1. CSC 411/511: DBMS Design Dr. Nan Wang 1 Schema Refinement and Normal Forms Chapter 19

2. Dr. Nan Wang The Evils of Redundancy Redundancy causes several problems associated with relational schemas: – Redundant storage – Insert anomalies – Delete anomalies – Update anomalies – What are the problems in the following schema? Hourly_Emps (ssn, name, lot, rating, hourly_wages, hours_worked)

3. Dr. Nan Wang Example Hourly_Emps (ssn, name, lot, rating, hourly_wages, hours_worked) Abbreviate an attribute name to a single letter Refer to a relation schema by a string of letters, one per attribute. In the example, hourly_wages is determined by rating. This IC is an example of a functional dependency (R  W). What does it lead to?

4. Dr. Nan Wang Example Problems: – Redundant storage – Update anomaly: Can we change W in just the 1st tuple of SNLRWH? – Solutions to the anomalies?

5. Dr. Nan Wang Example Problems: – Insertion anomaly: What if we want to insert an employee and don’t know the hourly wage for his rating? – Solution? Null values What if the s is unknow?

6. Dr. Nan Wang Example Problems: – Deletion anomaly: If we delete all employees with rating 5, we lose the information about the wage for rating 5! – Solutions?

7. Dr. Nan Wang Fighting the Evils of Redundancy Redundancy is at the root of evil - true or not? Integrity constraints, in particular functional dependencies, can be used to identify schemas with such problems and to suggest refinements. R  W Decomposition(replace SNLRWH with SNLRH and RW) Hourly_Emps2 Wages

8. Dr. Nan Wang Fighting the Evils of Redundancy R  W Decomposition(replace SNLRWH with SNLRH and RW) Do we solve all this problems? – Redundant storage – Insert anomalies – Delete anomalies – Update anomalies Hourly_Emps2 Wages

9. Dr. Nan Wang Fighting the Evils of Redundancy Main refinement technique: decomposition (replacing ABCD with, say, AB and BCD, or ACD and ABD). Problems related to decomposition: – Is there reason to decompose a relation? Normal forms help us to decide whether or not to decompose a relation in further. How? – What problems (if any) does the decomposition cause? Lossless join decomposition Dependency-preservation decomposition

10. Dr. Nan Wang Properties of decomposition Lossless join decomposition – Enable us to recover any instance of the decomposed relation from corresponding instances of the smaller relations. Dependency-preservation decomposition – Enable us to enforce any constraints on the original relation by simple enforcing some constraints on each of smaller relations. – Key constrains Hourly_Emps2 Wages

11. Dr. Nan Wang Good? 11 – List N, W, H – How?

12. Dr. Nan Wang Good DB designer Normal forms and what problems they do or do not alleviate? Decomposition and what problems with decompositions. Questions: –Is a relation in a given normal form? –Is a decomposition dependency preserving? 12

13. Dr. Nan Wang Functional Dependencies (FDs) A functional dependency X Y holds over relation R if, for every allowable instance r of R: – t1 r, t2 r, t1.X =t2.X implies t1.Y =t2.Y – i.e., given two tuples in r, if the X values agree, then the Y values must also agree. (X and Y are sets of attributes.)

14. Dr. Nan Wang Functional Dependencies (FDs) 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! K is a candidate key for R means that K  R – However, K  R does not require K to be minimal!

15. Dr. Nan Wang FDs- example 15 FD: AB  C Question: Do all tuples in the relation satisfy the Functional Dependents AB  C Can we add a tuple to the relation? Why?

16. Dr. Nan Wang exercise List all the functional dependencies that this relation instance satisfies. Z → Y, X → Y, and XZ → Y 16

17. Dr. Nan Wang Example: Constraints on Entity Set Consider relation obtained from Hourly_Emps: – 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: S  SNLRWH – rating determines hrly_wages: R  W

18. Dr. Nan Wang Example (Contd.) Problems due to R  W : – Update anomaly: Can we change W in just the 1st tuple of SNLRWH? – Insertion anomaly: What if we want to insert an employee and don’t know the hourly wage for his rating? – Deletion anomaly: If we delete all employees with rating 5, we lose the information about the wage for rating 5! Hourly_Emps2 Wages Will 2 smaller tables be better?

19. Dr. Nan Wang Reasoning About FDs Given some FDs, we can usually infer additional FDs: –For example: –Workers(ssn,name,lot,did,since) – Ssn  did holds, – did  lot holds, implies ssn  lot An FD f is implied by a set of FDs F if f holds whenever all FDs in F hold. –F + = closure of F is the set of all FDs that are implied by a given set F of FDs.

20. Dr. Nan Wang Closure of a set of FDs Infer or compute the closure of a given set of F of FDs –Following three rules (Armstrong’s Axioms) Armstrong’s Axioms (X, Y, Z are sets of attributes): – Reflexivity: If X Y, then Y  X – Augmentation: If X  Y, then XZ  YZ for any Z – Transitivity: If X  Y and Y  Z, then X  Z These are sound and complete inference rules for FDs!

21. Dr. Nan Wang Soundness and Completeness Completeness: given F, the rules allows us to determine all dependencies in Soundness: we cannot generate FDs not in Addition rules: –Union: if X  Y, X  Z then X  YZ –Decomposition: if X  YZ, then X  Y and X  Z

22. Dr. Nan Wang Soundness and Completeness R=(A,B,C) and F={A  B, B  C} –Reflexivity: if Y is subset of X, then X  Y –Transitivity : if X  Y and Y  Z then X  Z A  C –Augmentation : if X  Y then XZ  YZ for any Z AC  BC AB  BC AB  AC –A  BC F + ={A  B, B  C, A  C, AC  BC,AB  AC. …}

23. Dr. Nan Wang Reasoning About FDs FD is a statement about the world as we understand it – Ssn  did, did  lot implies ssn  lot Is it correct to state that “Since X functionally determines Y, if we know X, we know Y.” Or “if X  Y, then X identifies Y”

24. Dr. Nan Wang Reasoning About FDs (Contd.) Couple of additional rules – Union: If X  Y and X  Z, then X  YZ – Decomposition: If X  YZ, then X  Y and X  Z Example: Contracts(cid,sid,jid,did,pid,qty,value), and: – C is the key: C  CSJDPQV – Project purchases each part using single contract: JP  C – Dept purchases at most one part from a supplier: SD  P

25. Dr. Nan Wang Reasoning About FDs (Contd.) C is the key: C  CSJDPQV Project purchases each part using single contract: JP  C Dept purchases at most one part from a supplier: SD  P JP  C, C  CSJDPQV imply JP  CSJDPQV SD  P implies SDJ  JP SDJ  JP, JP  CSJDPQV imply SDJ  CSJDPQV

26. Dr. Nan Wang Exercise Consider a relation R with five attributes ABCDE. You are given the following dependencies: A→B, BC→E, and ED→A. List all keys for R. CDE, ACD, BCD 26