Skip to main content
SpringerLink
Log in

A class of mechanically decidable problems beyond Tarski’s model

  • Published:
Science in China Series A: Mathematics Aims and scope Submit manuscript

Abstract

By means of dimension-decreasing method and cell-decomposition, a practical algorithm is proposed to decide the positivity of a certain class of symmetric polynomials, the numbers of whose elements are variable. This is a class of mechanically decidable problems beyond Tarski model. To implement the algorithm, a program nprove written in maple is developed which can decide the positivity of these polynomials rapidly.

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. Tarski A. A Decision Method for Elementary Algebra and Geometry, 2nd ed. Berkeldy: University of California Press, 1951

    MATH  Google Scholar 

  2. Wu W T. On the decision problem and the mechanization of theorem-proving in elementary geometry. Science in China, 21: 150–172 (1978)

    Google Scholar 

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

    MATH  Google Scholar 

  4. Buchberger B. Grobner bases: An algorithmic method in polynomial ideal theory. In: Multidimensional Systems Theory. Dordrecht: Reidel, 1985, 184–232

    Google Scholar 

  5. Kapur D. Geometry theorem proving using Hilbert’s Nullstellensata. In: Proc. SYMSAC’86. New York: ACM Press, 1986, 202–208

    Google Scholar 

  6. Kutzler B, Stifter S. Automated geometry theorem proving using Buchberger’s algorithm. In: Proc SYM-SAC’86. New York: ACM Press, 1986, 209–214

    Google Scholar 

  7. Yang L, Zhang J Z, Hou X R. Nonlinear Algebraic Equation System and Automated Theorem Proving. Shanghai: Shanghai Scientific and Technological Education Publishing House, 1996

    Google Scholar 

  8. Zhang J Z, Yang L, Deng M K. The parallel numerical method of mechanical theorem proving. Theor Comput Sci, 74(3): 253–271 (1990)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  10. Collins G E, Hong H. Partial cylindrical algebraic decomposition for quantifier elimination, Journal of Symbolic Computation, 12: 299–328 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  11. Timofte V. On the positivity of symmetric polynomial functions I. J Math Anal Appl, 284: 174–190 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  12. Timofte V. On the positivity of symmetric polynomial functions, Part II: Lattice general results and positivity criteria for degrees 4 and 5. J Math Anal Appl, 284: 652–667 (2005)

    Article  MathSciNet  Google Scholar 

  13. Choi M D, Lam T Y, Reznick B. Even symmetric sextics. Math Z, 195: 559–580 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  14. Yang L, Xia B C. Computational real algebraic geometry. In: Wang D M, ed. Selected Lectures in Symbolic Computation (in Chinese). Beijing: Tsinghua University Press, 2003

    Google Scholar 

  15. Dolzman A, Sturm T. REDLOG: Computer algebra meets computer logic. Acm Sigsam Bulletin, 31(2): 2–9 (1997)

    Article  Google Scholar 

  16. Yang L, Hou X R, Xia B C. A complete algorithm for automated discovering of a class of inequality-type theorems. Sci China Ser F-Info, 44: 33–49 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  17. Yang L, Xia S H. Automated proving for a class of constructive geometric inequalities. Chinese Journal of Computers (in Chinese), 26(7): 769–778 (2003)

    MathSciNet  Google Scholar 

  18. Yang L, Zhang J. A practical program of automated proving for a class of geometric inequalities, automated deduction in geometry. Lecture Notes in Artificial Intelligence 2061. Berlin Heidelberg: Springer-Verlag, 2001, 41–57

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratory for Automated Reasoning and Programming, Chengdu Institute of Computer Applications, Chinese Academy of Sciences, Chengdu, 610041, China

    Lu Yang, Yong Feng & Yong Yao

  2. Institute of Theoretical Computing, East China Normal University, Shanghai, 200062, China

    Lu Yang

Authors
  1. Lu Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yong Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yong Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yong Feng.

Additional information

This work was partially supported by China 973 Project NKBRPC (Grant No. 2004CB318003) and the Knowledge Innovation Program of the Chinese Academy of Sciences (Grant No. KJCX2-YW-S02)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yang, L., Feng, Y. & Yao, Y. A class of mechanically decidable problems beyond Tarski’s model. SCI CHINA SER A 50, 1611–1620 (2007). https://doi.org/10.1007/s11425-007-0090-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-007-0090-8

Keywords

MSC(2000)

Search

Navigation