Conceptual Treatment of Multivalued Dependencies

Transcription

1 Conceptual Treatment of Multivalued Dependencies Bernhard Thalheim Computer Science Institute, Brandenburg University of Technology at Cottbus, PostBox , D Cottbus 1 Abstract. Multivalued dependencies are considered to be difficult to teach, to handle and to model. This observation is true if multivalued dependencies are treated in the classical approach. We introduce another treatment of multivalued dependencies based on ER modeling techniques and show that multivalued dependenices can be handled in a more natural and intuitive way within our framework. Based on the concept of competing multivalued dependencies we can prove in which case a unique ER schema representation exists. If multivalued dependencies are competing then either one of the competing schemata is chosen or an approximation which combines the competing schemata can be used. 1 Introduction Multivalued dependencies (MVD) have been thoroughly investigated at the end of the 70ies and during the 80ies. The research led to a deep insight into the relational model and to five equivalent definitions of MVD s. [Thal91] surveys more than threescore papers directly concerned with MVD s. At the same time it surprises that multivalued dependencies did not find their place in the ER model, except[chnc81,ling85,ling85 ]. We claim that ER models handle MVD s in a more sophisticated, better understandable, and more natural form. This paper aims in proving this claim. We are able to provide a natural treatment for MVD s which more flexible and more natural than that one which has been developed for the relational model. 1.1 Multivalued Dependencies Are Difficult To Model, To Teach, To Learn And To Handle Multivalued dependencies are introduced in almost any database book. The classical introduction is based on the tuple-generating definition of 1 Published at ER 2003, Chicago, LNCS 2813, Springer,

2 the multivalued dependency Y Z. It requires that whenever two tuples have the same value on the left-hand side then there also exist a tuple in the relation which matches to the first tuple by the left-hand side and the first element of the right-hand side and which matches to the second tuple by the left-hand side and the second element of the right-hand side. More formally, given a type R, two sets, Y of components of R and the set Z of remaining components of R. The statement Y Z is called multivalued dependency. The multivalued dependency Y Z is valid in a class R C defined over the type R (denoted by R C = Y Z ) if for all objects t, t R C with t = t an object t R C can be found for which the equalities t = Y t and t = Y t are valid. This definition has the clarity of a mathematical definition and the problematic treatment often observed for other mathematical constructs. It has been stated (e.g., [Mood01,Sims94]) in a number of practitioner reports that modeling MVD is rather difficult and often confuses people in practice. Modeling MVD s in teams becomes a nightmare. For this reason, modelers try to stay away from MVD s. At the same time, MVD s are extensively used whenever normalization is concerned. In this case, they express that a certain normalization step is based on (vertical) decomposition, i.e. R C = R C [ Y ] R C [ Z]. Functional dependency normalization is mainly based on synthesis approaches. Currently, no synthesis approach is known for multivalued dependencies. Normalization based on multivalued dependencies is based on the decomposition approach. Unfortunately, the two normalization approaches are incompatible [Thal91]. Therefore, sets of multivalued and functional dependencies must be treated by the decomposition approach. 1.2 The Outline of The Paper In Section 2 we introduce multivalued dependencies through the extended entity-relationship model. It is demonstrated that this treatment is far more natural and much simpler. A logical implication theory for the ER treatment of sets of multivalued dependencies is provided. In Section 3 we introduce the notion of competing schemata. We show that competing equivalent schema naturally appear and outline a way how to find a unique schema. Finally, a schema approximation approach is proposed for the case that competition among schemata cannot be resolved.

3 2 The ER Approach To Dependencies 2.1 Multivalued Dependencies within ER Schemata J. Biskup [Bisk95] introduces three heuristical principles of conceptual modeling. The first principle is Separation of Application Aspects. Each type in the conceptual schema describes one and only one aspect. The type describes one and only one class of existing things. This principle allows to give another definition of validity of multivalued dependencies in the context of extended entity-relationship models [Thal00]. Definition 1 Given a type R, the partition of components of R into, Y and Z. The multivalued dependency Y Z is ER-valid in a class R C defined over the type R (denoted by R C = ER Y Z) if the type can be decomposed into three types representing, Y, and Z and two mandatory relationship types defined on Y and Z, respectively. The MVD can be represented by a decomposition of the type R displayed in Figure 1. We use the last figure whenever we want to use compact schemata. In this case, the relationship type with the components ()Z is based on the -components. It allows to show the direct decomposition imposed by the multivalued dependency. Y Y Y Z (1,n) (1,n) Y Z (1,n) (1,n) Y() ()Z (1,n) (1,n) Z Z -components form an entity type of the relationship type R -components form a relationship sub-type of the relationship type R abbreviated notation for -components that form an entity type of the relationship type R Fig. 1. Three ER representations of a MVD Example 1. (Running example) Let us consider a relationship type EmployeeAssociation defined on the entity types: StaffMember, DependentPerson, Project, Supplier, Product. We observe a number of MVDs, e.g. A staff member determines the department he is working for and members of his family in dependently on the projects and products and their suppliers. { StaffMember } { Department, DependentPerson } { Project, Product, Supplier } This MVD can be represented by following picture. We shall see later that

4 (Department, DependentPerson) StaffMember (Project, Product, Supplier) other MVD s allow to decompose the association of the type EmployeeAssociation into types representing the relationship between staff member and their dependent people, between staff member and departments and between staff members and projects, products and suppliers. 2.2 ER-Schema-Based Derivation Rules for Multivalued Dependencies It is now our interest to reason on ER-validity of multivalued dependencies within the ER schema. In this case we can directly display the results of the type decomposition within the ER schema. We introduce a deductive system in Figure 2 that allows to derive ERschema decompositions based on the knowledge of ER-validity of MVD s. The axiom enables in introducing any component clustering for any relationship type. For rules of our calculus 2 we use subsets,,, Y, Y, Z, Z, V, W R and assume that the sets in one multivalued dependency constitute a cover of R. The following three rules is based on the ER-validity of MVD s. We remember the following statement of [Thal91]: Proposition 1 The deductive system consisting of the trivial MVD, the root reduction rule, and the weakening rule is correct and complete for inference of multivalued dependencies. The tree restructuring rule is correct too but does not form together with the trivial MVD and the weakening rule a complete system. This proposition allows to deduce the following theorem 3. Theorem 1 The axiom and the weakening, root reduction and tree restructuring rules are sound and complete for inference of ER-validity of multivalued dependencies. 2 Derivation rules are displayed in the Hilbert style, i.e. using a dividing line. The formulas (in our case the sub-schemata) above the line are the prerequisites of the rule. The formula (in our case the sub-schema) below the line represents the conclusion of the rule. 3 Proofs of theorems in this paper are based on the classical relational theory and can be derived directly by applying classical approaches [PDGG89]. Further, the full paper is available [Thal03].

5 Axiom Z Y () V ()Z Root reduction rule Y Z() Y () ()V ()Z V Y () () Z Weakening rule Y ( ) Z ( ) Y Y () ()Z Z Tree restructuring rule Y Z() Y () ()Y Z ()Z Z Y Fig. 2. The deductive system for ER schema derivation based on MVD s

6 We observe that the axiomatization of functional and multivalued dependencies can be derived in a similar way. 2.3 ER Treatment of Horizontal Dependencies ER diagrams are not restricted to entity types which have not more than two associated relationship types. Therefore, multivalued dependencies are not powerful enough. We may use, however, hierarchical dependencies. Given a cover, Y 1,..., Y m of components of (R) where the sets Y 1,..., Y m are pairwise disjoint. The hierarchical dependency Y 1 Y 2... Y m is valid in R C if for object t 1, t 2,..., t m from R C which are equal on an object t exists in R C for which t[ Y i ] = t i [ Y i ] for all i (1 i m). Obviously, if m = 2 the hierarchical dependency is a multivalued dependency. The hierarchical dependency can be represented in the ER diagram. The following picture shows this representation for the case that the set of components forms an entity type. Y 2 ()... Y 1() ()Y m Fig. 3. The ER representation of a hierarchical dependency This star unfolding rule in Figure 4 is a very powerful rule. It leads directly to star or snowflake schemata [Thal02]. We conclude based on [Thal00]: Corollary 1 The star unfolding rule is sound. Star unfolding provides an insight into the finest separation of types associated with a type through a set of multivalued dependencies. Based on hierarchical dependencies we can generalize the the dependency basis defined for multivalued dependencies to the dependency basis for functional, multivalued and hierarchical dependencies. Define now + = { A U Σ = {A} }. The dependency basis of a set Σ of functional, multivalued and hierarchical dependencies is given by

