When Nominal Analogical Proportions Do Not Fail

Miguel Couceiro, Erkko Lehtonen, Laurent Miclet, Henri Prade, Gilles Richard

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

2 Citations (Scopus)

Abstract

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.

Original languageEnglish
Title of host publicationScalable Uncertainty Management - 14th International Conference, SUM 2020, Proceedings
EditorsJesse Davis, Karim Tabia
Place of PublicationCham
PublisherSpringer
Pages68-83
Number of pages16
ISBN (Electronic)978-3-030-58449-8
ISBN (Print)978-3-030-58448-1
DOIs
Publication statusPublished - 2020
Event14th International Conference on Scalable Uncertainty Management, SUM 2020 - Bolzano, Italy
Duration: 23 Sep 202025 Sep 2020

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
PublisherSpringer
Volume12322 LNAI
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference14th International Conference on Scalable Uncertainty Management, SUM 2020
Country/TerritoryItaly
CityBolzano
Period23/09/2025/09/20

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