Skip to main content
SpringerLink
Log in

A review and prospect of readable machine proofs for geometry theorems

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

After half a century research, the mechanical theorem proving in geometries has become an active research topic in the automated reasoning field. This review involves three approaches on automated generating readable machine proofs for geometry theorems which include search methods, coordinate-free methods, and formal logic methods. Some critical issues about these approaches are also discussed. Furthermore, the authors propose three further research directions for the readable machine proofs for geometry theorems, including geometry inequalities, intelligent geometry softwares and machine learning.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. D. Hilbert, Foundations of Geometry, Open Court Publishing Company, La Salla, Illinois, 1971.

    MATH  Google Scholar 

  2. W. T. Wu, Mathematics Mechanization, Science Press, Beijing, 2003.

    Google Scholar 

  3. A. Tarski, A Decision Method for Elementary Algebra and Geometry, The RAND Corporation Press, Santa Monica, 1948.

    MATH  Google Scholar 

  4. A. Seidenberg, A new decision method of elementary algebra, Annals of Mathematics, 1954, 60: 365–371.

    Article  MathSciNet  MATH  Google Scholar 

  5. G. Collins, Quantifier elimination for real closed fields by cylindrical algebraic decomposition, Lecture Notes Computer Sciences 33, Springer-Verlag, Berlin, Heidelberg, 1975.

    Google Scholar 

  6. D. Arnon, Geometric reasoning with logic and algebra, Artificial Intelligence, 1988, 37: 37–60.

    Article  MathSciNet  MATH  Google Scholar 

  7. W. T. Wu, On the decision problem and the mechanization of theorem proving in elementary geometry, Journal of Systems Science and Mathematical Science, 1978, 21(16): 157–179 (in Chinese).

    Google Scholar 

  8. W. T. Wu, Basic Principles of Mechanical Theorem Proving in Geometries, Science Press, Beijing, 1984 (in Chinese).

    Google Scholar 

  9. J. Ritt, Differential Algebra, American Mathematical Society, New York, 1950.

    MATH  Google Scholar 

  10. S. C. Chou, An introduction to Wu’s method for mechanical theorem proving in geometry, Journal of Automated Reasoning, 1988, 4: 237–267.

    Article  MathSciNet  MATH  Google Scholar 

  11. S. C. Chou, Proving Elementary Geometry Theorems Using Wu’s Algorithm, Automated Theorem Proving: After 25 Years (ed. by W. Bledsoe and D. Loveland), AMS Contemporary Mathematics Series, 1984, 29: 243–286.

  12. S. C. Chou, Mechanical Geometry Theorem Proving, D. Reidel Publishing Company, Dordrecht, Netherlands, 1988.

    MATH  Google Scholar 

  13. B. Kutzler and S. Stifter, On the application of Buchberger’s algorithm to automated geometry theorem proving, Journal of Symbolic Computation, 1986, 2: 389–397.

    Article  MathSciNet  MATH  Google Scholar 

  14. B. Buchberger, G. Collins, and B. Kutzler, Algebraic methods for geometric reasoning, Annual Review of Computer Sciences, 1995, 3(19): 85–119.

    Google Scholar 

  15. D. Kapur, T. Saxena, and L. Yang, Algebraic and geometric reasoning with dixon resultants, Proceedings of International Symposium on Symbolic and Algebraic Computation (Oxford, 1994), ACM Press, New York, 1994 99–107.

    Google Scholar 

  16. J. Z. Zhang, L. Yang, and X. R. Hou, The sub-resultant method for automated theorem proving, Journal of Systems Science and Mathematical Sciences, 1995, 15(1): 10–15. (in Chinese).

    MathSciNet  MATH  Google Scholar 

  17. D. M. Wang, Elimination procedures for mechanical theorem proving in geometry, Annals of Mathematics and Artificial Intelligence, 1995, 13: 1–24.

    Article  MathSciNet  MATH  Google Scholar 

  18. J. W. Hong, Can geometry be proved by an example? Scientia Sinica, 1986, 29: 824–834.

    MATH  Google Scholar 

  19. J. Z. Zhang and L. Yang, Principles of parallel numerical method and single-instance method of mechanical theorem proving, Mathematics in Practice and Theory, 1989, 1: 34–43.

    Google Scholar 

  20. J. Z. Zhang, L. Yang, and M. Deng, The parallel numerical method of mechanical theorem proving, Theoretical Computer Science, 1990, 74: 253–271.

