1 CS 430 Database Theory Winter 2005 Lecture 4: Relational Model

2 Domains A Domain is a set of atomic values  Atomic: the values are indivisible as far as the relational model is concerned  Easy examples: Numbers Character strings Dates  Questionable example: A set of Character Strings  Allowed by some theoreticians  Not allowed by many RDBMSs  Assumption: Domains come with appropriate operations for each domain

3 Relation Schema Denoted by: R(A 1, A 2, …, A n )  R is the name of the Relation  A 1, A 2, …, A n is the list of attributes of R Attributes are names  Each attribute has an associated Domain: dom(A i )  n is the Degree or Arity of the Relation Example:  PROJECT (PName, PNumber, PLocation, DNum) Domains: Appropriate strings and numbers

4 Tuples Relation or Relation State  Date: Relation Variable  r(R) = {t 1, t 2, …, t m } May use just r if the context is clear A mathematical set  Each t = An (ordered) tuple of values Each v i is either a member of dom(A i ) or the special null value Mathematically  r is a subset of the cartesian product of the dom(A i )

5 Relations The term Relation State is used to emphasize the potential time varying nature of the relation Characteristics  There is no order for the tuples of relation (the t i )  The ordering of values in a tuple is only important to associate each value with the corresponding domain It is possible to define relations as a mapping of attributes to values, thereby removing the order However, in SQL there are cases where the order is important  The null value is provided for representing unknown or not applicable data

6 Relations (Continued) A Relation can be viewed as a set of facts or assertions in a logic programming language  “Closed World” assumption: all true (at the current time) facts are in the database  “Open World” assumption: opposite of “Closed World”  Most database applications make the “closed world” assumption Notation  t[A i ] or t.A i or t[i] can be used to refer an attribute value  t[A u, A v, … A z ] or t.(A u, A v, … A z ) can be used to refer to the corresponding sub-tuple of t

7 Database A Relational Database Schema S is a set of Relations and a set of Integrity Constraints  S = ({R 1, R 2, …, R m }, IC)  Integrity Constraints are a collection of assertions or predicates that should be satisfied by the database A Database State DB is  A set of Relation States DB = {r 1 (R 1 ), r 2 (R 2 ), …, r m (R m )}  which satisfy the Integrity Constraints

8 Example Relational Schema  See Figure 5.5 of Text Book Relational State  See Figure 5.6 of Text Book

9 Constraints Internal  Ones enforced by the DBMS External  Ones true in the users view of the world  Some may be enforced by Application software

10 Types of Internal Constraints Inherent Model Based Constraints  Ones based on relational model, e.g. no repeated values Schema-Based Constraints  Domain Constraints Attributes must be in the corresponding domain  Key Constraints and Constraints on Null Values  Entity Integrity, Referential Integrity, and Foreign Keys  General Constraints and Triggers

11 General Constraints and Triggers Some DBMSs provide mechanisms for General Constraints and Triggers  General constraint: an arbitrary assertion attached to a Table (Relation) or Schema  Trigger: procedural code that is executed when certain kinds of events (e.g. updating the database) occur These can be used to enforce arbitrary constraints Example:  No employee makes more than their supervisor

12 Key Constraints Super Key  For a given relation R, a subset of the attributes {A u, A v, … A z } such that t[A u, A v, … A z ] uniquely determines t (Candidate) Key  A super key which is minimal or irreducible in the sense that no smaller subset of the attributes is also a Super Key Primary and Alternate Keys  Primary Key - some chosen (distinguished) Candidate Key  Alternate Key - any non-primary Candidate Key

13 Entity Integrity  All attributes of a primary key are non-null  Good practice: All attributes of an candidate key are non-null Question: If nulls are allowed how is it a candidate key?

14 Foreign Keys Foreign Key or Referential Integrity Constraint  Given relations R 1 and R 2, a subset FK of the attributes of R 1, FK = {A u, A v, … A z } is a foreign key referencing R 2 which has primary PK if given t 1 a tuple in R 1 either 1. t 1 [FK] is null, or 2. there exists a tuple t 2 in R 2 with t 1 [FK] = t 2 [PK] Notes 1. R 1 and R 2 can be the same relation 2. Circular foreign key dependencies are allowed Example:  See figure 5.7 in Text Book

15 Database Operations and Constraint Violations Three basic kinds of update operations in the relational model:  Insert  Update  Delete What happens if an operation causes a constraint violation Especially violation of a referential integrity constraint

16 Possible Remedies No Action  Don’t do anything, accept the action as is This may cause problems “down the road” Restrict  Refuse the action Cascade  Cascade the change to other tuples to correct violation Set Null and Set Default  Set offending attributes to null or a default value

17 Insert Violations Domain constraint  Insert value not in domain Key constraint  New tuple with same key Entity integrity  Primary key attribute is null Foreign Key  Foreign key refers to tuple that doesn’t exist Generally Restrict (i.e. Reject) such an Insert

18 Delete Violations Foreign Key  Delete primary key referred to by foreign key  Remedies: No Action: Accept operation  May result in subsequent integrity violations Restrict: Reject operation Cascade: delete referring records Set Null or Set Default: Modify referencing keys to some other value

19 Update Violations Domain, Key, Entity  Same as insert - usually Rejected Foreign Key  Modify primary key referred to by foreign key No action, Restrict, Set Null or Set Default: same as insert Cascade: modify referencing keys to new value