Fuzzy Functional Dependency and the Resolution Principle

. In this paper we establish equivalence between a theory of fuzzy functional dependences and a fragment of fuzzy logic. We give a way to interpret fuzzy functional dependences as formulas in fuzzy logic. This goal is realized in four steps. Truth assignment of attributes is deﬁned in terms of closeness between two tuples in a fuzzy relation. A corresponding fuzzy formula is associated to a fuzzy functional dependence. It is proved that if a relation satisﬁes a fuzzy functional dependence, then the corresponding fuzzy formula is satisﬁed and vice verse. Finally, equivalence of a fuzzy formulas and a set fuzzy functional dependence is demonstrated. Thus we are in position to apply the rule of resolution from fuzzy logic, while calculating fuzzy functional dependences.


Introduction
According to the classical relation database all the information in it, have to involve precisely defined values (atomic).So in a case that those values are not defined precisely then the imprecise values could be involved as one value, so called NULL.
Codd (Codd, 1970) considers the NULL value in a meaning "completely unknown", i.e., some values of attribute domain could have this meaning.
Lipski (Lipski, 1981) extended the of Codd's null value by considering that a value though unknown is in a specific subset of the attribute domain.
In some other study extension, variety of null values have been introduced to model unknow or not-applicable data values.As an alternative approach is the usage of first order predicate calculus where Skolem functions are used to represent null values.
The other way of considering this imprecise information is the involving of fuzzy value to the domain of attribute.These imprecise information have been focused on Zadeh's fuzzy set theory and fuzzy logic.The fuzzy set theory and fuzzy logic pro-vide mathematical framework to deal with the imprecise information in fuzzy relational databases.
Approaches to representation of inexact information in relation database theory, include fuzzy membership values (Buckles and Petry, 1982a;Chen, 1998;Zadeh, 1975), similarity relationships (Buckles and Petry, 1982b;Sozat and Yazici, 2001) and possibility distributions.This paper takes the similarity-based fuzzy relational database approach.
In a fuzzy set, each element of the set has an associated degree of membership.The degree of membership is a real number between zero and one and measure the extent to which an element is in a fuzzy set (Zadeh, 1975).
As an extension of the degree of membership concept for sets elements, we have similarity relationship.Here the domain elements are considered as having varying degrees of similarity, replacing the idea of exact equality/inequality.
To deal with fuzzy data constraint, Zadeh has introduced the concept of particularization (restriction) of fuzzy relation due to a fuzzy proposition.The formed formulas of first order calculus can be used to represent integrity constraints in a classical relational databases (Codd, 1970;Ullman, 1982), fuzzy integrity constraint can be represented by suitable fuzzy propositions.The particularization of fuzzy relational database due to a set of fuzzy integrity constraints can be computed by combining the fuzzy propositions associated with these integrity constraints according to the rules of fuzzy calculus.
Our primary aim in this paper is to establish a connection between theory of fuzzy functional dependence and one fragment of fuzzy logic.So it will be shown that if relation r satisfies fuzzy functional dependence then is truth value of the belonging fuzzy formula is greater or is equal to 0.5 and vice verse.
If we have some set of fuzzy functional dependences it will be possible to show whether or not some other fuzzy dependences will follow, in a way of using, the correponding axioms and inferences rules for fuzzy functional dependence.However such deduction could be very complicited, because it is not obvious which axioms has be to selected in its phase, and there isn't some globaly strategy for valid results.But in classic logic as in fuzzy logic there is effective procedure, which from its starting set of formulas as well as its logic consequence shows validity of given formula.Such a procedure is known as the rule of resolution (Habiballa, 2000;Lee, 1972;Mukaidono, 1986).Therefore it will be here established the equivalence of calculation of one part of fuzzy logic and fuzzy functional dependence.After establishing this equivalence, then is possible to apply the rules of deduction in fuzzy logic, on calculus of fuzzy functional dependence.