    Article  MathSciNet  MATH  Google Scholar 

  21. H. Gelernter and N. Rochester, Intelligent behavior in problem-solving machines, IBM Journal, 1958: 336–345.

  22. H. Gelernter, Realization of a Geometry Theorem Machine, Computers and Thought (ed. by E. Feigenbaum and J. Feldman), McGraw-Hill, New York, 1963.

    Google Scholar 

  23. H. Gelernter, J. Hansen, and D. Loveland, Empirical Explorations of the Geometry Theorem Proving Machine, Computers and Thought (ed. by E. Feigenbaum and J. Feldman), McGraw-Hill, New York, 1963.

    Google Scholar 

  24. S. C. Chou, X. S. Gao, and J. Z. Zhang, Automated Production of Traditional Proofs for Theorems in Euclidean Geometry, IV, A Collection of 400 Geometry Theorems, TR-92-7, Department of Computer Science, WSU, 1992.

  25. S. C. Chou, X. S. Gao, and J. Z. Zhang, Automated production of traditional proofs for constructive geometry theorems, Proceedings of Eighth IEEE Symposium on Logic in Computer Science, IEEE Computer Society Press, 1993.

  26. J. Z. Zhang, X. S. Gao, and S. C. Chou, The geometric information search system by forward reasoning, Chinese Journal of Computers, 1996, 19(10): 721–727 (in Chinese).

    Google Scholar 

  27. S. C. Chou, X. S. Gao, and J. Z. Zhang, A deductive database approach to automated geometry theorem proving and discovering, Journal of Automated Reasoning, 2000, 25(3): 219–246.

    Article  MathSciNet  MATH  Google Scholar 

  28. H. Goldstein, Elementary Geometry Theorem Proving, AIM-280, MIT, Cambridge, Massachusetts, 1973.

    Google Scholar 

  29. J. Anderson, C. Boyle, A. Corbett, and M. Lewis, The geometry tutor, Proceedings of IJCAI’85, Los Angles, 1985.

  30. P. Gilmore, An examination of the geometry theorem proving machine, Artificial Intelligence, 1970, 1(2): 171–187.

    Article  MathSciNet  MATH  Google Scholar 

  31. A. Nevins, Plane geometry theorem proving using forward chaining, Artificial Intelligence, 1975, 6(1): 1–23.

    Article  MathSciNet  MATH  Google Scholar 

  32. S. Ullman, Model-Driven Geometry Theorem Prover, AIM-321, MIT, Cambridge, Massachusetts, 1975.

    Google Scholar 

  33. H. Coelho and L. Pereira, Automated reasoning in geometry theorem proving with prolog, Journal of Automated Reasoning, 1986, 2(4): 329–390.

    Article  MathSciNet  MATH  Google Scholar 

  34. T. Evans, A Heuristic Program to Solve Geometry Analogy Problems, Semantic Information Processing (ed. by M. Minsky), MIT Press, Cambridge, 1969.

    Google Scholar 

  35. J. Anderson, Tuning of Search of the Problem Space for Geometry Proofs, Proceeding of IJCAI’81, Vancouver, 1981.

  36. Y. Zheng, S. C. Chou, and X. S. Gao, Visually dynamic presentation of proofs in plane geometry — part 2. Automated generation of visually dynamic presentation with the Full-Angle method and the deductive database method, Journal of Automated Reasoning, 2010, 45(3): 243–266

    Article  MathSciNet  MATH  Google Scholar 

  37. X. S. Gao, J. Z. Zhang, and S. C. Chou, Geometry Expert, Nine Chapters Publishing, TaiPei, Taiwan, 1998 (in Chinese).

    Google Scholar 

  38. J. G. Jiang and J. Z. Zhang, The automated geometry reasoning network based on equivalent class reasoning, Patttern Recognition and Artificial Intelligence, 2006, 19(5): 617–622 (in Chinese).

    Google Scholar 

  39. J. G. Jiang and J. Z. Zhang, The automated geometry reasoning system based on RETE algorithm, Journal of Sichuan University: Engineering Science Edition, 2006, 38(3): 135–139 (in Chinese).

    Google Scholar 

  40. J. G. Jiang, J. Z. Zhang, and X. J. Wang, Readable proving for a class of geometric theorems of polynomial equality type, Chinese Journal of Computers, 2008, 31(2): 207–213 (in Chinese).