7 Y Y () ()Z Z Star unfolding rule Y Z() Y () ()Y Z Z()... Y () ()Z Fig. 4. The star unfolding rule for hierarchical dependencies Dep M,H (, Σ) = { Y i Σ = Y i, Y i + =, Y i Y i(y i Y i Σ = Y i ) } Dep M,H,F (, Σ) = Dep M (, Σ) { + \ }. It defines the finest separation among components of the type. Similar to the proof for the classical dependency basis in [Thal91] we can proof the following statement. Proposition 2 For any set Σ of functional, multivalued and hierarchical dependencies and any multivalued or hierarchical dependency Z 1... Z m, the set Σ implies Z 1... Z r if and only if either Y i Z j or Y i Z j = for all i, j(1 i n, 1 j m) and for the dependency basis Dep M,H (, Σ) = {Y 1,..., Y n } of. Furthermore, Σ = Z if and only if Z +. 3 Competing Schemata Due To MVD s So far only local decomposition based on one component set has been investigated. Another component set may lead to another decomposition of the same type. Example 2. (Continuation of the running example by additional MVD s) { StaffMember } { DependentPerson } { Department, Project, Product, Supplier } { Project } { StaffMember, Department, DependentPerson } { Product, Supplier } { Product } { Department, StaffMember, DependentPerson, Project } { Supplier }

8 The dependency basis for StaffMember is the set Dep M,H ( StaffMember,Σ) = {{ Department }, { DependentPerson }, { Project, Product, Supplier }} Therefore, the following ER schema can be derived. Staff Member Point of View (Project, Product, Supplier) Department StaffMember DependentPerson Computing the dependency basis for Project Dep M,H ( Product, Σ) = {{ Supplier }, { StaffMember, Department, DependentPerson }} we derive the point of view in the following figure: Project Point of View (StaffMember, Department, DependentPerson) Product Project Supplier Which point is the correct one? Is there any unifying point of view? The schemata are somehow competing. Therefore, we need a competition resolution technique. Let us consider the reduced cover of all multivalued and hierarchical dependencies by Σ = { Y Z Σ = Y Z, Y = Z = Y Z =, Y, Z }. Definition 2 Given two partitions, Z, V and Y, U, W of components of R. Two multivalued dependencies Z V and Y U W from Σ are competing if (1) Σ = Y Z V W for Z = U or (2) U and W.

9 Competing schemata offer different points of views. It might be the case that we find a unifying point of view that resolves competition. One solution for competing schemata is the introduction of an artificial separator type. [Scio81] gave it in the context of conflicting dependency bases. Definition 3 Given a type R and two partitions {, V 1,..., V n, 1,..., m } and {Y, V 1,..., V n, Y 1,..., Y k } of components of R which both form a dependency basis, i.e. Dep M (, Σ) = {V 1,..., V n, 1,..., m } Dep M (Y, Σ) = {V 1,..., V n, Y 1,..., Y k }. The two dependency bases are conflicting if Σ = Y n i=1 V i. The solution given by [Scio81] is based on the introduction of an artificial separator type P,Y into R combined with the additional functional dependencies P,Y and Y P,Y and the hierarchical dependencies ( Y ) {P,Y } V 1... V n, {P,Y } V 1... V n 1... m and Y {P,Y } V 1... V n Y 1... Y k. This solution may be adequate in some cases. In most practical situations the solutions is not adequate for conceptual modeling since conceptual modeling aims in developing conceptually minimal schemata which use only types that are meaningful in the application. The solution may be however used as an implementation trick. This idea can be however further developed. We introduce a new rule 4 displayed in the picture 5. Proposition 3 The tree separation rule is sound. Proof sketch. First we observe that root reduction allows to derive a convenient generalized root split rule in Figure 5. Another rule that can be derived from the theory of MVD s is the second root reduction rule shows in Figure 5. The combination of the three rules proves the proposition. We finally obtain a very serious argument in favor of non-competing sets of multivalued dependencies. Theorem 2 If a set of multivalued dependencies is not competing then there exists a unique decomposition which can be pictured entirely by a higher-order entity-relationship schema without additional multivalued or path dependencies. 4 The tree separation rules uses a property of the extended entity-relationship model: Entity types are of order 0, relationship types are ordered by orders 1, 2, 3,... The lowest order in any schema is 0. Therefore, the orders can be adapted whenever we wish to have a dense order.

10 Y Z() ()U Tree separation rule U (Y) Y (Y)Z Z P,Y Y U () Y () V ()Z Root split rule Y Z() Y () ()V ()Z V () Y U W () ()Z V Second root reduction rule U ( Y Z) U ( Y ) Y Z ( Y Z)V W Y ( Y )Z V W Fig. 5. Three derived rules for resolution of competition

11 Example 3. (Continuation of the running example) We can now resolve the competition among schemata. The application of the tree separation rules leads to the schema in the following picture which is the most detailed view and lies underneath of the two competing schemata discussed before. Let us use the name Working for denoting the new type P P roduct, Department StaffMember. Department Project Working StaffMember Product Supplier Dependent 4 Approximation and Weakening of Competing Schemata So far we considered only the case that multivalued dependency sets can be represented by a unique minimal schema. There are however cases in which competing dependency sets cannot be resolved to a unique minimal schema. Example 4. (Example from [LevL99]: Competing dependency set without resolution) Given a schema with a relationship type Engaged = ( Employee, Project, Manager, Location, Σ) and the multivalued dependencies { Employee, Manager, } { Location } { Project }, { Project, Location } { Employee} { Manager}. Location Manager Engaged Project Employee

12 The relationship type can be represented by two competing sub-schemata. The competition stresses two different points of view: The Engaged relationship type is differentiated by the working association and the leadership association. The last one related Mangers with employees at different project. The first one associates workers with the location and with the leadership. Additionally, the decomposition generates an inclusion constraint Works[Employee, Manager] Leads [Employee, Manager]. The Engaged relationship type is decomposed into the relationship types WorkingAt and ManagingAt. We observe the same kind of inclusion constraint WorkingAt[Project, Location] MangagingAt [Project, Location]. This inclusion constraint show again that the decomposition is not the most appropriate one. Second-order relationship types improve the situation. The inclusion constraints directly lead to another decomposition which is more appropriate and much simpler to maintain. These decompositions are more appropriate and better reflect the sense of the multivalued dependencies. We may, however, find stronger decompositions which are not entirely supported by the multivalued dependencies and weaker schemata which do not reflect the full power of the multivalued dependencies. Definition 4 Given a set of multivalued dependencies Σ for a relationship type R. Let weaker M (Σ) = { Σ MV D Σ = Σ Σ = Σ and stronger M (Σ) = { Σ MV D Σ = Σ Σ = Σ } the sets of weaker or stronger sets of multivalued dependencies for a set of multivalued dependencies. We define now the set approx M (Σ) of maximal elements of weaker M (Σ) and the set approx M (Σ) of minimal elements of stronger M (Σ) as approximations of Σ. If we discover in an application sets of competing schemata then we might either stress whether one of the elements of approx M (Σ) is valid as well (strengthening the specification) or one of sets in approx M (Σ) can be used instead of Σ (weakening the specification). Both approaches have their merits. Proposition 4 Approximations have the following regularity properties: approx M (Σ 1 Σ 2 ) approx M (Σ 1 ) approx M (Σ) Σ Σ for some Σ approx M (Σ) approx M (approx M (Σ)) = approx M (Σ)

