Skip to main content
SpringerLink
Log in

Analyses and Implementations of Chordality-Preserving Top-Down Algorithms for Triangular Decomposition

  • Conference paper
  • First Online:
Computer Algebra in Scientific Computing (CASC 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13366))

Included in the following conference series:

  • 509 Accesses

Abstract

When the input polynomial set has a chordal associated graph, top-down algorithms for triangular decomposition are proved to preserve the chordal structure. Based on these theoretical results, sparse algorithms for triangular decomposition were proposed and demonstrated with experiments to be more efficient in case of sparse polynomial sets. However, existing implementations of top-down triangular decomposition are not guaranteed to be chordality-preserving due to operations which potentially destroy the chordality. In this paper, we first analyze the current implementations of typical top-down algorithms for triangular decomposition in the Epsilon package to identify these chordality-destroying operations. Then modifications are made accordingly to guarantee new implementations of such algorithms are chordality-preserving. In particular, the technique of dynamic checking is introduced to ensure that the modifications also keep the computational efficiency. Experimental results with polynomial sets from biological systems are also reported.

This work was partially supported by the National Natural Science Foundation of China (NSFC 11971050) and Beijing Natural Science Foundation (Z180005).

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 69.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 89.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    https://odebase.cs.uni-bonn.de/ODEModelApp.

References

  1. Aubry, P., Lazard, D., Moreno Maza, M.: On the theories of triangular sets. J. Symb. Comput. 28(1–2), 105–124 (1999)

    Article  MathSciNet  Google Scholar 

  2. Aubry, P., Moreno Maza, M.: Triangular sets for solving polynomial systems: a comparative implementation of four methods. J. Symb. Comput. 28(1), 125–154 (1999)

    Article  MathSciNet  Google Scholar 

  3. Bächler, T., Gerdt, V., Lange-Hegermann, M., Robertz, D.: Algorithmic Thomas decomposition of algebraic and differential systems. J. Symb. Comput. 47(10), 1233–1266 (2012)

    Article  MathSciNet  Google Scholar 

  4. Berry, A., Blair, J., Heggernes, P., Peyton, B.: Maximum cardinality search for computing minimal triangulations of graphs. Algorithmica 39(4), 287–298 (2004)

    Article  MathSciNet  Google Scholar 

  5. Chai, F., Gao, X.S., Yuan, C.: A characteristic set method for solving Boolean equations and applications in cryptanalysis of stream ciphers. J. Syst. Sci. Complex. 21(2), 191–208 (2008)

    Article  MathSciNet  Google Scholar 

  6. Chen, C.: Chordality preserving incremental triangular decomposition and its implementation. In: Bigatti, A.M., Carette, J., Davenport, J.H., Joswig, M., de Wolff, T. (eds.) ICMS 2020. LNCS, vol. 12097, pp. 27–36. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-52200-1_3

    Chapter  Google Scholar 

  7. Chen, C., Golubitsky, O., Lemaire, F., Moreno Maza, M., Pan, W.: Comprehensive triangular decomposition. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2007. LNCS, vol. 4770, pp. 73–101. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-75187-8_7

    Chapter  Google Scholar 

  8. Chen, C., Moreno Maza, M.: Algorithms for computing triangular decompositions of polynomial systems. J. Symb. Comput. 47(6), 610–642 (2012)

    Article  MathSciNet  Google Scholar 

  9. Chou, S.-C., Gao, X.-S.: Ritt-Wu’s decomposition algorithm and geometry theorem proving. In: Stickel, M.E. (ed.) CADE 1990. LNCS, vol. 449, pp. 207–220. Springer, Heidelberg (1990). https://doi.org/10.1007/3-540-52885-7_89

    Chapter  Google Scholar 

  10. Cifuentes, D., Parrilo, P.: Chordal networks of polynomial ideals. SIAM J. Appl. Algebra Geom. 1(1), 73–110 (2017)

    Article  MathSciNet  Google Scholar 

  11. Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Undergraduate Texts in Mathematics, Springer, New York (1997). https://doi.org/10.1007/978-3-319-16721-3

    Book  MATH  Google Scholar 

  12. Della Dora, J., Dicrescenzo, C., Duval, D.: About a new method for computing in algebraic number fields. In: Caviness, B.F. (ed.) EUROCAL 1985. LNCS, vol. 204, pp. 289–290. Springer, Heidelberg (1985). https://doi.org/10.1007/3-540-15984-3_279

    Chapter  Google Scholar 

  13. Gao, X.S., Huang, Z.: Characteristic set algorithms for equation solving in finite fields. J. Symb. Comput. 47(6), 655–679 (2012)

    Article  MathSciNet  Google Scholar 

  14. Greuel, G.M., Pfister, G., Bachmann, O., Lossen, C., Schönemann, H.: A Singular Introduction to Commutative Algebra. Springer, Heidelberg (2002). https://doi.org/10.1007/978-3-662-04963-1

    Book  Google Scholar 

  15. Huang, Z., Lin, D.: Attacking bivium and trivium with the characteristic set method. In: Nitaj, A., Pointcheval, D. (eds.) AFRICACRYPT 2011. LNCS, vol. 6737, pp. 77–91. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-21969-6_5

    Chapter  Google Scholar 

  16. Hubert, E.: Notes on triangular sets and triangulation-decomposition algorithms I: polynomial systems. In: Winkler, F., Langer, U. (eds.) SNSC 2001. LNCS, vol. 2630, pp. 1–39. Springer, Heidelberg (2003). https://doi.org/10.1007/3-540-45084-X_1

