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Mathematics mechanization and applications after thirty years

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Abstract

The aim of mathematics mechanization is to develop symbolic algorithms for manipulating mathematical objects, proving and discovering theorems in a mechanical way. This paper gives a brief review of the major advances in the field over the past thirty years. The characteristic set method for symbolic solution of algebraic, differential, and difference equation systems are first introduced. Methods for automated proving and discovering geometry theorems are then reviewed. Finally, applications in computer-aided geometric design, computer vision, intelligent computer-aided design, and robotics are surveyed.

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References

  1. Pólya G. Mathematical Discovery. Vol 1. John Wiley & Sons, 1962

  2. Wu W T. Mathematics Machenization. Beijing: Science Press/Kluwer, 2001

    Google Scholar 

  3. Ritt J F. Differential Algebra. New York: AMS Press, 1950

    MATH  Google Scholar 

  4. Wu W T. Basic principles of mechanical theorem-proving in elementary geometries. Sys Sci & Math Scis, 1984, 4: 207–235; also in Journal of Automated Reasoning, 1986, 2: 221–252

    Google Scholar 

  5. Wu W T. Basic Principle of Mechanical Theorem Proving in Geometries. Beijing: Science Press, 1984 (in Chinese); English translation, Wien: Springer, 1994

    Google Scholar 

  6. Wu W T. On the decision problem and the mechanization of theorem-proving in elementary geometry. Scientia Sinica, 1978, 21: 159–172

    MATH  MathSciNet  Google Scholar 

  7. Hsiang J. Herbrand Award for Distinguished Constributions to Automated Reasoning, vi–vii. Automated Deduction-CADE-14. LNAI 1249. Berlin: Springer, 1997

    Google Scholar 

  8. Gao X S, Hou X R, Tang J, et al. Complete solution classification for the perspective-three-point problem. IEEE Tran on PAMI, 2003, 25: 930–943

    Google Scholar 

  9. Kapur D, Mundy J L. Wu’s method and its applications to perspective viewing. Artificial Intelligence, 1988, 37: 15–36

    Article  MATH  MathSciNet  Google Scholar 

  10. Su C, Xu Y, Li H, et al. Application of Wu’s method in computer animation. In: Proceedings of Fifth Int’l Conf. CAD/CG Vol 1. 1997, 211–215

    Google Scholar 

  11. Zhi L, Reid G, Tang J. A complete symbolic-numeric linear method for camera pose determination. In: Proceedings of ISSAC’03. New York: ACM Press, 2003, 215–223

    Google Scholar 

  12. Gao X S, Lin Q, Zhang G. A C-tree decomposition algorithm for 2D and 3D geometric constraint solving. Computer-Aided Design, 2006, 38: 1–13

    Article  Google Scholar 

  13. Chen F, Deng J, Feng Y. Algebraic surface blending using Wu’s method. Computer Mathematics. Singapore: World Scientific, 2000, 172–181

    Google Scholar 

  14. Wu T, Lei N, Cheng J. Wu Wen-tsun formulae for the blending of pipe surfaces. Northeast Math J, 2002, 17: 383–386

    MathSciNet  Google Scholar 

  15. Gao X S, Chou S C. Implicitization of rational parametric equations. Journal of Symbolic Computation, 1992, 14: 459–470

    Article  MATH  MathSciNet  Google Scholar 

  16. Mao W, Wu J. Application of Wu’s method to symbolic model checking. In: Proceedings of ISSAC’05. New York: ACM Press, 2005, 237–244

    Google Scholar 

  17. Gao X S, Lei D, Liao Q, et al. Generalized Stewart-Gough platforms and their direct kinematics. IEEE Trans Robotics, 2005, 21: 141–151

    Article  Google Scholar 

  18. Chou S C, Gao X S. Ritt-Wu’s decomposition algorithm and geometry theorem proving. In: Proceedings of CADE’10. LNCS, No 449. Berlin: Springer-Verlag, 1990, 207–220

    Google Scholar 

  19. Gao X S, Yuan C. Resolvent systems of difference polynomial ideals. In: Proceedings of ISSAC’06. New York: ACM Press, 2006, 101–108

    Google Scholar 

  20. Wu W T. Mechanical theorem proving inelementary differential geometry. Scientia Sinica, 1979, 94–102 (in Chinese)

  21. Boulier F, Lazard D, Ollivier F, et al. Representation for the radical of a finitely generated differential ideal. In: Proceedings of ISSAC’95. New York: ACM Press, 1995, 158–166.

    Google Scholar 

  22. Bouziane D, Kandri Rody A, Maârouf H. Unmixed decomposition of a finitely generated perfect differential ideal. Journal of Symbolic Computation, 2001, 31: 631–649

    Article  MATH  MathSciNet  Google Scholar 

  23. Gao X S, Luo Y. A characteristic set method for difference polynomial systems. In: Inter Conf on Poly Sys Sol, Nov 24–26, Paris, 2004; also in MM-Preprints, 2004, 23: 66–91

    Google Scholar 

  24. Aubry P, Lazard D, Maza M M. On the theory of triangular sets. Journal of Symbolic Computation, 1999, 25: 105–124

    Article  Google Scholar 

  25. Wang D. Elimination Methods. Berlin: Springer, 2000

    Google Scholar 

  26. Dahan X, Maza M M, Schost E, et al. Lifting techniques for triangular decompositions. In: Proceedings of ISSAC’05. New York: ACM Press, 2005, 108–115

    Google Scholar 

  27. Wu W T. On a projection theorem of quasi-varieties in elimination theory. Chinese Annals of Math B, 1990, 11: 220–226

    MATH  Google Scholar 

  28. Gallo G, Mishra B. Efficient algorithms and bounds for Wu-Ritt characteristic sets. In: Progress in Mathematics, 94: Boston: Birkhäuser, 1991, 119–142

    Google Scholar 

  29. Kalkbrener M. A generalized Euclidean algorithm for computing triangular representations of algebraic varieties. Journal of Symbolic Computation, 1993, 15: 143–167

    Article  MATH  MathSciNet  Google Scholar 

  30. Yang L, Zhang J Z, Hou X R. Non-linear Algebraic Equations and Automated Theorem Proving. Shanghai: ShangHai Science and Education Pub, 1996 (in Chinese)

    Google Scholar 

  31. Chen F, Yang W. Applications of interval arithmetic in solving polynomial equations by Wu’s elimination method. Science in China, Ser A, 2005, 48: 1260–1273

    Article  MathSciNet  MATH  Google Scholar 

  32. Wu W T. On a hybrid method of polynomial equations solving. MM-Preprints, 1993, 9: 1–10

    Google Scholar 

  33. Kapur D, Wan H K. Refutational proofs of geometry theorems via characteristic sets. In: Proceedings of ISSAC’90. New York: ACM Press, 1990, 277–284

    Google Scholar 

  34. Li B. An algorithem to decompose a polynomial ascending set into irredncible ones. Acta Anal Funct Appl, 2005, 7: 97–105

    MATH  MathSciNet  Google Scholar 

  35. Wang D. An elimination method for polynomial systems. Journal of Symbolic Computation, 1993, 16(2): 83–114

    Article  MATH  MathSciNet  Google Scholar 

  36. Chou S C, Gao X S. Automated reasoning in differential geometry and mechanics. Journal of Automated Reasoning, 1993, 10: 161–172

    Article  MATH  MathSciNet  Google Scholar 

  37. Hubert E. Factorization-free decomposition algorithms in differential algebra. Journal of Symbolic Computation, 2000, 29: 641–662

    Article  MATH  MathSciNet  Google Scholar 

  38. Wang D. A method for proving theorems in differential geometry and mechanics. J Univ Comput Sci, 1995, 9: 658–673

    Google Scholar 

  39. Richardson D. Wu’s method and the Khovanskii finiteness theorem. Journal of Symbolic Computation, 1991, 12: 127–141

    Article  MATH  MathSciNet  Google Scholar 

  40. Gao X S, Wang D K, Qiao Z, et al. Equation Solvings and Theorem Provings-Problem Solvings with MMP. Beijing: Science Press, 2006 (in Chinese)

    Google Scholar 

  41. Wang D K. Wsolve: A Maple Package for Solving System of Polynomial Equations. http://www.mmrc.iss.ac.cn/dwang/wsolve.htm. 2004

  42. Wang D. Elimination Practice: Software Tools and Applications. London: Imperial College Press, 2004

    MATH  Google Scholar 

  43. Lu Z, He B, Luo Y. Real Roots Isolating for Polynomial Systems and Applications. Beijing: Science Press, 2004 (in Chinese).

