Skip to main content

Interval Additive Generators of Interval T-Norms

  • Conference paper
Logic, Language, Information and Computation (WoLLIC 2008)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 5110))

Abstract

The aim of this paper is to introduce the notion of interval additive generators of interval t-norms as interval representations of additive generators of t-norms, considering both the correctness and the optimality criteria, in order to provide a more systematic methodology for the selection of interval t-norms in the various applications. We prove that interval additive generators satisfy the main properties of punctual additive generators discussed in the literature.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 74.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 99.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Zadeh, L.A.: Fuzzy sets. Information and Control, 338–353 (1965)

    Google Scholar 

  2. Mitra, S., Pal, S.K.: Fuzzy sets in pattern recognition and machine intelligence. Fuzzy Sets and Systems 156, 381–386 (2005)

    Article  MathSciNet  Google Scholar 

  3. Chen, G., Pham, T.T.: Fuzzy Sets, Fuzzy Logic, and Fuzzy Control Systems. CRC Press, Boca Raton (2001)

    Google Scholar 

  4. Carlsson, C., Fuller, R.: Fuzzy Reasoning in Decision Making and Optimization. Physicva-Verlag, Springer, Heidelberg (2002)

    MATH  Google Scholar 

  5. Siler, W., Buckley, J.J.: Fuzzy Expert Systems and Fuzzy Reasoning. John Wiley, NY (2004)

    Google Scholar 

  6. Bedregal, B.C., Costa, A.C.R., Dimuro, G.P.: Fuzzy rule-based hand gesture recognition. In: Bramer, M. (ed.) Artificial Intelligence in Theory And Practice. IFIP Series, vol. 271, pp. 285–294. Springer, Boston (2006)

    Chapter  Google Scholar 

  7. Schweizer, B., Sklar, A.: Probabilistic Metric Spaces. North-Holland, NY (1983)

    MATH  Google Scholar 

  8. Fodor, J., Roubens, M.: Fuzzy Preference Modelling and Multicriteria Decision Support. Kluwer Academic Publisher, Dordrecht (1994)

    MATH  Google Scholar 

  9. Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer Academic Publishers, Boston (1998)

    MATH  Google Scholar 

  10. Klement, E., Mesiar, R., Pap, E.: Triangular norms. position paper I: basic analytical and algebraic properties. Fuzzy Sets and Systems 143(1), 5–26 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Klement, E., Mesiar, R., Pap, E.: Triangular norms. position paper II: general constructions and parameterized families. Fuzzy Sets and Systems 145(3), 411–438 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Klement, E., Mesiar, R., Pap, E.: Triangular Norms. Kluwer, Dordrecht (2000)

    MATH  Google Scholar 

  13. Klement, E., Navara, M.: A survey on different triangular norm-based fuzzy logics. Fuzzy Sets and Systems 101, 241–251 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  14. Klement, E.P., Mesiar, R., Pap, E.: Triangular norms: Basic notions and properties. In: Klement, E.P., Mesiar, R. (eds.) Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms, pp. 17–60. Elsevier, Amsterdam (2005)

    Chapter  Google Scholar 

  15. Mesiarová, A.: Generators of triangular norms. In: Klement, E.P., Mesiar, R. (eds.) Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms, pp. 95–111. Elsevier, Amsterdam (2005)

    Chapter  Google Scholar 

  16. Leventides, J., Bounas, A.: An approach to the selection of fuzzy connectives in terms of their additive generators. Fuzzy Sets and Systems 126, 219–224 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  17. Mas, M., Monserrat, M., Torrens, J.: Two types of implications derived from uninorms. Fuzzy Sets and Systems 158(3), 2612–2626 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Faucett, W.M.: Compact semigroups irreducibly connected between two idempotents. Proc. American Mathematics Society 6, 741–747 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  19. Klement, E.P., Mesiar, R., Pap, E.: Quasi- and pseudo-inverses of monotone functions, and the construction of t-norms. Fuzzy Sets and Systems 104, 3–13 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ling, C.H.: Representation of associative functions. Publ. Math. Debrecen 12, 189–212 (1965)

    MathSciNet  Google Scholar 

  21. Mesiarová, A.: H-transformation of t-norms. Information Sciences 176, 1531–1545 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mostert, P.S., Shields, A.L.: On the structure of semigroups on a compact manifold with boundary. Ann. Math. 65, 117–143 (1957)

    Article  MathSciNet  Google Scholar 

  23. Viceník, P.: Additive generators of associative functions. Fuzzy Sets and Systems 153, 137–160 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. Alefeld, G., Herzberger, J.: Introduction to Interval Computations. Academic Press, New York (1983)

    MATH  Google Scholar 

  25. Moore, R.: Methods and Applications of Interval Analysis. SIAM, Philadelphia (1979)

    MATH  Google Scholar 

  26. Cornelis, G., Deschrijver, G., Kerre, E.: Implication in intuitionistic fuzzy and interval-valued fuzzy set theory: construction, classification, application. Int. Journal Approximate Reason 35, 55–95 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  27. Deschrijver, G., Kerre, E.E.: Implicators based on binary aggregation operators in interval-valued fuzzy set theory. Fuzzy Sets and Systems 153(2), 229–248 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  28. Dubois, D., Prade, H.: Interval-valued fuzzy sets, possibility theory and imprecise probability. In: Proc. Intl. Conf. Fuzzy Logic and Technology, Barcelona, pp. 314–319 (2005)

    Google Scholar 

  29. Gehrke, M., Walker, C., Walker, E.: Some comments on interval valued fuzzy sets. Intl. Journal of Intelligent Systems 11, 751–759 (1996)

    Article  MATH  Google Scholar 

  30. Gorzalczany, M.B.: A method of inference in approximate reasoning based on interval-valued fuzzy sets. Fuzzy Sets and Systems 21(1), 1–17 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  31. Grattan-Guiness, I.: Fuzzy membership mapped onto interval and many-valued quantities. Z. Math. Logik. Grundladen Math. 22, 149–160 (1975)

    Article  Google Scholar 

  32. Moore, R., Lodwick, W.: Interval analysis and fuzzy set theory. Fuzzy Sets and Systems 135(1), 5–9 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  33. Nguyen, H., Kreinovich, V., Zuo, Q.: Interval-valued degrees of belief: applications of interval computations to expert systems and intelligent control. Int. Journal of Uncertainty, Fuzziness, and Knowledge-Based Systems 5(3), 317–358 (1997)

    Article  MathSciNet  Google Scholar 

  34. Sambuc, R.: Fonctions φ-floues. Application l’aide au diagnostic en pathologie thyroidienne. PhD thesis, Univ. Marseille, Marseille (1975)

    Google Scholar 

  35. Turksen, I.: Interval valued fuzzy sets based on normal forms. Fuzzy Sets and Systems 20, 191–210 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  36. Gasse, B.V., Cornelis, G., Deschrijver, G., Kerre, E.: On the properties of a generalized class of t-norms in interval-valued fuzzy logics. New Math. and Natural Comput. 2, 29–42 (2006)

    Article  MATH  Google Scholar 

  37. Lodwick, W.A.: Preface. Reliable Computing 10(4), 247–248 (2004)

    Article  MathSciNet  Google Scholar 

  38. Moore, R.E.: Interval Arithmetic and Automatic Error Analysis in Digital Computing. PhD thesis, Stanford University, Stanford (1962)

    Google Scholar 

  39. Kearfort, R.B., Kreinovich, V. (eds.): Applications of Interval Computations. Kluwer Academic Publishers, Boston (1996)

    Google Scholar 

  40. Walley, P.: Statistical Reasoning with Imprecise Probabilities. Chapman&Hall, London (1991)

    MATH  Google Scholar 

  41. Walley, P.: Measures of uncertainty. Artificial Intelligence 83, 1–58 (1996)

    Article  MathSciNet  Google Scholar 

  42. Bedregal, B.C., Santiago, R.H.N., Reiser, R.H.S., Dimuro, G.P.: The best interval representation of fuzzy Simplications and automorphisms. In: Proc. of the IEEE Intl. Conf. on Fuzzy Systems, Londres, pp. 3220–3230. IEEE, Los Alamitos (2007)

