Fuzzy Implication and Functional Dependency on Formal Context

Fuzzy formal concept analysis(FFCA) is a development of formal concept analysis(FCA) with the degree of relation between objects and attributes. Using FCA approach, we will investigate the condition logical implication for fuzzy functional dependency. We also use Armstrong’s rule to deﬁne soundness and completeness of our implication and fuzzy functional dependency model. We show diﬀerence and equivalence condition between fuzzy implication and fuzzy functional dependency. This condition can be used to develop the algorithm for ﬁnding attribute dependency.


Introduction
Dependency theory is an essential theory in data analysis. A notion of functional dependency, which is a constraint between two sets of attributes has been introduced by Codd [10]. Model of attribute dependency has been developed by Belohlvek et al. [7,8,6].
In the real condition, it is difficult to give "certain" value for representing the condition. We have to deal "uncertain" condition with the "degree". For example, we give value from 0 to 1 to define the condition "Healthy Food", "Regularly Exercise", "Cholesterol" From table 1, we use "degree" value to define high possibility of the condition. We would like to find the correlations between attributes. We have to use fuzzy system to define the correlation. From table 1, we can suspect that "Healthy Food" and "Regularly Exercise" have impact to the "Cholesterol ((Healthy F ood, Regularly Exercise) → Cholesterol). In this research, we would like to investigate the dependency and implication in fuzzy case.
In the first section, we formalize fuzzy relation from fuzzy theory that has been introduced by Zadeh [19]. The Fuzzy relational database is a development model of the relational database to process imprecise data. Fuzzy relational database and its operation have been developed by Nakata et al. [14], Raju [16], and Umano et al. [17]. Formal operation of the fuzzy relational database using relational calculus has been developed by Okuma et al. [15], Akbar et al. [1,2]. After defining model of the fuzzy relational database, fuzzy functional dependency and lossless decomposition have been defined by Raju [16], and Bhuniya [9]. Class dependency is a new dependency using fuzzy partition and conditional probability. Fuzzy class dependency has been developed by Akbar et al. [3].
Ganter and Wille defined logical implication for a formal concept analysis [12]. Formal concept analysis is a method in data mining using lattice theory proposed by Wille [18]. Ishida et al. have been developed a formalization of completeness and soundness for functional dependencies based on [5] using relational calculus [13]. They showed a relation-algebraic proof of the completeness theorem for Armstrong's inference rules in a Schröder category. Comparing formalization to a functional dependency and a logical implication of FCA was also investigated. We extend its formalization using a fuzzy relational concept. A notion of fuzzy was initially proposed by Zadeh [19]. In section 5, We define fuzzy equivalence relation using an indiscernibility relation by Düntsch and Günther [11] to construct a fuzzy functional dependency in section 7. In section 6, we formalized a fuzzy implication. Then in section 8, we give the Completeness theorem for fuzzy implication and fuzzy functional dependency. In section 9, The main purpose of our formalization is investigating correlations between a functional dependency and a logical implication of FCA. We follow the Ishida's approach to defining Armstrong's inference rules that explain in section 4, and show the soundness and completeness in our formalization using a fuzzy relation.
We introduce the formalization of the fuzzy equivalence relation, fuzzy implication dependency, and functional dependency. The formalization can be used to analyze the equivalent condition between functional and implication. We prove theorems in our formalization of fuzzy concepts using relational calculus. Since our proof is using relational calculus, it is simple and its correctness can be easily verified. We also show the logical comparison between fuzzy implication dependency and functional dependency. Further, the comparison is important aspects for making the algorithm of fuzzy functional and implication dependency. We show some examples of the application of data analysis. Since formal concept analysis is essential theory in data analysis, we believe our formalization useful for data analysis. Future work includes to constructing a theory of fuzzy relational database theory with computer verified formal proofs using relational calculus.

Fuzzy Relations
In this section, we summarize basic notations for fuzzy relations. We denote the set {x ∈ IR|0 ≤ x ≤ 1} as [0,1]. The supremum and infimum of a family {x λ } λ∈Λ of elements x λ ∈ [0, 1] is denoted by λ∈Λ x λ and λ∈Λ x λ , respectively. In particular, x ∨ x = max{x, x }, We extend to define fuzzy operations (union), (intersection), (subset), ⇒(the relative pseudo-complement), and constants 0 AB (least), ∇ AB (greatest) in F Rel, as follows: The fuzzy power set ℘ f (Y ) of a set Y is the set of all fuzzy relations ρ : I Y , where I denotes a singleton set { * }. A fuzzy relation ρ in ℘ f (Y ) is called a fuzzy relation into Y . We will identify a point y of Y with a (crisp) fuzzy relationŷ : I Y such thatŷ( * , y ) = 1 if y = y andŷ( * , y ) = 0 otherwise.
Let B : I Y be a fuzzy relation and y ∈ Y . The restriction B y of B on y is a fuzzy relation into Y such that

Fuzzy Context
Attribute values in ordinary databases may allow to be characters or strings as well as numbers. A context is a special database whose all attribute values are the truth values 0 and 1. A fuzzy context is a fuzzy relation α : X Y , which is equivalent to its intent function α @ : X → ℘(Y ) defined as α @ (x) = xα for all x ∈ X. Thus the fuzzy context α : X Y is equivalent to an X-indexed set τ = {xα|x ∈ X} of fuzzy relations into Y , where xα denotes the composite of fuzzy relations x : I → X and α : X Y .
x n 0.4 1 0 · · · x n α Table 2. Fuzzy Context Table In what follows a subset τ of ℘(Y ) will be called a fuzzy context on Y .

