Automated Theorem Proving in GeoGebra: Current Achievements

GeoGebra is an open-source educational mathematics software tool, with millions of users worldwide. It has a number of features (integration of computer algebra, dynamic geometry, spreadsheet, etc.), primarily focused on facilitating student experiments, and not on formal reasoning. Since including automated deduction tools in GeoGebra could bring a whole new range of teaching and learning scenarios, and since automated theorem proving and discovery in geometry has reached a rather mature stage, we embarked on a project of incorporating and testing a number of different automated provers for geometry in GeoGebra. In this paper, we present the current achievements and status of this project, and discuss various relevant challenges that this project raises in the educational, mathematical and software contexts. We will describe, first, the recent and forthcoming changes demanded by our project, regarding the implementation and the user interface of GeoGebra. Then we present our vision of the educational scenarios that could be supported by automated reasoning features, and how teachers and students could benefit from the present work. In fact, current performance of GeoGebra, extended with automated deduction tools, is already very promising—many complex theorems can be proved in less than 1 second. Thus, we believe that many new and exciting ways of using GeoGebra in the classroom are on their way

FormalGeo: An Extensible Formalized Framework for Olympiad Geometric Problem Solving

This is the first paper in a series of work we have accomplished over the past three years. In this paper, we have constructed a consistent formal plane geometry system. This will serve as a crucial bridge between IMO-level plane geometry challenges and readable AI automated reasoning. Within this formal framework, we have been able to seamlessly integrate modern AI models with our formal system. AI is now capable of providing deductive reasoning solutions to IMO-level plane geometry problems, just like handling other natural languages, and these proofs are readable, traceable, and verifiable. We propose the geometry formalization theory (GFT) to guide the development of the geometry formal system. Based on the GFT, we have established the FormalGeo, which consists of 88 geometric predicates and 196 theorems. It can represent, validate, and solve IMO-level geometry problems. we also have crafted the FGPS (formal geometry problem solver) in Python. It serves as both an interactive assistant for verifying problem-solving processes and an automated problem solver. We've annotated the formalgeo7k and formalgeo-imo datasets. The former contains 6,981 (expand to 133,818 through data augmentation) geometry problems, while the latter includes 18 (expand to 2,627 and continuously increasing) IMO-level challenging geometry problems. All annotated problems include detailed formal language descriptions and solutions. Implementation of the formal system and experiments validate the correctness and utility of the GFT. The backward depth-first search method only yields a 2.42% problem-solving failure rate, and we can incorporate deep learning techniques to achieve lower one. The source code of FGPS and datasets are available at https://github.com/BitSecret/FGPS.Comment: 44 page

Verifying safety and persistence in hybrid systems using flowpipes and continuous invariants

We describe a method for verifying the temporal property of persistence in non-linear hybrid systems. Given some system and an initial set of states, the method establishes that system trajectories always eventually evolve into some specified target subset of the states of one of the discrete modes of the system, and always remain within this target region. The method also computes a time-bound within which the target region is always reached. The approach combines flowpipe computation with deductive reasoning about invariants and is more general than each technique alone. We illustrate the method with a case study showing that potentially destructive stick-slip oscillations of an oil-well drill eventually die away for a certain choice of drill control parameters. The case study demonstrates how just using flowpipes or just reasoning about invariants alone can be insufficient and shows the richness of systems that one can handle with the proposed method, since the systems features modes with non-polynomial ODEs. We also propose an alternative method for proving persistence that relies solely on flowpipe computation

Advancing mathematical reasoning with deep learning : from numerical insights to geometrical understanding

