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On the Decision Problem and the Mechanization of Theorem-Proving in Elementary Geometry
... If we order each support coordinate as ðOðP ð1Þ Þ; OðP ð2Þ Þ; OðP ð3Þ ÞÞ, then Eqs. (13) have the following supports: which we have ordered by descending power. If we plot the support coordinates, we find the convex hulls, Fig. 3. ...
... Finding solutions of multivariate polynomial systems has been studied for many years, with recent advances occurring in the age of computational algebraic geometry. Common symbolic techniques include Gröbner Basis, [11] which can be viewed as a multivariate, nonlinear generalization of both Euclid's algorithm and Gaussian elimination for linear systems; the Resultant Method, [12] which performs variable elimination via construction of matrices and determinants; and Wu's Method, [13] which is based on polynomial division. Though capable of finding accurate zeros, these methods are computationally expensive. ...
... ROURKE et al. · ON THE ROOT STRUCTURE OF THE COUPLED ASSEMBLY PROBABILITY Equation (19) again tells us that there are B À 1 distinct nontrivial roots of the three coupled assembly POI equations given by Eqs.(13) and that there are then B roots total when including the trivial root. Obviously, we know that there is a trivial root because there are no monomials with support ð0; 0; 0Þ (i.e., there are no constants in the POI equations). ...
... One of the key milestones in the field of artificial intelligence is the capability to reason (Pearl 1998) and prove theorems (Wu 1978;Chou et al. 2000;Trinh et al. 2024). However, theorem proving often involves long reasoning chains, complex mathematical structures, intricate calculations, and infinite reasoning spaces. ...
... Automatic Theorem Proving. Automatic theorem proving has been a focus of artificial intelligence since the 1950s (Harrison et al. 2014;Wu 1978). Modern theorem provers, based on tactic and premise selection, search for proofs by interacting with proof assistants such as Lean (De Moura et al. 2015), Coq (Barras et al. 1999) and Isabelle (Nipkow et al. 2002). ...
... We provide more information on AIPS' deductive engine and the training process for the value network. To highlight the reasoning ability and maintain readability of proofs, we avoid using brute-force methods such as augmentation-substitution and Wu's method Wu (1978). ...
Solving Olympiad-level mathematical problems represents a significant advancement in machine intelligence and automated reasoning. Current machine learning methods, however, struggle to solve Olympiad-level problems beyond Euclidean plane geometry due to a lack of large-scale, high-quality datasets. The challenge is even greater in algebraic systems, which involve infinite reasoning spaces within finite conditions. To address these issues, we propose AIPS, an Algebraic Inequality Proving System capable of autonomously generating complex inequality theorems and effectively solving Olympiad-level inequality problems without requiring human demonstrations. During proof search in a mixed reasoning manner, a value curriculum learning strategy on generated datasets is implemented to improve proving performance, demonstrating strong mathematical intuitions. On a test set of 20 International Mathematical Olympiad-level inequality problems, AIPS successfully solved 10, outperforming state-of-the-art methods. Furthermore, AIPS automatically generated a vast array of non-trivial theorems without human intervention, some of which have been evaluated by professional contestants and deemed to reach the level of the International Mathematical Olympiad. Notably, one theorem was selected as a competition problem in a major city 2024 Mathematical Olympiad.
... Synthesis methods, such as the backward search method [9], forward chaining method [10] and deductive database method [11], is essentially a search-based method. Algebraic methods are based on coordinates, such as Wu's method [12], transforming GPS into a computational problem. Invariant-based point elimination methods [13] find that the solution methods for geometric problems are embedded in their geometric shape construction. ...
... Wen-Tsun proposed Wu's method [12], which transforms geometric problem into a system of algebraic equations consisting of polynomials and inequalities. Then, by utilizing the properties of algebraic computation, these algebraic expressions are simplified and solved, transforming complex algebraic expressions into forms understandable by humans, and thus interpreting their geometric meanings. ...
... Under the condition of satisfying Equation (11), adding x t should introduce as few new unknown variables as possible, as depicted in Equation (12). The closer the number of t and |M t | are, the higher the likelihood of solving G t . ...
Geometric problem solving (GPS) has always been a long-standing challenge in the fields of automated reasoning. Its problem representation and solution process embody rich symmetry. This paper is the second in a series of our works. Based on the Geometry Formalization Theory and the FormalGeo geometric formal system, we have developed the Formal Geometric Problem Solver (FGPS) in Python 3.10, which can serve as an interactive assistant or as an automated problem solver. FGPS is capable of executing geometric predicate logic and performing relational reasoning and algebraic computation, ultimately achieving readable, traceable, and verifiable automated solutions for geometric problems. We observed that symmetry phenomena exist at various levels within FGPS and utilized these symmetries to further refine the system’s design. FGPS employs symbols to represent geometric shapes and transforms various geometric patterns into a set of symbolic operation rules. This maintains symmetry in basic transformations, shape constructions, and the application of theorems. Moreover, we also have annotated the formalgeo7k dataset, which contains 6981 geometry problems with detailed formal language descriptions and solutions. Experiments on formalgeo7k validate the correctness and utility of the FGPS. The forward search method with random strategy achieved a 39.71% problem-solving success rate.
... Schreck developed Progé [7] -a general framework implemented in PROLOG where different kind of geometric objects (not only triangles) can automatically be constructed. Gao and Chou applied algebraic approach and used Wu's method [8] or Gröbner bases [9] to find locations of unknown objects from the locations of known objects and determine RC-constructibility [10]. Schreck also focused on the algebraic approach and compared Gao and Chou's method with the Lesbegue's method using them to show non-constructibility, but also to extract some RC-constructions from algebraic methods [11]. ...
... The hyperbolic lines (abbr. h-lines) are the arcs of the circles orthogonal 7 to the absolute lying in the interior of the absolute (arc through points A and B in Fig. 6) or the line segments passing through the center of the absolute (segment through points A and C in Fig. 6) 8 . ...
... When considering only hypebolic geometry, and there is no danger of confusion, the prefix h is omitted and standard terms (e.g., points, lines, circles) are used.7 Recall that the two circles are orthogonal if their tangents are orthogonal at the point of intersection.8 If the Poincaré disc is defined within the projective complex line CP 1 instead of the Cartesian plane R 2 , then there is no difference between circles and lines and these segments can also be considered to be circular arcs. ...
We describe a system for automated ruler and compass triangle constructions in hyperbolic geometry. We discuss key differences between constructions in Euclidean and hyperbolic setting, compile a list of primitive constructions and lemmas used for constructions in hyperbolic geometry, build an automated system for solving construction problems, and test it on a corpus of triangle-construction problems. We extend the list of primitive constructions for hyperbolic geometry by several constructions that cannot be done by ruler and compass, but can be done by using algebraic calculations and show that such extended system solves more problems. We also use a dynamic geometry library to build an online compendium containing construction descriptions, illustrations, and step-by-step animations.
... Computational geometry is a branch of computer science [1]. In the late 1970s, Wen-tsun Wu established Wu's method [2][3][4], which has greatly promoted the development of mathematics mechanization. Over the past few decades, experts and scholars have made significant progress in this field [5]. ...
... In order to identify these equivalents, normalization processing is required, which is mainly performed by sorting the letters representing points of the geometry unit into alphabetical order, so that there is only one writing form for each geometry unit in the system. Taking Triangle (A,B,C) and Triangle (C,A,B) as an example, the processing comprises the following: representing Triangle (A,B,C) with Triangle (1,2,3), and representing Triangle (C,A,B) with Triangle (3,2,1), according to the alphabetical orders of the corresponding letters; respectively sorting Triangle (1,2,3) and Triangle (3,2,1) to Triangle (1,2,3) and Triangle (1,2,3); then comparing the sorted forms. Thus, the system can recognize that Triangle (A,B,C) and Triangle (C,A,B) are equivalents. ...
... Parallel (Segment (A,B), Segment (C,D)) Segment AB is parallel to segment CD Perpendicular (Segment (E,F), Segment (G,H)) Segment EF is perpendicular to segment GH Equal (Angle (A,B,C), Angle (A,C,B)) Angle ABC is equal to angle ACB Collinear (Point(A), Point(B), Point(C)) Point A, point B and point C are collinear Area (Triangle (A,B,C), 3) Area of triangle ABC is 3 ...
Predicates and rules are usually enclosed as built-in functions in automated geometry reasoning systems, meaning users cannot add any predicate or rule, thus resulting in a limited reasoning capability of the systems. A method for expanding predicates and rules in automated geometry reasoning systems is, thus, proposed. Specifically, predicate and rule descriptions are transformed to knowledge trees and forests based on formal representations of geometric knowledge, and executable codes are dynamically and automatically generated by using “code templates”. Thus, a transformation from controlled natural language descriptions to mechanization algorithms is completed, and finally, the dynamic expansion of predicates and rules in the reasoning system is achieved. Moreover, the method has been implemented in an automated geometry reasoning system for Chinese college entrance examination questions, and the practicality and effectiveness of the method were tested. In conclusion, the enclosed setting, which is a shortcoming of traditional reasoning systems, is avoided, the user-defined dynamic expansion of predicates and rules is realized, the application scope of the reasoning system is extended, and the reasoning capability is improved.
... In recent years we are witnessing a rich interaction between two geometric instruments which were initially born with different intentions in mind, but are converging and joining forces nowadays to create a new generation of software for doing Geometry. On the one hand there were the Geometric Automatic Theorem Provers (GATPs), which can be traced back to the seminal work by Wu [1], with an increasing variety of methods and approaches developed along the years, such as the Gröbner Basis Method, the Area Method, the Full-Angle Method. . . . These methods have succeeded to automatically prove a large collection of geometric statements, as can be seen for instance [2]. ...
... The GATPs based on algebraic techniques, such as Wu's Method ( [1,3]) or the Gröbner Basis method ( [4,5]), require frequently the manipulation of a large set of polynomials, which represents a given geometric construction. Unfortunately, this translation of geometric properties into algebraic equations can be a tedious process. ...
... The first two points of the construction are identified with (0,0) and (1,0). This reduces the number of variables required, and avoids some degenerate configurations. ...
Recently developed GeoGebra tools for the automated deduction and discovery of geometric statements combine in a unique way computational (real and complex) algebraic geometry algorithms and graphic features for the introduction and visualization of geometric statements. In our paper we will explore the capabilities and limitations of these new tools, through the case study of a classic geometric inequality, showing how to overcome, by means of a double approach, the difficulties that might arise attempting to ‘discover’ it automatically. On the one hand, through the introduction of the dynamic color scanning method, which allows to visualize on GeoGebra the set of real solutions of a given equation and to shed light on its geometry. On the other hand, via a symbolic computation approach which currently requires the (tricky) use of a variety of real geometry concepts (determining the real roots of a bivariate polynomial p(x,y) by reducing it to a univariate case through discriminants and Sturm sequences, etc.), which leads to a complete resolution of the initial problem. As the algorithmic basis for both instruments (scanning, real solving) are already internally available in GeoGebra (e.g., via the Tarski package), we conclude proposing the development and merging of such features in the future progress of GeoGebra automated reasoning tools.
... This paper deals with some issues related to the successful approach to the automated verification and derivation of theorems in elementary geometry through computational algebraic geometry techniques, initiated more than forty years ago by Wu [1]. ...
... This example might look artificial and, indeed, when dealing with geometric statements, it is often assumed in most cases that the 'intuitively' maximal set of independent variables is maximum-size, but there are examples in which this is not the case. See Example 2 in ( [12], p. 72): the number of coordinates of free points in the chosen geometric construction is 5, but the Hilbert dimension of H is 6; or Example 7 in the Reference [20], about Euler's formula concerning the distance between the centers and the radii of the inner and outer circles of a triangle with vertices (−1, 0), (1, 0), (u [1], u [2]). Here the 'intuitive' dimension of the hypotheses variety should be 2 (the coordinates of the only free vertex of the triangle), but applying the algebraic definition it turns out to be three, unless it is explicitly added to the hypotheses the (obvious) fact that (u [1], u[2]) should not lie in the x-axis. ...
... See Example 2 in ( [12], p. 72): the number of coordinates of free points in the chosen geometric construction is 5, but the Hilbert dimension of H is 6; or Example 7 in the Reference [20], about Euler's formula concerning the distance between the centers and the radii of the inner and outer circles of a triangle with vertices (−1, 0), (1, 0), (u [1], u [2]). Here the 'intuitive' dimension of the hypotheses variety should be 2 (the coordinates of the only free vertex of the triangle), but applying the algebraic definition it turns out to be three, unless it is explicitly added to the hypotheses the (obvious) fact that (u [1], u[2]) should not lie in the x-axis. This quite common problem-related to the a priori control and detail of all geometric degeneracies-is already considered in the basic Reference [7], mentioning, in particular (cf. ...
We report, through different examples, the current development in GeoGebra, a widespread Dynamic Geometry software, of geometric automated reasoning tools by means of computational algebraic geometry algorithms. Then we introduce and analyze the case of the degeneracy conditions that so often arise in the automated deduction in geometry context, proposing two different ways for dealing with them. One is working with the saturation of the hypotheses ideal with respect to the ring of geometrically independent variables, as a way to globally handle the statement over all non-degenerate components. The second is considering the reformulation of the given hypotheses ideal—considering the independent variables as invertible parameters—and developing and exploiting the specific properties of this zero-dimensional case to analyze individually the truth of the statement over the different non-degenerate components.
... Traditional methods, such as search-based methods [4,5], algebraic methods [6][7][8], and point elimination methods [9], require manual preliminary formalization of geometric problems, with the solving process involving numerous manually defined rules. The development of deep learning technologies has introduced new avenues for geometric problem-solving. ...
... The Wu method [6] turns geometric problems into an equation system with algebraic equalities and inequalities, making the solution similar to that of algebraic equations. Its appearance has propelled geometric problem-solving research, spawning methods like Gröbner bases [7], numerical parallel [20], polynomial system triangulation elimination [21], and dimensionality reduction [22]. ...
Automatic geometric problem-solving is an active and challenging subfield at the intersection of AI and mathematics, where geometric problem parsing plays a critical role. It involves converting geometric diagram and text into certain formal language. Due to the complexity of geometric shapes and the diversity of geometric relationships, geometric problem parsing demands that the parser exhibit cross-modal comprehension and reasoning capabilities. In this paper, we propose an enhanced geometric problem parsing method called FGeo-Parser, which converts problem diagrams and text into the formal language of the FormalGeo. It also supports reverse formalization to generate human-like solutions, reflecting the symmetry between parsing and generating. Specifically, diagram parser leverages the BLIP to generate the construction CDL and image CDL, while text parser employs the T5 to produce the text CDL and goal CDL where these neural networks are both based on a symmetric encoder–decoder architecture. With the assistance of a theorem predictor, these CDLs were automatically parsed and step-by-step reasoning was executed within FGPS. Finally, the reasoning process was input into a solution generator, which subsequently produced a human-like solution process. Additionally, we re-annotated problem diagrams and text based on the FormalGeo7K dataset. The formalization experiments on the new dataset achieved a match accuracy of 91.51% and a perfect accuracy of 56.47%, while the combination with the theorem predictor achieved a problem-solving accuracy of 63.45%.
... However it did not generate classic, readable, synthetic construction proofs. In her PhD thesis [4], Marinković describes how theorem provers, based on algebraic methods such as Wu's method [5] and Gröbner basis method [6], and semi-synthetic methods such as area method [7], integrated within GCLC tool [8] and OpenGeoProver [9], could be employed to check the construction correctness. The problem with this approach is that generated proofs are not human-readable. ...
Although there are several systems that successfully generate construction steps for ruler and compass construction problems, none of them provides readable synthetic correctness proofs for generated constructions. In this paper, we demonstrate how our triangle construction solver ArgoTriCS can cooperate with automated theorem provers for first-order logic and coherent logic so that it generates construction correctness proofs, that are both human-readable and formal (can be checked by interactive theorem provers such as Isabelle/HOL or Coq). For this purpose we identified a set of relevant lemmas and developed a coherent logic prover GCProver customized for geometry construction problems. Our experiments show that results are much better than with general purpose theorem provers.
... To address these limitations, mechanical symbolic reasoning is still essential. Unlike LLMs, symbolic methods leverage domain-specific knowledge to achieve greater efficiency and generalization without relying on extensive training data (Wu, 2008;Heule et al., 2016). Integrating LLMs with symbolic methods presents a promising strategy for tactic generation and theorem proving. ...
Large language models (LLMs) can prove mathematical theorems formally by generating proof steps (\textit{a.k.a.} tactics) within a proof system. However, the space of possible tactics is vast and complex, while the available training data for formal proofs is limited, posing a significant challenge to LLM-based tactic generation. To address this, we introduce a neuro-symbolic tactic generator that synergizes the mathematical intuition learned by LLMs with domain-specific insights encoded by symbolic methods. The key aspect of this integration is identifying which parts of mathematical reasoning are best suited to LLMs and which to symbolic methods. While the high-level idea of neuro-symbolic integration is broadly applicable to various mathematical problems, in this paper, we focus specifically on Olympiad inequalities (Figure~1). We analyze how humans solve these problems and distill the techniques into two types of tactics: (1) scaling, handled by symbolic methods, and (2) rewriting, handled by LLMs. In addition, we combine symbolic tools with LLMs to prune and rank the proof goals for efficient proof search. We evaluate our framework on 161 challenging inequalities from multiple mathematics competitions, achieving state-of-the-art performance and significantly outperforming existing LLM and symbolic approaches without requiring additional training data.
... Once the point locations are accurate, constructing the geometry becomes straightforward, such as connecting points with lines or drawing circles. Drawing inspiration from computational geometry methods used in geometry theorem provers (Wu, 2008), we model dia-gram generation as a set of polynomial equations based on point coordinates. ...
Geometric diagrams are critical in conveying mathematical and scientific concepts, yet traditional diagram generation methods are often manual and resource-intensive. While text-to-image generation has made strides in photorealistic imagery, creating accurate geometric diagrams remains a challenge due to the need for precise spatial relationships and the scarcity of geometry-specific datasets. This paper presents MagicGeo, a training-free framework for generating geometric diagrams from textual descriptions. MagicGeo formulates the diagram generation process as a coordinate optimization problem, ensuring geometric correctness through a formal language solver, and then employs coordinate-aware generation. The framework leverages the strong language translation capability of large language models, while formal mathematical solving ensures geometric correctness. We further introduce MagicGeoBench, a benchmark dataset of 220 geometric diagram descriptions, and demonstrate that MagicGeo outperforms current methods in both qualitative and quantitative evaluations. This work provides a scalable, accurate solution for automated diagram generation, with significant implications for educational and academic applications.
... There has been two main approaches for automatically solving geometry problems. One is bashing the problems algebraically with Wu's method Chou (1985); Wu (2008), Area method Chou et al. (1993Chou et al. ( , 1994, or Gröbner bases Kapur (1986a,b), and second approach relies in synthetic techniques such as Deduction database Chou et al. (2000), or Full angle method Chou et al. (1996). We focus on the latter as a more human-like approach suitable for transferring the research knowledge to other domains. ...
We present AlphaGeometry2, a significantly improved version of AlphaGeometry introduced in Trinh et al. (2024), which has now surpassed an average gold medalist in solving Olympiad geometry problems. To achieve this, we first extend the original AlphaGeometry language to tackle harder problems involving movements of objects, and problems containing linear equations of angles, ratios, and distances. This, together with other additions, has markedly improved the coverage rate of the AlphaGeometry language on International Math Olympiads (IMO) 2000-2024 geometry problems from 66% to 88%. The search process of AlphaGeometry2 has also been greatly improved through the use of Gemini architecture for better language modeling, and a novel knowledge-sharing mechanism that combines multiple search trees. Together with further enhancements to the symbolic engine and synthetic data generation, we have significantly boosted the overall solving rate of AlphaGeometry2 to 84% for geometry problems over the last 25 years, compared to 54% previously. AlphaGeometry2 was also part of the system that achieved silver-medal standard at IMO 2024 https://dpmd.ai/imo-silver. Last but not least, we report progress towards using AlphaGeometry2 as a part of a fully automated system that reliably solves geometry problems directly from natural language input.
... As far as we know, nineteen years passed before the next work in computerizing geometry; it was 1978 when Wen-Tsün Wu [45] began a series of papers on the subject, culminating in his 1984 book, published only in Chinese, and not available in English until 1994 [46]. He was soon assisted by S. C. Chou [12,13]. ...
We used computer proof-checking methods to verify the correctness of our proofs of the propositions in Euclid Book I. We used axioms as close as possible to those of Euclid, in a language closely related to that used in Tarski's formal geometry. We used proofs as close as possible to those given by Euclid, but filling Euclid's gaps and correcting errors. Euclid Book I has 48 propositions, we proved 235 theorems. The extras were partly "Book Zero", preliminaries of a very fundamental nature, partly propositions that Euclid omitted but were used implicitly, partly advanced theorems that we found necessary to fill Euclid's gaps, and partly just variants of Euclid's propositions. We wrote these proofs in a simple fragment of first-order logic corresponding to Euclid's logic, debugged them using a custom software tool, and then checked them in the well-known and trusted proof checkers HOL Light and Coq.
... The study of differential algebras is an algebraic approach to differential equations replacing analytic notions like differential quotients by abstract operations, initiated by Ritt [36,37] in 1930s and developed by Kolchin and his school [20]. After the fundamental work of Ritt and Kolchin, differential algebra has evolved into a vast area of mathematics that is important in both theory and applications, including differential Galois theory, differential algebraic geometry, differential algebraic groups [20,32,39] and mechanic theorem proving [41,42]. Guo and Keigher posed the concept of weighted differential algebra, including the usual differential algebra of weight zero and the difference algebra of weight one [16] as examples. ...
In this paper, we first introduce a weighted derivation on algebras over an operad [Formula: see text], and prove that for the free [Formula: see text]-algebra, its weighted derivation is determined by the restriction on the generators. As applications, we propose the concept of weighted differential ([Formula: see text]-tri)dendriform algebras and study some basic properties of them. Then Novikov-(tri)dendriform algebras are initiated, which can be induced from differential ([Formula: see text]-tri)dendriform of weight zero. Finally, the corresponding free objects are constructed, in both the commutative and noncommutative contexts.
... After the fundamental work of Ritt and Kolchin, differential algebra has evolved into a vast area of mathematics that is important in both theory and applications, including differential Galois theory, differential algebraic geometry, differential algebraic groups [20,32,39] and mechanic theorem proving [41,42]. Guo and Keigher posed the concept of weighted differential algebra, including the usual differential algebra of weight zero and the difference algebra of weight one [16] as examples. ...
In this paper, we first introduce a weighted derivation on algebras over an operad , and prove that for the free -algebra, its weighted derivation is determined by the restriction on the generators. As applications, we propose the concept of weighted differential (q-tri)dendriform algebras and study some basic properties of them. Then Novikov-(tri)dendriform algebras are initiated, which can be induced from differential (q-tri) dendriform of weight zero. Finally, the corresponding free objects are constructed, in both the commutative and noncommutative contexts.
... The representation model serves as the foundation for our method, aiming to comprehensively represent the premises, the conclusion to be proven, and their relationships in a geometry proof problem. Based on point geometry, we propose a model called point geometry identity, inspired by Hilbert's Nullstellensatz [19] and Wu's method [20,21]. Let {R 1 , R 2 , ..., R n−1 } and R n respectively represent the set of premises and the conclusion to be proven in a geometric proof problem. ...
The automated generation of geometry proof problems represents a burgeoning research domain in the realm of artificial intelligence, with significant practical implications for mathematics education. In this study, we present a method for automatically generating fresh geometry proof questions by adapting existing ones. The core of the approach is a novel representation model for geometry proof problems, which we term “point geometry identity". According to this model, the premises and conclusion to be proven in a geometry proof problem, as well as their relationships, can be expressed as an expression. Subsequently, by applying equivalent deformation principles of algebraic expressions, we can modify the premises and conclusions of existing problems to generate new ones. Experimental analysis and expert evaluations indicate that our approach can efficiently generate a substantial number of diverse problems suitable for mathematics education within a short span of time. Moreover, under the same input conditions, our method can generate novel problems that existing techniques cannot.
... Search-based methods often prove only a limited number of planar geometry theorems due to their high computational complexity. Wen-Tsun proposed Wu's Method [14], which transforms geometry problems into algebraic equation-solving problems but is confined to the algebraic domain. Zhang introduced the point elimination method based on geometric invariants [15], generating concise and meaningful readable proofs for a large number of geometry problems. ...
The application of contemporary artificial intelligence techniques to address geometric problems and automated deductive proofs has always been a grand challenge to the interdisciplinary field of mathematics and artificial intelligence. This is the fourth article in a series of our works, in our previous work, we established a geometric formalized system known as FormalGeo. Moreover, we annotated approximately 7000 geometric problems, forming the FormalGeo7k dataset. Despite the fact that FGPS (Formal Geometry Problem Solver) can achieve interpretable algebraic equation solving and human-like deductive reasoning, it often experiences timeouts due to the complexity of the search strategy. In this paper, we introduced FGeo-TP (theorem predictor), which utilizes the language model to predict the theorem sequences for solving geometry problems. The encoder and decoder components in the transformer architecture naturally establish a mapping between the sequences and embedding vectors, exhibiting inherent symmetry. We compare the effectiveness of various transformer architectures, such as BART or T5, in theorem prediction, and implement pruning in the search process of FGPS, thereby improving its performance when solving geometry problems. Our results demonstrate a significant increase in the problem-solving rate of the language model-enhanced FGeo-TP on the FormalGeo7k dataset, rising from 39.7% to 80.86%. Furthermore, FGeo-TP exhibits notable reductions in solution times and search steps across problems of varying difficulty levels.
