The Web Geometry Laboratory Project

Transcription

1 The Web Geometry Laboratory Project (Intelligent Geometric Tools) Pedro Quaresma CISUC / University of Coimbra Progress in Decision Procedures, 30 March 2013, University of Belgrade, Serbia 1 / 32

2 Intelligent Geometric Tools Dynamic Geometry Systems. 2 / 32

3 Intelligent Geometric Tools Dynamic Geometry Systems. Geometry Automated Theorem Provers. 3 / 32

4 Intelligent Geometric Tools Dynamic Geometry Systems. Geometry Automated Theorem Provers. Repositories of Geometric Constructions. 4 / 32

5 Intelligent Geometric Tools Dynamic Geometry Systems. Geometry Automated Theorem Provers. Repositories of Geometric Constructions. Application on Learning Environments. Wide availability of their corpora. 5 / 32

6 WGL/TGTP/GeoThms WGL/TGTP/GeoThms WGL an adaptative and collaborative blended-learning Web-environment, integrating DGSs and GATPs. TGTP to support the testing and evaluation of GATPs. GeoThms a Web-based environment integrating DGSs, GATPs, and a repository of geometrical constructions. 6 / 32

7 WGL/TGTP/GeoThms WGL/TGTP/GeoThms - Questions raised Web-based environment - wide availability; Adaptative - geometric knowledge; Geometric Search - formal search; geometric inferences on the search; DGSs and GATPs loose integration - interchange formats; Proofs as objects of study - synthetic/visual proofs. 7 / 32

8 Web Geometry Laboratory Web Geometry Laboratory WGL Santos, Vanda and Quaresma, Pedro, Collaborative Aspects of the WGL Project, The Electronic Journal of Mathematics & Technology, LLC, USA, (to appear) 8 / 32

9 Web Geometry Laboratory Web Geometry Laboratory - Question Ahead Adaptative module - The system should be able to infer the geometric knowledge of the users or to use plan recognition, in terms of geometric knowledge, to infer the actual plan or information goal of the users. 9 / 32

10 Web Geometry Laboratory Web Geometry Laboratory - Question Ahead Adaptative module - The system should be able to infer the geometric knowledge of the users or to use plan recognition, in terms of geometric knowledge, to infer the actual plan or information goal of the users. GATPs loose integration with the DGS; usefullness in a learning environment: Construction Soundness; human-readeable proofs; visual proofs. 10 / 32

11 TGTP TGTP TGTP Quaresma, Pedro, Thousands of Geometric Problems for Geometric Theorem Provers (TGTP), Automated Deduction in Geometry, ADG 2010, Munich, Germany, July 22-24, 2010, Revised Selected Papers, pp , 2011, LNCS 6877, Springer 11 / 32

12 TGTP TGTP - Question Ahead taxonomy of methods; common conjecture/proof formats; loose integration of GATPs; conjecture searching looking for lemmas and theorems. 12 / 32

13 GeoThms GeoThms GeoThms Quaresma, Pedro and Janičić, Predrag, GeoThms - a Web System for Euclidean Constructive Geometry, Proceedings of the 7th Workshop on User Interfaces for Theorem Provers (UITP 2006), ENTCS 174-2, pp 35448, 2007, Elsevier 13 / 32

14 GeoThms GeoThms - Question Ahead taxonomy of methods; common conjecture/proof formats; loose integration of DGSs and GATPs; conjecture searching looking for lemmas and theorems. 14 / 32

15 Geometric Search Geometric Search The adaptative module of WGL: geometric knowledge of the users; plan recognition. Geometric Search (databases queries): formal search; geometric inferences on the queries. Given a formal language we need to be able to look for similar construction, sub-constructions or even different construction sharing a common property, e.g., a set of constructions about right angled triangles. 15 / 32

16 Geometric Search Geometric Search Our approach 4 is to transform the geometric construction into a semantic graph representation of the construction, in a given ontology. Graph pattern recognition algorithms can then be used to search for the similarities we need and the results brought back to the geometric setting. 4 Quaresma, Pedro and Haralambous, Yannis, Geometry Constructions Recognition by the Use of Semantic Graphs, Proceedings of RecPad 2012, Coimbra, / 32

17 Geometric Search Geometric Constructions and their Semantic Graph Representations A C B P B A C 17 / 32

18 Geometric Search Geometric Constructions and their Semantic Graph Representations P r CC C A B r BB r AB r AC B BC r C AA r A points (represented by ) and line segments (repr. by ) as concepts, and relations belongs to ( ) and is middle of ( ). 18 / 32

19 Geometric Search Geometric Queries Find a construction containing a triangle. r r r The ontology point ( ), line segment ( ), belongs to ( ) and the semantic graph 19 / 32

20 Geometric Search Geometric Queries Find all constructions containing an intersection of three line segments (or three lines, in a slightly different ontology). r r r The ontology point ( ), line segment ( ), belongs to ( ) and the semantic graph. 20 / 32

21 Common Format i2gatp An XML-Format for Conjectures in Geometry The i2gatp format 5 5 Quaresma, Pedro, An XML-Format for Conjectures in Geometry, Joint Proceedings of the 24 th OpenMath Workshop, the 7 th MathUI Workshop, and the Work in Progress Section of CICM2012, pp , CEUR Workshop Proceedings, 921, Aachen, Germany. 21 / 32

22 Common Format i2gatp An XML-Format for Conjectures in Geometry The i2gatp format 5 An extension of the i2g format; 5 Quaresma, Pedro, An XML-Format for Conjectures in Geometry, Joint Proceedings of the 24 th OpenMath Workshop, the 7 th MathUI Workshop, and the Work in Progress Section of CICM2012, pp , CEUR Workshop Proceedings, 921, Aachen, Germany. 22 / 32

23 Common Format i2gatp An XML-Format for Conjectures in Geometry The i2gatp format 5 An extension of the i2g format; support for geometric conjectures and proofs. 5 Quaresma, Pedro, An XML-Format for Conjectures in Geometry, Joint Proceedings of the 24 th OpenMath Workshop, the 7 th MathUI Workshop, and the Work in Progress Section of CICM2012, pp , CEUR Workshop Proceedings, 921, Aachen, Germany. 23 / 32

24 Common Format i2gatp An XML-Format for Conjectures in Geometry The i2gatp format 5 An extension of the i2g format; support for geometric conjectures and proofs. Converters from/to the format; 5 Quaresma, Pedro, An XML-Format for Conjectures in Geometry, Joint Proceedings of the 24 th OpenMath Workshop, the 7 th MathUI Workshop, and the Work in Progress Section of CICM2012, pp , CEUR Workshop Proceedings, 921, Aachen, Germany. 24 / 32

25 Common Format i2gatp An XML-Format for Conjectures in Geometry The i2gatp format 5 An extension of the i2g format; support for geometric conjectures and proofs. Converters from/to the format; Integration with GeoThms & TGTP & Web Geometry Laboratory; 5 Quaresma, Pedro, An XML-Format for Conjectures in Geometry, Joint Proceedings of the 24 th OpenMath Workshop, the 7 th MathUI Workshop, and the Work in Progress Section of CICM2012, pp , CEUR Workshop Proceedings, 921, Aachen, Germany. 25 / 32

26 Common Format i2gatp An XML-Format for Conjectures in Geometry The i2gatp format 5 An extension of the i2g format; support for geometric conjectures and proofs. Converters from/to the format; Integration with GeoThms & TGTP & Web Geometry Laboratory; Supported by other tools DGSs, GATPs, Quaresma, Pedro, An XML-Format for Conjectures in Geometry, Joint Proceedings of the 24 th OpenMath Workshop, the 7 th MathUI Workshop, and the Work in Progress Section of CICM2012, pp , CEUR Workshop Proceedings, 921, Aachen, Germany. 26 / 32

27 Overall Architecture Overall Architecture I2GATP Format information construction (I2G) name description statement[bibrefs][keywords] elements constraints display conjecture bibentry keyword proofinfo hypothesis ndg conclusion method status limits measures platform i2gatp container: a zip file, superset of the i2g container. 27 / 32

28 Implementation Implementation Graphical Rendering (SVG) Human Language Rendering (HTML) others DGS code (GCLC) GATP code (GCLC AM) DGS code (GeoGebra) others 2 3 Container I2GATP XML files TGTP WGL GeoThms 8 9 GATP code (Coq AM) others 1 From/to GCLC to/from I2G(ATP) 4 SVG rendering 2 From/to GeoGebra to/from I2G(ATP) 3 From/to DGS to/from I2G(ATP) 5 HTML rendering 6 other: proofs; bibrefs., etc. 7 From/to I2GATP to/from GCLC AM 8 From/to I2GATP to/from Coq AM 9 From/to I2GATP to/from GATP 28 / 32

29 Implementation Proofs as Objects of Study The readable theorem proving in geometries has high application value on education 6 How to generate readable machine proofs for geometry theorems automatically is quite challenging and interesting research topic (ibidem) 6 Jiang, Jianguo and Zhang Jingzhong, A Review and Prospect of Readable Machine Proofs for Geometry Theorems, Journal of Systems Science and Complexity, Springer, / 32

30 Implementation Proofs with Geometric Invariants The Area Method for Euclidean constructive geometry (...) can efficiently prove many non-trivial geometry theorems and is one of the most interesting and most successful methods for automated theorem proving in geometry. The method produces proofs that are often very concise and human-readable 7 Full-Angle Method can produce elegant proofs for many extremely difficult geometry theorems. Furthermore, the proofs produced with full-angles are more like the way people solving geometry theorems 8 7 Janičić, Predrag and Narboux, Julien and Quaresma, Pedro, The Area Method: a Recapitulation, JAR, 48-4, pp , Springer 8 Chou, Shang-Ching and Gao, Xiao-Shan and Zhang, Ji, Automated Generation of Readable Proofs with Geometric Invariants, II. Theorem Proving With Full-Angles, JAR, 17, pp , / 32

31 More Questions/Future Research Directions The i2gatp format does not address proofs. Should we try to create common format for proofs? how inference used in proofs by GATPs, interacts with the semantic representation? Geometric Inequalities - up to now, the success of readable machine proving (...) has not been reported! 6 Intelligent Geometry Softwares - The prover of the intelligent geometry softwares had better to be interactive! 6 Machine Learning - provers with analogy learning capability; provers with inductive learning capability! 6 31 / 32

32 Introduction WGL/TGTP/GeoThms Search/Common Format/Readable Proofs More Questions Thank You Thank You 32 / 32