Skip to main content
SpringerLink
Log in

Boolean-Arithmetic Equations: Acquisition and Uses

  • Conference paper
  • First Online:
Integration of Constraint Programming, Artificial Intelligence, and Operations Research (CPAIOR 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13884))

  • 777 Accesses

Abstract

Motivated by identifying equations to automate the discovery of conjectures about sharp bounds on combinatorial objects, we introduce a CP model to acquire Boolean-arithmetic equations (BAE) from a table providing sharp bounds for various combinations of parameters.

Boolean-arithmetic expressions consist of simple arithmetic conditions (SAC) connected by a single commutative operator such as ‘\(\wedge \)’, ‘\(\vee \)’, ‘\({{\,\mathrm{\oplus }\,}}\)’ or ‘\(+\)’. Each SAC can use up to three variables, two coefficients, and an arithmetic function such as ‘\(+\)’, ‘−’, ‘\(\times \)’, ‘floor’, ‘\(\bmod \)’ or ‘\(\min \)’. We enhance our CP model in the following way to limit the search space: (i) We break the symmetries linked to multiple instances of similar SACs in the same expression. (ii) We prevent the creation of SAC that could be simplified away. We identify several use cases of our CP model for acquiring BAE and show its applicability for learning sharp bounds for eight types of combinatorial objects as digraphs, forests, and partitions.

R. Gindulling is supported by the EU-funded ASSISTANT project no. 101000165.

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 79.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 99.99
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.

    The expression \(( cond \,?\,x:y)\) denotes x if condition \( cond \) holds, y otherwise.

References

  1. Alur, R., Singh, R., Fisman, D., Solar-Lezama, A.: Search-based program synthesis. Commun. ACM 61(12), 84–93 (2018). https://doi.org/10.1145/3208071

    Article  Google Scholar 

  2. Au, W.H., Chan, K.C.: Mining fuzzy association rules in a bank-account database. IEEE Trans. Fuzzy Syst. 11(2), 238–248 (2003)

    Article  Google Scholar 

  3. Aung, M.S.H., et al.: Comparing analytical decision support models through Boolean rule extraction: a case study of ovarian Tumour malignancy. In: Liu, D., Fei, S., Hou, Z., Zhang, H., Sun, C. (eds.) ISNN 2007. LNCS, vol. 4492, pp. 1177–1186. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-72393-6_139

    Chapter  Google Scholar 

  4. Barbareschi, M., Barone, S., Mazzocca, N.: Advancing synthesis of decision tree-based multiple classifier systems: an approximate computing case study. Knowl. Inf. Syst. 63(6), 1577–1596 (2021). https://doi.org/10.1007/s10115-021-01565-5

    Article  Google Scholar 

  5. Beldiceanu, N.: Global constraints as graph properties on a structured network of elementary constraints of the same type. In: Dechter, R. (ed.) CP 2000. LNCS, vol. 1894, pp. 52–66. Springer, Heidelberg (2000). https://doi.org/10.1007/3-540-45349-0_6

    Chapter  MATH  Google Scholar 

  6. Beldiceanu, N., Carlsson, M., Petit, T.: Deriving filtering algorithms from constraint checkers. In: Wallace, M. (ed.) CP 2004. LNCS, vol. 3258, pp. 107–122. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-30201-8_11

    Chapter  MATH  Google Scholar 

  7. Beldiceanu, N., Carlsson, M., Rampon, J.X.: Global Constraint Catalog, 2nd Edition (revision a). Technical report T2012–03, Swedish Institute of Computer Science (2012). http://ri.diva-portal.org/smash/get/diva2:1043063/FULLTEXT01.pdf

  8. Beldiceanu, N., Cheukam-Ngouonou, J., Douence, R., Gindullin, R., Quimper, C.G.: Acquiring maps of interrelated conjectures on sharp bounds. In: 28th International Conference on Principles and Practice of Constraint Programming (CP 2022). Schloss Dagstuhl-Leibniz-Zentrum für Informatik (2022)

    Google Scholar 

  9. Bessiere, C., et al.: The balance constraint family. In: O’Sullivan, B. (ed.) CP 2014. LNCS, vol. 8656, pp. 174–189. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-10428-7_15

    Chapter  Google Scholar 

  10. Blum, A.: Relevant examples and relevant features: thoughts from computational learning theory. In: AAAI Fall Symposium on Relevance, vol. 5, p. 1 (1994)

    Google Scholar 

  11. Blum, A.L., Langley, P.: Selection of relevant features and examples in machine learning. Artif. Intell. 97(1–2), 245–271 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. Brence, J., Todorovski, L., Džeroski, S.: Probabilistic grammars for equation discovery. Knowl.-Based Syst. 224 (2021). https://doi.org/10.1016/j.knosys.2021.107077

  13. Forman, G., Kirshenbaum, E.: Extremely fast text feature extraction for classification and indexing. In: Proceedings of the 17th ACM Conference on Information and Knowledge Management, pp. 1221–1230 (2008)

    Google Scholar 

  14. Golia, P., Slivovsky, F., Roy, S., Meel, K.S.: Engineering an efficient Boolean functional synthesis engine. In: 2021 IEEE/ACM International Conference On Computer Aided Design (ICCAD), pp. 1–9. IEEE (2021)

    Google Scholar 

  15. Guyon, I., Elisseeff, A.: An introduction to feature extraction. In: Guyon, I., Nikravesh, M., Gunn, S., Zadeh, L.A. (eds.) Feature Extraction. Studies in Fuzziness and Soft Computing, vol. 207, pp. 1–25. Springer, Berlin, Heidelberg (2006). https://doi.org/10.1007/978-3-540-35488-8_1

  16. Jakobovic, D., Picek, S., Martins, M.S., Wagner, M.: Toward more efficient heuristic construction of Boolean functions. Appl. Soft Comput. 107, 107327 (2021)

    Article  Google Scholar 

  17. Jun, S., Lee, S., Chun, H.: Learning dispatching rules using random forest in flexible job shop scheduling problems. Int. J. Prod. Res. 57(10), 3290–3310 (2019)

    Article  Google Scholar 

  18. Knuth, D.: Art of Computer Programming, Volume 4, Generating All Trees, pp. 461–462. Addison-Wesley, Boston (2006)

    Google Scholar 

  19. Kosman, E., Kolchinsky, I., Schuster, A.: Mining logical arithmetic expressions from proper representations. In: Proceedings of the 2022 SIAM International Conference on Data Mining (SDM), pp. 621–629. SIAM (2022)

    Google Scholar 

  20. Lambert-Torres, G.: Application of rough sets in power system control center data mining. In: 2002 IEEE Power Engineering Society Winter Meeting. Conference Proceedings (Cat. No. 02CH37309). vol. 1, pp. 627–631. IEEE (2002)

    Google Scholar 

  21. Larson, C.E., Van Cleemput, N.: Automated conjecturing iii. Ann. Math. Artif. Intell. 81(3), 315–327 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mereani, F., Howe, J.M.: Exact and approximate rule extraction from neural networks with Boolean features. In: Proceedings of the 11th International Joint Conference on Computational Intelligence, pp. 424–433. SCITEPRESS (2019)

    Google Scholar 

  23. Mossel, E., O’Donnell, R., Servedio, R.A.: Learning functions of k relevant variables. J. Comput. Syst. Sci. 69(3), 421–434 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Mutlu, E.C., Oghaz, T.A.: Review on graph feature learning and feature extraction techniques for link prediction. arXiv preprint arXiv:1901.03425 (2019)

  25. Nguifo, E.M., Njiwoua, P.: Using lattice-based framework as a tool for feature extraction. In: Nédellec, C., Rouveirol, C. (eds.) ECML 1998. LNCS, vol. 1398, pp. 304–309. Springer, Heidelberg (1998). https://doi.org/10.1007/BFb0026700

    Chapter  Google Scholar 

  26. Pachet, F., Roy, P.: Automatic generation of music programs. In: Jaffar, J. (ed.) CP 1999. LNCS, vol. 1713, pp. 331–345. Springer, Heidelberg (1999). https://doi.org/10.1007/978-3-540-48085-3_24

    Chapter  Google Scholar 

  27. Pesant, G.: A filtering algorithm for the stretch constraint. In: Walsh, T. (ed.) CP 2001. LNCS, vol. 2239, pp. 183–195. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-45578-7_13

    Chapter  MATH  Google Scholar 

  28. Schelldorfer, J., Wuthrich, M.V.: Nesting classical actuarial models into neural networks (2019). Available at SSRN 3320525

    Google Scholar 

  29. Todorovski, L.: Equation discovery. In: Sammut, C., Webb, G.I. (eds.) Encyclopedia of Machine Learning, pp. 327–330. Springer, Boston, MA (2011). https://doi.org/10.1007/978-0-387-30164-8_258

  30. Yang, H., Rudin, C., Seltzer, M.: Scalable Bayesian rule lists. In: International Conference on Machine Learning, pp. 3921–3930. PMLR (2017)

    Google Scholar 

  31. Yu, J., Ignatiev, A., Stuckey, P.J., Le Bodic, P.: Computing optimal decision sets with SAT. In: Simonis, H. (ed.) CP 2020. LNCS, vol. 12333, pp. 952–970. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-58475-7_55

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. IMT Atlantique, Nantes, France

    R. Gindullin, N. Beldiceanu, J. Cheukam Ngouonou & R. Douence

  2. LS2N, Nantes, France

    R. Gindullin, N. Beldiceanu, J. Cheukam Ngouonou & R. Douence

  3. INRIA, Paris, France

    R. Douence

  4. Université Laval, Quebec, Canada

    J. Cheukam Ngouonou & C. -G. Quimper

Authors
  1. R. Gindullin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. Beldiceanu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. Cheukam Ngouonou
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. R. Douence
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. C. -G. Quimper
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. Gindullin .

Editor information

Editors and Affiliations

  1. Department of Management, University of Toronto Scarborough and Rotman School of Management, University of Toronto, Toronto, ON, Canada

    Andre A. Cire

Rights and permissions

Reprints and permissions

Copyright information

© 2023 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

Gindullin, R., Beldiceanu, N., Ngouonou, J.C., Douence, R., Quimper, C.G. (2023). Boolean-Arithmetic Equations: Acquisition and Uses. In: Cire, A.A. (eds) Integration of Constraint Programming, Artificial Intelligence, and Operations Research. CPAIOR 2023. Lecture Notes in Computer Science, vol 13884. Springer, Cham. https://doi.org/10.1007/978-3-031-33271-5_25

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-33271-5_25

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-33270-8

  • Online ISBN: 978-3-031-33271-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation