Document Open Access Logo

Fuzzy Algebraic Theories

Authors Davide Castelnovo, Marino Miculan

PDF

File

Thumbnail PDF
PDF
LIPIcs.CSL.2022.13.pdf
  • Filesize: 0.88 MB
  • 17 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic
Keywords
  • categorical logic
  • fuzzy sets
  • algebraic reasoning
  • equational axiomatisations
  • monads
  • Eilenberg-Moore algebras

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

In this work we propose a formal system for fuzzy algebraic reasoning. The sequent calculus we define is based on two kinds of propositions, capturing equality and existence of terms as members of a fuzzy set. We provide a sound semantics for this calculus and show that there is a notion of free model for any theory in this system, allowing us (with some restrictions) to recover models as Eilenberg-Moore algebras for some monad. We will also prove a completeness result: a formula is derivable from a given theory if and only if it is satisfied by all models of the theory. Finally, leveraging results by Milius and Urbat, we give HSP-like characterizations of subcategories of algebras which are categories of models of particular kinds of theories.

Cite As Get BibTex

Davide Castelnovo and Marino Miculan. Fuzzy Algebraic Theories. In 30th EACSL Annual Conference on Computer Science Logic (CSL 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 216, pp. 13:1-13:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.CSL.2022.13

Author Details

Davide Castelnovo
  • Department of Mathematics, Computer Science and Physics, University of Udine, Italy
Marino Miculan
  • Department of Mathematics, Computer Science and Physics, University of Udine, Italy

Funding

  • Miculan, Marino: Supported by the Italian MIUR PRIN 2017FTXR7S IT MATTERS.

References

  1. N. Ajmal. Homomorphism of fuzzy groups, correspondence theorem and fuzzy quotient groups. Fuzzy sets and systems, 61(3):329-339, 1994. Google Scholar
  2. N. Ajmal and A. S. Prajapati. Fuzzy cosets and fuzzy normal subgroups. Information sciences, 64(1-2):17-25, 1992. Google Scholar
  3. G. Bacci, R. Mardare, P. Panangaden, and G. Plotkin. An algebraic theory of Markov processes. In 33rd Symposium on Logic in Computer Science (LICS), pages 679-688, 2018. Google Scholar
  4. G. Bacci, R. Mardare, P. Panangaden, and G. Plotkin. Quantitative equational reasoning. Foundations of Probabilistic Programming, page 333, 2020. Google Scholar
  5. G. Birkhoff. On the structure of abstract algebras. Proceedings of the Cambridge Philosophical Society, 10:433-454, 1935. Google Scholar
  6. Filippo Bonchi, Barbara König, and Daniela Petrisan. Up-to techniques for behavioural metrics via fibrations. In CONCUR, volume 118 of LIPIcs, pages 17:1-17:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. Google Scholar
  7. F. Borceux. Handbook of Categorical Algebra: Volume 2, Categories and Structures, volume 2. Cambridge University Press, 1994. Google Scholar
  8. Davide Castelnovo and Marino Miculan. Fuzzy algebraic theories. CoRR, abs/2110.10970, 2021. URL: http://arxiv.org/abs/2110.10970.
  9. M. P Fourman and D. S. Scott. Sheaves and logic. In Applications of sheaves, pages 302-401. Springer, 1979. Google Scholar
  10. Martin Hyland and John Power. Discrete lawvere theories and computational effects. Theoretical Computer Science, 366(1-2):144-162, 2006. Google Scholar
  11. Martin Hyland and John Power. The category theoretic understanding of universal algebra: Lawvere theories and monads. Electron. Notes Theor. Comput. Sci., 172:437-458, 2007. URL: https://doi.org/10.1016/j.entcs.2007.02.019.
  12. Peter T Johnstone. Stone spaces, volume 3. Cambridge University Press, 1982. Google Scholar
  13. Gregory Maxwell Kelly. A note on relations relative to a factorization system. In Category Theory, pages 249-261. Springer, 1991. Google Scholar
  14. S. MacLane. Categories for the working mathematician, volume 5. Springer Science & Business Media, 2013. Google Scholar
  15. R. Mardare, P. Panangaden, and G. Plotkin. On the axiomatizability of quantitative algebras. In 32nd Symposium on Logic in Computer Science (LICS), pages 1-12. IEEE, 2017. Google Scholar
  16. A. S. Mashour, M. H. Ghanim, and F. I. Sidky. Normal fuzzy subgroups. Information Sciences, 20:53-59, 1990. Google Scholar
  17. S. Milius and H. Urbat. Equational axiomatization of algebras with structure. In International Conference on Foundations of Software Science and Computation Structures, pages 400-417. Springer, 2019. Google Scholar
  18. Eugenio Moggi. Notions of computation and monads. Inf. Comput., 93(1):55-92, 1991. URL: https://doi.org/10.1016/0890-5401(91)90052-4.
  19. J. N. Mordeson, D. S. Malik, and N. Kuroki. Fuzzy semigroups, volume 131. Springer, 2012. Google Scholar
  20. K. Nishizawa and J. Power. Lawvere theories enriched over a general base. Journal of Pure and Applied Algebra, 213(3):377-386, 2009. Google Scholar
  21. Gordon D. Plotkin and John Power. Notions of computation determine monads. In FoSSaCS, volume 2303 of Lecture Notes in Computer Science, pages 342-356. Springer, 2002. Google Scholar
  22. Gordon D. Plotkin and John Power. Algebraic operations and generic effects. Appl. Categorical Struct., 11(1):69-94, 2003. URL: https://doi.org/10.1023/A:1023064908962.
  23. John Power. Discrete lawvere theories. In International Conference on Algebra and Coalgebra in Computer Science, pages 348-363. Springer, 2005. Google Scholar
  24. A. Rosenfeld. Fuzzy groups. Journal of mathematical analysis and applications, 35(3):512-517, 1971. Google Scholar
  25. N. H. Williams. On Grothendieck universes. Compositio Mathematica, 21(1):1-3, 1969. Google Scholar
  26. O. Wyler. Lecture notes on topoi and quasitopoi. World Scientific, 1991. Google Scholar
  27. O. Wyler. Fuzzy logic and categories of fuzzy sets. In Non-Classical Logics and Their Applications to Fuzzy Subsets, pages 235-268. Springer, 1995. Google Scholar
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact