1. Ivan Petrović Computer Science Department Faculty of Mathematics University of Belgrade February 5 th, 2011 Java implementation of Wu's method for Automated Theorem Proving in Geometry

2. Geometry Theorem Provers 1/11 _________________________  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)

3. Geometry Theorem Provers 2/11 _________________________ 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:  u s – independent (parametric) variables  x s – dependent variables  step 2 – triangulation of polynomial system (each next equation introduces exactly one new dependent variable) by using pseudo division 

4. Geometry Theorem Provers 3/11 _________________________  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)

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. Geometry Theorem Provers 6/11 _________________________ 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).”

8. Geometry Theorem Provers 7/11 _________________________ 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}

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

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

11. Geometry Theorem Provers 10/11 _________________________ 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

12. Geometry Theorem Provers 11/11 _________________________ 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

13. Geometry Theorem Provers The End _________________________ Thank you.