Xiao-Shan GaoMMRC

Institute of System Science, Academia Sinica

MMP/GeometerAutomated Generation for

Geometric Diagrams and Theorem Proofs

Outline of the talk

• MMP: A Brief Introduction

• MMP/Geometer: Diagram Generation

• MMP/Geometer: Proof Generation

• Demonstration

MMP: Mathmatics Mechanization Platform

• Is a standalone software system (Windows/C)

• With Wu-Ritt characteristic set (CS) method as the Core Method

aims to mechanize

• Geometry theorem proving and algebraic and differential equation solving

with applications in

• science and engineering

Long integerPolynomial operations

Linear algebraGUI and program lanugae

Suppor t i ng

Algebraic CaseAlgebraic Ordinary DEs

Algebraic PDEs

CS Al gor i t hms

Automated Geometric Theorem ProvingRobotics

Surface Fitting...

Appl i cat i on Modul es

MMP

Characteristic set method

Wu-Ritt’s zero decompostion theorem

Zero(PS) = Zero(CS∪ k/Ik)=Zero(Sat(CSk))

Equation Systems Triangular Form

P1(x1,…,xn)=0 T1(x1)=0

P2 (x1,…,xn)=0 T2 (x1,x2)=0

… …

Ps (x1,…,xn)=0 Tn (x1,…,xn)=0

WSOLVE Package by Dingkang Wang

• MMP/Geometer. Geometric theorem proving, discovering, and diagram generation in Euclidean geometries and differential geometry.

• MPP/Solition. Find the soliton and traveling-wave solutions for non-linear PDEs and approximate analytical solutions.

•  MMP/RealRoot. Find the number of real solutions for a system of algebraic equations

• MMP/Linkage. Linkage synthesis

• MMP/Robots. Simulate 6R serial robotic arms

• MMP/Blending. Blend surfaces automatically.

MMP Application Modules

MMP/Geometer

• Geometric diagram generation

• Geometric theorem proving

• Geometric theorem discovering

Goal: automate basic geometric activities:

To make geometry alive!

AGDG -Automated geometric diagram generation

• “A picture is more than one thousand words."

• In reality, it is still difficult to generate pictures with computer software, especially for pictures with exact geometric relations

Dynamic Geometry Software

• Geometric models built by software that can be changed dynamically.

• Basic Operations: dynamic transformation, dynamic measurement, free dragging, and animation.

• DG Software: Gabri, Geometer's Sketchpad, Geometry Expert, Cinderella

Limitation of DG

• Ruler and compass construction

Difficult to find ruler compass construction

Ruler compass construction does not exit

Intelligent Dynamic Geometry

• Combine idea of dynamic geometry and AGDG methods

• Basic Features: automated generation of ruler and compass construction, general methods for diagram construction. (Intelligent Dragging)

• Manipulate geometric diagrams interactively as DG software and does not have the limitation of ruler and compass construction.

AGDG Methods: phase 1 Find a Ruler and Compass construction

• Repeatedly remove those geometric objects that can be constructed explicitly.

DEG(v) DOF(V)• This is a linear algorithm• Solves about eighty percent of the problems in geo

metry textbooks.

Algorithm LIM0

AGDG Methods: phase 1 Find a Ruler and Compass construction

• Use Rigid Body Tran, Angle Tran, Parallelogram Tran to solve the problem.

• This is a quadratic algorithm

• Complete for drawing problems of simple polygons

Algorithm TRANS

An Example of Parallel Transformation

AGDG Methods: phase 2 Numerical Computation

• Use graph theory to decompose the problem into general construction sequence (GCS):

C1,C2,…,Cm

Ci are sets of geometric objects such that

• Ci can be constructed from C1…Ci-1

• C1…Ci form a rigid

Step 1: Generate a GCS:

AGDG Methods: phase 2 Numerical Computation

• Solving a set of algebraic equations:

f1(X)=0, … ,fm(X)=0

• Let S(X) = fi2

• Use optimization method to find a minimal value: S(X0): minimal

• If S(X0) =0, we found a set of solution

Step 2: Compute the GCS:

An Examples

Level of cirs # of Quad Eqs Time (Sec)

2 30 0.228

3 54 0.965

4 86 3.379

5 126 11.58

6 174 23.75

Automated Geometry Reasoning

• Geometry theorem proving is considered as one of the hardest mental labor.