The field of automated mathematical reasoning has captured the interest of the AI community since the last century, acknowledged as a key step towards achieving true artificial intelligence. This research domain’s evolution transits through rule-based approaches, semantic parsing, statistical machine learning, and recently, deep learning techniques. Moreover, automated mathematical reasoning has found extensive commercial applications. Educational enterprises have begun leveraging AI models for intelligent tutoring systems to assist students with mathematical problems. In the financial sector, it aids in analysing complex financial reports, with firms like JP Morgan incorporating AI to enhance their analysis capabilities. This thesis concentrates on two distinct tasks within automated mathematical reasoning: text-based numerical reasoning and automated geometry maths problem solving. Current methods face challenges in addressing complex mathematical reasoning tasks, evident in the lengthy and diverse solutions required. Additionally, in solving geometry maths problems, there is a noticeable deficiency in models’ abilities to accurately interpret geometric relationships from diagrams, which compromises their effectiveness. Furthermore, the advent of large language models (LLMs) and multi-modal models (MMs) underscores the need for a standardised benchmark to evaluate these models’ abilities in geometry problem-solving. To address these issues, we introduce the ELASTIC model in this thesis, designed for text-based numerical reasoning task. ELASTIC uniquely separates the generation of operators and operands to minimise errors from complex reasoning chains and is versatile enough to accommodate a varying number of operands per operator. This makes it broadly applicable across different domains. Our experimental results show ELASTIC’s superior performance, significantly outperforming prior models. Furthermore, we extend the application of the ELASTIC model to tackle geometry maths problems, which are inherently more complex due to the inclusion of geometric diagrams and a broader variety of problem types. To navigate these complexities, we propose the Geometry-Aware Problem Solver (GAPS), a model specifically crafted to solve diverse types of geometric maths problems by generating tailored solution programs. Our experiments validate GAPS’s advancement over existing methods. However, we observed that direct vector representation of geometric diagrams fails to capture the complex geometric relationships, which are critical in solving geometry maths problems. To overcome this, we propose converting geometric relationships into natural language, integrating them with the textual problem descriptions. This method not only improves the interpretability and effectiveness of the models but also allows for the utilisation of LLMs in generating reasoning programs. Lastly, despite the impressive capabilities of recent LLMs and MMs, their proficiency in solving geometry problems, requiring an integrated understanding of textual and visual information, remains unexplored. To fill this gap, we introduce the GeoEval benchmark in this thesis. Through extensive evaluation with GeoEval, we provide a comprehensive quantitative evaluation of the latest LLMs and MMs in geometry problem-solving task. This research marks a significant step forward in assessing the capabilities of state-of-the-art AI models in the realm of geometry problem-solving task.The field of automated mathematical reasoning has captured the interest of the AI community since the last century, acknowledged as a key step towards achieving true artificial intelligence. This research domain’s evolution transits through rule-based approaches, semantic parsing, statistical machine learning, and recently, deep learning techniques. Moreover, automated mathematical reasoning has found extensive commercial applications. Educational enterprises have begun leveraging AI models for intelligent tutoring systems to assist students with mathematical problems. In the financial sector, it aids in analysing complex financial reports, with firms like JP Morgan incorporating AI to enhance their analysis capabilities. This thesis concentrates on two distinct tasks within automated mathematical reasoning: text-based numerical reasoning and automated geometry maths problem solving. Current methods face challenges in addressing complex mathematical reasoning tasks, evident in the lengthy and diverse solutions required. Additionally, in solving geometry maths problems, there is a noticeable deficiency in models’ abilities to accurately interpret geometric relationships from diagrams, which compromises their effectiveness. Furthermore, the advent of large language models (LLMs) and multi-modal models (MMs) underscores the need for a standardised benchmark to evaluate these models’ abilities in geometry problem-solving. To address these issues, we introduce the ELASTIC model in this thesis, designed for text-based numerical reasoning task. ELASTIC uniquely separates the generation of operators and operands to minimise errors from complex reasoning chains and is versatile enough to accommodate a varying number of operands per operator. This makes it broadly applicable across different domains. Our experimental results show ELASTIC’s superior performance, significantly outperforming prior models. Furthermore, we extend the application of the ELASTIC model to tackle geometry maths problems, which are inherently more complex due to the inclusion of geometric diagrams and a broader variety of problem types. To navigate these complexities, we propose the Geometry-Aware Problem Solver (GAPS), a model specifically crafted to solve diverse types of geometric maths problems by generating tailored solution programs. Our experiments validate GAPS’s advancement over existing methods. However, we observed that direct vector representation of geometric diagrams fails to capture the complex geometric relationships, which are critical in solving geometry maths problems. To overcome this, we propose converting geometric relationships into natural language, integrating them with the textual problem descriptions. This method not only improves the interpretability and effectiveness of the models but also allows for the utilisation of LLMs in generating reasoning programs. Lastly, despite the impressive capabilities of recent LLMs and MMs, their proficiency in solving geometry problems, requiring an integrated understanding of textual and visual information, remains unexplored. To fill this gap, we introduce the GeoEval benchmark in this thesis. Through extensive evaluation with GeoEval, we provide a comprehensive quantitative evaluation of the latest LLMs and MMs in geometry problem-solving task. This research marks a significant step forward in assessing the capabilities of state-of-the-art AI models in the realm of geometry problem-solving task