    Chapter  Google Scholar 

  17. Kalkbrener, M.: A generalized Euclidean algorithm for computing triangular representations of algebraic varieties. J. Symb. Comput. 15(2), 143–167 (1993)

    Article  MathSciNet  Google Scholar 

  18. Lazard, D.: A new method for solving algebraic systems of positive dimension. Discret. Appl. Math. 33(1–3), 147–160 (1991)

    Article  MathSciNet  Google Scholar 

  19. Lazard, D.: Solving zero-dimensional algebraic systems. J. Symb. Comput. 13(2), 117–131 (1992)

    Article  MathSciNet  Google Scholar 

  20. Lemaire, F., Moreno Maza, M., Xie, Y.: The RegularChains library in MAPLE. ACM SIGSAM Bull. 39(3), 96–97 (2005)

    Article  Google Scholar 

  21. Li, H., Xia, B., Zhang, H., Zheng, T.: Choosing the variable ordering for cylindrical algebraic decomposition via exploiting chordal structure. In: Proceedings of ISSAC 2021, pp. 281–288 (2021)

    Google Scholar 

  22. Mou, C., Bai, Y.: On the chordality of polynomial sets in triangular decomposition in top-down style. In: Proceedings ISSAC 2018, pp. 287–294 (2018)

    Google Scholar 

  23. Mou, C., Bai, Y., Lai, J.: Chordal graphs in triangular decomposition in top-down style. J. Symb. Comput. 102, 108–131 (2021)

    Article  MathSciNet  Google Scholar 

  24. Mou, C., Ju, W.: Sparse triangular decomposition for computing equilibria of biological dynamic systems based on chordal graphs. In: IEEE/ACM Transactions Computational Biology and Bioinformatics (2022)

    Google Scholar 

  25. Mou, C., Lai, J.: On the chordality of simple decomposition in top-down style. In: Slamanig, D., Tsigaridas, E., Zafeirakopoulos, Z. (eds.) MACIS 2019. LNCS, vol. 11989, pp. 138–152. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-43120-4_12

    Chapter  MATH  Google Scholar 

  26. Mou, C., Wang, D., Li, X.: Decomposing polynomial sets into simple sets over finite fields: the positive-dimensional case. Theoret. Comput. Sci. 468, 102–113 (2013)

    Article  MathSciNet  Google Scholar 

  27. Niu, W., Wang, D.: Algebraic approaches to stability analysis of biological systems. Math. Comput. Sci. 1(3), 507–539 (2008)

    Article  MathSciNet  Google Scholar 

  28. Ritt, J.: Differential Equations from the Algebraic Standpoint. AMS (1932)

    Google Scholar 

  29. Ritt, J.: Differential Algebra. AMS (1950)

    Google Scholar 

  30. Wang, D.: An elimination method for polynomial systems. J. Symb. Comput. 16(2), 83–114 (1993)

    Article  MathSciNet  Google Scholar 

  31. Wang, D.: Decomposing polynomial systems into simple systems. J. Symb. Comput. 25(3), 295–314 (1998)

    Article  MathSciNet  Google Scholar 

  32. Wang, D.: Elimination Methods. Texts and Monographs in Symbolic Computation, Springer Science & Business Media, New York (2001). https://doi.org/10.1007/978-3-7091-6202-6

    Book  MATH  Google Scholar 

  33. Wang, D.: Epsilon: A library of software tools for polynomial elimination. In: Mathematical Software, pp. 379–389. World Scientific (2002)

    Google Scholar 

  34. Wang, D.: wsolve: A Maple package for solving system of polynomial equations (2004). http://www.mmrc.iss.ac.cn

  35. Wang, D., Xia, B.: Stability analysis of biological systems with real solution classification. In: Proceedings of ISSAC 2005, pp. 354–361 (2005)

    Google Scholar 

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

    Article  Google Scholar 

  37. Wu, W.T.: A zero structure theorem for polynomial-equations-solving and its applications. In: Davenport, J.H. (ed.) EUROCAL 1987. LNCS, vol. 378, pp. 44–44. Springer, Heidelberg (1989). https://doi.org/10.1007/3-540-51517-8_84

    Chapter  Google Scholar 

  38. Yang, L., Zhang, J.: Searching dependency between algebraic equations: an algorithm applied to automated reasoning. In: Artificial Intelligence in Mathematics, pp. 147–156 (1994)

    Google Scholar 

Download references

Acknowledgments

The authors would like to thank Prof. Dongming Wang for his insightful comments on the implementations in Epsilon package and the referees for their helpful comments resulting in improvements on the previous version of this paper.

Author information

Authors and Affiliations

  1. LMIB–School of Mathematical Sciences, Beihang University, Beijing, 100191, China

    Mingyu Dong & Chenqi Mou

Authors
  1. Mingyu Dong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chenqi Mou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chenqi Mou .

Editor information

Editors and Affiliations

  1. Université de Lille, Villeneuve d'Ascq, France

    François Boulier

  2. Coventry University, Coventry, UK

    Matthew England

  3. Plekhanov Russian University of Economics, Moscow, Russia

    Timur M. Sadykov

  4. Institute of Theoretical and Applied Mechanics, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Dong, M., Mou, C. (2022). Analyses and Implementations of Chordality-Preserving Top-Down Algorithms for Triangular Decomposition. In: Boulier, F., England, M., Sadykov, T.M., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2022. Lecture Notes in Computer Science, vol 13366. Springer, Cham. https://doi.org/10.1007/978-3-031-14788-3_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-14788-3_8

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-14787-6

  • Online ISBN: 978-3-031-14788-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation