ON THE SEMI-AUTOMATIC VALIDATION AND

DECOMPOSITION OF TERNARY RELATIONSHIPS WITH

OPTIONAL ELEMENTS

Ignacio-J. Santos, Paloma Martínez Fernandez and Dolores Cuadra

Departamento de Informática de la Universidad Carlos III de Madrid Av. Universidad 30 – 28911 Leganés (Madrid)

Keywords: Cardinality Constraint, Entity-Relationship Model, Relational Model, Semantic and Syntactic Anomalies,

CASE (Computer Aided Software/ Engineering) Tools.

Abstract: This paper analyzes the problems that concern the design of databases. CASE tools supply a resources kit

for the design and creation of database in a DBMS (Database Management System). Sometimes, these tools

only help to draw diagrams. Ideally, they would verify and validate DB design and transform it from

Conceptual to Logical Model. In a last step, they would transform the Logical Model to a specific DBMS.

Currently, commercial tools do not verify or validate the model in an optimal way. This paper is focused on

the validation and checking of database schemas. This work specially analyzes the ternary or higher-order

relationships when there are optional components.

1 INTRODUCTION

When a Project Leader develops an application from

the beginning, he or she has to think in the data.

Once the designer has created the Conceptual

Schema, the designer has to transform the

Conceptual to Logic schemas, because the Logic

Model is nearest to a DBMS. This paper is focused

on Entity-Relationship and Relational models. A

CASE tool of Database helps to realize the scheme

in a particular DBMS. These tools start helping to

design the Conceptual Model from UD and after

they transform this Conceptual to Logic Model. At

the end, they create a schema in a SGBD.

A scheme has to be verified syntactically and

semantically. The syntactical verification has to

check the rules of building. The semantic

ure 1: Valida

ion and Verification of a Schema.

465

Santos I., Martínez Fernandez P. and Cuadra D. (2007).

ON THE SEMI-AUTOMATIC VALIDATION AND DECOMPOSITION OF TERNARY RELATIONSHIPS WITH OPTIONAL ELEMENTS.

In Proceedings of the Ninth International Conference on Enterprise Information Systems - DISI, pages 465-472

DOI: 10.5220/0002386804650472

 SciTePress

verification tries to find the inconsistency between

the semantic constrains and the user (Bouzeghoub,

M. et al., 2000). The figure 1 shows the steps of

validation and verification of a scheme. A good

Conceptual scheme has to have Formal Properties,

Quality Factors and Conformance with the user

necessities (validation). Formal properties mean that

the scheme has to be consistent, complete and

irredundant. With respect to formal properties the

majority of commercial CASE tools do a good

syntactic validation, but no semantic. For example,

they check if there is, at least, an entity and that the

entities have different names. However, they do not

check if there are contradictions among the schema

concepts. These tools do not also verify the

redundant elements. The completeness of a

Conceptual scheme can be defined with respect to

the meta-model or the UD represented. The first part

the metamodel concerns the mandatory elements

that constitute a conceptual schema. The commercial

tools do this verification. The represented UD means

the validation with respect to the user requirements.

Checking whether a Conceptual schema represents

all the necessary knowledge for a given information

system, which refers to conformance of the

Conceptual schema to the real world. With respect to

quality factors, we have to look at the things the

readability and the reusability. Readability is a

desirable property, but it is a subjective valuation.

The reusability is far away in commercial tools. For

the last, the validation of a schema means that if the

schema is adapted to the user requirements. In this

topic, some tools have developed the Paraphrasing

(NLDB, 2000). This technique generates a textual

description from a Conceptual schema and the user

can validate the model.

This paper analyzes the validation and

correctness of the ternary or higher order

relationships with optional elements. The majority of

tools do not well implement this type of

relationships.

Next section will describe the necessary

definitions for this paper. The third section will look

over some research works about this topic. The

fourth section will explain our contribution and it

will analyze and validate the relationships with

optionality. The fifth section will observe the

semantic anomalies in the ternary relationships with

optional elements. The sixth section will show with

an example our proposal. The seventh section will

show the conclusions.

2 SOME REQUIRED

DEFINITIONS

We show in this section the necessary definitions for

developing this paper. We begin by defining the

Entity and the Relationship element according to

Thalheim (2000).

Let be

(

)

{

}

(

)

njid

≤≤Ε

1/,...,

an entity

with attributes

,...,

, each attribute is defined in

a domain, where

(

)

{

}

njid

≤≤

is the set of

candidate keys of the entity E, and this property

characterizes in a univocal way, every one of the

instances of E. We define

as a set of instances of

E. An element

is a vector of n values, where

the component i is denoted by

()

dome Α∈

, which

verifies that everywhere instance

and

(

)

()

(

)

≠Π

()

(

)

where

()

ΕΠ

is the

projection of A in E.

We define the set of key instances associated to

an entity as

(

)

Ε=Ε

ΙΡ

. A relationship of order n

with s attributes is defined as

(

)

sjn

r ΑΑΕΕ ,...,,,...,r

111

, where each

is the role

and, where the entity

participates in R. We

define

as the set of instances of R. An

element

of this set is a vector of n components,

where each component depicts a role that contains a

key instance of the entity that it takes part with this

role. Then, the set of instances

of a relationship R

is a subset of the product contained of the key

instances of the entities that participate in R and

domains of attributes that participate in R.

() ()

sjn

domdomErErR Α××Α×××⊆ ......

111

We define the participation cardinality

constraint of an entity

(

)

RrC

,Ε

= {0 or 1} as the

optional or mandatory participation, respectively, of

the key instances

with role

in the relationship

(

)

rrR ΕΕ= ,...,

where the relationship has order n.

Optional constraint is depicted with a white circle,

and the mandatory with a black circle, but both by

the side of the relationship.

The Merise’s cardinality for a relationship is

defined as CMerise

),( REr

= (n, m) where

is the

role in

(

)

rrR ΕΕ= ,...,

≤

, and n, m

∈

. This

means that a key instance of

REr ∈

is in

minimum and maximum n, m times. We depict this

cardinality with a label at the end of the line that

ICEIS 2007 - International Conference on Enterprise Information Systems

466

links the entity and the relationship but by the side

of the relationship.

The Chen’s cardinality (Cuadra, 2003) for a

role

into

(

)

rrR ΕΕ= ,...,

as CChen

(

)

,Ε

= (n, m),

where n, m

∈ and

≤

≤1

. This means that for

any combinations of key instances

nii

Rdeaaaa ,...,,,...,

111 +−

is in

, n and m times as

minimum and maximum, respectively. We depict

this cardinality in a label at the line that links the

entity and the relationship but by the side of the

entity.

Let be R=

(

)

ErEr ,...,

a relationship of grade

n. We define a Complementary Relationship of R

(Cuadra, 2003) as

, where a shows the roles in

which participates this relationship and it has to

carry out:

 a<n, that is to say, the number of roles which

are applied, it has to be less than the grade of

the relationship.

 Let C(aE, R)=0 be the cardinality constraint

for every entity, which participates with every

one of the roles in E has to be optional.



⊄

, that is to say, the instances which

belong to the relationship, they can not be a

subset of the complete relationship.

Let be R=

(

)

ErEr ,...,

a relationship of order

n. We define a Complete Relationship of R as

with the same definition of the Complementary but

it does not carry out the third point.

We use some definitions from the work of

Trevor H. Jones and Il Yeol Song (Trevor H. Jones

et al., 1996). If a binary relationship is semantically

a subset of the ternary relationship and constraints

the instances of the ternary relationship, then the

binary relationship is a Semantically Constraining

Binary (SBC) relationship. If not the binary

relationships is Semantically Unrelated Binary

(SUB) relationship. We do not analyze the SUB

because it has not an effect on the ternary. They

define the Implicit Binary Cardinality (IBC) rule as

in any given ternary relationship, regardless of

ternary cardinality, the implicit cardinalities between

any two entities must be considered M:N, provided

that there are no explicit restrictions on the number

of instances than can occur. They define the Explicit

Binary Permission (EBP) rule for any given ternary

relationship, a binary relationship cannot be imposed

where the binary cardinality is less than the

cardinality specified by the ternary, for any specific

entity. In addition, the Implicit Binary Override

(IBO) rule is given the imposition of a permitted

binary relationship on a ternary relationship, the

cardinality of the binary relationship determines the

final binary cardinality between the two entities

involved.