    MathSciNet  Google Scholar 

  41. R. Welham R, Geometry Problem Solving, D.A.I. Research Report No.14, 1976.

  42. R. Wong, Construction Heuristics for Geometry and a Vector Algebra Representation of Geometry, Technical Memorandum 28, Project MAC, MIT, Cambridge, Massachusetts, 1973.

    Google Scholar 

  43. R. Reiter, A semantically guided deductive system for automatic theorem proving, IEEE Transactions on Computers, 1977, C-25(4): 328–334.

    Article  MathSciNet  Google Scholar 

  44. A. Robinson, Proving a theorem (as done by Man, Logician, or Machine), Automation of Reasoning (ed. by J. Siekmann and G. Wrightson), Springer-Verlag, Berlin, Heidelberg, 1983.

    Google Scholar 

  45. W. Elcock, Representation of knowledge in geometry machine, Machine Intelligence (ed. by W. Elcock and D. Michie), John Wiley and Sons, 1977, 8: 11–29.

  46. G. Greeno, M. Magone, and S. Chaiklin, Theory of constructions and set in problem solving, Memory and Cognition, 1979, 7(6): 445–461.

    Article  Google Scholar 

  47. N. Matsuda and K. Van Lehn, GRAMY: A geometry theorem prover capable of construction, Journal of Automated Reasoning, 2004. 32(1): 3–33.

    Article  MathSciNet  Google Scholar 

  48. J. McCharen, R. Overbeek, and L. Wos, Problems and experiments for and with automated theorem-proving programs, IEEE Transactions on Computers, 1976, C-25: 773–782.

    Article  Google Scholar 

  49. A. Quaife, Automated Development of Tarski’s geometry, Journal of Automated Reasoning, 1989, 5: 97–118.

    Article  MathSciNet  MATH  Google Scholar 

  50. L. Meikle and J. Fleuriot, Formalizing Hilbert’s grundlagen in Isabelle/Isar, Theorem Proving in Higher Order Logics, 2003: 319–334.

  51. P. Scott, Mechanizing Hilbert’s Foundations of Geometry in Isabelle, Master Thesis, School of Informatics, University of Edinburgh, 2008.

  52. C. Dehlinger, J. Dufourd, and P. Schreck, Higher-order intuitionistic formalization and proofs in Hilbert’s elementary geometry, Automated Deduction in Geometry (2000), LNAI 2061 (ed. by J. Richter-Gebert and D. M. Wang), Springer-Verlag, Berlin, Heidelberg, 2001.

    Chapter  Google Scholar 

  53. J. Narboux, A decision procedure for geometry in Coq, Proceedings of TPHOLs’ 2004, LNCS 3223 (ed. by S. Konrad, B. Annett, and G. Genesh), 2004.

  54. J. Z. Zhang, S. C. Chou, and X. S. Gao, Automated production of traditional proofs for theorems in Euclidean geometry, I: The Hilbert intersection point theorems, Annals of Mathematics and Artificial Intelligence, 1995, 13: 109–137.

    Article  MathSciNet  MATH  Google Scholar 

  55. J. Narboux, A graphical user interface for formal proofs in geometry, Journal of Automated Reasoning, 2007, 39: 161–180.

    Article  MATH  Google Scholar 

  56. R. Caferra, N. Peltier, and F. Puitg, Emphasizing human techniques in automated geometry theorem proving: A practical realization, Automated Deduction in Geometry (2000), LNAI 2061 (ed. by J. Richter-Gebert and D. M. Wang), Springer-Verlag, Berlin, Heidelberg, 2001.

    Google Scholar 

  57. S. Fèvre, Integration of reasoning and algebraic calculus in geometry, Automated Deduction in Geometry (1996), Lecture Notes in Artificial Intelligence 1360, Springer-Verlag, Berlin, Heidelberg, 1998.

    Google Scholar 

  58. S. F`evre and D. M. Wang, Combining algebraic computing and term-rewriting for geometry theorem proving. Proceedings of AISC’98, Pittsburgh, USA, 1998.

  59. G. Défourneaux, C. Bourely, and N. Peltier, Semantic generalizations for proving and disproving conjectures by analogy, Journal of Automated Reasoning, 1998, 20(1–2): 27–45.

