Java implementation of Wu's method for Automated Theorem Proving in Geometry

1. Java implementation of Wu's method for Automated Theorem Proving in Geometry Ivan PetrovićComputer Science DepartmentFaculty of MathematicsUniversity of BelgradeFebruary 5th, 2011

2. Two categories of provers: algebraic (coordinate-based) methods coordinate-free methods Main algebraic methods: Wu's method (Wen-Tsun Wu) Gröbner bases method (Bruno Buchberger) Main coordinate-free methods: Area method (Shang-Ching Chou, Xiao-Shan Gao, Jing-Zhong Zhang) Full-Angle method (same authors) Geometry Theorem Provers 1/11_________________________

3. Wu's method is powerful mechanism for proving geometry theorems in elementary geometry. It is complete decision procedure for some classes of geometry problems. How Wu's method works? step 1 – translate geometry problem into multivariate polynomial system two types of variables: us – independent (parametric) variables xs– dependent variables step 2 – triangulation of polynomial system (each next equation introduces exactly one new dependent variable) by using pseudo division Geometry Theorem Provers 2/11_________________________

4. step 3 – calculating final reminder of polynomial that represents statement with each polynomial from triangulated system, by using pseudo division of polynomials step 4 – producing answer on the basis of final reminder and obtained non-degenerative conditions (zero reminder means proved theorem) Geometry Theorem Provers 3/11_________________________

5. Geometry Theorem Provers 4/11_________________________ • main operation – pseudo division:

6. Geometry Theorem Provers 5/11_________________________ Wu's method in WinGCLC application (screen shot of Euler's line theorem)

7. Simple example of Wu's method: [Theorem about circumcenter of a triangle] “The tree perpendicular bisectors of a triangle's sides meet in a single point (they are concurrent lines).” Geometry Theorem Provers 6/11_________________________

8. Construction written in GCLC: point A 20 20 cmark_b A point B 50 20 cmark_b B point C 40 70 cmark_t C drawsegment A B drawsegment B C drawsegment C A med mab A B med mac A C med mbc B C drawline mab drawline mac drawline mbc intersec M_1 mab mac intersec M_2 mab mbc cmark_rt M_1 cmark_lb M_2 prove {identical M_1 M_2} Geometry Theorem Provers 7/11_________________________

9. Prover output clipping for this example Geometry Theorem Provers 8/11_________________________

10. Prover result for this example Geometry Theorem Provers 9/11_________________________

11. Reimplementation in Java programming language (based on C++ version by Goran Predović and Predrag Janičić) Main objectives of this project: greater portability ability of integration in other systems for mechanical theorem proving and geometry related software (GeoGebra, Geo Thms etc) Directions for further work: possible improvements of current implementation by usage of concurrency implementing Gröbner bases prover Geometry Theorem Provers 10/11_________________________

12. Current state of this project: Classes for algebraic primitives are almost completed Prepared utilities for prover output to LaTeX and XML format Implemented pseudo reminder algorithm; after implementation of simple triangulation algorithm, Wu's method is almost completed At the end dealing with transformation of GCLC input into polynomial form Geometry Theorem Provers 11/11_________________________

13. Thank you. Geometry Theorem Provers The End_________________________