Xiao-Shan Gao MMRC Institute of System Science, Academia Sinica

1. MMP/Geometer Automated Generation for Geometric Diagrams and Theorem Proofs Xiao-Shan Gao MMRC Institute of System Science, Academia Sinica

2. Outline of the talk • MMP: A Brief Introduction • MMP/Geometer: Diagram Generation • MMP/Geometer: Proof Generation • Demonstration

3. 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

5. Characteristic set method Wu-Ritt’s zero decompostion theorem Zero(PS) = ∪Zero(CSk/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

6. MMP Application Modules • 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.

7. MMP/Geometer Goal: automate basic geometric activities: • Geometric diagram generation • Geometric theorem proving • Geometric theorem discovering To make geometry alive!

8. 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

9. 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

10. Limitation of DG • Ruler and compass construction Ruler compass construction does not exit Difficult to find ruler compass construction

11. 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.

12. 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 geometry textbooks. Algorithm LIM0

13. AGDG Methods: phase 1 Find a Ruler and Compass construction Algorithm TRANS • Use Rigid Body Tran, Angle Tran, Parallelogram Tran to solve the problem. • This is a quadratic algorithm • Complete for drawing problems of simple polygons

14. An Example of Parallel Transformation

15. AGDG Methods: phase 2 Numerical Computation Step 1: Generate a GCS: • 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

16. AGDG Methods: phase 2 Numerical Computation Step 2: Compute the GCS: • 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

17. An Examples

18. 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

19. An Application: Intelligent CAD Ruler and Compass Construction: Appolonius problem and CAD Open Problem：Geometric Solution to RC construction

20. 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

21. 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.

22. Wu’s Method Geometry Theorem: HYP => C coordinates Algebraic Statement: PS=0 => G=0 characteristic set method Automated Proof

23. 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

24. 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 => FG ，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.

25. Area Method • Automated Produced Proof: AO/CO = SDBA /SDCB=SCBA /SCBA= 1 Chou,Gao,Zhang: Machine Proof in Geometry, World Scientific, 1994.

26. Deductive Database Method R: Geometric Axiom/Rules D0: Hypotheses of a geometric theorem Forward Chaining R R R D0 D1 D2 Dk = Dk+1 D1＝All properties obtained from D0 with R R(Dk)= DkFIXPOIT of reasoning Chou,Gao,Zhang: A Deductive Geometry Database, JAR, 2000.

27. Experiment Results

28. Why Use More Methods? • Each method has advantage and limitation 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)

29. An Application: Stewart Platform • Positive solution to Stewart Platform is still open (Locus) • Has applications in: • NC Machine, • Nano technology, • Large scale telescope

30. An Application: Stewart Platform Virtual NC Machine • “NC machine of the 21 century” • “Machine made of mathematics”

31. Demonstration

32. What is the Locus of the OrthocenterWhen 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]], [], []]);

33. 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");

34. 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.

35. 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)]]);

36. A space curve satisfies t=k'=0 is a circle. curve(); wprove_curve([[],[],[],[t,diff(k,s)],[], [[FIX_PLANE,C],[FIX_SPHERE,C]]]);

37. Thanks！