13 approx M (Σ 1 Σ 2 ) approx M (Σ 1 ) approx M (Σ) Σ Σ for some Σ approx M (Σ) approx M (approx M (Σ)) = approx M (Σ) The proof is based on the properties of approximations and the monotonicity of derivable sets. These regularity properties allow to weaken or strengthen one of the multivalued dependencies. Therefore, we can restrict our attention to those multivalued dependencies which do not have strong support or which are a bit too strict. This idea invokes the idea of Lukasiewicz of assigning fractional truth values to formulas and sets of formulas depending on their support. Especially, weakening leads to singleton sets of multivalued dependencies in most cases. Since the theory is rather complex in this case we demonstrate this property for second example. Example 5. (Resolving the example from [LevL99] by approximation) The example has a convincing weakening which is based on a fact-association based modeling approach [Thal00] and uses the association by relating the left-hand sides of the two multivalued dependencies. Weakening the Set of Multivalued Dependencies for the Point of View that Stresses Work Association Employee Manages AssignedTo Location PerformedAt Manager Project We note that the multivalued dependencies may be easily maintained by triggers in our weaker schema. Furthermore, cardinality constraints such as card(manages, Employee) = (1, 1) can easily be integrated. The set of upper approximations has a similar convenient property. It is characterized by strengthening one of the dependencies used in the decomposition. For instance, if we are more interested in work association then strengthening the second multivalued dependency to the functional dependency {Project, Location} {Manager} then we obtain a unique decomposition through applying results in the previous section.

14 5 Conclusion Multivalued dependencies are a stepchild of database practice. Their specification is often error-prone. Sets of multivalued dependencies are difficult to survey and to understand. In reality, multivalued dependencies specify relative separation of aspects of concern. Therefore, the most natural way to reflect them is their usage for decomposition of types in ER schemata. In this case we obtain a view on the set of multivalued dependencies that is simple and which is easy to survey, to maintain and to extend. Therefore, multivalued dependencies should be treated at the level of ER schemata rather than at the level of logical schemata. It is an open problem whether competing schemata can be unified by critical pair resolution similar to term rewriting systems. References [Bisk95] J. Biskup: Foundations of Information Systems. Vieweg, Braunschweig, 1995 (in German). [ChNC81] I. Chung, F. Nakamura and P.P. Chen: A Decomposition of Relations Using the Entity-Relationship Approach. Proc. ER 1981: [LevL99] M. Levene and G. Loizou: A Guided Tour of Relational Databases and Beyond. Springer, Berlin, 1999 [Ling85] T.W. Ling: An Analysis of Multivalued and Join Dependencies Based on the Entity-Relationship Approach. Data and Knowledge Engineering, 1985,1: 3. [Ling85 ] T.W. Ling: A Normal Form for Entity-Relationship Diagram. Proc. 4th ER Conference, IEEE Computer Science Press, Silver Spring, 1985: [Mood01] D.L. Moody: Dealing with Complexity: A Practical Method for Representing Large Entity-Relationship Models. PhD., Dept. of Information Systems, University of Melbourne, 2001 [PDGG89] J. Paredaens, P. De Bra, M. Gyssens, and D. Van Gucht: The Structure of the Relational Database Model. Springer, Berlin, [Scio81] E. Sciore: Real-world MVD s. Proc. Int. Conf. on Data Management, 1981: [Sims94] G.C. Simsion, Data Modeling Essential - Analysis, Design, and Innovation. Van Nostrand Reinhold, New York 1994 [Thal91] B. Thalheim: Dependencies in Relational Databases. Teubner, Leipzig, 1991 [Thal00] B. Thalheim: Entity-Relationship Modeling - Fundamentals of Database Technology. Springer, Berlin, [Thal02] B. Thalheim: Component Construction of Database Schemes. Proc. ER 02, LNCS 2503, Springer, 2002: [Thal03] B. Thalheim: Scrutinizing Multivalued Dependencies through the Higher- Order Entity-Relationship Model. Preprint BTU Cottbus, Informatik