1. Relational Algebra

2. Relational Query Languages
Query languages: allow manipulation and retrieval of data from a database
Relational model supports simple, powerful QLs:
Strong formal foundation
Allows for much optimization
Query languages are NOT programming languages
QLs not expected to be “turing complete”
QLs not intended to be used for complex calculations
QLs support easy, efficient and sophisticated access to large data sets

3. Formal Relational Query Languages
Mathematical query languages form the basis for “real” languages (e.g. SQL) and for implementation:
Relational Algebra:
Operational - a query is a sequence of operations on data
Very useful for representing execution plans, i.e., to describe how a SQL query is executed internally
Relational Calculus:
descriptive - a query describes how the data to be retrieved looks like
Understanding Relational Algebra is key to understanding SQL.

4. Preliminaries
A query is applied to a relation instance, and the result of a query is also a relation instance (i.e., input = relation instances, output = relation instance)
Schemas of input relations for a query are fixed (but query will run regardless of instance)
The schema for the result of a given query is also fixed! Determined by the definition of query language constructs

5. Example Instances
We assume that names of fields in query results are ‘inherited’ from names of fields in query input relations
Sailors S1
sid
sname
rating
age
28 yuppy
31 debby
22 conny
58 lilly
Reserves R
Boat B
sid
bid
day
bid
bname
color
101 Interlake blue Snapper red Bingo green
/24/07
/30/07
/01/07
/23/07 S2
sid
sname
rating
age
44 robby
31 debby
58 lilly

6. Relational Algebra Basic Operations Additional operations
Selection : Selects a subset of rows from a relation
Projection ∏: projects to a subset of columns from a relation
Cross Product 5: Combines two relations
Set Difference : Tuples that are in the first but not the second relation
Union : tuples in any of the relations to be unified
Additional operations
Intersection (), join ( ), division, renaming
Not essential but very useful
Relational algebra is closed: since each operation has input relation(s), and returns a relation, operations can be composed

7. Projection S2 Projection: ∏ (S2) ∏ (S2) Notation: ∏L(Rin)
Returns a subset of the columns of the input relation Rin, i.e, it ignores the attributes that are not in the projection list L
Schema of result relation contains exactly the attributes of the projection list, with the same attribute names as in Rin
Projection operator has to eliminate duplicates!
Note: real systems typically do NOT eliminate duplicates unless the user explicitly asks for it; eliminating duplicates is a very costly operation!
∏ (S2) S2
∏ (S2)
sname,rating
age
sid
sname
rating
age
sname
rating
age
28 yuppy
31 debby
22 conny
58 lilly
yuppy 9
debby 8
conny 5
lilly 35 55

8. Selection and Operator Composition
Selection: C(Rin)
Returns the subset of the rows of the input relation Rin that fulfill the condition C
Condition C involves the attributes of Rin
Schema of result relation identical to schema of Rin
No duplicates (obviously)
Operation Composition: result relation of one operation can be input for another relational algebra operation S2
sid
sname
rating
age
28 yuppy
31 debby
22 conny
58 lilly
 (S2)
rating > 8
∏ ( (S2))
rating > 8
sname,rating
sid
sname
rating
age
sname
rating
28 yuppy
58 lilly
yuppy 9
lilly

9. Union, Intersection, Set-Difference
Notation:
Rin1  Rin2 (Union),
Rin1  Rin2 (Intersection),
Rin1 - Rin2 (Difference),
Usual operations on sets
Rin1 and Rin2 must be union-compatible, i.e., they must have the same number of attributes and corresponding attributes must have the same type (note that attribute names need not be the same)
The result schema is the same as the schema of the input relations (with possible renamed attributes)

10. Union, Intersection, Set-Difference: Examples
S1  S2
sid
sname
rating
age
sid
sname
rating
age
44 robby
31 debby lilly yuppy
22 conny
44 robby
31 debby
58 lilly S2
S1  S2
sid
sname
rating
age
sid
sname
rating
age
28 yuppy
31 debby
22 conny
58 lilly
31 debby
58 lilly
S1 - S2
sid
sname
rating
age
44 robby

