1. Relational Calculus Zachary G. Ives November 15, 2018
University of Pennsylvania
CIS 550 – Database & Information Systems
November 15, 2018
Some slide content courtesy of Susan Davidson & Raghu Ramakrishnan

2. Administrivia Reminder: Homework 1 due 9/28 (next Monday)
Upcoming research talks of possible interest:
Amol Deshpande, U Maryland, Monday 10/19, 3PM – adding probabilities to databases

3. The Relational Algebra as “Virtual Machine” Instructions
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
Intersection R1 Å R2
STUDENT
Takes
COURSE
Calculus
SELECT * FROM STUDENT, Takes, COURSE
WHERE STUDENT.sid = Takes.sID
AND Takes.cID = cid

4. Our Example Data Instance
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 11
Ives 12
Taskar 18
Martin
fid
cid 11 12 18

5. Relational Calculus, Variant I: Domain Relational Calculus (DRC)
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

6. Complex Predicates in the Calculus
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

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

8. 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

9. Terminology: 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

10. 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”…

11. 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

12. 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

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

14. 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}

15. 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}

16. 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}

17. 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

18. 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 }

19. What about the Tuple Relational Calculus (TRC)?
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)}
The use of the “output” variable is a bit unintuitive, but otherwise it should be very similar

20. Limitations of the Relational Algebra / Calculus
Can’t do:
Aggregate operations (average, sum, count)
Track the number of duplicate elements (bag semantics)
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

21. 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!