1. Relational Model & Algebra
Zachary G. Ives
University of Pennsylvania
CIS 550 – Database & Information Systems
September 21, 2018
Some slide content courtesy of Susan Davidson & Raghu Ramakrishnan

2. Recall Our Initial Discussion…
There are a variety of ways of representing data, each with trade-offs
Free text [often need a human]
Shapes/points in space …
“Objects” with “properties”
In general, our emphasis will be on the last item
… though there are spatial databases, OO databases, text databases, and the like…

3. The Relational Data Model (1970)
Lessons from the Codd paper
Let’s separate physical implementation from logical
Model the data independently from how it will be used (accessed, printed, etc.)
Describe the data minimally and mathematically
A relation describes an association between data items – tuples with attributes
We generally think of tables and rows, but that’s somewhat imprecise
Use standard mathematical (logical) operations over the data – these are the relational algebra or relational calculus
How does this model relate to objects, properties? What are its abilities and limitations?

4. Why Did It Take So Many Years to Implement Relational Databases?
Codd’s original work:
Earliest relational database research: ~1976
Oracle “2.0”: 1979
Why the gap?
“You could do the same thing in other ways”
“Nobody wants to write math formulas”
“Why would I turn my data into tables?”
“It won’t perform well”
What do you think?

5. Getting More Concrete: Building a Database and Application
Start with a conceptual model
“On paper” using certain techniques we’ll discuss next week
We ignore low-level details – focus on logical representation
Design & implement schema
Design and codify (in SQL) the relations/tables
Do physical layout – indexes, etc.
Import the data
Write applications using DBMS and other tools
Many of the hard problems are taken care of by other people (DBMS, API writers, library authors, web server, etc.)

6. Conceptual Design for CIS Student Course Survey
“Who’s taking what, and what grade do they expect?”
fid
PROFESSOR
name
This design is independent of the final form of the report!
Teaches
Takes
STUDENT
COURSE
sid
cid
name
semester
name
exp-grade

7. Example Schema STUDENT Takes COURSE
Our focus now: relational schema – set of tables
Can have other kinds of schemas – XML, object, …
sid
name 1
Jill 2
Qun 3
Nitin
sid
exp-grade
cid 1 A 3 C
cid
subj
sem DB
F09 AI
S09
Arch
PROFESSOR
Teaches
fid
name 1
Ives 2
Taskar 8
Martin
fid
cid 1 2 8

8. Some Terminology Columns of a relation are called attributes or fields
The number of these columns is the arity of the relation
The rows of a relation are called tuples
Each attribute has values taken from a domain, e.g., subj has domain string
Theoretically: a relation is a set of tuples; no tuple can occur more than once
Real systems may allow duplicates for efficiency or other reasons – we’ll ignore this for now
Objects and XML may also have the same content with different “identity”

9. Describing Relations A schema can be represented many ways
In relational DBs, we use relation(attribute:domain)
To the DBMS, use data definition language (DDL) – like programming language type definitions
STUDENT(sid:int, name:string)
Takes(sid:int, exp-grade:char[2], cid:string)
COURSE(cid:string, subj:string, sem:char[3])
Teaches(fid:int, cid:string)
PROFESSOR(fid:int, name:string)

10. More on Attribute Domains
Relational DBMSs have very limited “built-in” domains: either tables or scalar attributes – int, string, byte sequence, date, etc.
But more generally:
We can have “nested relations”
Object-oriented, object-relational systems allow complex, user-defined domains – lists, classes, etc.
XML systems allow for XML trees (or lists of trees) that follow certain structural constraints
Database people, when they are discussing design, often assume domains are evident to the reader: STUDENT(sid, name)

11. Integrity Constraints
Domains and schemas are one form of constraint on a valid data instance
Other important constraints include:
Key constraints:
Subset of fields that uniquely identifies a tuple, and for which no subset of the key has this property
May have several candidate keys; one is chosen as the primary key
A superkey is a subset of fields that includes a key
Inclusion dependencies (referential integrity constraints):
A field in one relation may refer to a tuple in another relation by including its key
The referenced tuple must exist in the other relation for the database instance to be valid

12. SQL: Structured Query Language
The standard language for relational data
Invented by folks at IBM, esp. Don Chamberlin
Actually not a particularly elegant language…
Beat a more elegant competing standard, QUEL, from Berkeley
Separated into a DML (data manipulation language) & DDL
DML based on relational algebra & (mostly) calculus, which we discuss this week
Later we’ll see how it’s embedded in a host language

13. Table Definition: SQL-92 DDL and Constraints
CREATE TABLE STUDENT
(sid INTEGER,
name CHAR(20), )
CREATE TABLE Takes
(sid INTEGER,
exp-grade CHAR(2),
cid STRING(8),
PRIMARY KEY (sid, cid),
FOREIGN KEY (sid)
REFERENCES STUDENT,
FOREIGN KEY (cid)
REFERENCES COURSE )

14. Example Data Instance STUDENT Takes COURSE PROFESSOR Teaches sid name1
Jill 2
Qun 3
Nitin
sid
exp-grade
cid 1 A 3 C
cid
subj
sem DB
F09 AI
S09
Arch
PROFESSOR
Teaches
fid
name 1
Ives 2
Taskar 8
Martin
fid
cid 1 2 8

15. From Tables  SQL  Web Application
<html>
<body>
<!-- hypotheticalEmbeddedSQL:
SELECT * FROM STUDENT, Takes, COURSE
WHERE STUDENT.sid = Takes.sID
AND Takes.cID = cid
-->
</body>
</html>
C -> machine code sequence -> microprocessor
Java -> bytecode sequence -> JVM
SQL -> relational algebra expression -> query execution engine

16. Codd’s Relational Algebra
A set of mathematical operators that compose, modify, and combine tuples within different relations
Relational algebra operations operate on relations and produce relations (“closure”)
f: Relation  Relation f: Relation x Relation  Relation

17. Codd’s Logical Operations: The Relational Algebra
Six basic operations:
Projection  (R)
Selection  (R)
Union R1 [ R2
Difference R1 – R2
Product R1 £ R2
(Rename) b (R)
And some other useful ones:
Join R1 ⋈ R2
Semijoin R1 ⋉ R2
Intersection R1 Å R2
Division R1 ¥ R2

18. Data Instance for Operator Examples
STUDENT
Takes
COURSE
sid
name 1
Jill 2
Qun 3
Nitin 4
Marty
sid
exp-grade
cid 1 A 3 C 4
cid
subj
sem DB
F09 AI
S09
Arch
PROFESSOR
Teaches
fid
name 1
Ives 2
Taskar 8
Martin
fid
cid 1 2 8

19. Projection, 

20. Selection, 

21. Product X

22. Join, ⋈: A Combination of Product and Selection

23. Union 

24. Difference –

25. Rename, ab The rename operator can be expressed several ways:
The book has a very odd definition that’s not algebraic
An alternate definition:
ab(x) Takes the relation with schema  Returns a relation with the attribute list 
Rename isn’t all that useful, except if you join a relation with itself
Why would it be useful here?

26. Mini-Quiz
This completes the basic operations of the relational algebra. We shall soon find out in what sense this is an adequate set of operations.
Try writing queries for these:
The names of students named “Bob”
The names of students expecting an “A”
The names of students in Milo Martin’s 501 class
The sids and names of students not enrolled

27. Deriving Intersection
Intersection: as with set operations, derivable from difference
A Å B
≡ (A [ B) – (A – B) – (B – A)
≡ (A – B) – (B – A)
A-B
B-A
A B

28. Division
A somewhat messy operation that can be expressed in terms of the operations we have already defined
Used to express queries such as “The fid's of faculty who have taught all subjects”
Paraphrased: “The fid’s of professors for which there does not exist a subject that they haven’t taught”

29. Division: R1 ¸ R2 Requirement: schema(R1) ¾ schema(R2)
Result schema: schema(R1) – schema(R2)
“Professors who have taught all courses”:
What about “Courses that have been taught by all faculty”?
fid (fid,subj(Teaches ⋈ COURSE) ¸ subj(COURSE))

30. Division Using Our Existing Operators
All possible teaching assignments: Allpairs:
NotTaught, all (fid,subj) pairs for which professor fid has not taught subj:
Answer is all faculty not in NotTaught:
fid,subj (PROFESSOR £ subj(COURSE))
Allpairs - fid,subj(Teaches ⋈ COURSE)
fid(PROFESSOR) - fid(NotTaught)
´ fid(PROFESSOR) - fid(
fid,subj (PROFESSOR £ subj(COURSE)) -
fid,subj(Teaches ⋈ COURSE))

31. The Big Picture: SQL to Algebra to Query Plan to Web Page
Web Server /
UI / etc
STUDENT
Takes
COURSE
Merge
Hash
by cid
Query Plan – an operator tree
Execution
Engine
Optimizer
Storage
Subsystem
SELECT * FROM STUDENT, Takes, COURSE
WHERE STUDENT.sid = Takes.sID
AND Takes.cID = cid

32. Hint of Future Things: Optimization Is Based on Algebraic Equivalences
Relational algebra has laws of commutativity, associativity, etc. that imply certain expressions are equivalent in semantics
They may be different in cost of evaluation!
c Ç d(R) ´ c(R) [ d(R)
c (R1 £ R2) ´ R1 ⋈c R2
c Ç d (R) ´ c (d (R))
Query optimization finds the most efficient representation to evaluate (or one that’s not bad)

33. Next Time: An Equivalent, But Very Different, Formalism
Codd invented a relational calculus that he proved was equivalent in expressiveness
Based on a subset of first-order logic – declarative, without an implicit order of evaluation
More convenient for describing certain things, and for certain kinds of manipulations
… And, in fact, the basis of SQL!