Similirarity-Based Fuzzy Relational Database
As in an ordinary relational database, the constituent parts of a fuzzy relational database are a set of relations comprised of tuples.Although tuples are not ordered with respect to a relation, for convenience, let t i represent the ith tuple.Tuple t i takes the form (d i1 , . . ., d im ), where d ij , a domain value, is selected from a given domain set, D j .
In an ordinary relational databas, d ij ∈ D j .In the fuzzy relational databas, d ij is not constrained to be a singleton, that is A second feature of the fuzzy relational database (Buckles and Petry, 1982a;Buckles and Petry, 1982b;Chen, 1998) is that for each domen set, D j , a similarity relation, s j , is defined over the set of elements: Clearly, the identity relation is a special case of the similarity relations.
In the fuzzy relational database domain values need not to be atomic.A domain value, d ij is defined to be a subset of its domain base set, D j .That is, any member of the powerset, 2 Dj , may be a domain value except the null set.
A fuzzy relation instance, r, in the fuzzy database model is defined as a subset of the set cross product of the power sets (2 D1 × . . .× 2 Dm ) of the domains attributes (Buckles and Petry, 1982a;Buckles and Petry, 1982b).
A fuzzy tuple, t, is any member of both fuzzy relation r and 2 D1 × . . .× 2 Dm .An arbitrary tuple, t i , is of form t i = (d i1 , . . ., d im ), where d ij is either a nonempty subset of D j or an element such d ij ⊆ D j .

Introduction in Fuzzy Functional Dependency
In the classical relation database functional dependency (Codd, 1970;Ullman, 1982) is a statement that describes a semantic constraint on data.
Let r be any relation instance on scheme R(A 1 , . . ., A n ), U be the universal set of attributes A 1 , . . ., A n , and both X and Y be subset of U .Relation instance r is said to satisfy the functional dependency X → Y if, for ever pair of tuples t 1 and t 2 in r, But the definition of functional dependency is not directly applicable to fuzzy relational database because it is based on the concept of equality.Functional dependency There is no clear way of checking whether two imprecise values are equal.Therefore the definiton of functional dependency have to be extended namely to be generalized and this generalization version of functional dependency is said to be the fuzzy functional dependency (FFD).
There are several way in corrected definition of fuzzy functional dependency (Wei-Y, 1993;Raju and Majumdar, 1988;Shenoi and Melton, 1990;Sozat and Yazici, 2001;Yazici et al., 1993).One of the important definition for fuzzy functional dependences was presented in paper (Sozat and Yazici, 2001).In that paper firstly was defined conformance of two tuples in relation.
DEFINITION 3.1.The conformance of attribute A k defined on domain D k for any two tuples t i and t j present in relation instance r and denoted by ϕ(A k [t i , t j ]) is given as where d i is the value of attribute A k for tuple t i , d j is the value of attribute A k for tuple t j , s(x, y) is a similarity relation for values x and y, and s is mapping of every pair of elements in the domain The definition of conformance is also extended to describe the closeness of two tuples on set of attributes.DEFINITION 3.2.The conformance of attribute set X for any two tuples t i and t j present in relation instance r and denote by ϕ(X[t i , t j ]) is given as

Fuzzy Functional Dependencis
DEFINITION 3.2.1.Let r be any fuzzy relation instance on scheme R(A 1 , . . ., A n ), U be the universal set of attributes A 1 , . . ., A n , and both X and Y be subsets of U .Fuzzy relation instance r is said to satisfy the fuzzy functional depedency (FFD) Here θ is a real number within the range [0,1], describing the linguistic strength (Sozat and Yazici, 2001;Yazici et al., 1993).

Inference Rules for Fuzzy Functional Dependency
IR1 Inclusive rule for fuzzy functional dependency: IR2 Reflexive rule for fuzzy functional dependency: IR3 Augmentation rule for fuzzy functional dependency: IR4 Transitivity rule for fuzzy functional dependency:

Additional Inference Rules for Fuzzy Functional Dependency
IR5 Union rule for fuzzy functional dependency: IR6 Pseudotransitivity rule for fuzzy functional dependency: IR7 Decomposition rule for fuzzy functional dependency:
In fuzzy logic, the truth value of a formula, can assume any value in the interval [0,1] and is used to indicate the degree of truth represented by the formula.

Satisfiability in Fuzzy Logic
DEFINITION 4.1.1.A formula f ∈ S, where is S set of a fuzzy formulas, is said to satisfy in interpretation I, if truth value of a formula T (f ) 0.5 under I.An interpretation I is said to falsity S if T (f ) 0.5.
A formula is said to be unsatisfiable if it is falsified by every interpretation of it (Lee, 1972).DEFINITION 4.1.2.Let D 1 : L 1 ∨ D 1 ' and D 2 : L 2 ∨ D 2 ' be two disjuncts, and L 1 and L 2 , contra pair of literals, i.e., L 2 : ¬L 1 and let D 1 ' and D 2 ' do not contain any such pair.Then, disjunct D 1 ' ∨ D 2 ' is said to be resolvent disjuncts D 1 and D 2 with the key word L 1 .
Let S be a set of clauses.The resolution of S, denoted Res(S), is the set consisting of members of S together with all the resolvents of the pairs of members of S. The nth resoluton of S, denoted Res n (S), is defined for n 0 as follows: Res 0 (S) = S and Res n+1 (S) = Res(Res n (S)).

Main Results: Fuzzy Functional Dependency and Fuzzy Formulas
In this section we establish a connection between fuzzy logic and the theory of fuzzy functional dependencies.We give a way to interpret fuzzy functional dependencies as formulas in fuzzy logic.For a set of fuzzy dependencies F and single fuzzy functional dependency f , we show that F implies f as fuzzy functional dependencies if and only if F implies f under the logic interpretation.
The correspondence between fuzzy functional dependencies and fuzzy formulas is direct.Let X θ → F Y be an fuzzy functional dependencies where X = A 1 A 2 . . .A m and Y = B 1 B 2 . . .B n .The corresponding logical formula is For determination of truth assignment attribute in relation r, we take definition of conformance the two tuples on attribute.
Let r be a fuzzy relation over schema R with exactly two tuples.Fuzzy relation r can be used to define a truth assignment, for attributes in R when they are considered as fuzzy variables.
DEFINITION 5.1.Let R = {A 1 , A 2 , . . ., A m } be a relation schema and let r = {t 1 , t 2 } be a two tuple relation on R. The truth assignment for r, denoted i r , is the function from R to [0, 1] defined by where d i is the value of attribute A k for tuples t i , d j is the value of attribute A k for tuple t j , s(x, y) is a similarity relation for values x and y, s is mapping of every pair of elements in the domain D k onto interval [0, 1] and θ is strenght of the dependency.
The following theorem enables equivalence between fuzzy functional dependence and fuzzy formulas.So by that theorem will be proved the mentioned equivalence when for the fuzzy formulas are taken the following Let assume, as first, that relation r satisfies FFD where is Let assume contra to theorem assertion that assigments F : Then follows that in interpretation i r truth validness of i r (F ) 0.5, respectively .., m and, i r (B j ) 0.5 ∃j = 1, 2, ..., n.
If is valid i r (A i ) 0.5 ∀i = 1, 2, ..., m then according to Definition 5.1 is Based on the Definition 3.2 we have ϕ( Because of theorem assumption that FFD is satisfied, we have This results that ϕ(B j [t 1 , t 2 ]) θ for each j = 1, 2, . . ., n.So follows i r (B j ) 0.5, what is contrary to i r (B j ) 0.5.Therefore the assertion is valid if relation r satisfies Y , then its assigment fuzzy fomula is satisfy in the interpretation i r .
Let be proved, now, vice verse of theorem.Assume that F satisfy in interpretation i r .Then then i r (A j ) 0.5 for some j from {1, 2, . . ., m}, from which follow ϕ(A j [t 1 , t 2 ] ≺ θ for some j from {1, 2, . . ., m}.Then From this follows that relation satisfies FFD Let be valid ii), i.e., i r (B 1 ∧ B 2 ∧, . . ., ∧B n ) 0.5, then for each i = 1, 2, . . ., n Hence it follows that r satisfies the FFD Let assume, as first, that relation r satisfies FFD where is Let assume contra to theorem assertion that assigments F : Then follows so, we have . ., m, and and i r (B j ) 0.5, ∃ j = 1, 2, . . ., n.
Then according to Definition 5.
Because of theorem assumption that FFD is satisfied, we have This results that ϕ(B j [t 1 , t 2 ]) θ for each j = 1, 2, . . ., n.So follows i r (B j ) 0.5, what is contrary to i r (B j ) 0.5.
Let be proved, now, vice verse of theorem.Assume that F satisfy in interpretation i r .Then Then ∃ j = 1, 2, . . ., m for which hold i r (A j ) 0.5.Then according to Definition 5.1 Based on the Definition 3.2 we have From this follows that relation satisfies FFD Hence it follows that r satisfies the FFD By this is proved the theorem.
In the following theorem we are going to show that if relation r satisfies a set of fuzzy functional dependece F and does not satisfy dependency X θ → F Y then exists two tuples subrelation, of relation r, which satisfies all the fuzzy functional dependece from set F , and does not satisfy dependency Y in the world of two tuple relations.
Let prove the reverse of theorem 2) implies 1).Let assumed a contra to the theorem that is not valid In that case some relation r satisfied all the fuzzy functional dependencies from F , and do not satisfy dependency X θ → F Y .This means that exists the elements t 1 and t 2 from r, for which hold The opposite to contraposition of this claim is the claim that 2) implies 1).Proof.Let assume that i r : R → [0, 1] be such interpretation where every formulas are satisfied, which are generated FFDs from set F , at let formula which is generated by dependency X θ → F Y be falsify.Let we consider that Let r z be fuzzy relation instance with two tuples t 1 and t 2 as shown in Table 1.We choose the set {a, b} as the domain of each attributes in R, where a = a 1 , . . ., a p , and b = b 1 , . . ., b q (p 1, q 1).Let s(a i , a j ) = θ, (which implies that ϕ(A[t 1 , t 2 ]) θ, for any attribute set A in r z ), and where s is similiraty relation.
Namely r z = {t 1 , t 2 } where t 1 = a, . . ., a for each attribute A from R, and let t 2 be defined as Let prove that relation, r z defined in such way is satisfying each fuzzy functional dependencies from F .To be able to prove this, let U θ → F V , any fuzzy functional dependency from F for which then holds Due to the definition t 1 , now it have to be and t 2 = a, . . ., a for each attribute A from U , namely ϕ(A[t 1 , t 2 ]) θ.This means that i r (A) 0.5, for each A from U .From this hold U ⊆ Z, i.e., ( * ) i r (U ) 0.5.
θ would not hold, then would be t 1 = a, . . ., a and t 2 = b, . . ., b for some attribute A from V , namely ϕ(A[t 1 , t 2 ]) ≺ θ.From this we have that A does not belong set Z, and would hold i r (A) ≺ 0.5, and also i r (V ) ≺ 0.5.
Based on this and (*) we have that for Kleens-Diens implication and Zadeh implication hold and this is would be in contra to first assumption.Let prowe that r z not satisfy fuzzy functional dependency As it is by assumption that the fuzzy formula is falsify in the interpretation i r , then must be that ( * * ) i r (X) 0.5 and i r (Y ) 0.5.
Let we prove vice verse of Theorem.Let assume contra, i.e., that does not hold that from set of FFDs F follows and FFDX When would not be i) then i r (U i ) 0.5 and i r (V j ) 0.5, for some Q from V .This first would mean that ϕ(U [t, t ]) θ, and the second that ϕ(V [t, t ]) ≺ θ.Therefore these together is contradiction with start assumption that r satisfies each fuzzy functional dependencies from F .By it is proved i).
If would not be ii ) then would be iii) i r (X i ) 0.5 or iv) i r (Y j ) 0.5.Y , what is also contradiction with the beginning assumption.
The right proved theorems enable the application of resolution rules in fuzzy logic as the rule of inference on calculation of fuzzy functional dependencies.EXAMPLE 5.1.Let R = {Name, Intelligence, Capability, Job, Success} be a relation scheme, and let be set a FFDs over scheme R, where is noted by Prove that holds where is θ = min(min(θ 1 , θ 2 ), θ 3 ).
Lets prove in two ways that this examples holds, using following a) calculus of fuzzy functional dependences.b) the resolution principle in fuzzy logic.
b) According to the previous theorems it is enough to prove that hold Γ ⇒ A 1 ∧A 2 → A 5 .Let's assert, as first, to FFDs the corresponding formulas: According to the definition logical consequence and already said mentioned, it is enought to show that unsatisfiable, where is G: To be able to apply a rule of resolution, it is needed, at first transform F in conjuctive normal form so to get a set F * , as a represent of F .

Conclusion
In this paper we proved the equivalence between theory of fuzzy functional dependencies for fuzzy database and the part theory of fuzzy logic.
To achive such an aim, we introduced the definition of truth assignment of attributes in relation r over the relation scheme R. Based on this definition of FFD was attached to the fuzzy formula and was proved that if relation r satisfies FFD then this fuzzy formula is satisfied in the given interpretation and vice verse.The equivalence between set of the FFDS and fuzzy formulas was proved as well.This equalence makes possible an application of the resolution principle.With this equivalence, we may substitute calculation of fuzzy functional dependencies by calculation of fuzzy formulas, applying the resolution principle as inference rules.The resolution principle in fuzzy logic enables a complete automatic proving, what is significant advantage over to the classic approach.
It is a progress a further study that will prove an equivalence of implication of fuzzy multivalued dependencies and of fuzzy logic.

FYFY
be a FFD over relation scheme R and let r be relation on R with two typles.A FFD X θ → is satisfied by relation r if and only if X → Y is satisfy under the truth assignments i r .Proof.a) For Kleens-Diens implication X → Y = max(1 − X, Y ).

FYFYFY
obvioes that r * satisfies all the FFDs from F , but does not satisfty this dependency X θ → .By this is shown that following Lemma 5.1.Let r be a relation, let F be set of FFDs on R, and let X θ → be a single FFD on R. If relation r satisfies all the FFDs from set F and violates fuzzy dependency X θ → , then some two tuple subrelation r * of r satisfies F and violates X θ → F Y .

FYFY
be an FFD over relation scheme R and let F be a set of FFDs over R. Then holds, F implies X θ → in the world of two tuple relations, if and only if F implies X → Y when FFDs are interpreted as fuzzy formulas.

FY
tuples relation r = {t, t } which satisfies each FFDs from F , but does not satisfy and FFD X θ → .By the above mentioned description it is defined the interpretation i r , by the relation r, formulas U 1

Table 1
The fuzzy relation instance rz It is obvious that r satisfies fuzzy functional dependency X If iii) hold, then ϕ(A i [t, t ]) ≺ θ, for some j = 1, 2, . .., m, A i ∈ X and from these ϕ(X[t, t ]) ≺ θ. i [t, t ])θ, for each j = 1, 2, . . ., n, B j ∈ Y and from these ϕ(Y [t, t ]) θ.From this, we would conclusion that and in this case r satisfied fuzzy functional dependency X θ → F