11. Cross-Product S2 S1 X R1 Cross-Product: Rin1 X Rin2
Each row of Rin1 is paired with each row of Rin2
Result schema has one attribute per attribute of Rin1 and one attribute per attribute Rin2 with field names inherited if possible
Conflict if both relations have an attribute with the same name: e.g., attribute names of result relation concatenate input relation name with input attribute name S2
sid
bid
day
sid
sname
rating
age
/24/07
/30/07
/01/07
/23/07
44 robby
31 debby
58 lilly
S1 X R1
S1.
sid
sname
rating
age
R1.
sid
bid
day
44 robby
31 debby
31 debby lilly
58 lilly
/24/ /30/00
/24/ /30/00
/24/ /30/00

12. Joins
Condition Join (Theta-Join): Rout = Rin1 C Rin2 = C(Rin1 X Rin2)
Result schema the same as for cross-product
Fewer tuples than cross-product
sid
bid
day S2
/24/07
/30/07
/01/07
/23/07
sid
sname
rating
age
44 robby
31 debby
58 lilly
S R
S2.sid > R.sid
S2.
sid
sname R.
sid
rating
age
bid
day
44 robby
31 debby
/24/07
lilly /24/07

13. Self-Joins give S2 a second name S2’ S2 S2’ S2 sid sname rating age
44 robby
31 debby
58 lilly
give S2 a second name S2’
S S2’
S2.age > S2’.sid
S2.
sid
S2.
sname
S2.
rating
S2.
age
S2’.
sname
S2’.
rating
S2’.
age
S2’.
sid
robby lilly
31 debby lilly

14. Equi-Join
Equi-Join: Rout = Rin1 Rin1.a1 = Rin2.b1, … Rin1.an = Rin2.bn Rin2 =
A special case of condition join where the condition C contains only equalities.
Result schema similar to cross-product, but only one copy of attributes for which equality is specified
Natural Join: Equijoin on all common attributes, i.e., on all attributes with the same name
sid
bid
day S2
/24/07
/30/07
/01/07
/23/07
sid
sname
rating
age
44 robby
31 debby
58 lilly
S R
S2.sid = R.sid
S2.
sid
sname
rating
age
bid
day
lilly
58 lilly
/30/07
/01/07
/23/07
58 lilly

15. Division
Not supported as a primitive operation, but useful for expressing queries like
Find sailors who have reserved all boats
Let A have 2 fields, x and y; B have only field y:
A/B = {<x> |  <y>  B  <x,y>  A}
I.e., A/B contains all x tuples (sailors) such that for every y tuple (boat) in B, there is an xy tuple in A
Or: if the set of y values (boats) associated with an x value (sailor) in A contains all y values in B, the x value is in A/B
In general, x and y can be any lists of fields; y is the list of fields in B, x y is the list of fields of A.
Reserves R1
sid
bid
101
103
Boat B1
bid
101
103
104

16. Examples of Division A/by y y y
x y1
x y2
x y3
x y4
x y1
x y2
x y2
x y2
x y4 y2 y2 y4 y1 y2 y4 B1 B2 x B3 x1 x2 x3 x4 x x1 x4 x x1 A
A/B1
A/B3
A/B2

17. Expressing A/B using Basic Operators
Division is not an essential operation; just a useful shorthand
(Also true of joins, but joins are so common that systems implement joins specially)
Idea: For A/B, compute all x values that are not ‘disqualified’ by some y value in B.
x value is disqualified if by attaching y value from B, we obtain an xy tuple that is not in A
Disqualified x values: ∏X((∏X (A) x B)-A)
A/B: ∏X (A) - all disqualified tuples

18. Renaming Renaming : (Rout(B1,…Bn), Rin(A1, …An)) (Temp, S1),
Produces a relation identical to Rin but named Rout and with attributes A1, … An of Rin renamed to B1, … Bn
(Temp, S1),
(Temp1(sid1,rating1), S1(sid,rating))

19. Examples (discussed in class)
Relations
Sailors(sid,sname,rating,age)
Reserves(sid,bid,day)
Boats(bid,bname,color)
Queries
Find names of sailors who have reserved boat #103 (three solutions)
Find names of sailors who have reserved a red boat (2 solutions)
Find names of sailors who have reserved a red or a green boat (2 solutions)
Find names of sailors who have reserved a red and a green boat (1 solution)
Find names of sailors who have reserved all boats (solution with division)

