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Building a Framework of Rough Inclusion Functions by Means of Computerized Proof Assistant

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Rough Sets (IJCRS 2019)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 11499))

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Abstract

The paper describes some of the issues concerning the development of automated formal framework for the reasoning about rough inclusion functions, starting with the classical one, and generalizations thereof. We work with the Mizar system; the viewpoint of the rough set theory, and especially mereology by Leśniewski, can allow for the creation of new foundations for the Mizar Mathematical Library, or at least for fresh branch of this formal database, originally based on Tarski-Grothendieck axioms.

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Notes

  1. 1.

    All items from the Mizar Mathematical Library which are automatically hyperlinked can be browsed online from the page http://mizar.org/version/current/html/.

  2. 2.

    Radix type together with the cluster of attributes makes a new type of an object.

  3. 3.

    http://mizar.uwb.edu.pl/library/roughif1/.

  4. 4.

    The author wishes to thank one of the referees for this valuable remark. Of course, we are interested in the development of quasi RIFs and weak quasi RIFs, so this distinction is important even if it is meaningless in the theory of RIFs.

References

  1. Bancerek, G., et al.: Mizar: state-of-the-art and beyond. In: Kerber, M., Carette, J., Kaliszyk, C., Rabe, F., Sorge, V. (eds.) CICM 2015. LNCS (LNAI), vol. 9150, pp. 261–279. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-20615-8_17

    Chapter  Google Scholar 

  2. Gomolińska, A.: A comparative study of some generalized rough approximations. Fundamenta Informaticae 51, 103–119 (2002)

    MathSciNet  MATH  Google Scholar 

  3. Gomolińska, A.: On certain rough inclusion functions. In: Peters, J.F., Skowron, A., Rybiński, H. (eds.) Transactions on Rough Sets IX. LNCS, vol. 5390, pp. 35–55. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-89876-4_3

    Chapter  MATH  Google Scholar 

  4. Gomolińska, A.: On three closely related rough inclusion functions. In: Kryszkiewicz, M., Peters, J.F., Rybinski, H., Skowron, A. (eds.) RSEISP 2007. LNCS (LNAI), vol. 4585, pp. 142–151. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-73451-2_16

    Chapter  Google Scholar 

  5. Gomolińska, A.: Rough approximation based on weak q-RIFs. In: Peters, J.F., Skowron, A., Wolski, M., Chakraborty, M.K., Wu, W.-Z. (eds.) Transactions on Rough Sets X. LNCS, vol. 5656, pp. 117–135. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-03281-3_4

    Chapter  MATH  Google Scholar 

  6. Grabowski, A.: Mechanizing complemented lattices within Mizar type system. J. Autom. Reasoning 55, 211–221 (2015). https://doi.org/10.1007/s10817-015-9333-5

    Article  MathSciNet  MATH  Google Scholar 

  7. Grabowski, A.: Binary relations-based rough sets - an automated approach. Formalized Math. 24(2), 143–155 (2016). https://doi.org/10.1515/forma-2016-0011

    Article  MATH  Google Scholar 

  8. Grabowski, A.: Automated discovery of properties of rough sets. Fundamenta Informaticae 128(1–2), 65–79 (2013). https://doi.org/10.3233/FI-2013-933

    Article  MathSciNet  MATH  Google Scholar 

  9. Grabowski, A.: Efficient rough set theory merging. Fundamenta Informaticae 135(4), 371–385 (2014). https://doi.org/10.3233/FI-2014-1129

    Article  MathSciNet  MATH  Google Scholar 

  10. Grabowski, A.: Computer certification of generalized rough sets based on relations. In: Polkowski, L., et al. (eds.) IJCRS 2017. LNCS (LNAI), vol. 10313, pp. 83–94. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-60837-2_7

    Chapter  Google Scholar 

  11. Grabowski, A.: Automated comparative study of some generalized rough approximations. In: Proceedings of CS&P 2018, pp. 39–45 (2018)

    Google Scholar 

  12. Grabowski, A., Korniłowicz, A., Naumowicz, A.: Four decades of Mizar. J. Autom. Reasoning 55, 191 (2015). https://doi.org/10.1007/s10817-015-9345-1

    Article  MathSciNet  MATH  Google Scholar 

  13. Grabowski, A.: Lattice theory for rough sets - a case study with Mizar. Fundamenta Informaticae 147(2–3), 223–240 (2016). https://doi.org/10.3233/FI-2016-1406

    Article  MathSciNet  MATH  Google Scholar 

  14. Grabowski, A.: On the computer-assisted reasoning about rough sets. In: Dunin-Kȩplicz, B., Jankowski, A., Szczuka, M. (eds.) Monitoring, Security and Rescue Techniques in Multiagent Systems. AISC, vol. 28, pp. 215–226 (2005). https://doi.org/10.1007/3-540-32370-8_15

  15. Grabowski, A., Jastrzȩbska, M.: Rough set theory from a math-assistant perspective. In: Kryszkiewicz, M., Peters, J.F., Rybinski, H., Skowron, A. (eds.) RSEISP 2007. LNCS (LNAI), vol. 4585, pp. 152–161. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-73451-2_17

    Chapter  Google Scholar 

  16. Järvinen, J.: Lattice theory for rough sets. In: Peters, J.F., Skowron, A., Düntsch, I., Grzymała-Busse, J., Orłowska, E., Polkowski, L. (eds.) Transactions on Rough Sets VI. LNCS, vol. 4374, pp. 400–498. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-71200-8_22

    Chapter  MATH  Google Scholar 

  17. Korniłowicz, A.: Flexary connectives in Mizar. Comput. Lang. Syst. Struct. 44(Part C), 238–250 (2015). https://doi.org/10.1016/j.cl.2015.07.002

    Article  Google Scholar 

  18. Naumowicz, A.: Automating boolean set operations in Mizar proof checking with the aid of an external SAT solver. J. Autom. Reasoning 55, 285 (2015). https://doi.org/10.1007/s10817-015-9332-6

    Article  MathSciNet  MATH  Google Scholar 

  19. Pawlak, Z.: Theoretical Aspects of Reasoning about Data. Kluwer, Dordrecht (1991). https://doi.org/10.1007/978-94-011-3534-4

    Book  MATH  Google Scholar 

  20. Pąk, K.: Improving legibility of formal proofs based on the close reference principle is NP-hard. J. Autom. Reasoning 55, 295–306 (2015). https://doi.org/10.1007/s10817-015-9337-1

    Article  MathSciNet  MATH  Google Scholar 

  21. Polkowski, L.: Approximate Reasoning by Parts: An Introduction to Rough Mereology. Intelligent Systems Reference Library, vol. 20. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22279-5

    Book  Google Scholar 

  22. Polkowski, L., Skowron, A.: Rough mereology: a new paradigm for approximate reasoning. Int. J. Approximate Reasoning 15(4), 333–365 (1996). https://doi.org/10.1016/S0888-613X(96)00072-2

    Article  MathSciNet  MATH  Google Scholar 

  23. Rudnicki, P.: Obvious inferences. J. Autom. Reasoning 3(4), 383–393 (1987). https://doi.org/10.1007/BF00247436

    Article  MathSciNet  MATH  Google Scholar 

  24. Skowron, A., Stepaniuk, J.: Tolerance approximation spaces. Fundamenta Informaticae 27(2–3), 245–253 (1996). https://doi.org/10.3233/FI-1996-272311

    Article  MathSciNet  MATH  Google Scholar 

  25. Urban, J., Sutcliffe, G.: Automated reasoning and presentation support for formalizing mathematics in Mizar. In: Autexier, S., et al. (eds.) CICM 2010. LNCS (LNAI), vol. 6167, pp. 132–146. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-14128-7_12

    Chapter  MATH  Google Scholar 

  26. Wiedijk, F.: Formal proof - getting started. Not. AMS 55(11), 1408–1414 (2008)

    MathSciNet  MATH  Google Scholar 

  27. Yao, Y.Y.: Two views of the rough set theory in finite universes. Int. J. Approximate Reasoning 15(4), 291–317 (1996). https://doi.org/10.1016/S0888-613X(96)00071-0

    Article  MathSciNet  MATH  Google Scholar 

  28. Zadeh, L.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965). https://doi.org/10.1016/S0019-9958(65)90241-X

    Article  MATH  Google Scholar 

  29. Zhu, W.: Generalized rough sets based on relations. Inf. Sci. 177(22), 4997–5011 (2007). https://doi.org/10.1016/j.ins.2007.05.037

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Informatics, University of Białystok, Konstantego Ciołkowskiego 1M, 15-245, Białystok, Poland

    Adam Grabowski

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  1. Adam Grabowski
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Correspondence to Adam Grabowski .

Editor information

Editors and Affiliations

  1. University of Debrecen, Debrecen, Hungary

    Tamás Mihálydeák

  2. Southwest Petroleum University, Chengdu, China

    Fan Min

  3. Chongqing University of Posts and Telecommunications, Chongqing, China

    Guoyin Wang

  4. Indian Institute of Technology Kanpur, Kanpur, India

    Mohua Banerjee

  5. Fujian Normal University, Fuzhou, China

    Ivo Düntsch

  6. University of Rzeszów, Rzeszow, Poland

    Zbigniew Suraj

  7. University of Milano-Bicocca, Milan, Italy

    Davide Ciucci

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Grabowski, A. (2019). Building a Framework of Rough Inclusion Functions by Means of Computerized Proof Assistant. In: Mihálydeák, T., et al. Rough Sets. IJCRS 2019. Lecture Notes in Computer Science(), vol 11499. Springer, Cham. https://doi.org/10.1007/978-3-030-22815-6_18

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  • DOI: https://doi.org/10.1007/978-3-030-22815-6_18

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  • Online ISBN: 978-3-030-22815-6

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