... Our proposed method is on the base of Wu's algorithm to solve FFPEs systems [35]. The theory of Ritt and some efficient algorithms for zero decomposition of arbitrary systems of polynomials have been considerably improved by Wu Wen-Tsun since 1980 [34,36]. ...
This article introduces a productive algebraic approach to identifying positive solutions for a system of fully fuzzy polynomial equations (FFPEs). To achieve this, the FFPEs system is transformed into a comparable system of crisp polynomial equations. The Wu's algorithm is then employed to solve the set of crisp polynomial equations as the solution method. This algorithm results in the solution of characteristic sets that are readily solvable. A key benefit of the proposed method is that all the solutions are obtained simultaneously. The article concludes by presenting some practical examples to demonstrate the efficacy of the proposed method.
... Proving is accomplished with specialized transformations of large polynomials. Gröbner bases 20 and Wu's method 21 are representative approaches in this category, with theoretical guarantees to successfully decide the truth value of all geometry theorems in IMO-AG-30, albeit without a human-readable proof. Because these methods often have large time and memory complexity, especially when processing IMO-sized problems, we report their result by assigning success to any problem that can be decided within 48 h using one of their existing implementations 17 . ...
Proving mathematical theorems at the olympiad level represents a notable milestone in human-level automated reasoning1–4, owing to their reputed difficulty among the world’s best talents in pre-university mathematics. Current machine-learning approaches, however, are not applicable to most mathematical domains owing to the high cost of translating human proofs into machine-verifiable format. The problem is even worse for geometry because of its unique translation challenges1,5, resulting in severe scarcity of training data. We propose AlphaGeometry, a theorem prover for Euclidean plane geometry that sidesteps the need for human demonstrations by synthesizing millions of theorems and proofs across different levels of complexity. AlphaGeometry is a neuro-symbolic system that uses a neural language model, trained from scratch on our large-scale synthetic data, to guide a symbolic deduction engine through infinite branching points in challenging problems. On a test set of 30 latest olympiad-level problems, AlphaGeometry solves 25, outperforming the previous best method that only solves ten problems and approaching the performance of an average International Mathematical Olympiad (IMO) gold medallist. Notably, AlphaGeometry produces human-readable proofs, solves all geometry problems in the IMO 2000 and 2015 under human expert evaluation and discovers a generalized version of a translated IMO theorem in 2004.
... An algebraic structure with a derivation is broadly called a differential algebra [20]. In fact, differential algebra has found important applications in arithmetic geometry, logic and computational algebra, especially in the profound work of Wu on mechanical proof of geometric theorems [30,31]. There are many instances of differential algebras, such as for fields [27], commutative algebras [28], noncommutative algebras [15], lattices [14], and MV-algbras [16]. ...
Let A be an MV-algebra. An -derivation on A is a map satisfying: for all . This paper initiates the study of -derivations on MV-algebras. Several families of -derivations on an MV-algebra are explicitly constructed to give realizations of the underlying lattice of an MV-algebra as lattices of -derivations. Furthermore, -derivations on a finite MV-chain are enumerated and the underlying lattice is described.
... The study of differential algebras is an algebraic approach to differential equations replacing analytic notions like differential quotients by abstract operations, initiated by Ritt [36,37] in 1930s and developed by Kolchin and his school [20]. After the fundamental work of Ritt and Kolchin, differential algebra has evolved into a vast area of mathematics that is important in both theory and applications, including differential Galois theory, differential algebraic geometry, differential algebraic groups [20,32,39] and mechanic theorem proving [41,42]. Guo and Keigher posed the concept of weighted differential algebra, including the usual differential algebra of weight zero and the difference algebra of weight one [16] as examples. ...
In the present paper, we propose the concepts of weighted differential (q-tri)dendriform algebras and give some basic properties of them. The corresponding free objects are constructed, in both the commutative and noncommutative contexts.
... The equations systems are not always linear (if circumferences or distances are involved, second degree equations arise and the corresponding equations system are algebraic but not linear). The best known solving methods in such case are Wu's pseudoremainder method [4,32,33] and Gröbner bases method [2,16] (both even allowing to prove new theorems [23][24][25]). ...
The authors were surprised by the number of articles that used or cited the computer algebra system DERIVE more than 10 years after it was discontinued and developed a small bibliographic study about it, published in 2019. Now they address in a similar way the very successful dynamic geometry system GeoGebra that, although created 20 years ago, later than the other great dynamic geometry systems ( Cabri Geometry II , The Geometer’s Sketchpad and Cinderella ), has now dozens of millions of users around the world. Not surprisingly, the cites to GeoGebra in the well known bibliographic databases Scopus , Web of Science and Google Scholar show an impressive growth.
... It should be mentioned that there are other effective symbolic methods available for the MP problem such as characteristic (triangular) sets [36] and involutive bases [20]. Among non-symbolic solvers for the binary field GF(2), we have well-developed and widely used algorithms such as SAT solvers, besides new promising methods as Quantum Annealing which are about to be fully exploited for the MP problem [34]. ...
In this paper we introduce a multistep generalization of the guess-and-determine or hybrid strategy for solving a system of multivariate polynomial equations over a finite field. In particular, we propose performing the exhaustive evaluation of a subset of variables stepwise, that is, by incrementing the size of such subset each time that an evaluation leads to a polynomial system which is possibly unfeasible to solve. The decision about which evaluation to extend is based on a preprocessing consisting in computing an incomplete Grobner basis after the current evaluation, which possibly generates linear polynomials that are used to eliminate further variables. If the number of remaining variables in the system is deemed still too high, the evaluation is extended and the preprocessing is iterated. Otherwise, we solve the system by a Grobner basis computation. Having in mind cryptanalytic applications, we present an implementation of this strategy in an algorithm called MultiSolve which is designed for polynomial systems having at most one solution. We prove explicit formulas for its complexity which are based on probability distributions that can be easily estimated by performing the proposed preprocessing on a testset of evaluations for different subsets of variables. We prove that an optimal complexity of MultiSolve is achieved by using a full multistep strategy with a maximum number of steps and in turn the classical guess-and-determine strategy, which essentially is a strategy consisting of a single step, is the worst choice. Finally, we extensively study the behaviour of MultiSolve when performing an algebraic attack on the well-known stream cipher Trivium.
... Examples of the synthetic approach can be found in Inter-GPS [18] and GEOS [8], which are solvers with built-in Euclidean formulas. The algebraic approach involves algebraic operations such as Wenjun Wu's method [19] that decomposes problems based on the well-ordering principle and successive pseudo-reduction. However, the algebraic approach has many limitations, and no geometric construction solver has adopted this method thus far. ...
In this paper, we present a visual reasoning framework driven by deep learning for solving constructible problems in geometry that is useful for automated geometry theorem proving. Constructible problems in geometry often ask for the sequence of straightedge-and-compass constructions to construct a given goal given some initial setup. Our EuclidNet framework leverages the neural network architecture Mask R-CNN to extract the visual features from the initial setup and goal configuration with extra points of intersection, and then generate possible construction steps as intermediary data models that are used as feedback in the training process for further refinement of the construction step sequence. This process is repeated recursively until either a solution is found, in which case we backtrack the path for a step-by-step construction guide, or the problem is identified as unsolvable. Our EuclidNet framework is validated on the problem set of Euclidea with an average of 75% accuracy without prior knowledge and complex Japanese Sangaku geometry problems, demonstrating its capacity to leverage backtracking for deep visual reasoning of challenging problems.
... • logical deduction from the axioms (using deduction rules), applicable to all fields of mathematics [22], and • automated theorem proving in geometry (using algebraic methods, like Gröbner bases or Wu's method [23][24][25][26][27][28][29]), also denoted mechanical theorem proving. CAS are the key tool for this line of research. ...
The author began working with computer algebra systems (CAS) in the 80s to perform effective computations for his Ph.D Thesis in algebra. He thought at that moment that there would be an explosion in the use of CAS for research and teaching (at all levels of education). Surprisingly, its use in secondary education is still scarce. This article details some personal reflections on elementary mathematics questions (from both the mathematical and the computational points of view) and proposes a classification of such questions, illustrated with several examples. It is focused on some of the present impressive capabilities of CAS, underlining their abstraction levels in some eye-catching examples. The article is mainly aimed at mathematics teachers who are not experts in CA. Nevertheless, it may also be of interest to CAS experts, as it includes reflections on a topic not usually treated: the abstraction level achieved by CAS and its impact in teaching and assessment.
... In our extension we build on the classical method for translating geometric constructions into algebraic systems, based on the revolutionary work of Wu [39], Chou [9], and improved later by many others (see, for example, the approach of Recio and Vélez [32] through elimination theory, and the references therein). ...
We introduce an experimental version of GeoGebra that successfully conjectures and proves a large scale of geometric inequalities by providing an easy-to-use graphical interface. GeoGebra Discovery includes an embedded version of the Tarski/QEPCAD B system which can solve a real quantifier elimination problem, so the input geometric construction can be translated into a semi-algebraic system, and after some algebraic manipulations, the obtained formula can be translated back to a precisely stated geometric inequality. We provide some non-trivial examples to illustrate the performance of GeoGebra Discovery when dealing with inequalities, as well as some technical difficulties.
... Furthermore, differential algebra has found important applications in arithmetic B Li Guo liguo@rutgers.edu Aiping Gan ganaiping78@163.com 1 geometry, logic and computational algebra, especially in the profound work of W.-T. Wu on mechanical proof of geometric theorems (Wu 1978(Wu , 1987. ...
This paper studies the differential lattice, defined to be a lattice L equipped with a map d:L→L that satisfies a lattice analog of the Leibniz rule for a derivation. Isomorphic differential lattices are studied and classifications of differential lattices are obtained for some basic lattices. Several families of derivations on a lattice are explicitly constructed, giving realizations of the lattice as lattices of derivations. Derivations on a finite distributive lattice are shown to have a natural structure of lattice. Moreover, derivations on a complete infinitely distributive lattice form a complete lattice. For a general lattice, it is conjectured that its poset of derivations is a lattice that uniquely determines the given lattice.
... The Wu's method [44] is based on pseudo-divisions and thus more efficient than the method of Gröbner bases. However the pseudo-divisions only yield the zero locus or radical ideal of the original ideal and hence lose too much algebraic information to solve algebraic problems like the ideal membership problem. ...
The new type of ideal basis introduced herein constitutes a compromise between the Gr\"obner bases based on the Buchberger's algorithm and the characteristic sets based on the Wu's method. It reduces the complexity of the traditional Gr\"obner bases and subdues the notorious intermediate expression swell problem and intermediate coefficient swell problem to a substantial extent. The computation of an S-polynomial for the new bases requires at most word operations whereas word operations are requisite in the Buchberger's algorithm. Here m denotes the upper bound for the numbers of terms both in the leading coefficients and for the rest of the polynomials. The new bases are for zero-dimensional polynomial ideals and based on univariate pseudo-divisions. However in contrast to the pseudo-divisions in the Wu's method for the characteristic sets, the new bases retain the algebraic information of the original ideal and in particular, solve the ideal membership problem. In order to determine the authentic factors of the eliminant, we analyze the multipliers of the pseudo-divisions and develop an algorithm over principal quotient rings with zero divisors.
... In this paper we describe our current work that goes one step forward. In our work we use partly the same algebro-geometric theory that plays a crucial role in GeoGebra's toolset, based on the revolutionary work of Wu [22], Chou [4], and improved later by Recio and Vélez [16] with elimination theory. This is the required algebraic basis for gaining a conjecture on a possibly fixed ratio of two quantities. ...
Comparison of geometric quantities usually means obtaining generally true equalities of different algebraic expressions of a given geometric figure. Today's technical possibilities already support symbolic proofs of a conjectured theorem, by exploiting computer algebra capabilities of some dynamic geometry systems as well. We introduce GeoGebra's new feature, the Compare command, that helps the users in experiments in planar geometry. We focus on automatically obtaining conjectures and their proofs at the same time, including not just equalities but inequalities too. Our contribution can already be successfully used to support teaching geometry classes at secondary level, by getting several well-known and some previously unpublished result within seconds on a modern personal computer.
... Decomposing a polynomial system to finitely many triangular sets with corresponding zero decomposition, called triangular decomposition of polynomial systems, is one of the fundamental tools in computational ideal theory. Wu first proposed a triangulardecomposition algorithm for computing characteristic sets in [28], which was applied to geometry theorem proving. Since then, triangular decomposition methods have been applied successfully to not only geometry theorem proving but also lots of problems with diverse backgrounds, such as automated reasoning, real solution isolation, real solution classification and computing the radical of a polynomial ideal [3, 8-10, 17, 28-35], to name a few. ...
