java implementation of wu's method for automated theorem proving in geometry

Ivan Petrović

Computer Science DepartmentFaculty of MathematicsUniversity of Belgrade

February 5th, 2011

Java implementation of Wu's method for

Automated Theorem Proving in Geometry

Geometry Theorem Provers 1/11

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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

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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

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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

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main operation – pseudo division:

Geometry Theorem Provers 5/11

_________________________Wu's method in WinGCLC application

(screen shot of Euler's line theorem)

Geometry Theorem Provers 6/11

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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 20cmark_b Apoint B 50 20cmark_b B point C 40 70cmark_t C

drawsegment A Bdrawsegment B Cdrawsegment C A

med mab A Bmed mac A Cmed mbc B C

drawline mabdrawline macdrawline mbc

intersec M_1 mab macintersec M_2 mab mbc

cmark_rt M_1cmark_lb M_2

prove {identical M_1 M_2}

Geometry Theorem Provers 8/11

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Prover output clipping for this example

Geometry Theorem Provers 9/11

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Prover result for this example

Geometry Theorem Provers 10/11

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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 11/11

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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 The End

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Thank you.