Pedro Quaresma | University of Coimbra (original) (raw)

Papers by Pedro Quaresma

arXiv (Cornell University), Mar 10, 2023

The introduction of automated deduction systems in secondary schools faces several bottlenecks. B... more The introduction of automated deduction systems in secondary schools faces several bottlenecks. Beyond the problems related with the curricula and the teachers, the dissonance between the outcomes of the geometry automated theorem provers and the normal practice of conjecturing and proving in schools is a major barrier to a wider use of such tools in an educational environment. Since the early implementations of geometry automated theorem provers, applications of artificial intelligence methods, synthetic provers based on inference rules and using forward chaining reasoning are considered to be best suited for education proposes. Choosing an appropriate set of rules and an automated method that can use those rules is a major challenge. We discuss one such rule set and its implementation using the geometry deductive databases method (GDDM). The approach is tested using some chosen geometric conjectures that could be the goal of a 7th year class (≈12-year-old students). A lesson plan is presented, its goal is the introduction of formal demonstration of proving geometric theorems, trying to motivate students to that goal.

Annals of Mathematics and Artificial Intelligence, May 24, 2023

The Geometry Automated-Theorem-Provers (GATP) based on the deductive database method use a data-b... more The Geometry Automated-Theorem-Provers (GATP) based on the deductive database method use a data-based search strategy to improve the efficiency of forward chaining. An implementation of such a method is expected to be able to efficiently prove a large set of geometric conjectures, producing readable proofs. The number of conjectures a given implementation can prove will depend on the set of inference rules chosen, the deductive database method is not a decision procedure. Using an approach based in an SQL database library and using an in-memory database, the implementation described in this paper tries to achieve the following goals. Efficiency in the management of the inference rules, the set of already known facts and the new facts discovered, by the use of the efficient data manipulation techniques of the SQL library. Flexibility, by transforming the inference rules in SQL data manipulation language queries, will open the possibility of meta-development of GATP based on a provided set of rules. Natural language and visual renderings, possible by the use of a synthetic forward chaining method. Implemented as an open source library, that will open its use by third-party programs, e.g. the dynamic geometry systems.

Journal of Automated Reasoning, Jan 10, 2023

Using an approach, inspired by our modernisation of Lemoine's Geometrography, this paper proposes... more Using an approach, inspired by our modernisation of Lemoine's Geometrography, this paper proposes a new readability criterion for formal proofs produced by automated theorem provers for geometry. We analyse two criteria to measure the readability of a proof: the criterion given by Chou et al. and the one given by Wiedijk. After discussing the limitations of these two criteria, we introduce a novel approach, which provides a new criterion. We conclude discussing some future work.

Mathematics in Computer Science, 2020

Scientific research and education at all levels are concerned primarily with the discovery, verif... more Scientific research and education at all levels are concerned primarily with the discovery, verification, communication, and application of scientific knowledge. Learning, reusing, inventing, and archiving are the four essential aspects of knowledge accumulation in mankind's civilisation process. In this cycle of knowledge accumulation, which has been supported for thousands of years by written books and other physical means, rigorous reasoning has always played an essential role. Nowadays this process is becoming more and more effective due to the availability of new paradigms based on computer applications. Geometric reasoning with such computer applications is one of the most attractive challenges for future accumulation and dissemination of knowledge.

Education and Information Technologies, 2017

The role of information and communication technologies (ICT) in education is nowadays well recogn... more The role of information and communication technologies (ICT) in education is nowadays well recognised. The Web Geometry Laboratory, is an e-learning, collaborative and adaptive, Web environment for geometry, integrating a well known dynamic geometry system. In a collaborative session, teachers and students, engaged in solving collaboratively a given set of problems, can exchange geometrical and textual information between them. In a normal work session (stand-alone mode), all the geometric steps done by the students are recorded, allowing, in a latter stage, their teachers to “play back” the students sessions. This information, alongside the navigation and chat information, can be used, later on, to assert the students level of geometric knowledge, adjusting the teaching strategies to each individual student. Teachers can register and begin using the public servers, defining students, preparing materials to be released to the students, open collaborative sessions, etc. Students can work in WGL, defining his/her own working space, sharing geometric constructions between themselves. From the case studies already conducted it was possible to conclude that, using WGL, the students improved their achievement in mathematics, in the classroom and doing homework. In this paper an in-depth, full description of the WGL system in its current version, is made, covering all the features and functioning modes, from the perspective of teachers and students.

