xiao-shan gao mmrc institute of system science, academia sinica
DESCRIPTION
MMP/Geometer Automated Generation for Geometric Diagrams and Theorem Proofs. Xiao-Shan Gao MMRC Institute of System Science, Academia Sinica. Outline of the talk. MMP: A Brief Introduction MMP/Geometer: Diagram Generation MMP/Geometer: Proof Generation Demonstration. - PowerPoint PPT PresentationTRANSCRIPT
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 !