Relational Algebra

Content based on Chapter 4Database Management Systems,

(Third Edition), by Raghu Ramakrishnan and Johannes

Gehrke. McGraw Hill, 2003

Relational Query Languages

Query languages: Allow manipulation and retrieval of data from a database.

Relational model supports simple, powerful QLs: Strong formal foundation based on logic. Allows for much optimization.

Query Languages != programming languages! QLs not expected to be “Turing complete”. QLs not intended to be used for complex calculations. QLs support easy, efficient access to large data sets.

Formal Relational Query Languages Two mathematical Query Languages

form the basis for “real” languages (e.g. SQL), and for implementation: Relational Algebra: More operational, very

useful for representing execution plans. Relational Calculus: Lets users describe

what they want, rather than how to compute it. (Non-operational, declarative.)

Preliminaries

A query is applied to relation instances, and the result of a query is also a 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 definition of query language constructs.

Positional vs. named-field notation: Positional notation easier for formal definitions,

named-field notation more readable. Both used in SQL

Example Instances

“Sailors” and “Reserves” relations for our examples.

We’ll use positional or named field notation, assume that names of fields in query results are `inherited’ from names of fields in query input relations.

Example Instances

sid sname rating age

22 dustin 7 45.0

31 lubber 8 55.558 rusty 10 35.0

sid sname rating age28 yuppy 9 35.031 lubber 8 55.544 guppy 5 35.058 rusty 10 35.0

sid bid day

22 101 10/10/9658 103 11/12/96

R1 S1

S2B1

bid bname color

101 Interlake blue

102 Interlake red

103 Clipper green

104 Marine red

Relational Algebra Basic operations:

Selection ( ) Selects a subset of rows from relation. Projection ( ) Deletes unwanted columns from relation. Cross-product ( ) Allows us to combine two relations. Set-difference ( ) Tuples in reln. 1, but not in reln. 2. Union ( ) Tuples in reln. 1 and in reln. 2.

Additional operations: Intersection, join, division, renaming: Not essential, but (very!)

useful. Since each operation returns a relation, operations can

be composed! (Algebra is “closed”.)

Projectionsname rating

yuppy 9lubber 8guppy 5rusty 10

sname rating

S,

( )2

age

35.055.5

age S( )2

Deletes attributes that are not in projection list.

Schema of result contains exactly the fields in the projection list, with the same names that they had in the (only) input relation.

Projection operator has to eliminate duplicates! (Why??) Note: real systems typically

don’t do duplicate elimination unless the user explicitly asks for it. (Why not?)

Selection

rating

S8 2( )

sid sname rating age28 yuppy 9 35.058 rusty 10 35.0

sname ratingyuppy 9rusty 10

sname rating rating

S,

( ( ))8 2

Selects rows that satisfy selection condition.

No duplicates in result! (Why?)

Schema of result identical to schema of (only) input relation.

Result relation can be the input for another relational algebra operation! (Operator composition.)

Union, Intersection, Set-Difference

All of these operations take two input relations, which must be union-compatible: Same number of

fields. `Corresponding’ fields

have the same type. What is the schema of

result?

sid sname rating age

22 dustin 7 45.031 lubber 8 55.558 rusty 10 35.044 guppy 5 35.028 yuppy 9 35.0

sid sname rating age31 lubber 8 55.558 rusty 10 35.0

S S1 2

S S1 2

sid sname rating age

22 dustin 7 45.0

S S1 2

Cross-Product Each row of S1 is paired with each row of

R1. Result schema has one field per field of S1

and R1, with field names `inherited’ if possible. Conflict: Both S1 and R1 have a field

called sid.

( ( , ), )C sid sid S R1 1 5 2 1 1

(sid) sname rating age (sid) bid day

22 dustin 7 45.0 22 101 10/ 10/ 96

22 dustin 7 45.0 58 103 11/ 12/ 96

31 lubber 8 55.5 22 101 10/ 10/ 96

31 lubber 8 55.5 58 103 11/ 12/ 96

58 rusty 10 35.0 22 101 10/ 10/ 96

58 rusty 10 35.0 58 103 11/ 12/ 96

Renaming operator:

Joins Condition Join:

Result schema same as that of cross-product. Fewer tuples than cross-product, might be able

to compute more efficiently Sometimes called a theta-join.

R c S c R S ( )

(sid) sname rating age (sid) bid day

22 dustin 7 45.0 58 103 11/ 12/ 9631 lubber 8 55.5 58 103 11/ 12/ 96

S RS sid R sid

1 11 1

. .

Joins

Equi-Join: A special case of condition join where the condition c contains only equalities.

Result schema similar to cross-product, but only one copy of fields for which equality is specified.

Natural Join: Equijoin on all common fields.

sid sname rating age bid day

22 dustin 7 45.0 101 10/ 10/ 9658 rusty 10 35.0 103 11/ 12/ 96

S Rsid

1 1

Division Not supported as a primitive operator, 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 = 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, and x y is the list of fields of A.

x x y A y B| ,

Examples of Division A/B

sno pnos1 p1s1 p2s1 p3s1 p4s2 p1s2 p2s3 p2s4 p2s4 p4

pnop2

pnop2p4

pnop1p2p4

snos1s2s3s4

snos1s4

snos1

A

B1B2

B3

A/B1 A/B2 A/B3

Expressing A/B Using Basic Operators

Division is not essential op; 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: )))((( ABAxxD

DAxBA )(/

Find names of sailors who’ve reserved boat #103

Solution 1: sname bidserves Sailors(( Re ) )103

Solution 2: ( , Re )Temp servesbid

1103

( , )Temp Temp Sailors2 1

sname Temp( )2

Solution 3: sname bidserves Sailors( (Re ))

103

Find names of sailors who’ve reserved a red boat

Information about boat color only available in Boats; so need an extra join: sname color red

Boats serves Sailors((' '

) Re )

A more efficient solution:

))Re)''

(((( SailorssBoatsredcolorbidsidsname

A query optimizer can find this, given the first solution!

Find sailors who’ve reserved a red or a green boat

Can identify all red or green boats, then find sailors who’ve reserved one of these boats: ( , (

' ' ' '))Tempboats

color red color greenBoats

sname Tempboats serves Sailors( Re )

Can also define Tempboats using union! (How?)

What happens if is replaced by in this query?

Find sailors who’ve reserved a red and a green boat

Previous approach won’t work! Must identify sailors who’ve reserved red boats, sailors who’ve reserved green boats, then find the intersection (note that sid is a key for Sailors): ( , ((

' ') Re ))Tempred

sid color redBoats serves

sname Tempred Tempgreen Sailors(( ) )

( , ((' '

) Re ))Tempgreensid color green

Boats serves

Find the names of sailors who’ve reserved all boats

Uses division; schemas of the input relations to / must be carefully chosen:

( , (,

Re ) / ( ))Tempsidssid bid

servesbidBoats

sname Tempsids Sailors( )

To find sailors who’ve reserved all ‘Interlake’ boats:

/ (' '

) bid bname Interlake

Boats.....

Relational Calculus

Comes in two flavors: Tuple relational calculus (TRC) and Domain relational calculus (DRC).

Calculus has variables, constants, comparison ops, logical connectives and quantifiers. TRC: Variables range over (i.e., get bound to) tuples. DRC: Variables range over domain elements (= field

values). Both TRC and DRC are simple subsets of first-order

logic. Expressions in the calculus are called formulas.

An answer tuple is essentially an assignment of constants to variables that make the formula evaluate to true.

Tuple Relational Calculus

Queries in tuple relational calculus : {t| (t)} t: tuple variable (t): well-formed formula (wff) = conditions t is the free variable in (t)

More intuitively,{t1| cond(t1, t2, …, tn)}, where t1: free variable and t2, .., tn : bound variables

Well-formed formula Atomic formulas connected by logical connectives,

AND, OR, NOT and quantifiers, (existential quantifier), (universal quantifier)

Atomic formula in TRC

1. R(s) R is a relation name s is a tuple variable

2. ti[A] tj[B] : comparison operator (=, <, >, , , ) ti, tj: tuple variables A, B: attributes

3. ti[A] constant

Example: TRC query Consider two relations

EMP(Name, MGR, DEPT, SAL) CHILDREN(Ename, Cname, Age)

Q1: Retrieve Salary and Children’s name of Employees whose manager is ‘white’

{r|(e)(c)(EMP(e) AND CHILDREN(c) AND/* initiate tuple variables */

e[Name] = c[Ename] AND /* join condition */e[MGR] = ‘white’ AND /* selection cond. */r[1st attr] = e[SAL] ANDr[2nd attr] = c[Cname] }/* projection */

Find the names and ages of sailors with a rating above 7

{p|(s)(Sailors(s) AND/* initiate tuple variables */

s[rating] > 7 AND /* selection condition */

p[1st attr] = s[sname] ANDp[2nd attr] = s[age]) } /*

projection */

Find the names of sailors who have reserved boat 103

{p|(r)(s)(Reserves(r) AND Sailors(s) AND/* initiate tuple variables */

r[sid] = s[sid] AND /* join condition */r[bid] = 103 AND /* selection condition */p[1st attr] = s[sname] } /* projection */

Domain Relational Calculus

Queries in domain relational calculus : {x1, x2, ..., xk| (t)} x1, x2, …, xk: domain variables; these are only

free variables in (t) (t): well-formed formula (wff) = conditions

Well-formed formula Atomic formulas connected by logical

connectives, AND, OR, NOT and quantifiers, (existential quantifier), (universal quantifier)

Atomic formula in DRC

R(x1, x2, …, xk) R is a k-ary relation xi : domain variable or constant

x y : comparison operator (=, <, >, , , ) x, y: domain variables A, B: attributes

x constant

DRC Examples

Consider two relations EMP(Name, MGR, DEPT, SAL) CHILDREN(Ename, Cname, Age)

Q1: Retrieve Salary and Children’s name of Employees whose manager is ‘white’

{q,s|(u)(v)(w)(x)(y)(EMP(u,v,w,q) AND CHILDREN(x,s,y) AND

/* initiate domain variables */u = x AND /* join condition */v = ‘white’ } /* selection condition */

/* projection is implied (q, s) */

Find all sailors with a rating above 7

{i,n,t,a| Sailors(i,n,t,a) AND t > 7}

Find sailors rated > 7 who’ve reserved boat #103

{i,n,t,a|(ir)(br)(d)(Sailors(i,n,t,a) AND Reserve(ir,br,d) AND ir = i AND /* join condition */t > 7 AND br = 103)} /* selection condition */

Alternative:{i,n,t,a|(br)(d)

(Sailors(i,n,t,a) AND Reserve(i,br,d) AND t > 7 AND br = 103)} /* selection condition */

Unsafe Queries, Expressive Power

It is possible to write syntactically correct calculus queries that have an infinite number of answers! Such queries are called unsafe. e.g.,

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: Query language (e.g., SQL) can express every query that is expressible in relational algebra/calculus.

S S Sailors|

Summary

The relational model has rigorously defined query languages that are simple and powerful.

Relational algebra is more operational; useful as internal representation for query evaluation plans.

Several ways of expressing a given query; a query optimizer should choose the most efficient version.

Summary

Relational calculus is non-operational, and users define queries in terms of what they want, not in terms of how to compute it. (Declarativeness.)

Algebra and safe calculus have same expressive power, leading to the notion of relational completeness.