Triangular decomposition with different properties has been used for various types of problem solving, e.g. geometry theorem proving, real solution isolation of zero-dimensional polynomial systems, etc. In this paper, the concepts of strong chain and square-free strong triangular decomposition (SFSTD) of zero-dimensional polynomial systems are defined. Because of its good properties, SFSTD may be a key way to many problems related to zero-dimensional polynomial systems, such as real solution isolation and computing radicals of zero-dimensional ideals. Inspired by the work of Wang and of Dong and Mou, we propose an algorithm for computing SFSTD based on Gr\"obner bases computation. The novelty of the algorithm is that we make use of saturated ideals and separant to ensure that the zero sets of any two strong chains have no intersection and every strong chain is square-free, respectively. On one hand, we prove that the arithmetic complexity of the new algorithm can be single exponential in the square of the number of variables, which seems to be among the rare complexity analysis results for triangular-decomposition methods. On the other hand, we show experimentally that, on a large number of examples in the literature, the new algorithm is far more efficient than a popular triangular-decomposition method based on pseudo-division. Furthermore, it is also shown that, on those examples, the methods based on SFSTD for real solution isolation and for computing radicals of zero-dimensional ideals are very efficient.
... In this paper we describe our current work that goes one step forward. In our work we use partly the same algebro-geometric theory that plays a crucial role in GeoGebra's toolset, based on the revolutionary work of Wu [22], Chou [4], and improved later by Recio and Vélez [16] with elimination theory. This is the required algebraic basis for gaining a conjecture on a possibly fixed ratio of two quantities. ...
Comparison of geometric quantities usually means obtaining generally true equalities of different algebraic expressions of a given geometric figure. Today's technical possibilities already support symbolic proofs of a conjectured theorem, by exploiting computer algebra capabilities of some dynamic geometry systems as well. We introduce GeoGebra's new feature, the Compare command, that helps the users in experiments in planar geometry. We focus on automatically obtaining conjectures and their proofs at the same time, including not just equalities but inequalities too. Our contribution can already be successfully used to support teaching geometry classes at secondary level, by getting several well-known and some previously unpublished result within seconds on a modern personal computer.
... In this extended abstract we describe our current work that focuses on inequalities. In our work we partly use the same algebro-geometric theory that plays a crucial role in GeoGebra's toolset, based on the revolutionary work of Wu [26], Chou [7], and improved later by Recio and Vélez [20] with elimination theory. This is the required algebraic basis for gaining a conjecture on a possibly fixed ratio of two quantities, if equality does not hold between them. ...
We introduce a system of software tools that can automatically prove or discover geometric inequalities. The system, called GeoGebra Discovery, consisting of an extended version of GeoGebra, a controller web service realgeom, and the computational tool Tarski (with the extensive help of the QEPCAD B system) successfully solves several non-trivial problems in Euclidean planar geometry related to inequalities.
... In this extended abstract we describe our current work that focuses on inequalities. In our work we partly use the same algebro-geometric theory that plays a crucial role in GeoGebra's toolset, based on the revolutionary work of Wu [26], Chou [7], and improved later by Recio and Vélez [20] with elimination theory. This is the required algebraic basis for gaining a conjecture on a possibly fixed ratio of two quantities, if equality does not hold between them. ...
We introduce a system of software tools that can automatically prove or discover geometric inequalities. The system, called GeoGebra Discovery, consisting of an extended version of GeoGebra, a controller web service realgeom, and the computational tool Tarski (with the extensive help of the QEPCAD B system) successfully solves several non-trivial problems in Euclidean planar geometry related to inequalities.
... Since Rota proposed his program, further exciting applications of the linear operators that Rota noted above have been found. Differential algebra, originated from an algebraic study of differential equation by Ritt [32] in the 1930's, have been developed by the school of Kolchin and many others into a vast area of research [8,22,37] with applications ranging from logic to arithmetic geometry and mechanical proof of geometric theorems [40,41]. After the promotion of Rota [33], Rota-Baxter algebra experienced a remarkable renaucence this century, most notably by its applications to renormalization of quantum field theory and Stotisctic process [5,9]. ...
Many years ago, Rota proposed a program on determining algebraic identities that can be satisfied by linear operators. After an extended period of dormant, progress on this program picked up speed in recent years, thanks to perspectives from operated algebras and Gr\"obner-Shirshov bases. These advances were achieved in a series of papers from special cases to more general situations. These perspectives also indicate that Rota's insight can be manifested very broadly, for other algebraic structures such as Lie algebras, and further in the context of operads. This paper gives a survey on the motivation, early developments and recent advances on Rota's program, for linear operators on associative algebras and Lie algebras. Emphasis will be given to the applications of rewriting systems and Gr\"obner-Shirshov bases. Problems, old and new, are proposed throughout the paper to prompt further developments on Rota's program.
... In our extension we build on the classical way of translating the geometric setup into an algebraic system, based on the revolutionary work of Wu [19], Chou [5], and improved later by Recio and Vélez [14] with elimination theory. On the other hand, we partly use general purpose real quantifier elimination (RQE) methods to find the best possible geometric constants to conjecture and prove sharp inequalities between two expressions. ...
Supporting automated reasoning in the classroom has a long history in the era of computer algebra. Several systems have been developed and introduced as prototypes at various school levels during the last decades. A breakthrough in using computers to obtain automated proofs is still expected, even if some freely available systems offer easy access to such technical means.
Although there are several systems that successfully generate construction steps for ruler and com-pass construction problems, none of them provides readable synthetic correctness proofs for generated constructions. In the present work, we demonstrate how our triangle construction solver ArgoTriCS can cooperate with automated theorem provers for first order logic and coherent logic so that it generates construction correctness proofs, that are both human-readable and formal (can be checked by interactive theorem provers such as Coq or Isabelle/HOL). These proofs currently rely on many high-level lemmas and our goal is to have them all formally shown from the basic axioms of geometry.
Let A be an MV-algebra. An -derivation on A is a map satisfying: for all . This paper initiates the study of -derivations on MV-algebras. Several families of -derivations on an MV-algebra are explicitly constructed to give realizations of the underlying lattice of an MV-algebra as lattices of -derivations. Furthermore, -derivations on a finite MV-chain are enumerated and the underlying lattice is described.