Mathematics in Computer Science, 2017

The pursuit of an Intelligent and Dynamic Geometry Book should involve the study of how currently... more The pursuit of an Intelligent and Dynamic Geometry Book should involve the study of how currently developing methodologies and technologies of geometry knowledge representation, management, deduction and discovery can be incorporated effectively into future education. Just as Doron Zeilberger pointed out in the Plane Geometry: An Elementary Textbook By Shalosh B. Ekhad (Circa 2050), a geometry book from the future would be a computer program, in which all the theorems can be automatically discovered (and of course proved) by computer and beautiful illustrations can be automatically generated and dynamically modified. Such a prospect motivates studies on how to represent and manage digitised geometric knowledge on computer. The geometry book of the future (the IntDynGeoBook) should be adaptive, collaborative, visual and intelligent. Adaptive because the contents should adapt itself to the curricula and readers. It will allow collaborative work and its contents would be collaboratively formed using a knowledge base open to contributions. Statements and proofs should be en-lighted by dynamic geometry sketches and diagrams, and the correctness of the proofs should be ensured by computer checking. The book will be intelligent, the reader should be able to ask closed or open questions, and can also for proof hints. The book should also provide interactive exercises with automatic correction. Such a cloud platform, freely available in all standard computational platforms and devices, collaborative, adaptive to each and every user's profiles, should bring together a whole new generation of mathematical tools with impact in all levels of education. To realise such a book, a network of experts must be built, increasing the connections between several research communities, such as: mathematical knowledge management; computer theorem proving and discovery; education, aggregating expertise in areas such as Proofs in a Learning Context; Interfaces and Searching; Tools Integration; Learning Environments in the Cloud. In this paper the author tries to make the case for such an endeavour.

Journal of Automated Reasoning

Using an approach, inspired by our modernisation of Lemoine’s Geometrography, this paper proposes... more Using an approach, inspired by our modernisation of Lemoine’s Geometrography, this paper proposes a new readability criterion for formal proofs produced by automated theorem provers for geometry. We analyse two criteria to measure the readability of a proof: the criterion given by Chou et al. and the one given by Wiedijk. After discussing the limitations of these two criteria, we introduce a novel approach, which provides a new criterion. We conclude discussing some future work.

Os ambientes de aprendizagem colaborativa são um recurso importante no ensino ao permitirem que o... more Os ambientes de aprendizagem colaborativa são um recurso importante no ensino ao permitirem que os alunos adquiram conhecimento, potenciando as suas capacidades individuais, pelo trabalho em grupo. Os objetivos do projeto "Laboratório de Geometria na Rede" (WGL do Inglês, Web Geometry Laboratory) passam pela criação de uma plataforma computacional para a geometria, colaborativa, adaptativa, incorporando programas computacionais para a geometria, num ambiente de aprendizagem misto (síncrono e/ou assíncrono). O módulo colaborativo está concebido de forma a permitir, aos professores, fazerem a gestão, e posterior avaliação, das aulas colaborativas e, aos alunos, organizados em grupos, a resolução das tarefas propostas de uma forma colaborativa.

A necessidade de desenvolver ferramentas educacionais de ensino para apoiar a aprendizagem da mat... more A necessidade de desenvolver ferramentas educacionais de ensino para apoiar a aprendizagem da matemaatica e bem reconhecida. As possibilidades abertas pelas utilizcao de ferramentas das tecnologias da informacao e comunicacao sao muitas, ambientes adaptativos, colaborativos, sincronos e assincronos, contribuem para reforcar a aprendizagem de materias complexas como a Matematica. A plataforma Laboratorio de Geometria na Rede (em Ingles, Web Geometry Laboratory, WGL) carateriza-se por ser um ambiente de ensino presencial/nao presencial, colaborativo, adaptativo e integrando um sistema de geometria dinâmica. A plataforma visa contribuir para o melhorar do nivel de raciocinio geometrico do aluno. Neste artigo descreve-se de forma breve a plataforma e, atraves da descricao de um estudo de caso, a sua utilizacao para a realizacao, de forma colaborativa, de trabalhos para casa.

Electronic Proceedings in Theoretical Computer Science, 2018

The 6th International Workshop on Theorem proving components for Educational software (ThEdu'... more The 6th International Workshop on Theorem proving components for Educational software (ThEdu'17) was held in Gothenburg, Sweden, on 6 Aug 2017. It was associated to the conference CADE26. Topics of interest include: methods of automated deduction applied to checking students' input; methods of automated deduction applied to prove post-conditions for particular problem solutions; combinations of deduction and computation enabling systems to propose next steps; automated provers specific for dynamic geometry systems; proof and proving in mathematics education. ThEdu'17 was a vibrant workshop, with one invited talk and eight contributions. It triggered the post-proceedings at hand.

Electronic Proceedings in Theoretical Computer Science, 2020

This EPTCS volume contains the proceedings of the ThEdu'19 workshop, promoted on August 25, 2... more This EPTCS volume contains the proceedings of the ThEdu'19 workshop, promoted on August 25, 2019, as a satellite event of CADE-27, in Natal, Brazil. Representing the eighth installment of the ThEdu series, ThEdu'19 was a vibrant workshop, with an invited talk by Sarah Winkler, four contributions, and the first edition of a Geometry Automated Provers Competition. After the workshop an open call for papers was issued and attracted seven submissions, six of which have been accepted by the reviewers, and collected in the present post-proceedings volume. The ThEdu series pursues the smooth transition from an intuitive way of doing mathematics at secondary school to a more formal approach to the subject in STEM education, while favoring software support for this transition by exploiting the power of theorem-proving technologies. The volume editors hope that this collection of papers will further promote the development of theorem-proving-based software, and that it will collaborate on improving mutual understanding between computer mathematicians and stakeholders in education.

Journal of Symbolic Computation, 2019

Abstract In the current Information Society the organisation of the information is key to ensure ... more Abstract In the current Information Society the organisation of the information is key to ensure the information safekeeping and retrieval. It is of utmost importance that each and every user can find the information he/she is looking for, presented in such a way that best fit his/her needs. Geometry is no exception, the servers of geometric information should be easily and successfully searchable. By classifying the information contained in the servers of geometric information accordingly to several taxonomies, it will be possible to begin applying filters to the users' queries, adjusting them to the perceived user's needs. Having that in mind, the introduction of an adaptive filtering mechanisms into servers of geometric information is considered. Different taxonomies for different goals are presented. For educational purposes, a classification like Common Core Standards should be considered, but other considerations like the complexity of the construction, the provability, by a geometry automatic theorem prover, of a given conjecture and the readability of the resulting proof, should be taken into account. For research in automated deduction purposes, other issues must be considered, e.g. efficiency and applicability of the available automated provers. To validate the usefulness of these taxonomies it will be used, as a case study, their application to a server of geometric information. In particular, Thousands of Geometric problems for geometric Theorem Provers will be considered. TGTP is a Web-based repository of geometric problems being developed to support the testing and evaluation of geometric automated theorem proving systems. Using this system it will be analysed how the taxonomies could help to tailor the search for information adapted to each and every geometer.

Electronic Proceedings in Theoretical Computer Science, 2017

The UITP workshop series brings together researchers interested in designing, developing and eval... more The UITP workshop series brings together researchers interested in designing, developing and evaluating user interfaces for automated reasoning tools, such as interactive proof assistants, automated theorem provers, model finders, tools for formal methods, and tools for visualising and manipulating logical formulas and proofs. The twelth edition of UITP took place in Coimbra, Portugal, and was part of the International Joint Conference on Automated Reasoning (IJCAR'16). The workshop consisted of an invited talk, six presentations of submitted papers and lively hands-on session for reasoning tools and their user-interface. These post-proceedings contain four contributed papers accepted for publication after a second round of reviewing after the workshop as well as the invited paper.

Electronic Proceedings in Theoretical Computer Science, 2022

The introduction of automated deduction systems in secondary schools face several bottlenecks, th... more The introduction of automated deduction systems in secondary schools face several bottlenecks, the absence of the subject of rigorous mathematical demonstrations in the curricula, the lack of knowledge by the teachers about the subject and the difficulty of tackling the task by automatic means. Despite those difficulties we claim that the subject of automated deduction in geometry can be introduced, by addressing it in particular cases: simple to manipulate by students and teachers and reasonably easy to be dealt by automatic deduction tools. The subject is discussed by addressing four secondary schools geometry problems: their rigorous proofs, visual proofs, numeric proofs, algebraic formal proofs, synthetic formal proofs, or the lack of them. For these problems we discuss a lesson plan to address them with the help of Information and Communications Technology, more specifically, automated deduction tools.

Abstract. The Web Geometry Laboratory (WGL) project’s goal is, to build an adaptive and collabora... more Abstract. The Web Geometry Laboratory (WGL) project’s goal is, to build an adaptive and collaborative blended-learning Web-environment for geometry. In its current version (1.2) the WGL is already a collaborative blended-learning Web-environment integrating a dynamic geometry system (DGS) and with some adaptive features. The building of the adaptive module is a two steps task, a first task, already almost completed, is the collection of data, textual, navigation and also geometric data. A second task will be, using the collected data, the construction of student profiles and/or learning paths. In this article the process of collection of the data and its visualisation will be described. The actual testing of the WGL platform by high-school teachers is un-derway and field-tests with high-school students are being prepared. The next steps in the development of this platform will be the construction of student profiles and/or learning paths. The integration of a geometric au-tomated th...

Electronic Proceedings in Theoretical Computer Science, 2021

Mathematical proof is undoubtedly the cornerstone of mathematics. The emergence, in the last year... more Mathematical proof is undoubtedly the cornerstone of mathematics. The emergence, in the last years, of computing and reasoning tools, in particular automated geometry theorem provers, has enriched our experience with mathematics immensely. To avoid disparate efforts, the Open Geometry Prover Community Project aims at the integration of the different efforts for the development of geometry automated theorem provers, under a common "umbrella". In this article the necessary steps to such integration are specified and the current implementation of some of those steps is described.

Proof Technology in Mathematics Research and Teaching, 2019