    Google Scholar 

  43. Bedregal, B.C., Santiago, R.H.N., Reiser, R.H.S., Dimuro, G.P.: Analyzing properties of fuzzy implications obtained via the interval constructor. In: IEEE Post-Proceedings of SCAN 2006: Revised Setected Papers of 12th GAMM - IMACS Intl. Symp. Scientific Computing, Computer Arithmetic and Validated Numerics, Duisburg, paper no. 13. IEEE, Los Alamitos (2007)

    Google Scholar 

  44. Bedregal, B.C., Santiago, R.H.N., Reiser, R.H.S., Dimuro, G.P.: Properties of fuzzy implications obtained via the interval constructor. TEMA 8(1), 33–42 (2007), http://www.sbmac.org.br/tema

    MathSciNet  Google Scholar 

  45. Bedregal, B.C., Santiago, R.H.N., Dimuro, G.P., Reiser, R.H.S.: Interval valued R-implications and automorphisms. In: Pre-Proceedings of the 2nd Work. on Logical and Semantic Frameworks, with Applications, Ouro Preto, pp. 82–97 (2007)

    Google Scholar 

  46. Reiser, R.H.S., Dimuro, G.P., Bedregal, B., Santiago, R.: Interval valued QL-implications. In: Leivant, D., de Queiroz, R. (eds.) WoLLIC 2007. LNCS, vol. 4576, pp. 307–321. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  47. Bedregal, B., Takahashi, A.: The best interval representation of t-norms and automorphisms. Fuzzy Sets and Systems 157(24), 3220–3230 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  48. Bedregal, B., Takahashi, A.: Interval valued versions of t-conorms, fuzzy negations and fuzzy implications. In: IEEE Proc. Intl. Conf. on Fuzzy Systems, Vancouver, Los Alamitos, pp. 9553–9559 (2006)

    Google Scholar 

  49. Santiago, R., Bedregal, B., Acióly, B.: Formal aspects of correctness and optimality in interval computations. Formal Aspects of Computing 18(2), 231–243 (2006)

    Article  MATH  Google Scholar 

  50. Hickey, T., Ju, Q., Emdem, M.: Interval arithmetic: from principles to implementation. Journal of the ACM 48(5), 1038–1068 (2001)

    Article  MathSciNet  Google Scholar 

  51. Callejas-Bedregal, R., Bedregal, B.C.: Intervals as a domain constructor. TEMA 2(1), 43–52 (2001), http://www.sbmac.org.br/tema

    MathSciNet  Google Scholar 

  52. Abramsky, S., Jung, A.: Domain theory. In: Abramsky, S., Gabbay, D., Maimbaum, T. (eds.) Handbook of Logic in Computer Science, vol. 3. Oxford Svience Publications (1994)

    Google Scholar 

  53. Gierz, G., Hoffman, K., Keimel, K., Lawson, J., Scott, D.: Continuous Lattices and Domains. Cambridge University Press, Cambridge (2003)

    MATH  Google Scholar 

  54. Kearfott, R.B.: Rigorous Global Search: Continuous problems. Kluwer, Dordrecht (1996)

    MATH  Google Scholar 

  55. Caprani, O., Madsen, K., Stauning, O.: Existence test for asynchronous interval iteration. Reliable Computing 3(3), 269–275 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  56. Jaulin, L., Kieffer, M., Didrit, O., Walter, E.: Applied Interval Analisys: With examples in parameter and state estimation, robust control and robotic. Springer, Heidelberg (2001)

    Google Scholar 

  57. Deschrijver, G.: Additive and multiplicative generators in interval-valued fuzzy set theory. IEEE Transactions on Fuzzy Systems 15(2), 1063–6706 (2007)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Wilfrid Hodges Ruy de Queiroz

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Dimuro, G.P., Bedregal, B.C., Reiser, R.H.S., Santiago, R.H.N. (2008). Interval Additive Generators of Interval T-Norms. In: Hodges, W., de Queiroz, R. (eds) Logic, Language, Information and Computation. WoLLIC 2008. Lecture Notes in Computer Science(), vol 5110. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69937-8_12

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-69937-8_12

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-69936-1

  • Online ISBN: 978-3-540-69937-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics