Generating complete set of implications for formal contexts

doi:10.1016/j.knosys.2008.03.001

Knowledge-Based Systems

Volume 21, Issue 5, July 2008, Pages 429-433

https://doi.org/10.1016/j.knosys.2008.03.001 Get rights and content

Abstract

In this paper, a necessary and sufficient condition on which a set of implications is complete is proposed with the help of the notion of model from logic. Besides, using the closure of an attribute subset to a set of implications, we present a formal method to remove the redundant implications from a complete set. Subsequently, we provide an algorithm to generate a complete set of implications and an illustrative example guarantees the availability of the algorithm.

Introduction

Formal concept analysis (FCA) is an order-theoretic method for the mathematical analysis of scientific data, pioneered by R.Wille [1] in mid 1980s. Over the past 20 years, FCA has been widely studied [2], [3], [4], [5] and become a powerful tool for machine learning [6], [7], software engineering [8], [9], [10] and information retrieval [11].

In essence, FCA is based on a formalization of the philosophical understanding of a concept as a unit of thought constituted by its extent and intent. The extent of a concept is understood as the collection of all objects belonging to the concept and the intent as the multitude of all attributes common to all those objects. The transformation from two-dimensional incidence tables to concept lattices structure is a crucial paradigm shift from which FCA derives much of its power and versatility as a modelling tool. The concept lattices obtained the way turn out to be exactly the complete lattices, and the particular way in which they structure and represent knowledge is very appealing and natural from the perspective of many scientific disciplines.

In addition to being a technique for classifying and defining concepts from data, FCA may be exploited to discover implications among the objects and the properties. On extracting implications from formal contexts, some fruitful results have been presented [12], [13], [14], [15]. However, there has been only little work relating whether a set of implications is complete and redundant. This paper serves to solve the problem of completeness and redundancy of implications. In the sequel, a necessary and sufficient condition on which a set of implications is complete is proposed with the help of the notion of model. In general, a set of implications is redundant, so we present a formal method to remove the redundant implications from a complete set using the closure of an attribute subset to a set of implications. By means of lectical order defined on attribute sets, we refine Titanic algorithm [16] and further propose a new algorithm TComGen to generate the complete set of implications. An illustrative example guarantees the availability of the algorithm.

Section snippets

Basic notions

This section provides a brief overview over FCA, in order to allow for a better understanding for the overall picture. We introduce the most basic notions of FCA, namely formal contexts, formal concepts, concept lattices and implications. For more extensive introduction refer to Ganter and Wille [2].

In FCA, an elementary form of the representation of data is defined mathematically as formal context.

Definition 1

A formal context is a triple: $K = (G, M, I)$ , where G and M are sets, and $I \subseteq G \times M$ is a binary relation.

Implications

One of the aspects of FCA thus is attribute logic, the study of possible attribute combinations. Most of the time, this will be very elementary. Dependencies between the attributes can be described by implications. An implication between attributes in M is a pair of subsets of M, denoted by $A \to B$ . The set A is the premise of the implication $A \to B$ , and B is its conclusion. Formally,

Definition 5

Let $K = (G, M, I)$ be a formal context, $A, B \subseteq M$ . $A \to B$ is true if each object which has all attributes from A has also all

Complete set of implications and implication deduction

In this section, with the help of the notion of model, we will discuss the necessary and sufficient condition on which a set of implications is complete.

Theorem 2

Let $K = (G, M, I)$ be a formal context, $A, B \in M$ . The following conditions are equivalent:

(i)
$A \to B$ is an implication of K;
(ii)
$B \subseteq A^{″}$ ;
(iii)
$\forall H \in B (K)$ , $in (H) ⊨ (A \to B)$ .

Proof 1

(i) $\Leftrightarrow$ (ii) We have, $A \to B$ is an implication of K iff $A^{'} \subseteq B^{'}$ iff $A^{″} \supseteq B^{″}$ iff $A^{″} \supseteq B$ .

(ii) $\Rightarrow$ (iii) Let $H \in B (K)$ . If $in (H) ⊉ A$ , then $in (H) ⊨ (A \to B)$ holds by the Definition 5. If $in (H) \supseteq A$ , then $in (H) \supseteq A^{″}$ . From the condition $B \subseteq A^{″}$ , we

Redundant implications

In general, a set of implications is redundant, so we need a method to remove the redundant implications from the set of implications. In this paper, by means of the notion of model and the attribute closure to a set of implication, we present a formal method to eliminate the redundant implications.

Definition 10

Let $K = (G, M, I)$ be a formal context and $Θ$ be a set of implications of K. We define:(i) an extension operator of attribute subset X to $Θ$ : $X^{Θ} = X \cup ⋃ ∣ (A \to B) \in Θ, A \subseteq X} .$ (ii) formally, $(X^{Θ})^{Θ} = X^{Θ^{2}},$ and due to the

Generation of a complete set of implications

By employing minimal generator, in the section, we propose a complete set $Σ$ of implications and bring forward an algorithm to generate the complete set.

An illustrative example

We demonstrate the algorithm TComGen by means of an illustrative example. First of all, a formal context concerning living things is presented in Table 1.

The complete set $Σ$ of implications is show in Table 2, where each implication has the form: $X \to X^{″} ⧹ X$ . Here X is a minimal generator satisfying $X \neq X^{″}$ .

After eliminating the redundant implications by means of Theorem 3, we can get a non-redundant complete set as Table 3.

With the help of Corollary 1, we can infer all implications of Table 1 using the

Conclusion

This paper presents a necessary and sufficient condition, on which a set of implications is complete, and a formal method is proposed to remove the redundant implications. On the other hand, it is tellable that, opposite to the present method of extracting implications from contexts, our approach dispenses with the process of building concept lattices and thus provides an alternative for implication extracting.

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^☆: This research is supported by National Natural Science Foundation of China (Nos. 60773133 and 207018) and Shanxi Provincial Natural Science Foundation of China (No. 2007011040).

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Knowledge-Based Systems

Generating complete set of implications for formal contexts☆

Abstract

Introduction

Section snippets

Basic notions

Implications

Complete set of implications and implication deduction

Redundant implications

Generation of a complete set of implications

An illustrative example

Conclusion

Restructuring lattice theory: an approach based on hierarchies of concepts

Formal Concept Analysis: Mathematical Foundations

The algebraic properties of concept lattice

Journal of Systems Science and Information

Study of decision implications based on formal concept analysis

International Journal of General Systems

Learning by discovering concept hierarchies

Artificial Intelligence

A lattice conceptual clustering system and its application to browsing retrieval

Machine Learning

Using a concept lattice of decomposition slices for program understanding and impact analysis

IEEE Transactions on Software Engineering