Implications from data with fuzzy attributes vs. scaled binary attributes

Abstract

The paper studies if and to what extent it is possible to reduce notions related to attribute implications from data with fuzzy attributes to corresponding notions related to attribute implications from data with crisp attributes. We provide a reduction (transformation) theorem. Still, working directly in fuzzy setting is beneficial. Namely, we prove that we can compute a complete and non-redundant basis of fuzzy attribute implications of data with fuzzy attributes such that the basis is at most as large as (and can be smaller than) any basis of implications of the corresponding data table with crisp attributes.

Attribute Implications in a Fuzzy Setting

“…For instance, it is shown in [9,13] that a data table T with fuzzy attributes can be transformed to a data table T with binary attributes in such a way that fuzzy attribute implications true in degree 1 in T correspond in a certain way to ordinary attribute implications which are true in T . The transformation of data tables and attribute implications makes it possible to obtain an ordinary non-redundant basis T for T and to obtain a corresponding set T of fuzzy attribute implications from T .…”

“…In what follows we describe an algorithm for computing this P. The algorithm is based on the ideas of Ganter's algorithm for computing ordinary pseudointents, see [21,22]. Details can be found in [7].…”

“…This can be done with polynomial time delay using a "fuzzy extension" of Ganter's algorithm for computing all fixed points of a closure operator, see [6]. We refer to [7] for details.…”

“…First, given a data table T = X, Y, I with fuzzy attributes, we can consider a degree ||A ⇒ B|| T to which A ⇒ B is true in T , see (4). Second, given a data table D over domains with similarities, we can consider a degree ||A ⇒ B|| D to which A ⇒ B is true in D, see (9). In the rest of this section, we focus on presenting the following relationship between the two kinds of semantics for our formulas A ⇒ B: The notion of semantic entailment based on data tables with fuzzy attributes coincides with the notion of semantic entailment based on data tables over domains with similarity relations.…”

Semantics of Propositional Fuzzy Modal Logic with Evaluated Syntax and its Application to Fuzzy Decision Implications

“…In fact, in Ref. [1][2][3][4][5]20 and Ref. 24, fuzzy implications or fuzzy decision implications are only considered by taking as a whole in fuzzy attribute logic.…”