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Minimally Many-Valued Extensions of the Monoidal t-Norm Based Logic MTL

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Fuzzy Logic and Soft Computing Applications (WILF 2016)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10147))

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Abstract

In this paper we shall deal with those axiomatic extensions of the monoidal t-norm based logic MTL which are minimally many-valued, that is, those logics extending MTL such that any further extension collapses them to Boolean Logic. We shall prove some characterisation results concerning the algebraic semantics of these logic, and completely classify the minimally many-valued logics extending Hájek’s Basic Logic and Weak Nilpotent Minimum Logic. For the latter logics, we shall use our results to evaluate the complexity of deciding whether a formula is Booleanising for some extensions of them, or whether it provides a non-classical extension.

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Notes

  1. 1.

    In [17] such structures are called basic semihoops, but we use the terminology of [15].

References

  1. Aglianò, P., Montagna, F.: Varieties of BL-algebras I: general properties. J. Pure Appl. Algebra 181(2–3), 105–129 (2003). doi:10.1016/S0022-4049(02)00329-8

    Article  MathSciNet  MATH  Google Scholar 

  2. Aguzzoli, S., Bianchi, M.: On Linear Varieties of MTL-Algebras (2017, in preparation)

    Google Scholar 

  3. Aguzzoli, S., Bianchi, M.: Single chain completeness and some related properties. Fuzzy Sets Syst. 301, 51–63 (2016). doi:10.1016/j.fss.2016.03.008

    Article  MathSciNet  Google Scholar 

  4. Aguzzoli, S., Bianchi, M., Valota, D.: A note on drastic product logic. In: Laurent, A., Strauss, O., Bouchon-Meunier, B., Yager, R.R. (eds.) IPMU 2014. CCIS, vol. 443, pp. 365–374. Springer, Heidelberg (2014). doi:10.1007/978-3-319-08855-6_37. arXiv:1406.7166

    Google Scholar 

  5. Aguzzoli, S., Gerla, B., Haniková, Z.: Complexity issues in basic logic. Soft Comput. 9(12), 919–934 (2005). doi:10.1007/s00500-004-0443-y

    Article  MATH  Google Scholar 

  6. Aguzzoli, S., Marra, V.: Two principles in many-valued logic. In: Montagna, F. (ed.) Petr Hájek on Mathematical Fuzzy Logic, pp. 159–174. Springer, Heidelberg (2015). doi:10.1007/978-3-319-06233-4_8

    Google Scholar 

  7. Bianchi, M.: The logic of the strongest and the weakest t-norms. Fuzzy Sets Syst. 276, 31–42 (2015). doi:10.1016/j.fss.2015.01.013

    Article  MathSciNet  Google Scholar 

  8. Blok, W., Pigozzi, D.: Algebraizable Logics, Memoirs of the American Mathematical Society, vol. 77. American Mathematical Society (1989). tinyurl.com/o89ug5o

  9. Burris, S., Sankappanavar, H.: A Course in Universal Algebra, vol. 78. Springer, Heidelberg (1981). 2012 Electronic Edition: http://tinyurl.com/zaxeopo

    MATH  Google Scholar 

  10. Cai, J., Gundermann, T., Hartmanis, J., Hemachandra, L.A., Sewelson, V., Wagner, K., Wechsung, G.: The boolean hierarchy I: structural properties. Siam J. Comput. 17(6), 1232–1252 (1988). doi:10.1137/0217078

  11. Cignoli, R., D’Ottaviano, I., Mundici, D.: Algebraic Foundations of Many-Valued Reasoning. Trends in Logic, vol. 7. Kluwer Academic Publishers, Dordrecht (1999)

    MATH  Google Scholar 

  12. Cignoli, R., Esteva, F., Godo, L., Torrens, A.: Basic fuzzy logic is the logic of continuous t-norms and their residua. Soft Comput. 4(2), 106–112 (2000). doi:10.1007/s005000000044

    Article  Google Scholar 

  13. Cignoli, R., Torrens, A.: An algebraic analysis of product logic. Mult. Valued Log. 5, 45–65 (2000)

    MathSciNet  MATH  Google Scholar 

  14. Cintula, P., Esteva, F., Gispert, J., Godo, L., Montagna, F., Noguera, C.: Distinguished algebraic semantics for t-norm based fuzzy logics: methods and algebraic equivalencies. Ann. Pure Appl. Log. 160(1), 53–81 (2009). doi:10.1016/j.apal.2009.01.012

    Article  MathSciNet  MATH  Google Scholar 

  15. Cintula, P., Hájek, P., Noguera, C.: Handbook of Mathematical Fuzzy Logic, vols. 1 and 2. College Publications (2011)

    Google Scholar 

  16. Esteva, F., Godo, L.: Monoidal t-norm based logic: towards a logic for left-continuous t-norms. Fuzzy Sets Syst. 124(3), 271–288 (2001). doi:10.1016/S0165-0114(01)00098-7

    Article  MathSciNet  MATH  Google Scholar 

  17. Esteva, F., Godo, L., Hájek, P., Montagna, F.: Hoops and fuzzy logic. J. Log. Comput. 13(4), 532–555 (2003). doi:10.1093/logcom/13.4.532

    Article  MathSciNet  MATH  Google Scholar 

  18. Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Studies in Logic and the Foundations of Mathematics, vol. 151. Elsevier, Hoboken (2007)

    MATH  Google Scholar 

  19. Gispert, J.: Axiomatic extensions of the nilpotent minimum logic. Rep. Math. Log. 37, 113–123 (2003). tinyurl.com/nqsle2f

    MathSciNet  MATH  Google Scholar 

  20. Hájek, P.: Metamathematics of Fuzzy Logic. Trends in Logic, Paperback edn., vol. 4. Kluwer Academic Publishers (1998)

    Google Scholar 

  21. Jenei, S., Montagna, F.: A proof of standard completeness for Esteva and Godo’s logic MTL. Stud. Log. 70(2), 183–192 (2002). doi:10.1023/A:1015122331293

    Article  MathSciNet  MATH  Google Scholar 

  22. Katoh, Y., Kowalski, T., Ueda, M.: Almost minimal varieties related to fuzzy logic. Rep. Math. Log., 173–194 (2006). tinyurl.com/h8sc6j3

  23. Kowalski, T.: Semisimplicity, EDPC and discriminator varieties of residuated lattices. Stud. Log. 77(2), 255–265 (2004). doi:10.1023/B:STUD.0000037129.58589.0c

    Article  MathSciNet  MATH  Google Scholar 

  24. Montagna, F.: Completeness with respect to a chain and universal models in fuzzy logic. Arch. Math. Log. 50(1–2), 161–183 (2011). doi:10.1007/s00153-010-0207-6

    Article  MathSciNet  MATH  Google Scholar 

  25. Mundici, D.: Satisfiability in many-valued sentential logic is NP-complete. Theor. Comput. Sci. 52(1), 145–153 (1987). doi:10.1016/0304-3975(87)90083-1

    Article  MathSciNet  MATH  Google Scholar 

  26. Noguera, C.: Algebraic study of axiomatic extensions of triangular norm based fuzzy logics. Ph.D. thesis, IIIA-CSIC (2006). tinyurl.com/ncxgolk

  27. Noguera, C., Esteva, F., Gispert, J.: On some varieties of MTL-algebras. Log. J. IGPL 13(4), 443–466 (2005). doi:10.1093/jigpal/jzi034

    Article  MathSciNet  MATH  Google Scholar 

  28. Papadimitriou, C., Yannakakis, M.: The complexity of facets (and some facets of complexity). J. Comput. Syst. Sci. 28(2), 244–259 (1984). doi:10.1016/0022-0000(84)90068-0

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, via Comelico 39/41, 20135, Milano, Italy

    Stefano Aguzzoli & Matteo Bianchi

Authors
  1. Stefano Aguzzoli
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  2. Matteo Bianchi
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Correspondence to Matteo Bianchi .

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Editors and Affiliations

  1. University of Naples “Parthenope”, Naples, Italy

    Alfredo Petrosino

  2. University of Salerno, Fisciano, (Salerno), Italy

    Vincenzo Loia

  3. University of Alberta, Edmonton, Alberta, Canada

    Witold Pedrycz

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Aguzzoli, S., Bianchi, M. (2017). Minimally Many-Valued Extensions of the Monoidal t-Norm Based Logic MTL. In: Petrosino, A., Loia, V., Pedrycz, W. (eds) Fuzzy Logic and Soft Computing Applications. WILF 2016. Lecture Notes in Computer Science(), vol 10147. Springer, Cham. https://doi.org/10.1007/978-3-319-52962-2_9

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  • DOI: https://doi.org/10.1007/978-3-319-52962-2_9

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