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

Automated generation of machine verifiable and readable proofs: A case study of Tarski’s geometry

The power of state-of-the-art automated and interactive theorem provers has reached the level at which a significant portion of non-trivial mathematical contents can be formalized almost fully automatically. In this paper we present our framework for the formalization of mathematical knowledge that can produce machine verifiable proofs (for different proof assistants) but also human-readable (nearly textbook-like) proofs. As a case study, we focus on one of the twentieth century classics – a book on Tarski’s geometry. We tried to automatically generate such proofs for the theorems from this book using resolution theorem provers and a coherent logic theorem prover. In the first experiment, we used only theorems from the book, in the second we used additional lemmas from the existing Coq formalization of the book, and in the third we used specific dependency lists from the Coq formalization for each theorem. The results show that 37 % of the theorems from the book can be automatically proven (with readable and machine verifiable proofs generated) without any guidance, and with additional lemmas this percentage rises to 42 %. These results give hope that the described framework and other forms of automation can significantly aid mathematicians in developing formal and informal mathematical knowledge

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

Defective Cytochrome P450-Catalysed Drug Metabolism in Niemann-Pick Type C Disease

Niemann-Pick type C (NPC) disease is a neurodegenerative lysosomal storage disease caused by mutations in either the NPC1 or NPC2 gene. NPC is characterised by storage of multiple lipids in the late endosomal/lysosomal compartment, resulting in cellular and organ system dysfunction. The underlying molecular mechanisms that lead to the range of clinical presentations in NPC are not fully understood. While evaluating potential small molecule therapies in Npc1-/- mice, we observed a consistent pattern of toxicity associated with drugs metabolised by the cytochrome P450 system, suggesting a potential drug metabolism defect in NPC1 disease. Investigation of the P450 system in the context of NPC1 dysfunction revealed significant changes in the gene expression of many P450 associated genes across the full lifespan of Npc1-/- mice, decreased activity of cytochrome P450 reductase, and a global decrease of multiple cytochrome P450 catalysed dealkylation reactions. In vivo drug metabolism studies using a prototypic P450 metabolised drug, midazolam, confirmed dysfunction in drug clearance in the Npc1-/- mouse. Expression of the Phase II enzyme uridinediphosphate-glucuronosyltransferase (UGT) was also significantly reduced in Npc1-/- mice. Interestingly, reduced activity within the P450 system was also observed in heterozygous Npc1+/- mice. The reduced activity of P450 enzymes may be the result of bile acid deficiency/imbalance in Npc1-/- mice, as bile acid treatment significantly rescued P450 enzyme activity in Npc1-/- mice and has the potential to be an adjunctive therapy for NPC disease patients. The dysfunction in the cytochrome P450 system were recapitulated in the NPC1 feline model. Additionally, we present the first evidence that there are alterations in the P450 system in NPC1 patients

The Area Method: a Recapitulation: A Recapitulation

The area method for Euclidean constructive geometry was proposed by Chou, Gao and Zhang in the early 1990’s. The method can efficiently prove many non-trivial geometry theorems and is one of the most interesting and most successful methods for automated theorem proving in geometry. The method produces proofs that are often very concise and human-readable. In this paper, we provide a first complete presentation of the method. We provide both algorithmic and implementation details that were omitted in the original presentations. We also give a variant of Chou, Gao and Zhang’s axiom system. Based on this axiom system, we proved formally all the lemmas needed by the method and its soundness using the Coq proof assistant. To our knowledge, apart from the original implementation by the authors who first proposed the method, there are only three implementations more. Although the basic idea of the method is simple, implementing it is a very challenging task because of a number of details that has to be dealt with. With the description of the method given in this paper, implementing the method should be still complex, but a straightforward task. In the paper we describe all these implementations and also some of their applications

Termination of Constraint Contextual Rewriting

The effective integration of decision procedures in formula simplication is a fundamental problem in mechanical verication. The main source of diculty occurs when the decision procedure is asked to solve goals containing symbols which are interpreted for the prover but uninterpreted for the decision procedure. To cope with the problem, Boyer & Moore proposed a technique, called augmentation, which extends the information available to the decision procedure with suitably selected facts. Constraint Contextual Rewriting (CCR, for short) is an extended form of contextual rewriting which generalizes the Boyer & Moore integration schema. In this paper we give a detailed account of the control issues related to the termination of CCR. These are particularly subtle and complicated since augmentation is mutually dependent from rewriting and it must be prevented from indefinitely extending the set of facts available to the decision procedure. A proof of termination of CCR is given