Fuzzy decision implications: interpretation within fuzzy decision context

: Fuzzy decision implication is an extension of decision implication in the fuzzy setting, serving to uncover the dependencies of fuzzy attributes. This study presents the interpretation of fuzzy decision implication in the fuzzy decision context. Specially, they will show that from fuzzy decision contexts one can obtain a closed fuzzy set of fuzzy decision implications, and the semantical characteristic of the obtained fuzzy set can be interpreted by fuzzy decision context and can be represented by some operators of fuzzy decision context. Conversely, starting from a fuzzy set of fuzzy decision implications, they can form a fuzzy decision context, from which the given fuzzy set can be derived. The result actually implies that they have constructed a correspondence between closed fuzzy sets of fuzzy decision implications and fuzzy decision contexts, and thus shows the equivalence of two interpretations of fuzzy decision implications.

implication canonical basis and proved that the canonical basis is complete, non-redundant and contains the least number of fuzzy decision implications among all complete sets of fuzzy decision implications, just as canonical basis with respect to attribute implications [1] and decision implication canonical basis with respect to decision implications [8,9]. Thus, a fuzzy decision implication canonical basis can be regarded as the most compact set of decision information without any information loss.
All the results, however, were obtained in a logical way and are not applied to data tables. Though the logical studies of fuzzy decision implications have provided a deep and theoretical insight into fuzzy decision implications, the studies of fuzzy decision implications with respect to data tables can link the properties of data tables with fuzzy decision implications and provide a practical insight into fuzzy decision implications [15]. Thus, this study aims to provide an interpretation of fuzzy decision implications within data tables, i.e. fuzzy decision contexts. We will show how to derive fuzzy decision implications from a fuzzy decision context and describe how to connect fuzzy decision implications with a fuzzy decision context. The results actually imply the equivalence of two interpretations of fuzzy decision implications, i.e. the logical way and the data-driven way. This paper will be organised as follows. Section 2 provides an overview of basic notions, in particular, some basic properties of a complete residuated lattice. We will introduce fuzzy decision implications and fuzzy decision contexts in Section 3. Afterwards, we connect fuzzy decision implications with fuzzy decision contexts in Sections 4 and 5. Section 6 concludes the paper and presents some further work.

Preliminaries
A complete residuated lattice [16] is an algebra (L,^, _ , ⊗ , , 0, 1) such that (i) (L,^, _ , 0, 1) is a complete lattice with 0 and 1 being the least and greatest elements of L, respectively; (ii) ⊗ is commutative, associative, and a ⊗ 1 = 1 ⊗ a = a, for each a [ L; (iii) ⊗ and satisfy the adjointness property: a ⊗ b ≤ c if and only if a ≤ b c, for each a, b, c [ L.
A truth-stressing hedge (hedge for short): * :L L such that for each a, b [ L,1 * = 1, a * ≤ a,( a b) * ≤ a * b * and a * * = a * . As for hedges, two commonly used ones are (i) Identity, i.e. for each a [ L, a * = a. (ii) Globalisation, i.e. a * = 1i f a = 1 0 otherwise Several properties of the residuated lattice are listed here for later use; for a, b, c, y i [ L, we have As a special case of complete residuated lattice, L = [0, 1] is commonly used, with 0 and 1 being the least and greatest elements, respectively, and^and _ being minimum and maximum, respectively. Several important pairs of adjoint operators are Gödel With L being the structure of truth degrees, we define a fuzzy set A (or L-set) as a mapping from a universe U to L, A:U L, whose value A(u) is the degree to which u is contained in A.A su s u a l ,w ea l s o denote a fuzzy set A by a 1 /u 1 , ..., a n /u i ,w h e r ea i = A(u i ). By L U , we denote the set of all L-sets in U. Several useful operators c a nb ed e fined for L-sets A and B accordingly, such as Other notions can, therefore, be adopted based on the two above operators; e.g. we say . As an extension of the classical subset-hood relation #, the subset-hood degree is given by

Fuzzy decision implications and fuzzy decision context
In this section, we will recall some basic notions and present some results of fuzzy decision implications and a fuzzy decision context [13,14].

Fuzzy decision implications
Let C, D be two finite universes and L C , L D be two systems of L-sets in C and D, respectively. A fuzzy decision implication on C and D is of the expression A ⇒ B, where A [ L C and B [ L D . Here A is the premise of the implication and B the consequence of the implication. All fuzzy decision implications on C and D are denoted by I. For a fuzzy set T [ L C<D , the degree to which T respects A ⇒ B is defined by where * is a hedge. The degree to which A ⇒ B holds in a set T= {T 1 , T 2 , ..., T n }i sd e fined by For a fuzzy set L of fuzzy decision implications, the set of models of L is given by where L(c) is the membership degree of c in the fuzzy set L.
Then the degree to which A ⇒ B semantically follows from L is defined by A fuzzy set L of fuzzy decision implications is closed if, for each

Fuzzy decision context
In this subsection, we will introduce a fuzzy decision context as an extension of the decision context under the setting of fuzzy attributes.  instance, I C (Uranus, small) = 0.5 says that the degree to which Uranus is small is equal to 0.5, while I C (Uranus, large) = 0.5 says that the degree to which Uranus is large is also equal to 0.5, which means that Uranus is not so small and is not so large either. Given a hedge *, the following notations from [13] are used in the study: and, for B [ L C and g [ G and, for B [ L D and g [ G )).
This limitation in Definition 2 ensures that stronger conditions This will be clearer when Definition 2 degenerates to the crisp case. Recall that in the crisp case, a decision context is consistent Observe that the condition of *-consistent is an extension of consistent.
In fact, Example 2: Table 1 is not *-consistent for any hedge * since, for the objects, Mercury and Pluto, we have

From fuzzy decision context to fuzzy decision implication
Now we show how to extract fuzzy decision implications from fuzzy decision contexts.

Definition 3: A fuzzy decision implication
Obviously, A ⇒ B kKl is just the degree to which A ⇒ B holds in the set I G = C g < D g |g [ G . We denote the fuzzy set of fuzzy decision implications by Since we have formed the fuzzy set K of fuzzy decision implications in Definition 3, it is natural to apply the notations such as complete, model, and non-redundant to the case of fuzzy decision context, if the fuzzy set K is closed.
To prove the other inequality K(A ⇒ B) ≥ A ⇒ B K , by the definitions of K(A ⇒ B) and A ⇒ B K , we need to prove that thus, it suffices to show I G # Mod(K). This is correct because, for where I g is a model of Mod(K). □ By Theorem 1, the fuzzy set K is closed; therefore, all the notations such as complete, model, and non-redundant, which are defined with respect to a closed fuzzy set, can now be applied to K and regarded as the notations with respect to fuzzy decision contexts. Now we say that a fuzzy set We denote the set of all models of K (i.e. the fuzzy set K) by Mod(K). A fuzzy set L is complete with respect to K if it is complete with respect to K; L is non-redundant if no proper subset of L is complete.
A special set of fuzzy decision implications can be formed by using only one row of fuzzy decision context one time, i.e.
At first glance, all of the fuzzy decision implications from I should fully hold in K, i.e. K(C g ⇒ D g ) = 1. However, it is not true in general. In fact, for the general case, we have the following result.
Proof: It is easily seen that I # K if and only if K(C g ⇒ D g ) = 1 for each g [ G. Now we have The last inequality is just the definition of * -consistent. □ Theorem 2 states that for a *-consistent fuzzy decision context K, all the fuzzy decision implications in I hold in K, and that if one wants the fuzzy decision implications in I to hold in K, K has to be * -consistent. Now we recall the notion of 'unite closure', which was first proposed in [13] and plays an important role in the semantical structures of decision implications [7], fuzzy decision implications [13], and variable decision implications [17].
For a fuzzy set L of fuzzy decision implications and an L-set A [ L C , the closure of A with respect to L is an L-set defined by The unite closure of A with respect to L is given by A < A L . Closure and unite closure have the following properties [13].
Lemma 1: For any L-set A and fuzzy set L of fuzzy decision implications, we have The first result of Lemma 1 shows that for each given L-set A [ L C , we can obtain a model of L, which is just the unite closure of A, whereas the second says that each fuzzy decision implication with the form A ⇒ A L can be fully followed from L.
Concerning the fuzzy decision context, we now obtain the following result.

Theorem 3: For any
Proof: For A [ L C , by the properties of residuated lattice, we have The last equation is true by Theorem 1 and (2) . Thus, we have S(A CD , A K ) = 1 and then A CD # A K , which completes the proof. □ This is the main result of the study, serving to connect fuzzy sets of fuzzy decision implications with fuzzy decision contexts, and showing that one can obtain closure and unite closure not only by means of the fuzzy set of fuzzy decision implications but also by using the operators C and D in fuzzy decision context. In other words, given a fuzzy decision context K and an L-set A, one can compute A CD by the operators C and D in K; one can also achieve this by (i) extracting all the fuzzy decision implications from K and forming the fuzzy set K, and (ii) computing the maximal consequence A K of premise A according to K. This is correct since Theorem 3 shows that the maximal consequence A K is equal to A CD .
By Theorem 3, many results from the logical study of fuzzy decision implications can be transferred to the data-driven study. For example, the following result shows how to represent models by means of the operators C and D .