    Article  MathSciNet  MATH  Google Scholar 

  60. S. C. Chou, X. S. Gao, and J. Z. Zhang, Machine Proofs in Geometry: Automated Production of Readable Proofs for Geometry Theorems, World Scientific, Singapore, 1994.

    MATH  Google Scholar 

  61. J. Z. Zhang, Points Elimination Methods for Geometric Problem Solving, Mathematics Mechanization and Applications (ed. by X. S. Gao and D. M. Wang), Academic Press, London, 2000.

    Google Scholar 

  62. P. Janičić, J. Narboux, and P. Quaresma, The area method: A recapitulation, Journal of Automated Reasoning, 2012, 48(4): 489–532

    Article  MATH  Google Scholar 

  63. J. Z. Zhang, How to Solve Geometry Problems Using Areas, Shanghai Educational Publishing Inc., Shanghai, 1982 (in Chinese).

    Google Scholar 

  64. J. Z. Zhang and P. Cao, From Education of Mathematics to Mathematics for Education, Sichun Educational Publishing Inc., Chengdu, 1988 (in Chinese).

    Google Scholar 

  65. J. Z. Zhang, A New Approach to Plane Geometry, Sichuan Educational Publishing Inc., Chengdu, 1992 (in Chinese).

    Google Scholar 

  66. S. C. Chou, X. S. Gao, and J. Z. Zhang, Automated generation of readable proofs with geometric invariants, I: Multiple and shortest proofs generation, Journal of Automated Reasoning, 1996, 17(3): 325–347.

    Article  MathSciNet  MATH  Google Scholar 

  67. S. C. Chou, X. S. Gao, and J. Z. Zhang, Automated generation of readable proofs with geometric invariants, II: Theorem proving with full-angles, Journal of Automated Reasoning, 1996, 17(3): 349–370.

    Article  MathSciNet  MATH  Google Scholar 

  68. S. C. Chou, X. S. Gao, and J. Z. Zhang, A Collection of 110 Geometry Theorems and Their Machine Proofs Based on Full-Angles, TR-94-4, Department of Computer Science, WSU, 1994.

  69. Y. Zou and J. Z. Zhang, Automated generation of readable proofs for constructive geometry statements with the mass point method, Automated Deduction in Geometry (2010), Lecture Notes in Artificial Intelligence 6877, Springer-Verlag, Berlin Heidelberg, 2011.

    Google Scholar 

  70. S. C. Chou, X. S. Gao, and J. Z. Zhang, Mechanical geometry theorem proving by vector calculation, Proceedings of International Symposium on Symbolic and Algebraic Computation (Kiev, Ukraine, July 6–8, 1993), ACM Press, New York, 1993.

    Google Scholar 

  71. S. C. Chou, X. S. Gao, and J. Z. Zhang, Automated production of traditional proofs in solid geometry, Journal of Automated Reasoning, 1995, 14: 257–291.

    Article  MathSciNet  MATH  Google Scholar 

  72. L. Yang, X. S. Gao, S. C. Chou, and J. Z. Zhang, Automated production of readable proofs for theorems in non-euclidean geometries, Automated Deduction in Geometry, Lecture Notes Computer Sciences 1360, Springer-Verlag, Berlin, Heidelberg, 1997.

    Chapter  Google Scholar 

  73. H. B. Li, Clifford algebra approaches to mechanical geometry theorem proving, Mathematics Mechanization (ed. by X. S. Gao and D. M. Wang), Academic Press, London, 2000.

    Google Scholar 

  74. B. Sturmfels, Computational algebraic geometry of projective configurations, Journal of Symbolic Computation, 1993, 11: 595–618.

    Article  MathSciNet  Google Scholar 

  75. J. Righter-Gebert, Mechanical theorem proving in projective geometry, Annals of Mathematics and Artificial Intelligence, 1995, 13: 139–172.

    Article  MathSciNet  Google Scholar 

  76. H. B. Li, New Explorations in Automated Theorem Proving in Geometries, Ph.D. Thesis, Peking University, China, 1994.

    Google Scholar 

  77. H. B. Li, Some applications of clifford algebra to geometries, Automated Deduction in Geometry (1989), Lecture Notes in Artificial Intelligence 1669, Springer-Verlag, Berlin, Heidelberg, 1998.

    Google Scholar 

  78. D. M. Wang, Clifford algebraic calculus for geometric reasoning with application to computer vision, Automated Deduction in Geometry (1996), Lecture Notes in Artificial Intelligence 1360, Springer-Verlag, Berlin, Heidelberg, 1998.

    Google Scholar 

  79. H. B. Li and H. Shi, On Erdös’ ten-point problem, Acta Mathematica Sinica, New Series, 1997, 13(2): 221–230.

    MathSciNet  MATH  Google Scholar 

  80. S. Fèvre and D. M. Wang, Proving geometric theorems using clifford algebra and rewrite rules, Lecture Notes in Artificial Intelligence 1421 (ed. by D. M. Wang), Springer-Verlag, Berlin, Heidelberg, 1997.

    Google Scholar 

  81. H. Q. Yang, S. G. Zhang, and G. C. Feng, A Clifford algebraic method for geometric reasoning, Automated Deduction in Geometry (Beijing, 1998), Lecture Notes in Artificial Intelligence 1669 (ed. by X. S. Gao, D. M. Wang, and L. Yang), Springer-Verlag, Berlin, Heidelberg, 1999.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  83. H. B. Li and M. Cheng, Clifford algebraic reduction method for automated theorem proving in differential geometries, Journal of Automated Reasoning, 1998, 21: 1–21.

    Article  MathSciNet  Google Scholar 

  84. H. B. Li, Mechanical theorem proving in differential geometry, Mathematics Mechanization (ed. by X. S. Gao and D. M. Wang), Academic Press, London, 2000.

    Google Scholar 

  85. A. Dolzmann, T. Sturm, and V. Weispfenning, A new approach for automatic theorem proving in real geometry, Journal of Automated Reasoning, 1998, 21(3): 357–380.

    Article  MathSciNet  MATH  Google Scholar 

  86. W. T. Wu, On problems involving inequalities, MM-Res, Preprints, MMRC, 1992, 7: 103–138.

    Google Scholar 

  87. W. T. Wu, On a finiteness theorem about problems involving inequalities, Journal of Systems Science and Mathematical Sciences, 1994, 7(2): 193–200.

    MATH  Google Scholar 

  88. D. M. Wang, Polynomial Equations Solving and Mechanical Geometric Theorem Proving, Ph.D. Thesis, Institute of Systems Science, Chinese Academy of Sciences, Beijing, 1993.

    Google Scholar 

  89. L. Yang, Recent advances in automated theorem proving on inequalities, Journal of Computer Science and Technology, 1999, 14(5): 434–446.

    Article  MathSciNet  MATH  Google Scholar 

  90. L. Yang, X. R. Hou, and B. C. Xia, A complete algorithm for automated discovering of a class of inequality-type theorems, Science in China (Series F), 2001, 44(1): 33–49.

    Article  MathSciNet  MATH  Google Scholar 

  91. L. Yang and J. Z. Zhang, A practical program of automated proving for a class of geometric inequalities, Automated Deduction in Geometry, Lecture Notes in Artificial Intelligence 2061 (ed. by J. Richter-Gebert and D. M. Wang), Springer-Verlag, Berlin, Heidelberg, 2001.

    Chapter  Google Scholar 

  92. O. Bottema, et al, Geometric Inequalities, Wolters-Noordhoff Publishing, Groningen, Netherlands, 1969.

    MATH  Google Scholar 

  93. L. Yang, Solving harder problems with lesser mathematics, Proceedings of the 10th Asian Technology Conference in Mathematics, ATCM Inc., Blacksburg, 2005.

    Google Scholar 

  94. L. Yang, Difference substitution and automated inequality proving, Journal of Guangzhou University: Natural Science Edition, 2006, 5(2): 1–7 (in Chinese).

    MathSciNet  Google Scholar 

  95. L. Yang and Y. Yao, Difference substitution matrices and decision on nonnegativity of polynomials, Journal of Systems Science and Mathematical Sciences, 2009, 29(9): 1169–1177 (in Chinese).

    MathSciNet  MATH  Google Scholar 

  96. Y. Yao, Infinite product convergence of column stochastic mean matrix and machine decision for positive semi-definite forms, Science China Mathematics (Series A), 2010, 53(3): 251–264 (in Chinese).

    Google Scholar 

  97. S. H. Xia, Automated Production of Readable Proof for a Class of Inequality-type Theorems, Ph.D, Thesis, Chinese Academy of Sciences, Beijing, 2002.

    Google Scholar 

  98. S. Chen and J. Z. Zhang, Automated production of elementary and readable proof of inequality, Journal of Sichuan University: Engineering Science Education, 2003, 35(4): 86–93 (in Chinese).

    Google Scholar 

  99. S. C. Chou, X. S. Gao, and J. Z. Zhang, Automated geometry theorem proving and geometry education, Proceedings of Asian Technology Conference in Mathematics, Association of Mathematics Educators, Singapore, 1995.

    Google Scholar 

  100. C. Z. Li and J. Z. Zhang, Readable machine solving in geometry and ICAI software MSG, Automated Deduction in Geometry (1998), Lecture Notes in Artificial Intelligence 1669, Springer-Verlag, Berlin, Heidelberg, 1998.

    Google Scholar 

  101. S. C. Chou, X. S. Gao, and J. Z. Zhang, An introduction to geometry expert, Proceedings of CADE-13, Lecture Notes in Artifficial Intelligence (ed. by M. McRobbie and J. Slaney), Springer-Verlag, Berlin, Heidelberg, 1996.

    Google Scholar 

  102. Y. Zheng, S. C. Chou, and X. S. Gao, An introduction to java geometry expert-(extended abstract), Automated Deduction in Geometry — 7th International Workshop (2008), LNCS 6301 (ed. by T. Sturm and C. Zengler), Springer, 2011.

  103. Y. Zheng, S. C. Chou, and X. S. Gao, Visually dynamic presentation of proofs in plane geometry — part 1. Basic features and the manual input method, Journal of Automated Reasoning, 2010, 45(3): 213–241.

    Article  MathSciNet  MATH  Google Scholar 

  104. X. S. Gao, Q. Lin, MMP/Geometer-A software package for automated geometry reasoning, Automated Deduction in Geometry (Hagenberg Castle, Austria, 2002) (ed. by F. Winkler), Springer-Verlag, Berlin, Heidelberg, 2004: 44–66.

    Google Scholar 

  105. C. Z. Li and J. Z. Zhang, Super Sketchpad, Beijing Normal University Press, Beijing, 2004 (in Chinese).

    Google Scholar 

  106. S. Wilson and J. Fleuriot, Geometry explorer: combining dynamic geometry, automated geometry theorem proving and diagrammatic proofs, Proceedings of the 12th Workshop on Automated Reasoning (ARW), Edinburgh, UK, 2005.

  107. P. Janicic and P. Quaresma, System description: GCLCprover + GeoThms, International Joint Conference on Automated Reasoning (IJCAR-2006), Lecture Notes in Artificial Intelligence 4130 (ed. by U. Furbach and N. Shankar), Springer-Verlag, Berlin, Heidelberg, 2006.

    Google Scholar 

  108. H. Zheng and J. Z. Zhang, Reasoning of algorithm of geometry automatic reasoning platform with sustainable development by user, Journal of Computer Applications, 2011, 31(8): 2101–2107 (in Chinese).

    Google Scholar 

  109. H. Zheng, The sustainable geometry automated reasoning platform, Journal of System Science and Mathematical Sciences, 2011, 31(12): 1622–1632 (in Chinese).

    Google Scholar 

  110. A. Samuel, Some studies in machine learning using the game of checkers, IBM Journal of Research and Development, 1959, 3(3): 210–229.

    Article  Google Scholar 

  111. A. Samuel, Some studies in machine learning using the game of checkers II — Recent progress, IBM Journal of Research and Development, 1967, 11(6): 601–617.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Liaoning Normal University, Dalian, 116029, China

    Jianguo Jiang

  2. Laboratory of Computer Reasoning and Trustworthy Compution, University of Electronic Science and Technology of China, Chengdu, 610054, China

    Jingzhong Zhang

  3. Chengdu Institute of Computer Applications, Chinese Academy of Sciences, Chengdu, 610041, China

    Jingzhong Zhang

Authors
  1. Jianguo Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jingzhong Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jianguo Jiang.

Additional information

This research was supported by the Funds of the Chinese Academy of Sciences for Key Topics in Innovation Engineering under Grant No. KJCX2-YW-S02.

This paper was recommended for publication by Editor Xiao-Shan GAO.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jiang, J., Zhang, J. A review and prospect of readable machine proofs for geometry theorems. J Syst Sci Complex 25, 802–820 (2012). https://doi.org/10.1007/s11424-012-2048-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-012-2048-3

Key words

Search

Navigation