A construction of an optimal base for conditional attribute and attributional condition implications in triadic contexts

Highlights A construction of an optimal base for conditional attribute and attributional condition implications in triadic contexts. Romuald Kwessy Mouona, Blaise Blériot Koguep Njionou, Etienne Romuald Temgoua Alomo, Rokia Missaoui, Leonard Kwuida • We construct an optimal set of implications for triadic contexts, by augmentation. • We analyze the complexity of our construction’s method. arXiv:2601.01467v1 [cs.AI] 4 Jan 2026 A construction of an optimal base for conditional attribute and attributional condition implications in triadic contexts. Romuald Kwessy Mouonaa, Blaise Blériot Koguep Njionoub, Etienne Romuald Temgoua Alomoc, Rokia Missaouid, Leonard Kwuidae,∗ aDepartment of Mathematics, Faculty of Science, University of Yaounde 1, Yaounde, Cameroon bDepartment of Mathematics and Computer Science, Faculty of Science, University of Dschang, Dschang, Cameroon cDepartment of Mathematics, Higher Teacher Training College, University of Yaounde 1, Yaounde, Cameroon dDepartment of Computer Science and Engineering, Université du Québec en Outaouais (UQO), 101, rue Saint-Jean-Bosco, Gatineau (Québec), J8X 3X7, Canada eSchool of Business, Bern University of Applied Sciences, Brückenstrasse 73, Bern, 3005, Switzerland Abstract This article studies implications in triadic contexts. Specifically, we focus on those intro- duced by Ganter and Obiedkov, namely conditional attribute and attributional condition implications. Our aim is to construct an optimal base for these implications. Keywords: Triadic context, triadic implication bases, pseudo-intent, quasi-feature, pseudo-feature, simplification logic. 2020 MSC: 06A15, 68T30, 03G10. 1. Introduction A formal context is a triple (G, M, I) formed by two sets G (of objects) and M (of attributes), and a binary relation I between them, i.e. I ⊆G × M. In formal contexts, attribute implications are used to extract information about the dependencies between attributes. Thus, an implication is a relation between two sets of attributes A and B, denoted by A →B, and is valid if, whenever an object has all attributes in A, then it also has all attributes in B. Implications have been the subject of several studies [2, 11], notably those of Duquenne and Guigues [11], which, for a given formal context, led to the construction of the canonical base of implications. By incorporating the condition for which an object has an attribute, the notion of a formal context is extended [6]. This has led to the development of Triadic Concept Analysis (TCA) as an extension of Formal Concept Analysis (FCA) [7, 23]. A triadic context is defined as a quadruple ∗Corresponding author Email addresses: romualdkwessy@gmail.com (Romuald Kwessy Mouona), blaise.koguep@univ-dschang.org (Blaise Blériot Koguep Njionou), retemgoua@gmail.com (Etienne Romuald Temgoua Alomo), rokia.missaoui@uqo.ca (Rokia Missaoui), leonard.kwuida@bfh.ch (Leonard Kwuida) K := (G, M, C, I), where G is a set of objects, M is a set of attributes, C is a set of conditions, and I is a relation between objects, attributes, and conditions (I ⊆G×M×C). In this article, we focus on implications of triadic contexts, which are specific connections between subsets of M and C [4, 7, 15, 16, 17, 20]; they were introduced in the triadic framework by Biedermann [4]. Ganter and Obiedkov [7] extended this work by defining other types of implications. Implications in triadic contexts fall into two categories, namely Biedermann, and Ganter & Obiedkov ones. Those defined by Biedermann are the following: ⋆Biedermann’s conditional attributes implications (or BCAI for short), denoted by (A1 →A2)C, where A1 ⊆M is the premise, A2 ⊆M is the conclusion, and C ⊆C is the set of conditions (constraint). They can be interpreted as ’if an object of G has all attributes in A1 under all conditions in C, then it also has all attributes in A2 under the same conditions’. They are seen as knowledge from the point of view of attributes. ⋆Biedermann’s attributional conditions implications (or BACI for short), denoted by (C1 →C2)A , where C1 ⊆C is the premise, C2 ⊆C is the conclusion and A ⊆M is the constraint. They can be interpreted as ’if an object of G has all attributes in A under all conditions in C1, then it also has all attributes in A under all conditions in C2’. They are seen as knowledge from the point of view of conditions. The ones defined by Ganter and Obiedkov are: ⋆Attribute×condition implications (or A×CI for short), are of the form E →F, where E (premise) and F (conclusion) are subsets of M×C, and interpreted as: “any object g ∈G in relation with all attribute-condition pairs in E is also in relation with all attribute-condition pairs in F”. ⋆Conditional attribute implications (CAI for short), denoted by A1 C→A2, with A1, A2 ⊆ M and C ⊆C, are interpreted as: “if an object g ∈G has all attributes in A1 under all set of conditions X ⊆C, then g also has all attributes in A2 under X”. ⋆Attributional condition implications (ACI for short), denoted by C1 A

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