SpringerLink
Log in

Stone algebras: 3-valued logic and rough sets

  • Foundations
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

In this article, we propose 3-valued semantics of the logics compatible with Stone and dual Stone algebras. We show that these logics can be considered as 3-valued by establishing soundness and completeness results. We also establish rough set semantics of these logics where the third value can be interpreted as not certain but possible.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (France)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  • Avron A, Konikowska B (2008) Rough sets and 3-valued logics. Stud Log 90:69–92

    Article  MathSciNet  Google Scholar 

  • Balbes R, Dwinger P (1974) Distributive lattices. University of Missouri Press, Columbia

    MATH  Google Scholar 

  • Banerjee M (1997) Rough sets and 3-valued Łukasiewicz logic. Fund Inf 31:213–220

    MATH  Google Scholar 

  • Birkhoff G (1995) Lattice theory, colloquium publications, vol XXV, 3rd edn. American Mathematical Society, Providence

  • Boicescu V, Filipoiu A, Georgescu G, Rudeanu S (1991) Łukasiewicz-Moisil algebras. North-Holland, Amsterdam

    MATH  Google Scholar 

  • Chen CC, Gratzer G (1969) Stone lattice I: construction theorems. Can J Math 21:884–894

    Article  MathSciNet  Google Scholar 

  • Chen CC, Gratzer G (1969) Stone lattice II: structure theorems. Can J Math 21:895–903

    Article  MathSciNet  Google Scholar 

  • Ciucci D, Dubois D (2014) Three-valued logics, uncertainty management and rough sets. Transactions on rough sets XVII, LNCS, vol 8375. Springer, Berlin, Heidelberg, pp 1–32

    Chapter  Google Scholar 

  • Comer S (1995) Perfect extensions of regular double Stone algebras. Algebra Univ 34(1):96–109

    Article  MathSciNet  Google Scholar 

  • Davey BA, Priestley HA (2002) Introduction to lattices and order. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Düntsch I (1997) A logic for rough sets. Theor Comput Sci 179:427–436

    Article  MathSciNet  Google Scholar 

  • Düntsch I, Orłowska E (2011) Discrete dualities for double Stone algebras. Stud Log 99(1):127–142

  • Dunn J (1994) Star and perp: two treatments of negation. In: Tomberlin J (ed) Philosophical perspectives, vol 7. Ridgeview Publishing Company, Atascadero, pp 331–357

    Google Scholar 

  • Dunn J (1995) Positive modal logic. Stud Log 55:301–317

    Article  Google Scholar 

  • Dunn J (1996) Generalised ortho negation. In: Wansing H (ed) Negation: a notion in focus. Walter de Gruyter, Berlin, pp 3–26

    Chapter  Google Scholar 

  • Dunn J (1999) A comparative study of various model-theoretic treatments of negation: a history of formal negations. In: Gabbay D, Wansing H (eds) What is Negation?. Kluwer Academic Publishers, The Netherlands, pp 23–51

    Chapter  Google Scholar 

  • Gratzer G, Schmidt ET (1957) On a problem of M.H. stone. Acta Math Acad Sci Hunger 8:455–460

    Article  MathSciNet  Google Scholar 

  • Gratzer G (1969) Stone algebras form an equational class. J Austr Math Soc 9:308–309

    Article  Google Scholar 

  • Gratzer G (2009) Lattice theory: first concepts and distributive lattices. Dover Publications Inc

  • Gehrke M, Walker E (1992) On the structure of rough sets. Bull Pol Acad Sci, Math 40(3):235–255

    MathSciNet  MATH  Google Scholar 

  • Iturrioz L (1999) Rough sets and three-valued structures. In: Orłowska E (ed) Logic at work: essays dedicated to the memory of Helena Rasiowa, volume 24 of studies in fuzziness and soft computing. Springer, New York, pp 596–603

    Google Scholar 

  • Katriňák T (1973) Construction of regular double p-algebras. Proc Am Math Soc 40:75–79

    Article  MathSciNet  Google Scholar 

  • Kumar A, Banerjee M (2017) Kleene algebras and logic: Boolean and rough set representations, 3-valued, rough set and perp semantics. Stud Log 105:439–469

    Article  MathSciNet  Google Scholar 

  • Kumar A (2020) A study of algebras and logics of rough sets based on classical and generalized approximation spaces. In: Peters J, Skowron A (eds) Transactions on rough sets XXII, LNCS, vol 12485. Springer, Germany, pp 123–251

    Chapter  Google Scholar 

  • Kumar A, Banerjee M (2017) A semantic analysis of stone and dual stone negations with regularity. In: Ghosh S, Prasad S (eds) Logic and its applications. Springer, Berlin, Heidelberg, pp 139–153

    Chapter  Google Scholar 

  • Nakayama Y, Akama S, Murai T (2020) Rough set logic for Kleene’s three-valued logic. In: 2020 Joint 11th international conference on soft computing and intelligent systems and 21st symposium on advanced intelligent systems (SCIS-ISIS). IEEE, pp 1–5

  • Pagliani P (1998) Rough set theory and logic-algebraic structures. In: Orłowska E (ed) Incomplete information: rough set analysis, volume 3 of studies in fuzziness and soft computing. Springer, New York, pp 109–190

    Chapter  Google Scholar 

  • Panicker G, Banerjee M (2019) Rough sets and the algebra of conditional logic. In: Mihálydeák T (ed) IJCRS, volume 11499 of LNCS. Springer, Cham, pp 28–39

    Google Scholar 

  • Pawlak Z (1982) Rough sets. Int J Comput Inf Sci 11:341–356

    Article  Google Scholar 

  • Pawlak Z (1991) Rough sets: theoretical aspects of reasoning about data. Kluwer Academic Publishers

  • Priestley H (1974) Stone lattices: a topological approach. Fund Math 84(2):127–143

    Article  MathSciNet  Google Scholar 

  • Pomykała J, Pomykała JA (1988) The stone algebra of rough sets. Bull Pol Acad Sci, Math 36:495–508

    MathSciNet  MATH  Google Scholar 

  • Zhou H, Zhao B (2011) Stone-like representation theorems and three-valued filters in \(R_{0}\)-algebras (nilpotent minimum algebras). Fuzzy Sets Syst 162:1–26

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi, India

    Arun Kumar & Shilpi Kumari

Authors
  1. Arun Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shilpi Kumari
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Arun Kumar.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kumar, A., Kumari, S. Stone algebras: 3-valued logic and rough sets. Soft Comput 25, 12685–12692 (2021). https://doi.org/10.1007/s00500-021-06068-7

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-021-06068-7

Keywords

Search

Navigation