McAllister, A. (1997 and 1998) defines the

MX2 rule, denoted as augmentation rule. He defines

r as the total set of roles for a relationship R. The

cardinality constraint Cmax(a, b) means that if we

fix a role [a] is the number maximum [a, b]

permitted in R. The augmentation rule (MX2)

defines that Cmax (a, b)

≤

Cmax (ac, b). The MX2

rule is equivalent to the Explicit Binary Permission

(EBP) rule.

On the other hand, we use some definitions of

the Relational Model. They are definitions of Millist,

W. V. (1994). Let t be a tuple of a relation R. Let

be a tuple to insert, update or delete. Let

∑

be the

set of key dependencies. The set of all relations that

they satisfy

∑

is denoted as SAT (

∑

Let R be a relation,

∑

a set of dependencies,

which apply to R and, r(R) a relation. A tuple

said to be compatible with r if

}{

tr ∪

is a relation

which is in SAT (

∑

3 RESEARCH ABOUT THE

VALIDATION OF TERNARY

RELATIONSHIPS

In this section, we analyze different proposals about

the validation and verification of relationships.

Trevor H J., et al. (1996) focus on cardinality

constraints associated to ternary relationships. They

analyze the SBC relationships and as they can affect

to cardinality constraints in the ternary relationship.

They analyze the implicit and explicit binary

relationships and demonstrate their theories through

the functional dependencies. The proposal does not

depict the syntactic or semantic anomalies, they only

study the semantic associated to ternary relationship

through binary relationships. The paper of James

Dullea (Dullea, J. et al., 1998) depicts when an E/R

diagram has not redundancy. They do an analysis

about the path (cycle path), which can be right or

wrong. They analyze the optionality and they study

its cardinality. However, they do not look at the E/R

model anomalies.

The papers of McAllister, A. (1997 and 1998)

describe an analysis about the minimum and

maximum cardinality constraints in the

relationships. He establishes rules for deducing

cardinalities in the schema. If we apply these rules is

ON THE SEMI-AUTOMATIC VALIDATION AND DECOMPOSITION OF TERNARY RELATIONSHIPS WITH

OPTIONAL ELEMENTS

467

possible to get a simplification or decomposition of

the original schema. However, these works do not

explain the problem from the UD. His work shows

the minimum cardinality, 0, but he does not resolve

the semantic problem of the optionality.

Rafael Camps (Camps, R. 2002) depicts an

excellent analysis about the transformation of

ternary relationships, with and without imposition

binaries from E/R to R Model. In his work he

establishes that the “Look across” cardinality

constrains with the Chen approximation is richer

semantically. We think also it. Furthermore, he

shows the problem that the transformation from E/R

to R using only functional dependencies has

semantic anomalies.

In the R Model Millest W. Vicent (Millist W.

Vincent, 1993, 1994, 1999) describes the semantic

anomalies that have the relationships.

Santos (Santos, I. et al., 2006) depicts the semi-

automatic validation and decomposition of ternary

relationships, however this work does not analyze

the optionality.

4 VERIFICATION AND

VALIDATION OF TERNARY

RELATIONSHIPS WITH

OPTIONALITY

We use the representation of “Look across”

cardinality constraints of Chen and Merise

approximations (Cuadra, 2003), because the

depicted semantic is very good for the automation in

a CASE tool. The Chen approximation can be use

for deriving the functional dependencies. We use the

MX2 rule of McAllister for validating the

Conceptual schema. On the other hand, the “Look

across” cardinality constraint with Merise

approximation shows us the primary key and, the

candidate keys, if they exist. Furthermore, with this

approximation we can get complex rules, because

the value of an attribute in a relationship for a

domain as minimum has to be n and as maximum m

times,

∈∀ mn,

(Al-Jumaily, T. H., 2006).

When there are optional elements in a ternary

relationship, we have problems in its transformation.

A solution is the Complementary binary relationship

(Cuadra, D., 2003). In this work, we propose also

the Complete binary relationship. Both solutions

were defined in the second section. The

Complementary relationship is a good solution,

because it has not redundancy. However, in the

Complete, there is redundancy, but it will be good

solution when there is decomposition.

Next, we show two algorithms of validation and

simplification of ternary relationships. Theses

algorithms are a modification of Santos, I, (Santos, I.

et al., 2006). We begin by checking the schema

semantic consistency. In a next step, we have to

verify if the concepts are according to the definition

and, there are not incompatibilities among the

concepts and the schema.

