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Automated Reducible Geometric Theorem Proving and Discovery by Gröbner Basis Method

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Abstract

In this paper, we investigate the problem that the conclusion is true on some components of the hypotheses for a geometric statement. In that case, the affine variety associated with the hypotheses is reducible. A polynomial vanishes on some but not all the components of a variety if and only if it is a zero divisor in a quotient ring with respect to the radical ideal defined by the variety. Based on this fact, we present an algorithm to decide if a geometric statement is generally true or generally true on components by the Gröbner basis method. This method can also be used in geometric theorem discovery, which can give the complementary conditions such that the geometric statement becomes true or true on components. Some reducible geometric statements are given to illustrate our method.

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Notes

  1. Programs for computing minimal comprehensive Gröbner systems, which is based on the algorithm in [23], are available at http://mmrc.iss.ac.cn/~dwang/software.html.

  2. The variables are divided in three blocks y, X, and U. A monomial ordering is chosen for each block. To compare two monomials, we firstly compare their y part. If the y part is equal, then we compare their X part. If the y and X parts are equal, then we compare their U part.

References

  1. Gelernter, H., Hanson, J.R., Loveland, D.W.: Empirical explorations of the geometry-theorem proving machine. In: Proceeding of the West Joint Computer Conference, pp. 143–147 (1960)

  2. Tarski, A.: A decision method for elementary algebra and geometry. Texts Monogr. Symb. Comput. 35, 24–84 (1951)

    MathSciNet  MATH  Google Scholar 

  3. Seidenberg, A.: A new decision method for elememtary algebra. Ann. Math. 60(2), 365–374 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  4. Collins, G.E.: Quantifier elimination for real closed fields by cylindrical algebraic decomposition. Quantifier Elimination and Cylindrical Algebraic Decomposition, pp. 85–121. Springer, Vienna (1998)

  5. Wu, W.T.: On the decision problem and the mechanization of theorem proving in elementary geometry. Sci. Sin. 21, 159–172 (1978)

    MathSciNet  MATH  Google Scholar 

  6. Wu, W.T.: Basic principles of mechanical theorem proving in elementary geometries. J. Autom. Reason. 2(3), 221–252 (1986)

    Article  MATH  Google Scholar 

  7. Chou, S.C.: Mechanical Geometry Theorem Proving, Mathematics and Its Applications. Reidel, Amsterdam (1988)

    Google Scholar 

  8. Ritt, J.F.: Differential equations from the algebraic standpoint. Am. Math. Soc. (1932)

  9. Chou, S.C.: Proving geometry theorems with rewrite rules. J. Autom. Reason. 2(3), 253–273 (1986)

    Article  MATH  Google Scholar 

  10. Kapur, D.: Geometry theorem proving using Hilbert’s Nullstellensatz. In: Proceedings of the ISSAC, pp. 202–208. ACM Press, New York (1986)

  11. Kutzler, B., Stifter, S.: Automated geometry theorem proving using Buchberger’s algorithm. In: Proceedings of ISSAC, pp. 209–214. ACM Press, New York (1986)

  12. Coxeter, H.S.M.: Introduction to Geometry. Wiley, New York (1961)

    MATH  Google Scholar 

  13. Taylor, K.B.: Three circles with collinear centers. Am. Math. Mon. 90, 484–486 (1983)

    Google Scholar 

  14. Dalzotto, G., Recio, T.: On protocols for the automated discovery of theorems in elementary geometry. J. Autom. Reason. 43(2), 203–236 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Botana, F., Montes, A., Recio, T.: An algorithm for automatic discovery of algebraic loci. In: Proceedings of the ADG, pp. 53–59 (2012)

  16. Chen, X.F., Li, P., Lin, L., Wang, D.K.: Proving geometric theorems by partitioned-parametric Gröbner bases. In: Proceedings of the ADG, pp. 34–43 (2004)

  17. Montes, A., Recio, T.: Automatic discovery of geometry theorems using minimal canonical comprehensive Gröbner systems. In: Proceedings of the ADG, pp. 113–138 (2006)

  18. Recio, T., Velez, M.P.: Automatic discovery of theorems in elementary geometry. J. Autom. Reason. 23(1), 63–82 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  19. Wang, D.K., Lin, L.: Automatic discovery of geometric theorem by computing Gröbner bases with parameters. In: Abstracts of Presentations of ACA, vol. 32, Japan (2005)

  20. Winkler, F.: Gröbner bases in geometry theorem proving and simplest degeneracy conditions. Math. Pannon. 1(1), 15–32 (1990)

    MathSciNet  MATH  Google Scholar 

  21. Weispfenning, V.: Comprehensive Gröbner bases. J. Symb. Comput. 14(1), 1–29 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kapur, D.: An approach for solving systems of parametric polynomial equations. In: Saraswat, V., Van Hentenryck, (eds.) Principles and Practices of Constraints Programming, pp. 217–244. MIT Press, Cambridge (1995)

  23. Kapur, D., Sun, Y., Wang, D.K.: A new algorithm for computing comprehensive Gröbner systems. In: Proceedings of the ISSAC, pp. 29–36. ACM Press, New York (2010)

  24. Montes, A.: A new algorithm for discussing Gröbner bases with parameters. J. Symb. Comput. 33(2), 183–208 (2002)

    Article  MATH  Google Scholar 

  25. Montes, A., Wibmer, M.: Gröbner bases for polynomial systems with parameters. J. Symb. Comput. 45(12), 1391–1425 (2010)

    Article  MATH  Google Scholar 

  26. Nabeshima, K.: A speed-up of the algorithm for computing comprehensive Gröbner systems. In: Proceedings of the ISSAC, pp. 299–306. ACM Press, New York (2007)

  27. Suzuki, A., Sato, Y.: An alternative approach to comprehensive Gröbner bases. J. Symb. Comput. 36(3), 649–667 (2003)

    Article  MATH  Google Scholar 

  28. Weispfenning, V.: Canonical comprehensive Gröbner bases. J. Symb. Comput. 36(3), 669–683 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  29. Manubens, M., Montes, A.: Minimal canonical comprehensive Groebner system. J. Symb. Comput. 44(5), 463–478 (2009)

    Article  MATH  Google Scholar 

  30. Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms, 3rd edn. Springer, New York (2007)

    Book  MATH  Google Scholar 

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Acknowledgements

The authors are grateful to Professors D. Kapur and B.C. Xia for their helpful discussions and suggestions.

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Authors and Affiliations

  1. Science and Technology on Communication Security Laboratory, Chengdu, 610041, China

    Jie Zhou

  2. Academy of Mathematics and Systems Science, CAS, Beijing, 100190, China

    Dingkang Wang

  3. Institute of Information Engineering, CAS, Beijing, 100093, China

    Yao Sun

Authors
  1. Jie Zhou
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  2. Dingkang Wang
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  3. Yao Sun
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Correspondence to Jie Zhou.

Additional information

This research was partially supported by National Nature Science Foundation of China (Nos. 11301523 and 11371356).

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Zhou, J., Wang, D. & Sun, Y. Automated Reducible Geometric Theorem Proving and Discovery by Gröbner Basis Method. J Autom Reasoning 59, 331–344 (2017). https://doi.org/10.1007/s10817-016-9395-z

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