Skip to main content
SpringerLink
Log in

Fuzzy Implications: Past, Present, and Future

  • Chapter
Springer Handbook of Computational Intelligence

Part of the book series: Springer Handbooks ((SHB))

Abstract

Fuzzy implications are a generalization of the classical two-valued implication to the multi-valued setting. They play a very important role both in the theory and applications, as can be seen from their use in, among others, multivalued mathematical logic, approximate reasoning, fuzzy control, image processing, and data analysis. The goal of this chapter is to present the evolution of fuzzy implications from their beginnings to the current days. From the theoretical point of view, we present the basic facts, as well as the main topics and lines of research around fuzzy implications. We also devote a specific section to state and recall a list of main application fields where fuzzy implications are employed, as well as another one to the main open problems on the topic.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 269.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 349.99
Price excludes VAT (USA)
  • Durable hardcover 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

Abbreviations

AR:

approximate reasoning

BKS:

Bandler–Kohout subproduct

CP:

contrapositive symmetry

CRI:

compositional rule of inference

EP:

exchange property

FIM:

fuzzy inference mechanism

FMM:

fuzzy mathematical morphology

FRI:

fuzzy relational inference

GMP:

generalized modus ponens

IP:

identity principle

LI:

law of importation

MISO:

multiple inputs-single output

MM:

mathematical morphology

NP:

neutrality principle

OP:

ordering property

RAF:

representable aggregation function

SBR:

similarity based reasoning

SISO:

single input single output

References

  1. M. Baczyński, B. Jayaram: $(S,N)$- and $R$-implications: A state-of-the-art survey, Fuzzy Sets Syst. 159(14), 1836–1859 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. M. Mas, M. Monserrat, J. Torrens, E. Trillas: A survey on fuzzy implication functions, IEEE Trans. Fuzzy Syst. 15(6), 1107–1121 (2007)

    Article  Google Scholar 

  3. M. Baczyński, B. Jayaram: Fuzzy Implications, Studies in Fuzziness and Soft Computing, Vol. 231 (Springer, Berlin, Heidelberg 2008)

    MATH  Google Scholar 

  4. M. Baczyński, G. Beliakov, H. Bustince, A. Pradera (Eds.): Advances in Fuzzy Implication Functions, Studies in Fuzziness and Soft Computing, Vol. 300 (Springer, Berlin, Heidelberg 2013)

    MATH  Google Scholar 

  5. E.P. Klement, R. Mesiar, E. Pap: Triangular norms (Kluwer, Dordrecht 2000)

    Book  MATH  Google Scholar 

  6. M. Mas, M. Monserrat, J. Torrens: Two types of implications derived from uninorms, Fuzzy Sets Syst. 158(23), 2612–2626 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. S. Massanet, J. Torrens: On the characterization of Yager's implications, Inf. Sci. 201, 1–18 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. I. Aguiló, J. Suñer, J. Torrens: A characterization of residual implications derived from left-continuous uninorms, Inf. Sci. 180(20), 3992–4005 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. E. Trillas, C. Alsina: On the law $[(p\wedge q)\to r]=[(p\to r)\vee(q\to r)]$ in fuzzy logic, IEEE Trans. Fuzzy Syst. 10(1), 84–88 (2002)

    Article  Google Scholar 

  10. J. Balasubramaniam, C.J.M. Rao: On the distributivity of implication operators over T and S norms, IEEE Trans. Fuzzy Syst. 12(2), 194–198 (2004)

    Article  Google Scholar 

  11. D. Ruiz-Aguilera, J. Torrens: Distributivity of strong implications over conjunctive and disjunctive uninorms, Kybernetika 42(3), 319–336 (2006)

    MathSciNet  MATH  Google Scholar 

  12. D. Ruiz-Aguilera, J. Torrens: Distributivity of residual implications over conjunctive and disjunctive uninorms, Fuzzy Sets Syst. 158(1), 23–37 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. S. Massanet, J. Torrens: On some properties of threshold generated implications, Fuzzy Sets Syst. 205(16), 30–49 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. M. Baczyński: On the distributivity of fuzzy implications over continuous and Archimedean triangular conorms, Fuzzy Sets Syst. 161(10), 1406–1419 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. M. Baczyński: On the distributivity of fuzzy implications over representable uninorms, Fuzzy Sets Syst. 161(17), 2256–2275 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. M. Baczyński, B. Jayaram: On the distributivity of fuzzy implications over nilpotent or strict triangular conorms, IEEE Trans. Fuzzy Syst. 17(3), 590–603 (2009)

    Article  Google Scholar 

  17. F. Qin, M. Baczyński, A. Xie: Distributive equations of implications based on continuous triangular norms (I), IEEE Trans. Fuzzy Syst. 20(1), 153–167 (2012)

    Article  Google Scholar 

  18. M. Baczyński, F. Qin: Some remarks on the distributive equation of fuzzy implication and the contrapositive symmetry for continuous, Archimedean t-norms, Int. J. Approx. Reason. 54(2), 290–296 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. B. Jayaram: On the law of importation $(x\wedge y)\to z\equiv(x\to(y\to z))$ in fuzzy logic, IEEE Trans. Fuzzy Syst. 16(1), 130–144 (2008)

    Article  Google Scholar 

  20. M. Štěpnička, B. Jayaram: On the suitability of the Bandler--Kohout subproduct as an inference mechanism, IEEE Trans. Fuzzy Syst. 18(2), 285–298 (2010)

    Article  Google Scholar 

  21. S. Massanet, J. Torrens: The law of importation versus the exchange principle on fuzzy implications, Fuzzy Sets Syst. 168(1), 47–69 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. H. Bustince, J. Fernandez, J. Sanz, M. Baczyński, R. Mesiar: Construction of strong equality index from implication operators, Fuzzy Sets Syst. 211(16), 15–33 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. P. Hájek, L. Kohout: Fuzzy implications and generalized quantifiers, Int. J. Uncertain. Fuzziness Knowl. Syst. 4(3), 225–233 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  24. P. Hájek, M.H. Chytil: The GUHA method of automatic hypotheses determination, Computing 1(4), 293–308 (1966)

    Article  MATH  Google Scholar 

  25. P. Hájek, T. Havránek: The GUHA method-its aims and techniques, Int. J. Man-Mach. Stud. 10(1), 3–22 (1977)

    Article  MATH  Google Scholar 

  26. P. Hájek, T. Havránek: Mechanizing Hypothesis Formation: Mathematical Foundations for a General Theory (Springer, Heidelberg 1978)

    Book  MATH  Google Scholar 

  27. B. Jayaram, R. Mesiar: On special fuzzy implications, Fuzzy Sets Syst. 160(14), 2063–2085 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. F. Durante, E. Klement, R. Mesiar, C. Sempi: Conjunctors and their residual implicators: Characterizations and construction methods, Mediterr. J. Math. 4(3), 343–356 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. M. Carbonell, J. Torrens: Continuous R-implications generated from representable aggregation functions, Fuzzy Sets Syst. 161(17), 2276–2289 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Y. Ouyang: On fuzzy implications determined by aggregation operators, Inf. Sci. 193, 153–162 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  31. V. Biba, D. Hliněná: Generated fuzzy implications and known classes of implications, Acta Univ. M. Belii Ser. Math. 16, 25–34 (2010)

    MathSciNet  MATH  Google Scholar 

  32. H. Bustince, J. Fernandez, A. Pradera, G. Beliakov: On $({TS},{N})$-fuzzy implications, Proc. AGOP 2011, Benevento, ed. by B. De Baets, R. Mesiar, L. Troiano (2011) pp. 93–98

    Google Scholar 

  33. I. Aguiló, M. Carbonell, J. Suñer, J. Torrens: Dual representable aggregation functions and their derived S-implications, Lect. Notes Comput. Sci. 6178, 408–417 (2010)

    Article  Google Scholar 

  34. S. Massanet, J. Torrens: An overview of construction methods of fuzzy implications. In: Advances in Fuzzy Implication Functions, Studies in Fuzziness and Soft Computing, Vol. 300, ed. by M. Baczyński, G. Beliakov, H. Bustince, A. Pradera (Springer, Berlin, Heidelberg 2013) pp. 1–30

    Chapter  Google Scholar 

  35. J. Balasubramaniam: Yager's new class of implications ${J}_{f}$ and some classical tautologies, Inf. Sci. 177(3), 930–946 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  36. M. Baczyński, B. Jayaram: Yager's classes of fuzzy implications: Some properties and intersections, Kybernetika 43(2), 157–182 (2007)

    MathSciNet  MATH  Google Scholar 

  37. A. Xie, H. Liu: A generalization of Yager's f-generated implications, Int. J. Approx. Reason. 54(1), 35–46 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  38. S. Massanet, J. Torrens: On a generalization of Yager's implications, Commun. Comput. Inf. Sci. Ser. 298, 315–324 (2012)

    Article  MATH  Google Scholar 

  39. S. Massanet, J. Torrens: On a new class of fuzzy implications: h-implications and generalizations, Inf. Sci. 181(11), 2111–2127 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  40. J.C. Fodor: Contrapositive symmetry of fuzzy implications, Fuzzy Sets Syst. 69(2), 141–156 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  41. S. Jenei: New family of triangular norms via contrapositive symmetrization of residuated implications, Fuzzy Sets Syst. 110(2), 157–174 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  42. B. Jayaram, R. Mesiar: I-Fuzzy equivalence relations and I-fuzzy partitions, Inf. Sci. 179(9), 1278–1297 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  43. Y. Shi, B.V. Gasse, D. Ruan, E.E. Kerre: On dependencies and independencies of fuzzy implication axioms, Fuzzy Sets Syst. 161(10), 1388–1405 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  44. S. Massanet, J. Torrens: Threshold generation method of construction of a new implication from two given ones, Fuzzy Sets Syst. 205, 50–75 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  45. S. Massanet, J. Torrens: On the vertical threshold generation method of fuzzy implication and its properties, Fuzzy Sets Syst. 226, 32–52 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  46. N.R. Vemuri, B. Jayaram: Fuzzy implications: Novel generation process and the consequent algebras, Commun. Comput. Inf. Sci. Ser. 298, 365–374 (2012)

    Article  MATH  Google Scholar 

  47. P. Grzegorzewski: Probabilistic implications, Proc. EUSFLAT-LFA 2011, ed. by S. Galichet, J. Montero, G. Mauris (Aix-les-Bains, France 2011) pp. 254–258

    Google Scholar 

  48. P. Grzegorzewski: Probabilistic implications, Fuzzy Sets Syst. 226, 53–66 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  49. P. Grzegorzewski: On the properties of probabilistic implications. In: Eurofuse 2011, Advances in Intelligent and Soft Computing, Vol. 107, ed. by P. Melo-Pinto, P. Couto, C. Serôdio, J. Fodor, B. De Baets (Springer, Berlin, Heidelberg 2012) pp. 67–78

    Google Scholar 

  50. A. Dolati, J. Fernández Sánchez, M. Úbeda-Flores: A copula-based family of fuzzy implication operators, Fuzzy Sets Syst. 211(16), 55–61 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  51. P. Grzegorzewski: Survival implications, Commun. Comput. Inf. Sci. Ser. 298, 335–344 (2012)

    Article  MATH  Google Scholar 

  52. G. Deschrijver, E. Kerre: On the relation between some extensions of fuzzy set theory, Fuzzy Sets Syst. 133(2), 227–235 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  53. G. Deschrijver, E. Kerre: Triangular norms and related operators in ${L}^{{*}}$-fuzzy set theory. In: Logical, Algebraic, Analytic, Probabilistic Aspects of Triangular Norms, ed. by E. Klement, R. Mesiar (Elsevier, Amsterdam 2005) pp. 231–259

    Chapter  Google Scholar 

  54. G. Deschrijver: Implication functions in interval-valued fuzzy set theory. In: Advances in Fuzzy Implication Functions , Studies in Fuzziness and Soft Computing, Vol. 300, ed. by M. Baczyński, G. Beliakov, H. Bustince, A. Pradera (Springer, Berlin, Heidelberg 2013) pp. 73–99

    Chapter  Google Scholar 

  55. C. Alcalde, A. Burusco, R. Fuentes-González: A constructive method for the definition of interval-valued fuzzy implication operators, Fuzzy Sets Syst. 153(2), 211–227 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  56. H. Bustince, E. Barrenechea, V. Mohedano: Intuitionistic fuzzy implication operators-an expression and main properties, Int. J. Uncertain. Fuzziness Knowl. Syst. 12(3), 387–406 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  58. R. Reiser, B. Bedregal: K-operators: An approach to the generation of interval-valued fuzzy implications from fuzzy implications and vice versa, Inf. Sci. 257, 286–300 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  59. G. Mayor, J. Torrens: Triangular norms in discrete settings. In: Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms, ed. by E.P. Klement, R. Mesiar (Elsevier, Amsterdam 2005) pp. 189–230

    Chapter  Google Scholar 

  60. M. Mas, M. Monserrat, J. Torrens: S-implications and R-implications on a finite chain, Kybernetika 40(1), 3–20 (2004)

    MathSciNet  MATH  Google Scholar 

  61. M. Mas, M. Monserrat, J. Torrens: On two types of discrete implications, Int. J. Approx. Reason. 40(3), 262–279 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  62. J. Casasnovas, J. Riera: S-implications in the set of discrete fuzzy numbers, Proc. IEEE-WCCI 2010, Barcelona (2010), pp. 2741–2747

    Google Scholar 

  63. J.V. Riera, J. Torrens: Fuzzy implications defined on the set of discrete fuzzy numbers, Proc. EUSFLAT-LFA 2011 (2011) pp. 259–266

    Google Scholar 

  64. J.V. Riera, J. Torrens: Residual implications in the set of discrete fuzzy numbers, Inf. Sci. 247, 131–143 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  65. P. Smets, P. Magrez: Implication in fuzzy logic, Int. J. Approx. Reason. 1(4), 327–347 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  66. J.C. Bezdek, D. Dubois, H. Prade: Fuzzy Sets in Approximate Reasoning and Information Systems (Kluwer, Dordrecht 1999)

    Book  MATH  Google Scholar 

  67. D. Driankov, H. Hellendoorn, M. Reinfrank: An Introduction to Fuzzy Control, 2nd edn. (Springer, London 1996)

    Book  MATH  Google Scholar 

  68. D. Dubois, H. Prade: Fuzzy sets in approximate reasoning, Part 1: Inference with possibility distributions, Fuzzy Sets Syst. 40(1), 143–202 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  69. G.J. Klir, B. Yuan: Fuzzy sets and fuzzy logic-theory and applications (Prentice Hall, Hoboken 1995)

    MATH  Google Scholar 

  70. L.A. Zadeh: Outline of a new approach to the analysis of complex systems and decision processes, IEEE Trans. Syst. Man Cybern. 3(1), 28–44 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  71. W. Bandler, L.J. Kohout: Semantics of implication operators and fuzzy relational products, Int. J. Man-Mach. Stud. 12(1), 89–116 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  72. W.E. Combs, J.E. Andrews: Combinatorial rule explosion eliminated by a fuzzy rule configuration, IEEE Trans. Fuzzy Syst. 6(1), 1–11 (1998)

    Article  Google Scholar 

  73. B. Jayaram: Rule reduction for efficient inferencing in similarity based reasoning, Int. J. Approx. Reason. 48(1), 156–173 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  74. L.A. Zadeh: Fuzzy sets, Inf. Control 8(3), 338–353 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  75. D. Sinha, E.R. Dougherty: Fuzzification of set inclusion: Theory and applications, Fuzzy Sets Syst. 55(1), 15–42 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  76. L. Kitainik: Fuzzy inclusions and fuzzy dichotomous decision procedures. In: Optimization models using fuzzy sets and possibility theory, ed. by J. Kacprzyk, S. Orlovski (Reidel, Dordrecht 1987) pp. 154–170

    Chapter  Google Scholar 

  77. W. Bandler, L. Kohout: Fuzzy power sets and fuzzy implication operators, Fuzzy Sets Syst. 4(1), 13–30 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  78. S. Mandal, B. Jayaram: Approximation capability of SISO SBR fuzzy systems based on fuzzy implications, Proc. AGOP 2011, ed. by B. De Baets, R. Mesiar, L. Troiano (University of Sannio, Benevento 2011) pp. 105–110

    Google Scholar 

  79. M. Štěpnička, B. De Baets: Monotonicity of implicative fuzzy models, Proc. FUZZ-IEEE, 2010 Barcelona (2010), pp. 1–7

    Google Scholar 

  80. D. Dubois, H. Prade, L. Ughetto: Checking the coherence and redundancy of fuzzy knowledge bases, IEEE Trans. Fuzzy Syst. 5(3), 398–417 (1997)

    Article  Google Scholar 

  81. B. De Baets: Idempotent closing and opening operations in fuzzy mathematical morphology, Proc. ISUMA-NAFIPS'95, Maryland (1995), pp. 228–233

    Google Scholar 

  82. B. De Baets: Fuzzy morphology: A logical approach. In: Uncertainty Analysis in Engineering and Science: Fuzzy Logic, Statistics, Neural Network Approach, ed. by B.M. Ayyub, M.M. Gupta (Kluwer, Dordrecht 1997) pp. 53–68

    Google Scholar 

  83. J. Serra: Image Analysis and Mathematical Morphology (Academic, London, New York 1988)

    Google Scholar 

  84. M. Nachtegael, E.E. Kerre: Connections between binary, gray-scale and fuzzy mathematical morphologies original, Fuzzy Sets Syst. 124(1), 73–85 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  85. I. Bloch: Duality vs. adjunction for fuzzy mathematical morphology and general form of fuzzy erosions and dilations, Fuzzy Sets Syst. 160(13), 1858–1867 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  86. M. González-Hidalgo, A. Mir Torres, D. Ruiz-Aguilera, J. Torrens Sastre: Edge-images using a uninorm-based fuzzy mathematical morphology: Opening and closing. In: Advances in Computational Vision and Medical Image Processing, Computational Methods in Applied Sciences, Vol. 13, ed. by J. Tavares, N. Jorge (Springer, Berlin, Heidelberg 2009) pp. 137–157

    Chapter  Google Scholar 

  87. M. González-Hidalgo, A. Mir Torres, J. Torrens Sastre: Noisy image edge detection using an uninorm fuzzy morphological gradient, Proc. ISDA 2009 (IEEE Computer Society, Los Alamitos 2009) pp. 1335–1340

    Google Scholar 

  88. M. Baczyński, B. Jayaram: Fuzzy implications: Some recently solved problems. In: Advances in Fuzzy Implication Functions, Studies in Fuzziness and Soft Computing, Vol. 300, ed. by M. Baczyński, G. Beliakov, H. Bustince, A. Pradera (Springer, Berlin, Heidelberg 2013) pp. 177–204

    Chapter  Google Scholar 

  89. B. Jayaram, M. Baczyński, R. Mesiar: R-implications and the exchange principle: The case of border continuous t-norms, Fuzzy Sets Syst. 224, 93–105 (2013)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Inst. Mathematics, University of Silesia, Bankowa 14, 40-007, Katowice, Poland

    Michał Baczynski

  2. Dep. Mathematics, Indian Institute of Technology Hyderabad, ODF Estate, Yeddumailaram, 502 205, Hyderabad, India

    Balasubramaniam Jayaram

  3. Dep. Mathematics and Computer Science, University of the Balearic Islands, Crta. Valldemossa km. 7,5, 07122, Palma de Mallorca, Spain

    Sebastia Massanet & Joan Torrens

Authors
  1. Michał Baczynski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Balasubramaniam Jayaram
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sebastia Massanet
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Joan Torrens
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michał Baczynski .

Editor information

Editors and Affiliations

  1. Systems Research Inst., Polish Academy of Sciences, ul. Newelska 6, 01-447, Warsaw, Poland

    Janusz Kacprzyk

  2. Dep. Electrical and Computer Engineering, University of Alberta, 116 Street 9107, T6J 2V4, Edmonton, Alberta, Canada

    Witold Pedrycz

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Baczynski, M., Jayaram, B., Massanet, S., Torrens, J. (2015). Fuzzy Implications: Past, Present, and Future. In: Kacprzyk, J., Pedrycz, W. (eds) Springer Handbook of Computational Intelligence. Springer Handbooks. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43505-2_12

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-43505-2_12

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-43504-5

  • Online ISBN: 978-3-662-43505-2

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation