Advertisement

Journal of Systems Science and Complexity

, Volume 32, Issue 1, pp 95–123 | Cite as

Automated Theorem Proving Practice with Null Geometric Algebra

  • Hongbo LiEmail author
Article

Abstract

This paper presents the practice of automated theorem proving in Euclidean geometry with null geometric algebra, a combination of Conformal Geometric Algebra and Grassmann-Cayley algebra. This algebra helps generating extremely short and readable proofs: The proofs generated are mostly one-termed or two-termed. Besides, the theorems are naturally extended from qualitative description to quantitative characterization by removing one or more geometric constraints from the hypotheses.

Keywords

Automated theorem discovering automated theorem extending automated theorem proving Clifford bracket algebra null geometric algebra 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Wu W T, Basic Principles of Mechanical Theorem Proving in Geometries I: Part of Elementary Geometries, Science Press, Beijing, 1984; Springer, Wien, 1994.Google Scholar
  2. [2]
    Kutzler B and Stifter S, On the application of Buchberger’s algorithm to automated geometry theorem proving, J. Symb. Comp., 1986, 2: 389–398.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Crapo H and Richter-Gebert J, Automatic proving of geometric theorems, Invariant Methods in Discrete and Computational Geometry, Ed. by White N, Kluwer Academic Publishers, 1994, 107–139.Google Scholar
  4. [4]
    Wang D, Elimination Methods, Springer, Wien, New York, 2001.CrossRefzbMATHGoogle Scholar
  5. [5]
    Chou S C, Mechanical Geometry Theorem Proving, D. Reidel, Dordrecht, 1988.zbMATHGoogle Scholar
  6. [6]
    Li H, Symbolic geometric reasoning with advanced invariant algebras, Mathematical Aspects of Computer and Information Sciences, Eds. by Kotsireas I S, et al., LNCS 9582, Springer Switzerland, 2015, 35–49.Google Scholar
  7. [7]
    Richter-Gebert J and Li H, Coordinate-free theorem proving in incidence geometry, Eds. by Sitharam M, et al., Handbook of Geometric Constraint Systems Principles, CRC Press, Boca Raton, 2019, 59–82.Google Scholar
  8. [8]
    Li H, A recipe for symbolic geometric computing: Long geometric product, breefs and Clifford factorization, Ed. by Brown C W, Proc. ISSAC 2007, ACM Press, New York, 2007, 261–268.Google Scholar
  9. [9]
    Havel T F and Li H, From molecular distance geometry to conformal geometric algebra, Eds. by Sitharam M, et al., Handbook of Geometric Constraint Systems Principles, CRC Press, Boca Raton, 2019, 106–136.Google Scholar
  10. [10]
    Cao N, Geometric decomposition based on conformal geometric algebra and its program realization, PhD Dissertation, Acad. Math. and Sys. Sci., Chinese Academy of Sciences, Beijing, 2006.Google Scholar
  11. [11]
    Li H and Cao Y, On geometric theorem proving with null geometric algebra, Eds. by Dorst L and Lasenby J, Guide to Geometric Algebra in Practice, Springer, London, 2011, 195–216.CrossRefGoogle Scholar
  12. [12]
    Chou S C, Gao X S, and Zhang J Z, Machine Proofs in Geometry, World Scientific, Singapore, 1994.CrossRefzbMATHGoogle Scholar
  13. [13]
    Li H, Symbolic computation in the homogeneous geometric model with Clifford algebra, Ed. by Gutierrez J, Proc. ISSAC 2004, ACM Press, New York, 2004, 221–228.Google Scholar
  14. [14]
    Li H, Geometric reasoning with invariant algebras, Proc. ATCM 2006, ATCM Inc., Blacksburg, USA, 2006, 6–19.Google Scholar
  15. [15]
    Cao Y and Li H, Elimination and simplification algorithms in geometric algebra for theorem proving, J. Sys. Sci. Math., 2009, 29(9): 1189–1199 (in Chinese).zbMATHGoogle Scholar
  16. [16]
    Li H, Automated geometric reasoning with geometric algebra: Theory and practice, Proc. ISSAC 2017, 2015, 7–8, Slides: http://www.issac-conference.org/2017/assets/tutorial slides/Li.pdf.Google Scholar
  17. [17]
    Hestenes D and Sobczyk G, Clifford Algebra to Geometric Calculus, Kluwer, Dordrecht, 1984.CrossRefzbMATHGoogle Scholar
  18. [18]
    Li H, Invariant Algebras and Geometric Reasoning, World Scientific, Singapore, 2008.CrossRefzbMATHGoogle Scholar
  19. [19]
    Brini A, Regonati F, and Teolis A G B, Grassmann Geometric Calculus, Invariant Theory and Superalgebras, Eds. by Crapo H and Senato D, Algebraic Combinatorics and Computer Science, Springer, Milano, 2001, 151–196.CrossRefGoogle Scholar
  20. [20]
    Caianiello E, Combinatorics and Renormalization in Quantum Field Theory, Benjamin, Reading, 1973.Google Scholar
  21. [21]
    Li H, Automated theorem proving in the homogeneous model with Clifford bracket algebra, Applications of Geometric Algebra in Computer Science and Engineering, Eds. by Dorst L, et al., Birkhäuser, Boston, 2002, 69–78.CrossRefGoogle Scholar
  22. [22]
    Li H, Automated geometric theorem proving, clifford bracket algebra and Clifford expansions, Eds. by Qian T, et al., Trends in Mathematics: Advances in Analysis and Geometry, Birkhäuser, Basel, 2004, 345–363.CrossRefGoogle Scholar
  23. [23]
    Havel T F, Geometric algebra and Möbius sphere geometry as a basis for Euclidean invariant theory, Ed. by White N, Invariant Methods in Discrete and Computational Geometry, D. Reidel, Dordrecht, 1995, 245–256.CrossRefGoogle Scholar
  24. [24]
    Li H, Hestenes D, and Rockwood A, Generalized homogeneous coordinates for computational geometry, Geometric Computing with Clifford Algebras, Ed. by Sommer G, Springer, Heidelberg, 2001, 27–60.CrossRefGoogle Scholar
  25. [25]
    Cecil T E, Lie Sphere Geometry, Springer, New York, 1992.CrossRefzbMATHGoogle Scholar
  26. [26]
    Li H and Zhang L, Line geometry in terms of the null geometric algebra over R3,3, and application to the inverse singularity analysis of generalized stewart platforms, Eds. by Dorst L and Lasenby J, Guide to Geometric Algebra in Practice, Springer, London, 2011, 253–272.CrossRefGoogle Scholar
  27. [27]
    Li H and Huang L, Complex brackets, balanced complex differences, and applications in symbolic geometric computing, Proc. ISSAC 2008, ACM Press, New York, 2008, 181–188.Google Scholar
  28. [28]
    Li H, From geometric algebra to advanced invariant computing, J. Sys. Sci. Math., 2008, 28(8): 915–929 (in Chinese).MathSciNetzbMATHGoogle Scholar
  29. [29]
    Li H, Huang L, Dong L, et al., Elements of line geometry with geometric algebra, Proc. AGACSE’15, Eds. by Descamps S X, et al., FIB Barcelona, 2015, 195–204. Full version: arXiv: 1507.06634 [math.MG].Google Scholar
  30. [30]
    Li H, The Lie model for Euclidean geometry, Eds. by Sommer G and Zeevi Y, Algebraic Frames for the Perception-Action Cycle, LNCS 1888, Springer, Berlin, 2000, 115–133.CrossRefGoogle Scholar

Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Systems Science, University of Chinese Academy of SciencesChinese Academy of SciencesBeijingChina

Personalised recommendations