Mathematical proof is undoubtedly the cornerstone of mathematics. The emergence, in the last years, of computing and reasoning tools, in particular automated geometry theorem provers, has enriched our experience with mathematics immensely. To avoid disparate efforts,the Open Geometry Prover Community Project aims at the integration of the different efforts for the development of geometry automated theorem provers, under a common"umbrella". In this article the necessary steps to such integration are specified and the current implementation of some of those steps is described.
{"title":"Open Geometry Prover Community Project","authors":"Nuno Baeta, P. Quaresma","doi":"10.4204/EPTCS.352.14","DOIUrl":"https://doi.org/10.4204/EPTCS.352.14","url":null,"abstract":"Mathematical proof is undoubtedly the cornerstone of mathematics. The emergence, in the last years, of computing and reasoning tools, in particular automated geometry theorem provers, has enriched our experience with mathematics immensely. To avoid disparate efforts,the Open Geometry Prover Community Project aims at the integration of the different efforts for the development of geometry automated theorem provers, under a common\"umbrella\". In this article the necessary steps to such integration are specified and the current implementation of some of those steps is described.","PeriodicalId":127390,"journal":{"name":"Automated Deduction in Geometry","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123084915","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study how to formalize in the Coq proof assistant the smallest projective space PG(3,2). We then describe formally the spreads and packings of PG(3,2), as well as some of their properties. The formalization is rather straightforward, however as the number of objects at stake increases rapidly, we need to exploit some symmetry arguments as well as smart proof techniques to make proof search and verification faster and thus tractable using the Coq proof assistant. This work can be viewed as a first step towards formalizing projective spaces of higher dimension, e.g. PG(4,2), or larger order, e.g. PG(3,3).
{"title":"Spreads and Packings of PG(3, 2), Formally!","authors":"Nicolas Magaud","doi":"10.4204/EPTCS.352.12","DOIUrl":"https://doi.org/10.4204/EPTCS.352.12","url":null,"abstract":"We study how to formalize in the Coq proof assistant the smallest projective space PG(3,2). We then describe formally the spreads and packings of PG(3,2), as well as some of their properties. The formalization is rather straightforward, however as the number of objects at stake increases rapidly, we need to exploit some symmetry arguments as well as smart proof techniques to make proof search and verification faster and thus tractable using the Coq proof assistant. This work can be viewed as a first step towards formalizing projective spaces of higher dimension, e.g. PG(4,2), or larger order, e.g. PG(3,3).","PeriodicalId":127390,"journal":{"name":"Automated Deduction in Geometry","volume":"138 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124330479","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we prove a geometrical inequality which states that for any four points on a hemisphere with the unit radius, the largest sum of distances between the points is 4+4*sqrt(2). In our method, we have constructed a rectangular neighborhood of the local maximum point in the feasible set, which size is explicitly determined, and proved that (1): the objective function is bounded by a quadratic polynomial which takes the local maximum point as the unique critical point in the neighborhood, and (2): the rest part of the feasible set can be partitioned into a finite union of a large number of very small cubes so that on each small cube the conjecture can be verified by estimating the objective function with exact numerical computation.
{"title":"Maximizing the Sum of the Distances between Four Points on the Unit Hemisphere","authors":"Zhenbing Zeng, Jian Lu, Yaochen Xu, Yuzheng Wang","doi":"10.4204/EPTCS.352.4","DOIUrl":"https://doi.org/10.4204/EPTCS.352.4","url":null,"abstract":"In this paper, we prove a geometrical inequality which states that for any four points on a hemisphere with the unit radius, the largest sum of distances between the points is 4+4*sqrt(2). In our method, we have constructed a rectangular neighborhood of the local maximum point in the feasible set, which size is explicitly determined, and proved that (1): the objective function is bounded by a quadratic polynomial which takes the local maximum point as the unique critical point in the neighborhood, and (2): the rest part of the feasible set can be partitioned into a finite union of a large number of very small cubes so that on each small cube the conjecture can be verified by estimating the objective function with exact numerical computation.","PeriodicalId":127390,"journal":{"name":"Automated Deduction in Geometry","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115865158","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In our contribution we will reflect, through a collection of selected examples, on the potential impact of the GeoGebra Discovery application on different social and educational contexts.
{"title":"GeoGebra Discovery in Context","authors":"Z. Kovács, T. Recio, M. Vélez","doi":"10.4204/EPTCS.352.16","DOIUrl":"https://doi.org/10.4204/EPTCS.352.16","url":null,"abstract":"In our contribution we will reflect, through a collection of selected examples, on the potential impact of the GeoGebra Discovery application on different social and educational contexts.","PeriodicalId":127390,"journal":{"name":"Automated Deduction in Geometry","volume":"66 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131457239","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
One approach to Geometric Discovery starts with a given geometry diagram, and hunts systematically, or unsystematically for provable statements about the geometric entities, or further derived geometric entities [1]. The given diagram can be in fact a parametrized family of diagrams [5]. Another approach [4] is to start with the statements one wants to prove, and discover supplementary conditions required to make the theorems true. Again, however, the geometric milieu is given. A problem for such systems is to determine the interestingness of generated theorems, metrics for which are an active topic of research[2]. In this paper, we consider working in reverse and generating the geometric diagram to match a more abstract form of the theorem, which guarantees both its solution, but also a certain level of interestingness. The abstract form is developed by analogy with known theorems, considered (by this author) to be aesthetically pleasing. We develop and automate here a method for generating many theorems of comparable structure but different geometry to our seed theorems. Hopefully this might lend us some control of the richness and tractability, even aesthetic appeal of our generated theorems. Having promised emergent geometry, we immediately limit the scope of our work, however, to consider theorems in the Naive Angle Method employed by Geometry Expressions [9] for angle specific problems. While the method accommodates a number of different constraint types, in the bulk of this paper, we focus solely on the angle bisector constraint, which can be disguised as an isosceles triangle, a circle chord, or a reflection. In any case, it contributes a row with 3 values -1,-1,2 to the constraint matrix. At one level, we can re-interpret the same matrix using different geometry: for example changing a circle chord into an angle bisector (figure 1). At another level, we consider matrices with non zero elements in the same places, but with different assignments within the row of the numerical values (they will still be -1, -1 , 2, only their order will be different). At a third level, we generalize to consider matrices with a similar pattern of non-zero positions. For a class of such matrices, we give structural conditions which determine the presence or absence of theorems of comparable interest to the prototype.
{"title":"A Method for the Automated Discovery of Angle Theorems","authors":"P. Todd","doi":"10.4204/EPTCS.352.17","DOIUrl":"https://doi.org/10.4204/EPTCS.352.17","url":null,"abstract":"One approach to Geometric Discovery starts with a given geometry diagram, and hunts systematically, or unsystematically for provable statements about the geometric entities, or further derived geometric entities [1]. The given diagram can be in fact a parametrized family of diagrams [5]. Another approach [4] is to start with the statements one wants to prove, and discover supplementary conditions required to make the theorems true. Again, however, the geometric milieu is given. A problem for such systems is to determine the interestingness of generated theorems, metrics for which are an active topic of research[2]. In this paper, we consider working in reverse and generating the geometric diagram to match a more abstract form of the theorem, which guarantees both its solution, but also a certain level of interestingness. The abstract form is developed by analogy with known theorems, considered (by this author) to be aesthetically pleasing. We develop and automate here a method for generating many theorems of comparable structure but different geometry to our seed theorems. Hopefully this might lend us some control of the richness and tractability, even aesthetic appeal of our generated theorems. Having promised emergent geometry, we immediately limit the scope of our work, however, to consider theorems in the Naive Angle Method employed by Geometry Expressions [9] for angle specific problems. While the method accommodates a number of different constraint types, in the bulk of this paper, we focus solely on the angle bisector constraint, which can be disguised as an isosceles triangle, a circle chord, or a reflection. In any case, it contributes a row with 3 values -1,-1,2 to the constraint matrix. At one level, we can re-interpret the same matrix using different geometry: for example changing a circle chord into an angle bisector (figure 1). At another level, we consider matrices with non zero elements in the same places, but with different assignments within the row of the numerical values (they will still be -1, -1 , 2, only their order will be different). At a third level, we generalize to consider matrices with a similar pattern of non-zero positions. For a class of such matrices, we give structural conditions which determine the presence or absence of theorems of comparable interest to the prototype.","PeriodicalId":127390,"journal":{"name":"Automated Deduction in Geometry","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128741569","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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.
{"title":"Supporting Proving and Discovering Geometric Inequalities in GeoGebra by using Tarski","authors":"Christopher W. Brown, Z. Kovács, Róbert Vajda","doi":"10.4204/EPTCS.352.18","DOIUrl":"https://doi.org/10.4204/EPTCS.352.18","url":null,"abstract":"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.","PeriodicalId":127390,"journal":{"name":"Automated Deduction in Geometry","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134044361","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Several tools have been developed to enhance automation of theorem proving in the 2D plane. However, in 3D, only a few approaches have been studied, and to our knowledge, nothing has been done in higher dimensions. In this paper, we present a few examples of incidence geometry theorems in dimensions 3, 4, and 5. We then prove them with the help of a combinatorial prover based on matroid theory applied to geometry.
{"title":"Mechanization of Incidence Projective Geometry in Higher Dimensions, a Combinatorial Approach","authors":"P. Schreck, Nicolas Magaud, David Braun","doi":"10.4204/EPTCS.352.8","DOIUrl":"https://doi.org/10.4204/EPTCS.352.8","url":null,"abstract":"Several tools have been developed to enhance automation of theorem proving in the 2D plane. However, in 3D, only a few approaches have been studied, and to our knowledge, nothing has been done in higher dimensions. In this paper, we present a few examples of incidence geometry theorems in dimensions 3, 4, and 5. We then prove them with the help of a combinatorial prover based on matroid theory applied to geometry.","PeriodicalId":127390,"journal":{"name":"Automated Deduction in Geometry","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125539235","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduced the package/subsystem GeoGebra Discovery to GeoGebra which supports the automated proving or discovering of elementary geometry inequalities. In this case study, for inequality exploration problems related to isosceles and right angle triangle subclasses, we demonstrate how our general real quantifier elimination (RQE) approach could be replaced by a parametric root finding (PRF) algorithm. The general RQE requires the full cell decomposition of a high dimensional space, while the new method can avoid this expensive computation and can lead to practical speedups. To obtain a solution for a 1D-exploration problem, we compute a Groebner basis for the discriminant variety of the 1-dimensional parametric system and solve finitely many nonlinear real (NRA) satisfiability (SAT) problems. We illustrate the needed computations by examples. Since Groebner basis algorithms are available in Giac (the underlying free computer algebra system in GeoGebra) and freely available efficient NRA-SAT solvers (SMT-RAT, Tarski, Z3, etc.) can be linked to GeoGebra, we hope that the method could be easily added to the existing reasoning tool set for educational purposes.
{"title":"Parametric Root Finding for Supporting Proving and Discovering Geometric Inequalities in GeoGebra","authors":"Z. Kovács, Róbert Vajda","doi":"10.4204/EPTCS.352.19","DOIUrl":"https://doi.org/10.4204/EPTCS.352.19","url":null,"abstract":"We introduced the package/subsystem GeoGebra Discovery to GeoGebra which supports the automated proving or discovering of elementary geometry inequalities. In this case study, for inequality exploration problems related to isosceles and right angle triangle subclasses, we demonstrate how our general real quantifier elimination (RQE) approach could be replaced by a parametric root finding (PRF) algorithm. The general RQE requires the full cell decomposition of a high dimensional space, while the new method can avoid this expensive computation and can lead to practical speedups. To obtain a solution for a 1D-exploration problem, we compute a Groebner basis for the discriminant variety of the 1-dimensional parametric system and solve finitely many nonlinear real (NRA) satisfiability (SAT) problems. We illustrate the needed computations by examples. Since Groebner basis algorithms are available in Giac (the underlying free computer algebra system in GeoGebra) and freely available efficient NRA-SAT solvers (SMT-RAT, Tarski, Z3, etc.) can be linked to GeoGebra, we hope that the method could be easily added to the existing reasoning tool set for educational purposes.","PeriodicalId":127390,"journal":{"name":"Automated Deduction in Geometry","volume":"68 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126220686","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The area method is a decision procedure for geometry developed by Chou et al. in the 1990's. The method aims to reduce the specified hypothesis to an algebraically verifiable form by applying elimination lemmas. The order in which the lemmas are applied is determined by the stated conjecture and the underlying geometric construction. In this paper we present our implementation of the area method for Euclidean geometry as a stand-alone Mathematica package.
{"title":"The Area Method in the Wolfram Language","authors":"Jack Heimrath","doi":"10.4204/EPTCS.352.7","DOIUrl":"https://doi.org/10.4204/EPTCS.352.7","url":null,"abstract":"The area method is a decision procedure for geometry developed by Chou et al. in the 1990's. The method aims to reduce the specified hypothesis to an algebraically verifiable form by applying elimination lemmas. The order in which the lemmas are applied is determined by the stated conjecture and the underlying geometric construction. In this paper we present our implementation of the area method for Euclidean geometry as a stand-alone Mathematica package.","PeriodicalId":127390,"journal":{"name":"Automated Deduction in Geometry","volume":"28 4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123593881","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We describe first steps towards a system for automated triangle constructions in absolute and hyperbolic geometry. We discuss key differences between constructions in Euclidean, absolute and hyperbolic geometry, compile a list of primitive constructions and lemmas used for constructions in absolute and hyperbolic geometry, build an automated system for solving construction problems and test it on a corpus of triangle-construction problems. We also provide an online compendium containing construction descriptions and illustrations.
{"title":"On Automating Triangle Constructions in Absolute and Hyperbolic Geometry","authors":"Vesna Marinković, T. Šukilović, Filip Marić","doi":"10.4204/EPTCS.352.3","DOIUrl":"https://doi.org/10.4204/EPTCS.352.3","url":null,"abstract":"We describe first steps towards a system for automated triangle constructions in absolute and hyperbolic geometry. We discuss key differences between constructions in Euclidean, absolute and hyperbolic geometry, compile a list of primitive constructions and lemmas used for constructions in absolute and hyperbolic geometry, build an automated system for solving construction problems and test it on a corpus of triangle-construction problems. We also provide an online compendium containing construction descriptions and illustrations.","PeriodicalId":127390,"journal":{"name":"Automated Deduction in Geometry","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133990560","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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