In this paper, we analyze only the ternary or

higher-grade relationship. For this when we find a

ternary relationship in our model,

, we have to

look for the SBC relationship with

. For each

entity

related to

, we have to find other

relationships

, with

≠

, and the rest of entities

{

}

nii

+−

,...,,,...,

111

which are related to

These relationships that we find, they are candidates

to be semantic related relationship to

, and for this,

they can restrict the cardinality of the

relationship

. When we have the relationships, we

have to ask to the designer, because he/she has to

decide the relationships, which are SBC.

The step next is to check the optional roles of

the entities in the ternary relationship. Let be

and

∈

. If

has an optional role, then we

build between

and

the Complementary binary

relationship. However, if between

and

there is

an implicit ternary relationship, then we can build

the Complete binary relationship and we delete the

implicit binary relationship.

Now we show the algorithm of validation of a

relationship with optionality.

The first algorithm depicted in the figure two

has the follow steps:

1. We get the ternary relationship and SBC

relationships with the ternary to check,

with the help of the designer.

2. Are there some optional elements in the

ternary relationship?

3. If there are optional roles then we build

the Complementary or Complete binary

relationships.

4. We verify the Conceptual Design with

the MX2 rule of McAllister. Do the

relationships carry out MX2? We have

to verify the rule

(

)

2/123

+−

−nn

times.

ICEIS 2007 - International Conference on Enterprise Information Systems

468

Figure 2 : Validation of Relationship with optionality.

Figure 3.- SIMPL IF ICATION ALGORITHM O F A RELATIONSHIP

Begin

1. -W e get the r el ati onship

Of t he al gorit hm one.

2 .-S ynthesi s or A nal ys i s

al gorithm

3.- Is i t in BC NF? And Has not

Information l osses and preserv e

F uncti onal d ependenci es?

End

4. - Decom position wit h semantic

anomal ies

6. - T he decompo si ti on has

I nforma tion is lo ss es

7. - The de com po si tion is we ll,

wit hout s emantic ano ma lies.

Yes

5. - bel o ng s to

Relationships of the

dec o mp osi t io n?

Ye s

RorR

Figure 3 : Simplification Algorithm of a Relationship.

5. Design is incorrect. The E/R Model has

to be redesigned.

6. We transform the design from E/R to R

Model getting the functional

dependencies. We can get the FD from

“Look Across” cardinality constrains

with the approximation of Chen. We

have a FD if maximum cardinality is

one. After, with the classic algorithms

we verify the normal form of the

schema.

7. Is the schema in the Boyce Codd

Normal Form?

8. If the schema is in BCNF, the design is

good and it has not semantic anomalies.

9. Is it in 3NF?

10. Design good, but with semantic

anomalies. We could use the second

algorithm.

11. Design with semantic anomalies. We

ought to use the second algorithm.

If a ternary relationship with optionality can be

decompose and the decomposition or simplification

is in BCNF, this decomposition will always have

information losses if the Complementary (

) or

the Complete (

) relationship are not in the

decomposition.

Theorem: Let be relation schema R and let be

a Complete binary relationship and let be

the

Complementary.

The decomposition is in BCNF.

Let

RRR ,...,,

be a set of relationships of the

decomposition of R. The

RRR ,...,,

R∈ is

information lossless and functional lossless.

RR ≠

, where

Ni ∈

, then the

decomposition is information loss.

Proof: If

RR ≠

will not be the original

tuples, because

RRRR ><><>< ,...,

The algorithm implemented in the figure three

shows the steps of simplification of a ternary

relationship with or without optional elements.

1. We get the relations from algorithm

first.

2. We apply the Analysis or Synthesis

classic algorithms.

3. Is the result in BCNF? Does not it

exists information losses and preserve

functional dependencies?

4. The decomposition has semantic

anomalies. It is not good solution.

5. Are there Complete relationships?

These relationships belong to

decomposition?

6. The decomposition has information

losses.

7. The decomposition is valid.

We prefer the Complete relationship to the

Complementary, only in this case, because if we use

Complementary then we have to do an union

between the Complementary and its

corresponding

Fi gure 2.- VAL IDATI ON OF A R EL ATI ONSHI P W HITH OPTI ONALI TY

Begi n

1.-We get the rel ationships

2.-Are there optional

elements?

3.-We c reat e Co m pl eme n ta r y

or Complete relationship.

4.- MX2?

6.-We get the Normal From.

7.- Is i t i n BCNF?

8.-The design is val id.

End

5.-The relationship is not

val i dat e d

9. Is it 3 NF?

10.-Design val i d but

Semanti c anomal ies

11.- Desig n is not

val i d

Yes

ON THE SEMI-AUTOMATIC VALIDATION AND DECOMPOSITION OF TERNARY RELATIONSHIPS WITH

OPTIONAL ELEMENTS

469

5 ANOMALIES IN THE

COMPLEMENTARY

RELATIONSHIPS

When there is a ternary relationship, with or without

imposition binaries but with at least a

Complementary binary relationship, the insertion,

deletion, updated and selection operations have

anomalies. We will analyze in this section these

problems. For this analysis we have use the

definitions of Millis W. Vincent (Millist W. Vincent,

1993, 1994, 1999).

Let be three entities E, P and T, a relationship R

with attributes e, p, t and a Complementary

relationship ought to T is optional. In the insertion,

we will have to distinguish when we insert a null or

not.

1. Let

=<e,p,null> be a tuple to insert

and let r(R) the ternary relationship

and let

(

)

be the Complementary

relationship. If

{

}

(

)

(

)

Σ∈∪ SATtr

and

{

}

(

)

()

Σ∈∪ SATtr

c *

then we

can insert

>=< pet

2. Let be t=<e,p,t>,

nullt ≠

and let r(R)

be the ternary relationship and

(

)

the Complementary. If

{

}

(

)

∈∪

()

ΣSAT

and

{

}

(

)

∈∪

(

)

SAT

then we can insert

in R.

The delete operations have two cases.

1. Let

=<e,p,t>, where t=null. If

{

}

(

)

(

)

Σ∈− SATtr

c *

, then we can delete

(

)

2. Let

=<e,p,t>, where

nullt ≠

. If

{

}

(

)

(

)

Σ∈− SATtr

, then we can

delete

of R.

However, the modification operations have four

cases.

1. Let be t=<e,p,t> and

=<e’, p’, t’>,

where t, t’

null≠

. If

{

}

(

)

(

)

∈−∪ trt

(

)

ΣSAT

, then the modification is

right and we update only the ternary.

2. Let be t=<e,p,t> and

=<e’, p’, t’>,

where t and t’= null. If

−∪

rt (}({

))

()

∈ SAT

then the modification

is possible, but in the Complementary.

3. Let be t=<e,p,t> and

=<e’,p’,t’>,

where t=null and t’

null≠

. If

{

}

(

)

()

Σ∈− SATtr

c *

and

{

}

(

)

∈∪

(

)

SAT

, then we can delete

of the

Complementary and to insert

in the

ternary.

4. Let be t=<e,p,t> and

=<e’,p’,t’>,

where t

null

≠

and t’=null. If

{

}

(

)

()

Σ∈− SATtr

and

{

}

(

)

∈∪

(

)

SAT

, then we can delete t of

ternary and to insert

in the

Complementary.

The operation of selection is more complex:

Select e, p, t from ternary-relationship

Union

Select e, p, null from Complementary;

When there is decomposition and a binary

relationship is overlapped by the Complementary

relationship is better to replace the relationships by

the Complete relationship.

Figure 4 : Case (i).

6 VALIDATION AND REFINING

OF TERNARY RELATIONSHIP

AND ITS DECOMPOSITION

WITH OPTIONALITY

Let be a company that wants to manger the jobs and

employees. The management has imposed the next

constraints:

• An employee that works in a project, he can

only use a technique.

• An employee that works at a technique, he

can only work at a project.

• An employee can only work at a unique

project.

Figure 4, . Case (i)

Tec hni q ue

Pr oj ec t

Pr oj ec t-

Tec hni q ue

Job

Empl o yee-

-Project

Empl o ye e

Empl o yee-

- Tec hni q ue

(1,1)

(1,1) (1,1)

(1,1)

(1,n

(1,n)

(0,n)(1,n)

(0,n)

(1,n)

(0,n)

(1,1)

(0,n)

(1,n)

ICEIS 2007 - International Conference on Enterprise Information Systems

470

• In a project is only possible use a

technique.

• The last constraint, we distribute in two

exclusive cases in our example:

(i) It can have employees with projects

that they do not have allocated

technique.

(ii) It can have employees that use a

technique, but they are not allocated to

any project.

Figure 5,. Case ( ii)

Te c h n iq u e

Pr oj ec t

Project-

Technique

Job

Employee-

-Project

Employee

Employee-

-Technique

(0,n)

(0,1)

(1,1)

(1,n)

(1,1)

(0,n)

(1,n)

(1,n)(0,n)

(1,n)

(0,n)

Figure 5 : Case (ii).

We show in the figure fourth and fifth the E/R

model of this example. In the figure sixth (case (i))

and seventh (case (ii)) is the solution to optional

elements, with the Complementary binary

relationship. From the figure forth, we depicts the

“Employee-Technique” relationship with double

rhombus, because it is a deduced relationship.

Figure 6 : We transform the optionally case (i).

Through the algorithm one, we verify the

relationships and we look at the redundancies.

We can notice that in both cases that carry out

the augmentation rule (MX2 rule).

If we get the functional dependencies, in both

cases we have:

= {(Employee, Project

→

Technique; (Employee, Technique)

→

Project;

Employee

→

Project; Project

→

Technique}. The

Key is

={Employee} and the minimum cover

= {Employee

→

Project, Project

→

Technique}. The

resulting schema is in the 2NF.

If we apply the second algorithm, then we will

get two relations;

={Employee, Project} with

= {Employee

→

Project} and

={Project,

Technique} with

={Project

→

Technique}.

If we go on the case (i), we replace

={Employee, Project} and the Complementary

relationship by the Complete relationship. In the

figure eight depicts this case.

However, in the case (ii), this is not right,

because, we lose data. The Complementary is not in

the decomposed relationships. Furthermore, we can

not select

={Employee, Project} with

{Employee

→

Project} and

= { Employee,

Technique} with

= { Employee

→

Technique}

because we have dependency loss in this

decomposition. Then, the decomposition is not valid,

as we depict in the figure nine.

Figure 7 : We transform the optionally Case(ii).

We can resume that the decomposition or

simplification of the relationships, although this is in

Figure 8 : Simplification of Case (i).

FNBC, if in the decomposition are not

complementary relationship, the decomposition is

not valid.

Fi gure 6,. We transfor m the optional l y case (i)

Tec hni q ue

Pr oj ect

Pr oj ect -

- Tec hni q ue

Job

Empl oyee

Empl oyee-

- Tec hni q ue

(1,1)

(1,n)

(0,n)

(1,n)

(1,1)

(1,n)

Emp l o yee-

-Project

Compl em ent ar y

Comple te

(1,n)

(1,1)

(1,n)

Figur e7, . We trans for m the o pti onal l y of Case (i i )

Tec hniqueProject

Project-

Technique

Job

Employee-

-Project

Employee

Employee-

-Tec hnique

(1,1)

(1,n)

(1,n)(1,n)

(1,n)

(1,n)(1,n)

(1,n)

Compl ementary

(1,n)

(1,1)

Figure 8, . Simplification of l Case (i)

Empl oyee

Pr oj ect

(1,1)

(1,n)

(1,1)

(1,n)

Compl et e

Pr oj ect -

Tec hni q ue

ON THE SEMI-AUTOMATIC VALIDATION AND DECOMPOSITION OF TERNARY RELATIONSHIPS WITH

OPTIONAL ELEMENTS

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7 SOME CONCLUSIONS

The design of a Database is a complex work. The

CASE tools help to simplify validation, verification

and simplification of a Database design. However,

these tools do not implement theses properties or

they are very far away.

This paper is based on the ternary relationships,

but it ought to be extended to higher grade ones.

Two algorithms for verifying, validating and

decomposing relationships with or without optional

elements have been shown. However, this work is

limited to functional dependencies and not to

multivalued (MVDs) or join (JDs) dependencies. On

Figure 9 : We transform the optionally case(i).

the other hand, we can conclude that sometimes

the simplification is not the better solution, because

of the anomalies.

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Figure 9, . Decomposit ion Case ( ii)

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Compl ement ar y

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