20. Summary
The relational model has rigorously defined query languages that are simple and powerful
Relational algebra is operational; useful as internal representation for query evaluation plans
Projection, selection, cross-product, difference and union are the minimal set of operators with which all operations of the relational algebra can be expressed
Several ways of expressing a given query; a query optimizer should choose the most efficient version.
Relational Completeness of a query language: A query language (e.g., SQL) can express every query that is expressible in relational algebra

21. Relational Calculus
Relational Calculus is non-operational but descriptive:
Users define queries in terms of what they want, not in terms of how to compute it. (Declarativeness).
Comes in two flavors: Tuple relational calculus (TRC) and Domain relational calculus (DRC)
Both TRC and DRC are simple subsets of first-order logic
Calculus has variables, constants, comparison operations, logical connectives and quantifiers.
TRC: variables range over tuples
DRC: variables range over domain elements (= field values)
Find the names of all sailors with a rating > 7
TRC:{T | S  Sailors(S.rating > 7  S.name = T.name)}
DRC:{<N> | S,R,A(<S,N,R,A>  Sailors  R7}

22. Domain Relational Calculus
Query has the form:
{<x1,x2,…,xn>} | p(<x1,x2,…xn>)}
The Answer includes all tuples <x1,x2,…,xn> that make the formula p(<x1,x2,…xn>} be true. (‘|’ should be read as “such that”).
Formulas are recursively defined, starting with simple atomic formulas (getting tuples from relations or making comparisons of values), and building bigger and better formulas using the logical connectives.

23. DRC Formulas
Variables X,Y, x1,… range over domain elements (= field values)
Atomic formula: Let op be one of ,,,,,
<x1,x2,…xn>  Relation-name, or
X op Y, or
X op constant
Formula
An atomic formula, or
p, pq, pq, where p and q are formulas, or
X(p(X)), where variable X is free in p(X), or
X(p(X)), where variable X is free in p(X)
The use of quantifiers X and X is said to bind X. A variable that is not bound is free.

24. Free and Bound Variables
The use of quantifiers X and X is said to bind X. A variable that is not bound is free.
Let us revisit the definition of a query:
{<x1,x2,…,xn>} | p(<x1,x2,…xn>)}
There is an important restriction: the variables x1,…,xn that appear to the left of ‘|’ must be the only free variables in the formula p(…).

25. Example Find all sailors with a rating above 7:
{<I,N,T,A> | <I,N,T,A>  Sailors  T7}
The condition <I,N,T,A>Sailors ensures that the domain variables I,N,T, and A are bound to fields of the same Sailors tuple.
The condition T7 ensures that all tuples in the answers have a rating above 7
The term <I,N,T,A> to the left of ‘|’ says that every tuple of the Sailors relation with a rating above 7 is in the answer

26. Further Examples (discussed in class)
Relations
Sailors(sid,sname,rating,age)
Reserves(sid,bid,day)
Boats(bid,bname,color)
Queries
Find sailors who are older than 18 or have a rating under 9, and are called ‘debby’’
Find sailors with a rating above 7 that have reserved boat #103 (use of )
Find sailors rated > 7 who have reserved a red boat (use of )
Find sailors who have reserved all boats (use of )
Transfer to “find all sailors I such that for each 3-tuple (B,BN,C> either it is not a tuple in Boats or there is a tuple in Reserves showing that sailor I has reserved the boat.

27. Unsafe Queries and Expressive Power
Unsafe queries: It is possible to write syntactically correct calculus queries that have an infinite number of answers! Such queries are called unsafe.
E.g., {<I,N,T,A> |  (<I,N,T,A>  Sailors)}
Expressive Power: It is known that every query that can be expressed in relational algebra can be expressed as a safe query in DRC/TRC; the converse is also true.
Relational Completeness of a query language: A query language (e.g., SQL) can express every query that is expressible in relational algebra/calculus

28. Some rules and definitions
Equivalence: Let R, S, T be relations; C, C1, C2 conditions; L projection lists of the relations R and S
Commutativity:
∏L(C(R)) = C(∏L(R)) if C only considers attributes of L
R1 R2 = R2 R1
Associativity:
R1 (R2 R3) = (R1 R2) R3
Idempotence:
∏L2 (∏L1(R)) = ∏L2 (R) if L2  L1
C2(C1(R)) = C1C2(R))