Research On Automatically Solving Plane Geometry Problems

Automatically solving plane geometry problems is a long-standing research problem in the field of artificial intelligence and intelligent education.This problem aims to develop algorithms that can fully automatically produce readable solutions to plane geometry problems.It has been a hot research problem in recent years with the improvement of technologies in natural language processing(NLP)and machine reasoning and the urgent need of more intelligent educational services.Automatic understanding and solving plane geometry problems is a key problem of mechanization of intellectual labor in artificial intelligence.The researches on automatic understanding and solving problems in plane geometry have great potential and perspective applications in education.With the development of education informationization and individualized intelligent education,plenty of intelligent tutoring systems have been used in the teaching service.As a key technology,the function of automatic solving will greatly improve the intelligence and personalization levels of many intelligent tutoring systems,and eventually improve the quality and effects of educational services.Since the great protential in research and applications,many mechanical geometry problem solving methods have been proposed and implemented in many automated geometry solving systems(e.g.geometry theorem provers,GTPs)to conduct automated reasoning to obtain geometric solutions.These studies in the area of mathematical mechanization greatly improve the efficiency of mathematicians.However,in comparison to the large numbers of research work in automated reasoning in geometry domain,the research on the problem of understanding and solving geometry problems in a nature language environment in basic education has not yet fully conducted.Answering such geometry problems requires a method that can fully understand the nature language problem text.Automatic understanding problems is the crucial step of solving them and it is also the foundation of many intelligent systems to provide educational service.On the one hand,the natural language description of a problem often states in various ways by different users,and the technologies of NLP for processing these statements are immature.There is no specific method for analyzing and understanding the nature language statements of problems in geometry domain.On the other hand,diagram and text are used complementary as effective means to state the problems clearly in geometry discipline.In some geometry problems,the diagrams contain the necessary information to solve the problems.Hence the fusion of technologies in multiple fields,such as computer vision and natural language processing,is needed to fully understand a plane geometry problem,which is also a key step for solving the problem.This paper aims at proposing approaches for understanding and solving problems in plane geometry.It first proposes a new theory of solving plane geometry problems,which models understanding plane geometry problems as a problem of relation extraction,instead of as the problem of semantic understanding of natural language.Based on this theory,it further proposes three approaches for automatic understanding and solving different geometry problems and an interactive geometry tutoring system.More concretely,our research content includes four aspects.Firstly,the new theory of fully automatically solving plane geometry problems.Secondly,solving plane geometry problems described in nature language.Thirdly,solving plane geometry problems stated by text and diagram.Fourthly,an interactive intelligent geomety tutoring system.It makes the following contributions:Firstly,in view of the lack of systematic and comprehensive theoretical framework in the automatic solving domain,it conducts the theoretical research of machine solving to enrich the theoretical support.It proposes a new theory of solving plane geometry problems,which includes the equivalent representation,the principle of equivalence transformation and humanoid explanation.This theory transforms a given problem into the equivalent set of relations and hence transforms the understanding of plane geometry problems into the problem of relation extraction.Secondly,it proposes a syntax-semantics based relation extraction method to extract the contained geometric relations in the problems described by natural language by using the syntax and semantic information of natural language.This method can extract the realtions in the text in high performance.Experiments are conducted on the datasets of geometry word problems and plane geometry proof problems.The results show the effectiveness of the proposed method in geometric relation extraction,thus making it obtain high accuracy in solving these problems.Thirdly,it proposes a supervised machine learning approach to understand the problems described by natural language to overcome the diffculty of understanding problems through semantics analysis.This method learns the latent patterns for extracting true geometric relations by using two procedures:relation candidate generation and geometric relation identification.By first identify and extract all the geometric entities and geometric relation words,it generates all the candidate relations using the various combination of them.Then machine learning algorithm is used to predict whether a relation constitutes an actual relation or not.After finding all the actual relations equivalently representing the given problem,geometric reasoning and resolution are conducted to solving the problem.Fourthly,it proposes a method for understanding plane geometry problems by integrating the information from text and diagram two modalities.This method can be used to understand the problems including both texts and geometric diagrams.High-confidence geometric relations are extracted for problem understanding through integrating the information separately extracted from text and diagram.And an integration process is used to couple the information from text and diagram to obtain the high-confidence geometric relations.The experiments are conducted on a dataset of plane geometry problems including both texts and geometric diagrams.The experimental.results show that the proposed method can mine geometric relations in high accuracy and it can understand some problems that cannot be understood by using text only or by using diagram only.Hence it further extends the scope of problem understanding and solving.Fifthly,it proposes an interactive intelligent geometry tutoring system.By using the learner-initiating instruction model,this system can solving the problems input by users and gives the solution of the problem in an interactive way.In order to interact more naturally with the users,this system adopts a sketch-based interactive interface to mimic pen and paper environment,and builds the correspondence between the entities in the problem text and the primitives in the geometric diagram.Moreover,it visually encodes the extracted relations into the diagram to interactively present the visual effects of problem understanding and solving results to improve the user experience and to better allow users to carry out personalized geometry learning.