By Lemma 1, it is easy to see that Mod
Conversely, let T [ Mod(K). Then, since we have K(T > C ⇒ (T > C) L ) = 1 by Theorem 1 and Lemma 1, and since T is a model of K, we obtain This completes the proof. □ Theorem 4 shows that the set of models of K can be represented by the operators C and D . This result also implies that for any A [ L C ,Ā = A CD is the least set to make A <Ā be a model of K.
In the case of * -consistent, the unite closures of the special fuzzy sets, {C g |g [ G}, have simpler forms.
Theorem 5: For a * -consistent fuzzy decision context and g [ G,we have (C g ) K = (C g ) CD = D g .
Proof: By Theorem 3, it suffices to show (C g ) K = D g .Bydefinition of K , we have ( By Theorem 2, we know K(C g ⇒ D g ) = 1 and thus Since the last equation is true, we have (C g ) K # D g and thus (C g ) K = D g . □ As another application of Theorem 3, Theorem 5 shows that in a * -consistent fuzzy decision context, the closures of the fuzzy sets {C g }, g [ G with respect to the fuzzy set K are just the fuzzy sets 5 From fuzzy decision implication to the fuzzy decision context We have shown how to obtain a closed fuzzy set of fuzzy decision implications from a fuzzy decision context. In this section, we want to form a fuzzy decision context from a given fuzzy set of fuzzy decision implications and prove that the obtained fuzzy decision context has the same closed fuzzy set of fuzzy decision implications as that of the given fuzzy set. By doing this, one can study the fuzzy decision implications by studying the corresponding fuzzy decision context. A sufficient and necessary condition for completeness based on the set of models is first adopted from [13] for later use. Lemma 3 [13] shows that one needs only one step to get a closed fuzzy set of fuzzy decision implications, i.e. settinḡ L(A ⇒ B) = A ⇒ B L ; moreover, the fuzzy set given is complete with respect to the closed fuzzy set obtained. Now, for a fuzzy set L of fuzzy decision implications on C and D, we denote  Proof: According to the definition of K D , we only need to prove that S(A 1 , Since we only need to prove S(A 1 , ). This is true since, by  m)).  Table 2, each row of which is generated by one of the fuzzy decision implications in D.
Now, by Theorem 6, K D is *-consistent, where * is the identity hedge because D is computed by using Łukasiewicz adjoint pair and identity hedge. This result can also be checked by the definition of * -consistent. For example, for the objects g 3 and g 4 , we have S(C g 3 , C g 4 ) * = ((0 0.5)^(1 0)) * = (1^0) * = 0 * = 0 ≤ S(D g 3 , D g 4 ) By Theorem 7, we have K D =L, where L is given by Example 4 andL is computed by Lemma 3. This result can be verified by checking whether K D (A ⇒ B) =L(A ⇒ B) for any fuzzy decision implication A ⇒ B. For example, let A ⇒ BW {0.5/x,0.5/y} ⇒ 1/z. Then, we have In summary, we have shown that for any given fuzzy set L of fuzzy decision implications, we can generate a complete fuzzy set D and then form a corresponding fuzzy decision context K D ,a n d furthermore, the closed fuzzy set K D derived from K D is equivalent to the closed fuzzy setL of L (Theorem 7). This implies that the fuzzy decision context obtained preserves all information from the given fuzzy set L and that one can study the fuzzy decision context instead of the given fuzzy set of fuzzy decision implications.

Conclusion and further work
The study intended to interpret fuzzy decision implications within fuzzy decision contexts. Thus, one can extract a fuzzy set of fuzzy decision implications from a fuzzy decision context and form a fuzzy decision context from the given fuzzy set of fuzzy decision implications. This actually establishes a correspondence between fuzzy decision contexts and fuzzy sets of fuzzy decision implications and furthermore implies the equivalence of the logic way and the data-driven way of interpreting fuzzy decision implications.
Important further works include: (i) using fuzzy concept lattice [16] to analyse the relationship between condition sub-context and decision sub-context; (ii) clarifying the effects of various hedges on the connections between fuzzy decision implications and fuzzy decision contexts; (iii) considering the data-driven way in variable decision implications [17].

Acknowledgments
The National Natural Science Foundation of China (no. 61806116); the Natural Science Foundation of Shanxi Province (nos. 201801D221175 and 201601D021076); and the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (STIP) (no. 201802014).