Urban Informatics

This open access book is the first to systematically introduce the principles of urban informatics and its application to every aspect of the city that involves its functioning, control, management, and future planning. It introduces new models and tools being developed to understand and implement these technologies that enable cities to function more efficiently – to become ‘smart’ and ‘sustainable’. The smart city has quickly emerged as computers have become ever smaller to the point where they can be embedded into the very fabric of the city, as well as being central to new ways in which the population can communicate and act. When cities are wired in this way, they have the potential to become sentient and responsive, generating massive streams of ‘big’ data in real time as well as providing immense opportunities for extracting new forms of urban data through crowdsourcing. This book offers a comprehensive review of the methods that form the core of urban informatics from various kinds of urban remote sensing to new approaches to machine learning and statistical modelling. It provides a detailed technical introduction to the wide array of tools information scientists need to develop the key urban analytics that are fundamental to learning about the smart city, and it outlines ways in which these tools can be used to inform design and policy so that cities can become more efficient with a greater concern for environment and equity

Urban Informatics

This open access book is the first to systematically introduce the principles of urban informatics and its application to every aspect of the city that involves its functioning, control, management, and future planning. It introduces new models and tools being developed to understand and implement these technologies that enable cities to function more efficiently – to become ‘smart’ and ‘sustainable’. The smart city has quickly emerged as computers have become ever smaller to the point where they can be embedded into the very fabric of the city, as well as being central to new ways in which the population can communicate and act. When cities are wired in this way, they have the potential to become sentient and responsive, generating massive streams of ‘big’ data in real time as well as providing immense opportunities for extracting new forms of urban data through crowdsourcing. This book offers a comprehensive review of the methods that form the core of urban informatics from various kinds of urban remote sensing to new approaches to machine learning and statistical modelling. It provides a detailed technical introduction to the wide array of tools information scientists need to develop the key urban analytics that are fundamental to learning about the smart city, and it outlines ways in which these tools can be used to inform design and policy so that cities can become more efficient with a greater concern for environment and equity

Urban Informatics

This open access book is the first to systematically introduce the principles of urban informatics and its application to every aspect of the city that involves its functioning, control, management, and future planning. It introduces new models and tools being developed to understand and implement these technologies that enable cities to function more efficiently – to become ‘smart’ and ‘sustainable’. The smart city has quickly emerged as computers have become ever smaller to the point where they can be embedded into the very fabric of the city, as well as being central to new ways in which the population can communicate and act. When cities are wired in this way, they have the potential to become sentient and responsive, generating massive streams of ‘big’ data in real time as well as providing immense opportunities for extracting new forms of urban data through crowdsourcing. This book offers a comprehensive review of the methods that form the core of urban informatics from various kinds of urban remote sensing to new approaches to machine learning and statistical modelling. It provides a detailed technical introduction to the wide array of tools information scientists need to develop the key urban analytics that are fundamental to learning about the smart city, and it outlines ways in which these tools can be used to inform design and policy so that cities can become more efficient with a greater concern for environment and equity

Applied Cognitive Sciences

Cognitive science is an interdisciplinary field in the study of the mind and intelligence. The term cognition refers to a variety of mental processes, including perception, problem solving, learning, decision making, language use, and emotional experience. The basis of the cognitive sciences is the contribution of philosophy and computing to the study of cognition. Computing is very important in the study of cognition because computer-aided research helps to develop mental processes, and computers are used to test scientific hypotheses about mental organization and functioning. This book provides a platform for reviewing these disciplines and presenting cognitive research as a separate discipline