Modular relaxed indistinguishability and the aggregation problem

M. D. M. Bibiloni-Femenias; O. Valero; M. D. M. Bibiloni-Femenias; O. Valero

doi:10.3934/math.20241047

AIMS Mathematics

2024, Volume 9, Issue 8: 21557-21579. doi: 10.3934/math.20241047

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Research article

Modular relaxed indistinguishability and the aggregation problem

M. D. M. Bibiloni-Femenias ^{1,2
,
,},
O. Valero ^1,2

1.
Mathematics and Computer Science Department, Universitat de les Illes Balears, Ctra. Valldemossa km. 7.5, 07122, Palma de Mallorca, Illes Balears, Spain
2.
Health Research Institute of the Balearic Islands (IdISBa), Hospital Universitari Son Espases, 07120, Palma de Mallorca, Illes Balears, Spain

Received: 19 April 2024 Revised: 25 June 2024 Accepted: 26 June 2024 Published: 05 July 2024
MSC : 03B50, 03B52, 03E72

The notion of indistinguishability operator plays a central role in a large number of problems that arise naturally in decision-making, artificial intelligence, and computer science. Among the different issues studied for these operators, the aggregation problem has been thoroughly explored. In some cases, the notion of indistinguishability operator can be too narrow and, for this reason, we can find two different extensions of such notion in the literature. On the one hand, modular indistinguishability operators make it possible to measure the degree of similarity or indistinguishability with respect to a parameter. On the other hand, relaxed indistinguishability operators delete the reflexivity condition of classical indistinguishability operators. In this paper, we introduced the notion of modular relaxed indistinguishability operator unifying under the same framework all previous notions. We focused our efforts on the study of the associated aggregation problem. Thus, we introduced the notion of modular relaxed indistinguishability operator aggregation function for a family of t-norms extending the counterpart formulated for classical non-modular relaxed indistinguishability operators. We provided characterizations of such functions in terms of triangle triplets with respect to a family of t-norms. Moreover, we addressed special cases where the operators fulfill a kind of monotony and a condition called small-self indistinguishability. The differences between the modular and the non-modular aggregation problem were specified and illustrated by means of suitable examples.
- aggregation,
- modular relaxed indistinguishability operator,
- small self-indistinguishability,
- triangular triplet,
- monotony,
- dominance
Citation: M. D. M. Bibiloni-Femenias, O. Valero. Modular relaxed indistinguishability and the aggregation problem[J]. AIMS Mathematics, 2024, 9(8): 21557-21579. doi: 10.3934/math.20241047

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Abstract

The notion of indistinguishability operator plays a central role in a large number of problems that arise naturally in decision-making, artificial intelligence, and computer science. Among the different issues studied for these operators, the aggregation problem has been thoroughly explored. In some cases, the notion of indistinguishability operator can be too narrow and, for this reason, we can find two different extensions of such notion in the literature. On the one hand, modular indistinguishability operators make it possible to measure the degree of similarity or indistinguishability with respect to a parameter. On the other hand, relaxed indistinguishability operators delete the reflexivity condition of classical indistinguishability operators. In this paper, we introduced the notion of modular relaxed indistinguishability operator unifying under the same framework all previous notions. We focused our efforts on the study of the associated aggregation problem. Thus, we introduced the notion of modular relaxed indistinguishability operator aggregation function for a family of t-norms extending the counterpart formulated for classical non-modular relaxed indistinguishability operators. We provided characterizations of such functions in terms of triangle triplets with respect to a family of t-norms. Moreover, we addressed special cases where the operators fulfill a kind of monotony and a condition called small-self indistinguishability. The differences between the modular and the non-modular aggregation problem were specified and illustrated by means of suitable examples.

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