TY - GEN

T1 - When Nominal Analogical Proportions Do Not Fail

AU - Couceiro, Miguel

AU - Lehtonen, Erkko

AU - Miclet, Laurent

AU - Prade, Henri

AU - Richard, Gilles

N1 - Funding Information:
The authors acknowledge a partial support of ANR-11-LABX-0040-CIMI (Cent. Int. de Math. et d?Informat.) within program ANR-11-IDEX-0002-02, project ISIPA, and a partial support of Funda??o para a Ci?ncia e a Tecnologia (Portuguese Foundation for Science and Technology) through project UID/MAT/00297/2019 (Centro de Matem?tica e Aplica??es) and project PTDC/MAT-PUR/31174/2017.
Funding Information:
The authors acknowledge a partial support of ANR-11-LABX-0040-CIMI (Cent. Int. de Math. et d’Informat.) within program ANR-11-IDEX-0002-02, project ISIPA, and a partial support of Funda¸cão para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through project UID/MAT/00297/2019 (Centro de Matemática e Aplica¸cões) and project PTDC/MAT-PUR/31174/2017.
Publisher Copyright:
© 2020, Springer Nature Switzerland AG.

PY - 2020

Y1 - 2020

N2 - Analogical proportions are statements of the form “ is to as is to”, where, are tuples of attribute values describing items. The mechanism of analogical inference, empirically proved to be efficient in classification and reasoning tasks, started to be better understood when the characterization of the class of classification functions with which the analogical inference always agrees was established for Boolean attributes. The purpose of this paper is to study the case of finite attribute domains that are not necessarily two-valued, i.e., when attributes are nominal. In particular, we describe the more stringent class of “hard” analogy preserving (HAP) functions over finite domains for binary classification purposes. This description is obtained in two steps. First we observe that such AP functions are almost affine, that is, their restriction to any, where and, can be turned into an affine function by renaming variable and function values. We then use this result together with some universal algebraic tools to show that they are essentially unary or quasi-linear, which provides a general representation of HAP functions. As a by-product, in the case when, it follows that this class of HAP functions constitutes a clone on X, thus generalizing several results by some of the authors in the Boolean case.

AB - Analogical proportions are statements of the form “ is to as is to”, where, are tuples of attribute values describing items. The mechanism of analogical inference, empirically proved to be efficient in classification and reasoning tasks, started to be better understood when the characterization of the class of classification functions with which the analogical inference always agrees was established for Boolean attributes. The purpose of this paper is to study the case of finite attribute domains that are not necessarily two-valued, i.e., when attributes are nominal. In particular, we describe the more stringent class of “hard” analogy preserving (HAP) functions over finite domains for binary classification purposes. This description is obtained in two steps. First we observe that such AP functions are almost affine, that is, their restriction to any, where and, can be turned into an affine function by renaming variable and function values. We then use this result together with some universal algebraic tools to show that they are essentially unary or quasi-linear, which provides a general representation of HAP functions. As a by-product, in the case when, it follows that this class of HAP functions constitutes a clone on X, thus generalizing several results by some of the authors in the Boolean case.

UR - http://www.scopus.com/inward/record.url?scp=85092085968&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-58449-8_5

DO - 10.1007/978-3-030-58449-8_5

M3 - Conference contribution

AN - SCOPUS:85092085968

SN - 978-3-030-58448-1

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 68

EP - 83

BT - Scalable Uncertainty Management - 14th International Conference, SUM 2020, Proceedings

A2 - Davis, Jesse

A2 - Tabia, Karim

PB - Springer

CY - Cham

T2 - 14th International Conference on Scalable Uncertainty Management, SUM 2020

Y2 - 23 September 2020 through 25 September 2020

ER -