans les sections centrales de ce document, qui permet de représenter sous forme de graphe toutes les preuves possibles pour un problème donné. Ainsi, il devient possible d’analyser finement l’avancement de l’élève dans sa résolution du problème, et ce quel que soit l’ordre dans lequel il a fourni les éléments de la preuve.----------ABSTRACT : The QED-Tutrix project has the ultimate goal of creating, improving, and expanding the eponym tutor software, to accompany high school students in the resolution of geometry problems. Based on proven mathematics education theories, it has several distinctive characteristics. First, as a tutor software, the tutoring aspect is paramount. The virtual tutor, aptly named Professor Turing, is tasked with helping the student through the entire resolution process, from the exploration of the problem to the redaction of the proof. Second, the software as a whole stays close to the way students would typically solve a problem with paper and pencil, while adding digital tools to improve this process. This includes a dynamic geometry interface to explore the problem, a listing of available justifications, such as definitions or theorems, and a semi-automated redaction engine. Further, the student is allowed, like in real life, to construct the proof by adding elements in any order, for example by noticing a particular geometrical result on the figure, that is neither a hypothesis of the problem or its conclusion. It is possible in QED-Tutrix to begin the proof “in the middle”, by stating this particular result and working from there.These educational goals create interesting issues from a computer science point of view. Indeed, the tutoring is tailored to the exact point of the resultion reached by the student, and this point can be quite discontinuous since the elements of the resolution may have been entered in any order, can be unrelated, or pertain to different possible proofs of the problem. As a logical consequence, in order for the software to “know” what the student is doing and, mostly, what he or she is missing to complete the resolution, it is necessary to know, beforehand, all the possible proofs for the problem