    Google Scholar 

  44. Wu W T. On the foundation of algebraic differential geometry. Sys Sci & Math Scis, 1989, 2: 289–312

    MATH  Google Scholar 

  45. Gao X S, Chou S C. Solving parametric algebraic systems. In: Proceedings of ISSAC’92. New York: ACM Press, 1992, 335–341

    Google Scholar 

  46. Wu W T. A mechanization method of geometry and its applications, I. Distances, areas, and volumes in Euclidean and non-Euclidean Geometries. Kuxue Tongbao, 1986, 32: 436–440

    Google Scholar 

  47. Chou S C. Mechanical Geometry Theorem Proving. Dordrecht: D Reidel, 1988

    MATH  Google Scholar 

  48. Li Z. Mechanical theorem proving of the local theory of surfaces. Ann Math Artif Intell, 1995, 13: 25–46

    Article  MATH  Google Scholar 

  49. Chou S C, Gao X S, Zhang J Z. Machine Proofs in Geometry. Singapore: World Scientific, 1994

    MATH  Google Scholar 

  50. Richter-Gebert J. Mechanical theorem proving in projective geometry. Ann Math and Al, 1995, 13: 139–172

    MATH  MathSciNet  Google Scholar 

  51. Li H, Cheng M. Clifford algebraic reduction method for mechanical theorem proving in differential geometry. Journal of Automated Reasoning, 1998, 21: 1–21

    Article  MathSciNet  Google Scholar 

  52. Li H. Vectorial equation-solving for mechanical geometry theorem proving. Journal of Automated Reasoning, 2000, 25: 83–121

    Article  MATH  MathSciNet  Google Scholar 

  53. Li H, Hestenes D, Rockwood A. Generalized homogeneous coordinates for computational geometry. Geometric Computing with Clifford Algebra. Berlin: Springer, 2000, 27–60

    Google Scholar 

  54. Li H, Wu Y. Automated theorem proving in projective geometry with Cayley and bracket algebras. Journal of Symbolic Computation, 2004, 36: 717–762

    Article  MathSciNet  Google Scholar 

  55. Gerlentner H, Hanson J R, Loveland D W. Empirical explorations of the geometry-theorem proving machine. In: Proceedings of West Joint Computer Conf. 1960, 143–147

  56. Chou S C, Gao X S, Zhang J Z. A deductive database approach to automated geometry theorem proving and discovering. Journal Automated Reasoning, 2000, 25: 219–246

    Article  MATH  MathSciNet  Google Scholar 

  57. Gao X S, Zhang J Z, Chou S C. Geometry Expert. Teipei: Nine Chapters Pub, 1998 (in Chinese)

    Google Scholar 

  58. Collins G E. Quantifier elimination for real closed fields by cylindrical algebraic decomposition. LNCS, No 33. Berlin: Springer-Verlag, 1975, 134–183

    Google Scholar 

  59. Dolzmann A, Sturm T, Weispfenning V. A new approach for automatic theorem proving in real geometry. Journal of Automated Reasoning, 1998, 21: 357–380

    Article  MATH  MathSciNet  Google Scholar 

  60. Wu W T. On a finiteness theorem about optimization problems. Sys Sci & Math Scis, 1994, 7: 193–200

    MATH  Google Scholar 

  61. Yang L, Hou X, Zeng Z. Complete discriminant systems. Science in China, Ser E, 1996, 39(6): 628–646

    MATH  MathSciNet  Google Scholar 

  62. Yang L, Hou X, Xia B. A complete algorithm for automated discovering of a class of inequality-type theorems. Science in China, Ser F, 2001, 44: 33–49

    MATH  MathSciNet  Google Scholar 

  63. Xu C, Shi Q, Cheng M. A global stereo vision method based on Wu-solver. In: Proceedings of GMICV’95. 1995, 198–205

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Authors and Affiliations

  1. Key Laboratory of Mathematics Mechanization, Institute of Systems Science, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, 100080, China

    Wu Wenjun & Gao Xiaoshan

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  1. Wu Wenjun
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  2. Gao Xiaoshan
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Correspondence to Gao Xiaoshan.

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Wu, W., Gao, X. Mathematics mechanization and applications after thirty years. Front. Comput. Sc. China 1, 1–8 (2007). https://doi.org/10.1007/s11704-007-0001-8

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  • DOI: https://doi.org/10.1007/s11704-007-0001-8

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