• Euclid: There is no royal road to geometry!

• Geometry is considered the model of axiomazition and rigorous reasoning.

• It is a benchmark to test a reasoning method

An Application: Intelligent CAD

Ruler and Compass Construction: Appolonius problem and CAD

Open Problem ： Geometric Solution to RC construction

Automated Geometry Reasoning

• Wu's method: a coordinate-based method. Applies to Euclidean and differential geoms.

• Area method: use geometric invariants to prove theorems; can be used to produce human-readable proofs.

• Deductive database method: Generate fixpoint for a geometric figure; produce proofs in traditional style

MMP/Geometer

• Proved thousands of geometry theorems in elementary, differential geometries and mechanics

• Automated discover of geometric properties• Generate human readable proofs• Generate multiple and shortest proofs

Invites comparison with the best of human geometry provers.

Wu’s Method

Geometry Theorem: HYP => C

Algebraic Statement: PS=0 => G=0

Automated Proof

coordinates

characteristic set method

Wu’s Method: Implementation

• WU-C: For constructive statement

• WU-G: General version of Wu’s method

• WU-F: Discover geometric formula

• WU-D: General version for differential geometry

First Order Theory for Geometry

• Basic Statement ： collinear ， parallel ，equal distance, etc

• F is a statement => (f) is also a statement

• F,G are statement => FG ， F G are also statements

• F is a statement => x(f) 对 x(f) are also statements

Wu’s method is complete for the first order theory of geometry over the complex numbers.

Area Method

• Automated Produced Proof:

AO/CO = SDBA /SDCB=SCBA /SCBA= 1

Chou,Gao,Zhang: Machine Proof in Geometry, World Scientific, 1994.

Deductive Database Method

R: Geometric Axiom/Rules

D0 D1 D2 Dk = Dk+1

D1 ＝ All properties obtained from D0 with R

R(Dk)= Dk FIXPOIT of reasoning

Forward Chaining

R R R

Chou,Gao,Zhang: A Deductive Geometry Database, JAR, 2000.

D0: Hypotheses of a geometric theorem

Experiment Results

Method Number of Theorems Proved

WU-C 500

WU-G 400

WU-D 100

WU-F 120

AREA 400

GDD 170

Why Use More Methods?

WU-C WU-G AREA DBASE

Prove Power: Decrease

Proof Quality: Increase

• Produce a variety of proofs with different styles for the same theorem (for CAI)

•Each method has advantage and limitation

An Application: Stewart Platform

• Positive solution to Stewart Platform is still open (Locus)

• Has applications in:• NC Machine, • Nano technology,• Large scale telescope

Virtual NC Machine

• “NC machine of the 21 century”

• “Machine made of mathematics”

An Application: Stewart Platform

Demonstration

What is the Locus of the Orthocenter

When a vertex moves on a circle?

wderive([[y1,x1,y],[x,a,u,v,r],

[A,[0,0],B,[a,0],C,[x1,y1],H,[x,y],O,[u,v]],

[[perp,A,H,B,C],[perp,B,H,A,C],[dis,O,C,r]], [], []]);

Orthocenter theorem in natrual language

wcprove("Example Orthocenter. Let ABC be a triangle. Point E is the foot from point A to line BC. Point F is the foot from point B to line AC. Point H is the intersection of line AE and line BF. Show that CH is perpendicular to AB");

Kepler's experimental laws imply Newton's gravitational law

Kepler's laws:

K1:Each planet describes an ellipse with the sun in one focus.

K2:The radius vector drawn from the sun to a planet sweeps out equal areas in equal times.

Newton’s gravitational law:

The force is proportional to the inverse of the square of the distance from the sun to the panet.

Kepler's experimental laws imply Newton's gravitational law

restart;

depend([a,r,y,x],[t]);

wdprove([[a,r,y,x,p,e],[],[], [r^2-x^2-y^2,

a^2-x[2]^2-y[2]^2, x*y[2]-x[2]*y, r-p-e*x], [p],[diff(a*r^2,t)]]);

A space curve satisfies t=k'=0 is a circle.

curve();

wprove_curve([[],[],[],[t,diff(k,s)],[],

[[FIX_PLANE,C],[FIX_SPHERE,C]]]);

Thanks ！