Skip to main content
SpringerLink
Log in

On Matrices and K-Relations

  • Conference paper
  • First Online:
Foundations of Information and Knowledge Systems (FoIKS 2020)

Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 12012))

Abstract

We show that the matrix query language \(\mathsf {MATLANG}\) corresponds to a natural fragment of the positive relational algebra on K-relations. The fragment is defined by introducing a composition operator and restricting K-relation arities to two. We then proceed to show that \(\mathsf {MATLANG}\) can express all matrix queries expressible in the positive relational algebra on K-relations, when intermediate arities are restricted to three. Thus we offer an analogue, in a model with numerical data, to the situation in classical logic, where the algebra of binary relations is equivalent to first-order logic with three variables.

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 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
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.

    \(\mathsf {ARA}\) stands for Annotated-Relation Algebra, as the elements from K that a K-relation assigns to its tuples are usually viewed as annotations.

  2. 2.

    The paper [5] received the PODS 2017 test-of-time award.

  3. 3.

    \(\mathcal {L}_\mathrm{Aggr}\) is a two-sorted logic with base variables and numerical variables.

References

  1. Abiteboul, S., Hull, R., Vianu, V.: Foundations of Databases. Addison-Wesley, Reading (1995)

    MATH  Google Scholar 

  2. Brijder, R., Geerts, F., Van den Bussche, J., Weerwag, T.: On the expressive power of query languages for matrices. In: Kimelfeld, B., Amsterdamer, Y. (eds.) Proceedings of 21st International Conference on Database Theory. LIPIcs, vol. 98, pp. 10:1–10:17. Schloss Dagstuhl-Leibniz Center for Informatics (2018)

    Google Scholar 

  3. Van den Bussche, J.: \(\rm FO^3\) and the algebra of binary relations. https://databasetheory.org/node/94. Accessed 22 July 2019

  4. Geerts, F.: On the expressive power of linear algebra on graphs. In: Barcelo, P., Calautti, M. (eds.) Proceedings of 22nd International Conference on Database Theory. LIPIcs, vol. 127, pp. 7:1–7:19. Schloss Dagstuhl-Leibniz Center for Informatics (2019)

    Google Scholar 

  5. Green, T., Karvounarakis, G., Tannen, V.: Provenance semirings. In: Proceedings of 26th ACM Symposium on Principles of Database Systems, pp. 31–40 (2007)

    Google Scholar 

  6. Hutchison, D., Howe, B., Suciu, D.: LaraDB: a minimalist kernel for linear and relational algebra computation. In: Afrati, F., Sroka, J. (eds.) Proceedings of 4th ACM SIGMOD Workshop on Algorithms and Systems for MapReduce and Beyond, pp. 2:1–2:10 (2017)

    Google Scholar 

  7. Jananthan, H., Zhou, Z., et al.: Polystore mathematics of relational algebra. In: Nie, J.Y., Obradovic, Z., Suzumura, T., et al. (eds.) Proceedings of IEEE International Conference on Big Data, pp. 3180–3189. IEEE (2017)

    Google Scholar 

  8. Joglekar, M., Puttagunta, R., Ré, C.: AJAR: aggregations and joins over annotated relations. In: Proceedings of 35th ACM Symposium on Principles of Databases, pp. 91–106. ACM (2016)

    Google Scholar 

  9. Libkin, L.: Expressive power of SQL. Theoret. Comput. Sci. 296, 379–404 (2003)

    Article  MathSciNet  Google Scholar 

  10. Luo, S., Gao, Z., Gubanov, M., Perez, L., Jermaine, C.: Scalable linear algebra on a relational database system. SIGMOD Rec. 47(1), 24–31 (2018)

    Article  Google Scholar 

  11. Abo Khamis, M., Ngo, H.Q., Rudra, A.: FAQ: questions asked frequently. In: Proceedings of 35th ACM Symposium on Principles of Databases, pp. 13–28. ACM (2016)

    Google Scholar 

  12. Abo Khamis, M., Ngo, H.Q., Rudra, A.: Juggling functions inside a database. SIGMOD Rec. 46(1), 6–13 (2017)

    Article  Google Scholar 

  13. Maddux, R.: The origin of relation algebras in the development and axiomatization of the calculus of relations. Stud. Logica. 50(3/4), 421–455 (1991)

    Article  MathSciNet  Google Scholar 

  14. Marx, M., Venema, Y.: Multi-Dimensional Modal Logic. APLS, vol. 4. Springer, Dordrecht (1997). https://doi.org/10.1007/978-94-011-5694-3

    Book  MATH  Google Scholar 

  15. Pratt, V.: Origins of the calculus of binary relations. In: Proceedings of 7th Annual IEEE Symposium on Logic in Computer Science, pp. 248–254 (1992)

    Google Scholar 

  16. Tarski, A.: On the calculus of relations. J. Symb. Log. 6, 73–89 (1941)

    Article  MathSciNet  Google Scholar 

  17. Tarski, A., Givant, S.: A Formalization of Set Theory Without Variables. AMS Colloquium Publications, vol. 41. American Mathematical Society, Providence (1987)

    MATH  Google Scholar 

  18. Yan, Z., Tannen, V., Ives, Z.: Fine-grained provenance for linear algebra operators. In: Proceedings of 8th USENIX Workshop on the Theory and Practice of Provenance (2016)

    Google Scholar 

Download references

Acknowledgments

We thank Floris Geerts for inspiring discussions. Robert Brijder has been a postdoctoral fellow of the Research Foundation - Flanders (FWO). Jan Van den Bussche was partially supported by the National Natural Science Foundation of China under grant# 61972455.

Author information

Authors and Affiliations

  1. Data Science Institute, Hasselt University, Martelarenlaan 42, 3500, Hasselt, Belgium

    Robert Brijder, Marc Gyssens & Jan Van den Bussche

Authors
  1. Robert Brijder
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marc Gyssens
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jan Van den Bussche
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marc Gyssens .

Editor information

Editors and Affiliations

  1. CNRS, University of Toulouse, Toulouse, France

    Andreas Herzig

  2. University of Helsinki, Helsinki, Finland

    Juha Kontinen

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Brijder, R., Gyssens, M., Van den Bussche, J. (2020). On Matrices and K-Relations. In: Herzig, A., Kontinen, J. (eds) Foundations of Information and Knowledge Systems. FoIKS 2020. Lecture Notes in Computer Science(), vol 12012. Springer, Cham. https://doi.org/10.1007/978-3-030-39951-1_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-39951-1_3

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-39950-4

  • Online ISBN: 978-3-030-39951-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation