1. Relational Algebra & Calculus
Zachary G. Ives
University of Pennsylvania
CIS 550 – Database & Information Systems
September 15, 2009
Some slide content courtesy of Susan Davidson & Raghu Ramakrishnan

2. 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

3. 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

4. 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

5. Last Time… We discussed:
Projection, (R), specified a set of attributes to include in a new relation

6. Selection, 

7. Product X

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

9. Union 

10. Difference –

11. 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?

12. 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

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

14. 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

15. 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)

16. Switching Gears: 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
Tuple relational calculus
Domain relational calculus
More convenient for describing certain things, and for certain kinds of manipulations
The database uses the relational algebra internally
But query languages (e.g., SQL) are mostly based on the relational calculus

17. Domain Relational Calculus
Queries have form:
{<x1,x2, …, xn>| p}
Predicate: boolean expression over x1,x2, …, xn
Precise operations depend on the domain and query language – may include special functions, etc.
Assume the following at minimum:
<xi,xj,…>  R X op Y X op const const op X
where op is , , , , , 
xi,xj,… are domain variables
domain variables
predicate

18. More Complex Predicates
Starting with these atomic predicates, build up new predicates by the following rules:
Logical connectives: If p and q are predicates, then so are p  q, p  q, p, and p  q
(x>2)  (x<4)
(x>2)  (x>0)
Existential quantification: If p is a predicate, then so is x.p
x. (x>2) (x<4)
Universal quantification: If p is a predicate, then so is x.p
x.x>2
x. y.y>x

19. Some Examples Faculty ids
Subjects for courses with students expecting a “C”
All course numbers for which there exists a smaller course number

20. Logical Equivalences
There are two logical equivalences that will be heavily used:
p  q  p  q (Whenever p is true, q must also be true.)
x. p(x)  x. p(x) (p is true for all x)
The second can be a lot easier to check!
Example:
The highest course number offered

21. Free and Bound Variables
A variable v is bound in a predicate p when p is of the form v… or v…
A variable occurs free in p if it occurs in a position where it is not bound by an enclosing  or 
Examples:
x is free in x > 2
x is bound in x. x > y

22. Can Rename Bound Variables Only
When a variable is bound one can replace it with some other variable without altering the meaning of the expression, providing there are no name clashes
Example: x. x > 2 is equivalent to y. y > 2
Otherwise, the variable is defined outside our “scope”…

23. Safety Pitfall in what we have done so far – how do we interpret:
{<sid,name>| <sid,name>  STUDENT}
Set of all binary tuples that are not students: an infinite set (and unsafe query)
A query is safe if no matter how we instantiate the relations, it always produces a finite answer
Domain independent: answer is the same regardless of the domain in which it is evaluated
Unfortunately, both this definition of safety and domain independence are semantic conditions, and are undecidable

24. Safety and Termination Guarantees
There are syntactic conditions that are used to guarantee “safe” formulas
The definition is complicated, and we won’t discuss it; you can find it in Ullman’s Principles of Database and Knowledge-Base Systems
The formulas that are expressible in real query languages based on relational calculus are all “safe”
Many DB languages include additional features, like recursion, that must be restricted in certain ways to guarantee termination and consistent answers

25. Mini-Quiz How do you write:
Which students have taken more than one course from the same professor?

26. Translating from RA to DRC
Core of relational algebra: , , , x, -
We need to work our way through the structure of an RA expression, translating each possible form.
Let TR[e] be the translation of RA expression e into DRC.
Relation names: For the RA expression R, the DRC expression is {<x1,x2, …, xn>| <x1,x2, …, xn>  R}

27. Selection: TR[ R]
Suppose we have (e’), where e’ is another RA expression that translates as:
TR[e’]= {<x1,x2, …, xn>| p}
Then the translation of c(e’) is
{<x1,x2, …, xn>| p’} where ’ is obtained from  by replacing each attribute with the corresponding variable
Example: TR[#1=#2 #4>2.5R] (if R has arity 4) is
{<x1,x2, x3, x4>| < x1,x2, x3, x4>  R  x1=x2  x4>2.5}

28. Projection: TR[i1,…,im(e)]
If TR[e]= {<x1,x2, …, xn>| p} then TR[i1,i2,…,im(e)]= {<x i1,x i2, …, x im >|  xj1,xj2, …, xjk.p}, where xj1,xj2, …, xjk are variables in x1,x2, …, xn that are not in x i1,x i2, …, x im
Example: With R as before, #1,#3 (R)={<x1,x3>| x2,x4. <x1,x2, x3,x4> R}

29. Union: TR[R1  R2] R1 and R2 must have the same arity
For e1  e2, where e1, e2 are algebra expressions
TR[e1]={<x1,…,xn>|p} and TR[e2]={<y1,…yn>|q}
Relabel the variables in the second:
TR[e2]={< x1,…,xn>|q’}
This may involve relabeling bound variables in q to avoid clashes
TR[e1e2]={<x1,…,xn>|pq’}.
Example: TR[R1  R2] = {< x1,x2, x3,x4>| <x1,x2, x3,x4>R1  <x1,x2, x3,x4>R2

30. Other Binary Operators
Difference: The same conditions hold as for union
If TR[e1]={<x1,…,xn>|p} and TR[e2]={< x1,…,xn>|q}
Then TR[e1- e2]= {<x1,…,xn>|pq}
Product:
If TR[e1]={<x1,…,xn>|p} and TR[e2]={< y1,…,ym>|q}
Then TR[e1 e2]= {<x1,…,xn, y1,…,ym >| pq}
Example: TR[RS]= {<x1,…,xn, y1,…,ym >| <x1,…,xn> R  <y1,…,ym > S }

31. What about the Tuple Relational Calculus?
We’ve been looking at the Domain Relational Calculus
The Tuple Relational Calculus is nearly the same, but variables are at the level of a tuple, not an attribute
{Q | 9 S  COURSES, 9 T 2 Takes (S.cid = T.cid Æ Q.cid = S.cid Æ Q.exp-grade = T.exp-grade)}

32. Limitations of the Relational Algebra / Calculus
Can’t do:
Aggregate operations
Recursive queries
Complex (non-tabular) structures
Most of these are expressible in SQL, OQL, XQuery – using other special operators
Sometimes we even need the power of a Turing-complete programming language

33. Summary Can translate relational algebra into relational calculus
DRC and TRC are slightly different syntaxes but equivalent
Given syntactic restrictions that guarantee safety of DRC query, can translate back to relational algebra
These are the principles behind initial development of relational databases
SQL is close to calculus; query plan is close to algebra
Great example of theory leading to practice!