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Representations for logics and algebras related to revised drastic product t-norm

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Abstract

We obtain a functional representation of RDP logic formulas and a combinatorial representation for finite RDP algebras. As the variety of RDP algebras includes the varieties of Gödel, DP, and EMTL algebras, we derive analogous representations for such varieties and their corresponding logics.

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Notes

  1. MTL algebras are commutative integral bounded residuated lattices satisfying prelinearity (Galatos et al. 2007).

  2. A multiset is a family whose members have multiple instances (a set is a multiset whose members have exactly one instance).

  3. Note that, if \(g :J \rightarrow J'\) is an open map such that \(g(\max (J))=\max (J')\), then \(|J'| \le |J|\).

References

  • Aguzzoli S, Gerla B, Marra V (2008) Defuzzifying formulas in Gödel logic through finitely additive measures. In: 2008 IEEE international conference on fuzzy systems (IEEE World Congress on Computational Intelligence), pp 1886–1893

  • Aguzzoli S, Bova S, Gerla B (2011) Free algebras and functional representation for fuzzy logics. In: Cintula P, Hájek P, Noguera C (eds) Handbook of mathematical fuzzy logic, vol 2. College Publications, Cambridge, pp 713–791

    Google Scholar 

  • Aguzzoli S, Bianchi M, Valota D (2014) A note on drastic product logic. In: Information processing and management of uncertainty, volume 443 of communications in computer and information science, pp 365–374. Springer

  • Aguzzoli S, Bova S, Valota D (2017a) Free weak nilpotent minimum algebras. Soft Comput 21(1):79–95

    Article  MATH  Google Scholar 

  • Aguzzoli S, Bianchi M, Gerla B, Valota D (2017b) Probability measures in Gödel\(_\varDelta \) logic. In Antonucci A, Cholvy L, Papini O (eds) Proceedings of Symbolic and quantitative approaches to reasoning with uncertainty: 14th European Conference, ECSQARU 2017, Lugano, Switzerland, July 10–14, 2017, pp 353–363. Springer

  • Aguzzoli S, Bianchi M, Gerla B, Valota D (2018) Free algebras, states and duality for the propositional Gödel \(\Delta \) and Drastic Product logics (Submitted manuscript)

  • Bianchi M (2015) The logic of the strongest and the weakest t-norms. Fuzzy Sets Syst 276:31–42

    Article  MathSciNet  MATH  Google Scholar 

  • Bova S, Valota D (2012) Finite RDP-algebras: duality, coproducts and logic. J Log Comput 22(3):417–450

    Article  MathSciNet  MATH  Google Scholar 

  • Burris S, Sankappanavar HP (1981) A course in universal algebra. Springer, Berlin

    Book  MATH  Google Scholar 

  • D’Antona OM, Marra V (2006) Computing coproducts of finitely presented Gödel algebras. Ann Pure Appl Logic 142(1):202–211

    Article  MathSciNet  MATH  Google Scholar 

  • Esakia L (1974) Topological Kripke models. Soviet Math Dokl 15:147–151

    MATH  Google Scholar 

  • Esteva F, Domingo X (1980) Sobre Negaciones Fuertes y Débiles en [0,1]. Stochastica 4:141–166

    MathSciNet  Google Scholar 

  • Esteva F, Godo L (2001) Monoidal t-norm based logic: towards a logic for left-continuous t-norms. Fuzzy Sets Syst 124(3):271–288

    Article  MathSciNet  MATH  Google Scholar 

  • Flaminio T, Godo L, Ugolini S (2018) Towards a probability theory for product logic: states, integral representation and reasoning. Int J Approx Reason 93:199–218

    Article  MathSciNet  MATH  Google Scholar 

  • Galatos N, Jipsen P, Kowalski T, Ono H (2007) Residuated lattices: an algebraic glimpse at substructural logics, volume 151 of studies in logic and the foundations of mathematics. Elsevier, Amsterdam

  • Gerla B (2000) A note on functions associated with Gödel formulas. Soft Comput 4(4):206–209

    Article  MATH  Google Scholar 

  • Hájek P (1998) Metamathematics of fuzzy logic, volume 4 of trends in logic. Kluwer Academic Publishers, London

  • Horn A (1969) Logic with truth values in a linearly ordered Heyting algebra. J Symb Log 34(3):395–408

    Article  MathSciNet  MATH  Google Scholar 

  • Jenei S (2002) A note on the ordinal sum theorem and its consequence for the construction of triangular norms. Fuzzy Sets Syst 126(2):199–205

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Klement EP, Mesiar R, Pap E (2000) Triangular norms, vol 8. trends in logic-studia logica library. Kluwer Academic Publishers, Dordrecht

  • Mundici D (1995) Averaging the truth-value in Łukasiewicz logic. Stud Log 55(1):113–127

    Article  MathSciNet  MATH  Google Scholar 

  • Noguera C (2006) Algebraic study of axiomatic extensions of triangular norm based fuzzy logics. Ph.D. thesis, IIIA-CSIC

  • Valota D (2010) Poset representation for free RDP-algebras. In: Hosni H, Montagna F (eds) Probability, uncertainty and rationality, volume 10 of CRM series. Edizioni della Scuola Normale Superiore, Pisa

    Google Scholar 

  • Valota D (2018) Spectra of Gödel Algebras (Submitted manuscript)

  • Wang S (2007) A fuzzy logic for the revised drastic product t-norm. Soft Comput 11(6):585–590

    Article  MATH  Google Scholar 

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Acknowledgements

The author is thankful to the anonymous reviewers for their helpful comments.

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  1. Dipartimento di Informatica, Università degli Studi di Milano, Via Celoria 18, 20133, Milan, MI, Italy

    Diego Valota

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  1. Diego Valota
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Correspondence to Diego Valota.

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Valota, D. Representations for logics and algebras related to revised drastic product t-norm. Soft Comput 23, 2331–2342 (2019). https://doi.org/10.1007/s00500-018-3541-y

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