Ivan Petrovi Computer Science Department Faculty of Mathematics

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

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)

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: 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 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)

Geometry Theorem Provers 4/11 _____________ main operation – pseudo division:

Geometry Theorem Provers 5/11 _____________ Wu's method in Win. GCLC application (screen shot of Euler's line theorem)

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). ”

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}

Geometry Theorem Provers 8/11 _____________ Prover output clipping for this example

Geometry Theorem Provers 9/11 _____________ Prover result for this example

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 of integration in other systems for mechanical theorem proving and geometry related software (Geo. Gebra, Geo Thms etc) Directions for further work: possible improvements of current implementation by usage of concurrency implementing Gröbner bases prover

Geometry Theorem Provers 11/11 _____________ Current state of this project: Classes for algebraic primitives are almost completed Prepared utilities for prover output to La. Te. X 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 The End _____________ Thank you.