Remark.
A is a fuzzy relation into Y

Armstrong's Inference Rules
Armstrong's inference rules gives a basic framework of databases to treat the logical structure of dependencies on an attribute set. Let A and B be fuzzy relations into Y . A formal expression A £ B, namely, an ordered pair of A and B, is called a dependency on the attribute set Y .
Armstrong's Inference Rules of dependencies such that, for all k = 0, 1, ..., m, one of the following holds: The [A2 ] [A1 ] For example, the union rule and [A2'] as follows:  Remark. The above lemma always holds if the attribute set Y is finite.

Equivalence Relation
In mathematical logic, equivalence relation is binary relation which is reflexive, symmetric, and transitive. In this definition, we use definition of indiscernibility relation that was introduced by Ganter and Wille [12]. We extend the definition of indiscernibility relation for fuzzy relation to define equivalence relation. We also show that the definition has the characteristics such as: reflexive, symmetric, and transitive. Definition of equivalence will be used to define fuzzy dependency in next section. The following definition is equivalence relation for general case(fuzzy and boolean case) Then we can conclude that θ

Fuzzy Implication
In formal concept analysis, implication is very important. Implication will construct concept latices of formal context. Using the definition logical implication in formal context, we will extend the definition in case of fuzzy formal context. We will define fuzzy implication then we will use it to show Armstrong's rule for fuzzy implication. We also show the soundness and completeness of fuzzy implication in the next section. For a fuzzy context T on Y we define another fuzzy context T * on Y by Note. For example, {xα|x ∈ X} * is the set of all formal concepts for a fuzzy relation α : X Y .
Proof. (←) It is trivial from T ⊆ T * . (→) Conversely assume T |= G A £ B and let U = C ∈ T * for C ⊆ T . Then Proof.
The implication dependencies also satisfy the Armstrong's inference rules. Next proposition, we would like to show the soundness of fuzzy implication on formal context. Proposition 6.3. Let T be a fuzzy context on Y . Then

Fuzzy Functional Dependency
In relational database theory, a functional dependency is a constraint between two sets of attributes in a relation from a database. Functional dependency is one of important topic in database theory. Using functional dependency, we can conclude all "superkeys" in database systems. Baixeries et al has been introduced model of functional dependency on FCA [4]. We would like to extend his definition to fuzzy case. Definition 7.1. Let T be a fuzzy context on Y and A £ B a dependency on Y , we define that fuzzy context table has fuzzy implication (T |= F ), with the definition:.
Using Definition7.1, we can investigate fuzzy functional dependency between attributes.
The functional dependencies satisfy the Armstrong's inference rules. Next proposition, we would like to show the soundness of fuzzy functional dependency.
Proof. (a) It is trivial.
.Then for all S, T ∈ T we have Then for all S, T ∈ T it holds that After we showed the soundness of fuzzy functional dependency, we would like to define some condition of fuzzy functional dependency. The definition is important to show the completeness. We will show the completeness in the next section using the definition.

Definition 7.4. Consider a particular fuzzy context
Then for all fuzzy relations C into Y the following holds:

Completeness
Now we will state the soundness and the completeness theorems of functional and implication dependencies for fuzzy contexts.
Theorem 8.1. Let L a set of dependencies and A £ B a dependency on a finite set Y . Then the following equivalence holds: (II) In the case of L A 6 = ∇ IY . By 5.6 and 6.4 we can choose a fuzzy context T 0 satisfying the following conditions:

Comparison
After we define fuzzy implication and fuzzy functional dependency. In this section, we would like to show the comparison between an fuzzy implication and a functional dependency. The following example will show the difference between fuzzy implication and functional dependency.  Table 3. Fuzzy Relation α : X Y From example we got: (1) T |= F A £ B and T |= G A £ B If we compare tuple in "x4", T 4 then we get A T 4 but B T 4 . It means T |= G A£B. But, all condition tuple will fulfill the condition of fuzzy functional dependency.
In this condition, we can compare tuple in "x4" with tuple in "x2" or "x3", then we have C T 2 = C T 4 but D T 2 = D T 4 . Also, tuple in x 3 then C T 3 = C T 4 but D T 3 = D T 4 . But for all condition will fulfill the condition of fuzzy implication.
Then we can conclude that:T |= F A £ B is not equivalent with T |= I A £ B. Also, T |= I C £ D is not equivalent with T |= F C £ D Next, we would like to observe the equivalent condition of fuzzy functional and implication dependency. We review and extend the equivalent condition in the case of boolean [13] to the general or fuzzy relation case. Let T be a fuzzy context on Y . Define another fuzzy context T on Y by T = {(S ⇒ T ) (T ⇒ S)|S, T ∈ T }. Using theorem 9.2 and 9.3 can be used to find dependency using the properties then we can reduced the dataset using the theorems.

Conclusion
In this paper, we extend the idea that Armstrong inference rules are soundness and completeness for fuzzy functional and implication dependencies. We proved some properties about a dependency and an implication. After that, we give an example to explain comparison between a fuzzy implication and a fuzzy functional dependency. In the common condition, we can not make fuzzy rules(implication) using attribute dependency, and also vice versa. But, we also gave a condition that a fuzzy implication and a functional dependency are equivalent and showed that a functional dependency can be reduced to an implication when the condition of table(data) is fulfill in theorem 9.2 and 9.3.
Future works include to construct a theory of fuzzy relational database theory with computer verified formal proofs using relational calculus.