8 research outputs found
Towards a Geometry Automated Provers Competition
The geometry automated theorem proving area distinguishes itself by a large
number of specific methods and implementations, different approaches
(synthetic, algebraic, semi-synthetic) and different goals and applications
(from research in the area of artificial intelligence to applications in
education).
Apart from the usual measures of efficiency (e.g. CPU time), the possibility
of visual and/or readable proofs is also an expected output against which the
geometry automated theorem provers (GATP) should be measured.
The implementation of a competition between GATP would allow to create a test
bench for GATP developers to improve the existing ones and to propose new ones.
It would also allow to establish a ranking for GATP that could be used by
"clients" (e.g. developers of educational e-learning systems) to choose the
best implementation for a given intended use.Comment: In Proceedings ThEdu'19, arXiv:2002.1189
Geometric Search in TGTP
International audienceIn this age of information the importance of retrieve the knowledge from the many sources of information is paramount. In Geometry, apart from textual approaches, common to other areas of mathematics, there is also the need for a geometric search approach, i.e., semantic searching in a corpus of geometric constructions. The Web-based repository of geometric problems Thousands of Geometric problems for geometric Theorem Provers (TGTP) has, from the start, some text search mechanisms. Since version 2.0 an implementation of the geometric search mechanism is integrated in it. Using a dynamic geometry system it is possible to build a geometric construction and then semantically search in the corpus for geometric constructions that are super-sets of the former, with regard to geometric properties. It should be noted that this is a semantic check, the selected constructions may not look like the query construction, but they will possess similar sets of geometric properties
Exchange of Geometric Information Between Applications
The Web Geometry Laboratory (WGL) is a collaborative and adaptive e-learning
Web platform integrating a well known dynamic geometry system. Thousands of
Geometric problems for Geometric Theorem Provers (TGTP) is a Web-based
repository of geometric problems to support the testing and evaluation of
geometric automated theorem proving systems.
The users of these systems should be able to profit from each other. The TGTP
corpus must be made available to the WGL user, allowing, in this way, the
exploration of TGTP problems and their proofs. On the other direction TGTP
could gain by the possibility of a wider users base submitting new problems.
Such information exchange between clients (e.g. WGL) and servers (e.g. TGTP)
raises many issues: geometric search - someone, working in a geometric problem,
must be able to ask for more information regarding that construction; levels of
geometric knowledge and interest - the problems in the servers must be
classified in such a way that, in response to a client query, only the problems
in the user's level and/or interest are returned; different aims of each tool -
e.g. WGL is about secondary school geometry, TGTP is about formal proofs in
semi-analytic and algebraic proof methods, not a perfect match indeed;
localisation issues, e.g. a Portuguese user obliged to make the query and
process the answer in English; technical issues-many technical issues need to
be addressed to make this exchange of geometric information possible and
useful.
Instead of a giant (difficult to maintain) tool, trying to cover all, the
interconnection of specialised tools seems much more promising. The challenges
to make that connection work are many and difficult, but, it is the authors
impression, not insurmountable.Comment: In Proceedings ThEdu'17, arXiv:1803.0072
O método do ângulo completo
Dissertação de Mestrado em Matemática, área de Especialização em Análise Aplicada e Computação, apresentada à Faculdade de Ciências e Tecnologia da Universidade de CoimbraEste trabalho descreve o Método do Ângulo Completo para a geometria euclidiana construtiva assim como a sua implementacão no âmbito do Open Geo Prover.
O Método do Ângulo Completo é baseado na noção de ângulo completo e num conjunto de axiomas e regras de inferência.
Apresentamos um conjunto de regras de inferência para o método do ângulo completo como sendo a base de demonstrações automatizadas de teoremas de geometria. Este método é uma extensão do método da área, obtendo-se a partir deste pela introdução de uma nova quantidade geométrica designada por ângulo completo.
Descreve-se também a implementação do método do ângulo completo no projecto Open Geo Prover.This paper describes the Full Angle Method for constructive Euclidean geometry and its implementation on the Open Geo Prover project.
The Full Angle Method is based on the notion of full angle and a set of axioms and inference rules.
We present a set of rules based on the full angle as being a basis to automatic demonstration of geometry theorems. This method is an extension of the area method that we can obtain by introducing a new geometric quantity, designated the Full Angle.
We also describe the implementation of the full angle method on the Open Geo Prover project
Automated Generation of Geometric Theorems from Images of Diagrams
We propose an approach to generate geometric theorems from electronic images
of diagrams automatically. The approach makes use of techniques of Hough
transform to recognize geometric objects and their labels and of numeric
verification to mine basic geometric relations. Candidate propositions are
generated from the retrieved information by using six strategies and geometric
theorems are obtained from the candidates via algebraic computation.
Experiments with a preliminary implementation illustrate the effectiveness and
efficiency of the proposed approach for generating nontrivial theorems from
images of diagrams. This work demonstrates the feasibility of automated
discovery of profound geometric knowledge from simple image data and has
potential applications in geometric knowledge management and education.Comment: 31 pages. Submitted to Annals of Mathematics and Artificial
Intelligence (special issue on Geometric Reasoning
Querying Geometric Figures Using a Controlled Language, Ontological Graphs and Dependency Lattices
Dynamic geometry systems (DGS) have become basic tools in many areas of
geometry as, for example, in education. Geometry Automated Theorem Provers
(GATP) are an active area of research and are considered as being basic tools
in future enhanced educational software as well as in a next generation of
mechanized mathematics assistants. Recently emerged Web repositories of
geometric knowledge, like TGTP and Intergeo, are an attempt to make the already
vast data set of geometric knowledge widely available. Considering the large
amount of geometric information already available, we face the need of a query
mechanism for descriptions of geometric constructions.
In this paper we discuss two approaches for describing geometric figures
(declarative and procedural), and present algorithms for querying geometric
figures in declaratively and procedurally described corpora, by using a DGS or
a dedicated controlled natural language for queries.Comment: 14 pages, 5 figures, accepted at CICM 201
Integrating DGSs and GATPs in an Adaptative and Collaborative Blended-Learning Web-Environment
The area of geometry with its very strong and appealing visual contents and
its also strong and appealing connection between the visual content and its
formal specification, is an area where computational tools can enhance, in a
significant way, the learning environments.
The dynamic geometry software systems (DGSs) can be used to explore the
visual contents of geometry. This already mature tools allows an easy
construction of geometric figures build from free objects and elementary
constructions. The geometric automated theorem provers (GATPs) allows formal
deductive reasoning about geometric constructions, extending the reasoning via
concrete instances in a given model to formal deductive reasoning in a
geometric theory.
An adaptative and collaborative blended-learning environment where the DGS
and GATP features could be fully explored would be, in our opinion a very rich
and challenging learning environment for teachers and students.
In this text we will describe the Web Geometry Laboratory a Web environment
incorporating a DGS and a repository of geometric problems, that can be used in
a synchronous and asynchronous fashion and with some adaptative and
collaborative features.
As future work we want to enhance the adaptative and collaborative aspects of
the environment and also to incorporate a GATP, constructing a dynamic and
individualised learning environment for geometry.Comment: In Proceedings THedu'11, arXiv:1202.453