Popular SMT solvers have achieved great success in tackling Nonlinear Real Arithmetic (NRA) problems, but they struggle when dealing with literals involving highly nonlinear polynomials. Current symbolic-numerical algorithms can efficiently handle the conjunction of highly nonlinear literals but are limited in addressing complex logical structures in satisfiability problems. This paper proposes a new algorithm for SMT(NRA), providing an efficient solution to satisfiability problems with highly nonlinear literals. When given an NRA formula, the new algorithm employs a random sampling algorithm first to obtain a floating-point sample that approximates formula satisfaction. Then, based on this sample, the formula is simplified according to some strategies. We apply a DPLL(T)-based process to all equalities in the formula, decomposing them into several groups of equalities. A fast symbolic algorithm is then used to obtain symbolic samples from the equality sets and verify whether the samples also satisfy the inequalities. It is important to note that we adopt a sampling and rapid verification approach instead of the sampling and conflict analysis steps in some complete algorithms. Consequently, if our algorithm fails to verify the satisfiability, it terminates and returns ‘unknown’. We validated the effectiveness of our algorithm on instances from SMTLIB and the literature. The results indicate that our algorithm exhibits significant advantages on SMT(NRA) formulas with high-degree polynomials, and thus can be a good complement to popular SMT solvers as well as other symbolic-numerical algorithms.
This chapter presents a state of the art in the design of digital environments for mathematics education, with a particular focus on artificial intelligence techniques. A review of the work done in this area over the last few decades highlights current challenges and distinguishes between symbolic approaches and machine learning. About symbolic approaches, we review automatic reasoning tools in geometry and their potential. We also consider the design and research work around the Casyopee environment, and the use of logic programming in the QED-Tutrix intelligent tutoring system. With respect to machine learning, four classes of techniques constitute contemporary AI in computer science. Two examples are discussed: a deep learning system of monument analysis for learning situations in mathematics, technology and art, and a computer classroom simulator that provides a new approach to training teachers.
In this paper a brief overview of tools for automated reasoning in geometry developed in Prolog is given. We argue that Prolog is as a good choice for automated reasoning applications and this argument is justified by the example of the tool ArgoTriCS for automated solving of geometry construction problems, developed by the author of the paper. We point out features which made Prolog suitable for development of the tool ArgoTriCS, and illustrate the important aspects of the tool: specification of the underlying knowledge base and the search procedure. The system ArgoTriCS can solve many different triangle construction problems and output formal specification of construction in GCLC language, as well as construction in JSON format which enables generation of dynamic illustrations.
Triangular decomposition with different properties has been used for various types of problem solving. In this paper, the concepts of pure chains and square-free pure triangular decomposition (SFPTD) of zero-dimensional polynomial systems are defined. Because of its good properties, SFPTD may be a key way to many problems related to zero-dimensional polynomial systems. Inspired by the work of Wang (2016) and of Dong and Mou (2019), the authors propose an algorithm for computing SFPTD based on Gröbner bases computation. The novelty of the algorithm is that the authors make use of saturated ideals and separant to ensure that the zero sets of any two pure chains are disjoint and every pure chain is square-free, respectively. On one hand, the authors prove the arithmetic complexity of the new algorithm can be single exponential in the square of the number of variables, which seems to be among the rare complexity analysis results for triangular-decomposition methods. On the other hand, the authors show experimentally that, on a large number of examples in the literature, the new algorithm is far more efficient than a popular triangular-decomposition method based on pseudodivision, and the methods based on SFPTD for real solution isolation and for computing radicals of zero-dimensional ideals are very efficient.
Preprint Mars 2023. Accepted for publication in “Handbook of Digital Resources in Mathematics Education" edited by Prof. Dr. Birgit Pepin, Prof. Ghislaine Gueudet, Prof. Jeffrey Choppin
This chapter presents a state of the art in the design of digital environments for mathematics education, with a particular focus on artificial intelligence techniques. A review of the work done in this area over the last few decades highlights current challenges and distinguishes between symbolic approaches and machine learning. About symbolic approaches, we review automatic reasoning tools in geometry and their potential. We also consider the design and research work around the Casyopee environment, and the use of logic programming in the QED-Tutrix intelligent tutoring system. With respect to machine learning, four classes of techniques constitute contemporary AI in computer science. Two examples are discussed: a deep learning system of monument analysis for learning situations in mathematics, technology and art, and a computer classroom simulator that provides a new approach to training teachers.
The complex system of linear equations (CSLE) has a wide range of applications in engineering, optimization, operational research, such as circuit analysis and wave equations in quantum mechanics. Since variables and/or parameters of the CSLE are generally unknown, uncertain, or imprecise in real-life applications, Fuzzy CSLE (FCSLE) arises. The fuzziness enables the modeling of the CSLE in a more natural and direct way, and thus, the FCSLE has attracted the attention of many researchers and become a significant area both in theory and application. With this motivation, a short review of the FCSLE has been conducted to guide the future studies of new researchers in this area. This review will give a general framework about the progression of the area, its solution approaches, and provide a bibliography on the topic.
Given its formal, logical and spatial properties, geometry allows an integrated framework where model theory and proof theory approaches can be explored. The development of geometry automatic theorem proving systems and dynamic geometry systems began as separated enterprises but its merging in integrated systems is currently doing its way. In this text, the history of automated deduction in geometry is traced, from the early development of automated theorem provers for geometry and from the emergence of the dynamic geometry systems, to the current status where different application systems combine dynamic geometry and automated deduction. These tools enable their users to explore existing geometry knowledge in addition to creating new constructions and testing new conjectures.KeywordsAutomated Deduction in GeometryDynamic Geometry
Most studies of the time-reversibility are limited to a linear or an affine involution. In this paper, the authors consider the case of a quadratic involution. For a polynomial differential system with a linear part in the standard form (−y, x) in ℝ2, by using the method of regular chains in a computer algebraic system, the computational procedure for finding the necessary and sufficient conditions of the system to be time-reversible with respect to a quadratic involution is given. When the system is quadratic, the necessary and sufficient conditions can be completely obtained by this procedure. For some cubic systems, the necessary and sufficient conditions for these systems to be time-reversible with respect to a quadratic involution are also obtained. These conditions can guarantee the corresponding systems to have a center. Meanwhile, a property of a center-focus system is discovered that if the system is time-reversible with respect to a quadratic involution, then its phase diagram is symmetric about a parabola.
The research field of automated geometry theorem proving has developed many new methods; but, all of them have not used the rings of vector. In the paper, the authors have proposed a new approach based on vector rings, implemented a machine proving program, which emphasis loop of vectors. This program could construct most common constructive geometry drawings very quickly, do automated reasoning with various vector methods according to different types of constructions which includes equal vectors, perpendicular vectors or definite proportional division points, and the proofs are concise and readable. The prover with vectors has been used to produce short and elegant proofs for some constructive constructions. Therefore, this new approach could be used in education. With many instances test, it shows automated reasoning with vectors is available, which also enhance the efficiency and readability.
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