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**primary key constraint foreign key constraint**

miniworld Requirements & collection analysis refinement Database Requirements Conceptual Design Conceptual Schema ( ER diagram ) DBMS independent Data Model Mapping DBMS specific Conceptual Schema ( Relations ) primary key constraint foreign key constraint

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**Schema Refinement and Normal Forms**

Conceptual database design gives us a set of relation schemas and integrity constraints Given a design, how do we know it is good or not? A design can be evaluated from various perspectives, our focus is on data redundancy Conceptual design Schemas ICs

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**The Evils of Redundancy**

Redundancy is at the root of several problems associated with relational schemas: redundant storage Insertion/update/deletion anomalies

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**Example Schema Constraints:**

Hourly_Emps (ssn, name, lot, rating, hrly_wages, hrs_worked) Constraints: ssn is the primary key If two tuples have the same value on rating, they have the same value on hrly_wages

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**Solution: Decomposition**

If we break Hourly_Emps into Hourly_Emps2 and Wages, then we don’t have updates, insertion, deletion anomalies. Wages Hourly_Emps2

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**Decomposition Concerns**

Should a relation be decomposed? If a relation is not in certain form, some problems (e.g., redundancy) will arise, are these problems tolerable? Aforementioned anomalies Potential performance loss: Queries over the original relation may required to join the decomposed relations How to decompose a relation? Two properties must be preserved: lossless-join: the data in the original relation can be recovered from the smaller relations dependency-preservation: all constraints on the original relation must still hold by enforcing some constraints on each of the small relations

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**Functional Dependencies (FDs)**

In a relation schema R, a set of attributes X functionally determines a set of attributes Y if and only if whenever two tuples of R agree on X value, they must necessarily agree on the Y value. XY where r(R) is an instance of R, XY: Y is functionally dependent on X, or X uniquely determines Y or X functionally determines Y, or X determines Y

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X Y Z X1 Y2 Z1 Z2 X2 Z3 Does this data set violate X->Y? Does this data set violate Z->Y? X Y Z X1 Y1 Z1 Z2 Y2 Does this data set violate X->Y? Does this data set violate XY->Z? Does this data set violate Z->X? No, No, Yes, Yes, No

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**An FD is a statement about all allowable relations.**

Must be identified based on semantics of application. Given some allowable instance r1 of R, we can check if it violates some FD f, but we cannot tell if f holds over R! A primary key constraint is a special case of an FD The attributes in the key play the role of X, and the set of all attributes in the relation plays the role of Y K is a candidate key for R means that KR. However, KR does not require K to be minimal!

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**Example 1 Hourly_Emps (ssn, name, lot, rating, hrly_wages, hrs_worked)**

Notation: We will denote this relation schema by listing the attributes: SNLRWH This is really the set of attributes {S,N,L,R,W,H}. Sometimes, we will refer to all attributes of a relation by using the relation name. (e.g., Hourly_Emps for SNLRWH) Some FDs on Hourly_Emps: ssn is the key: SSNLRWH (or {S}{S,N,L,R,W,H}) rating determines hrly_wages : RW

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**Example 2 FD: did->lot Works_for(ssn,name,did,since)**

dname lot name ssn did budget Works_for Employees Departments Additional Constraints: Employees are assigned parking lots based on their department. All employees in the same department is given the same lot. FD: did->lot Works_for(ssn,name,did,since) Department (did,dname,budget,lot);

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Dependency Reasoning A set of dependencies may imply some additional dependencies. EMP_DEPT(ENAME,SSN,BDATE,ADDRESS,DNUMBER,DNAME,DMGRSSN) F={SSN->{ENAME,BDATE,ADDRESS,DNUMBER}, DNUMBER->{DNAME,DMGRSSN} } F infers the following additional functional dependencies: F {SSN}->{DNAME,DMGRSSN} F {SSN}->{SSN} F {DNUMBER}->{DNAME}

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**Some important questions**

Dependency Reasoning A set of dependencies may imply some additional dependencies. Some important questions Given a set of attributes X, what attributes can be determined by X Given an FD set, what other dependencies are implied Given an FD set F, what is the minimum set of dependencies that is equivalent to F

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Armstrong’s Axioms Armstrong’s Axioms where X, Y, Z are sets of attributes: Reflexivity: If X Y, then XY. Augmentation: If XY, then XZ YZ for any Z. Transitivity: If X Y and YZ, then XZ.

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**PROOFS Reflexive rule: If X Y, then XY. Let {t1,t2}**

r(R) such that t1[X]=t2[X] Since Y X, t1[X]=t2[X] t1[Y]=t2[Y] XY.

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**PROOFS (Cont’d) Transitive rule: If XY and YZ, then XZ.**

Let XY and (1) YZ (2) such that t1[X]=t2[X], (3) we have: (1) t1[Y]=t2[Y] (4) (2)&(4) t1[Z]=t2[Z] (5) (3)&(5) XZ

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**PROOFS (Cont’d) Augmentation rule: If XY, then XZYZ.**

Assume that the Augmentation rule is not true. t1[X] = t2[X] (1) t1[Y] = t2[Y] (2) t1[XZ] = t2[XZ] (3) t1[YZ] != t2[YZ] (4) (1)&(3) t1[Z]=t2[Z] (5) (2)&(5) t1[YZ]=t2[YZ] (6) (6) Contradicts (4)

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**Additional Inference Rules for Functional Dependencies**

Union: If X Y and X Z, then X YZ. Decomposition: If XYZ, then XY and XZ. Pseudotransitive Rule: If XY and WYZ then WXZ.

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**PROOFS (Cont’d) Union rule: If XY and XZ, then XYZ.**

Given XY and (1) XZ (2) Applying Augmentation rule on (1), we have XXXY XXY (3) Applying Augmentation rule on (2), we have XYZY XYYZ (4) Applying Transitive rule on (3) and (4), we have XYZ.

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**PROOFS (Cont’d) Decomposition rule: If XYZ then XY and XZ.**

Given XYZ (1) Since Y YZ, reflexive rule gives YZY (2) Applying Transitive rule on (1) and (2), we have XY. XZ is derived in a similar way.

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**PROOFS (Cont’d) Pseudotransitive rule: If XY and WYZ, then WXZ.**

Given XY (1) and WYZ (2) Applying Augmentation rule on (1), we have WXWY (3) Applying Transitive rule on (3)&(2), we have WXZ.

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**Exercise Prove or disprove the following inference rules**

{WY,XZ} |= {WXY} {XY,XW,WYZ} |= {XZ} {XY} |= {XYZ} {XY, Z Y} |= {XZY} Prove using inference rules Disprove by showing a counter example

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