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Relational Algebra & SQL

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Why SQL & Relational Algebra?

SQL is a very-high-level language. Say “what to do” rather than “how to do it.” Avoid a lot of data-manipulation details needed

in procedural languages like C++ or Java. Database management system figures out

“best” way to execute query. Called “query optimization.”

Relational Algebra used in the Query Engine

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Select-From-Where Statements

SELECT desired attributesFROM one or more tablesWHERE condition about tuples of

the tables

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Our Running Example

All our SQL queries will be based on the following database schema. Underline indicates key attributes.

Beers(name, manf)Bars(name, addr, license)Drinkers(name, addr, phone)Likes(drinker, beer)Sells(bar, beer, price)Frequents(drinker, bar)

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Example: Selection R1 := σC (R2)

Relation Sells:bar beer priceJoe’s Bud 2.50Joe’s Miller 2.75Sue’s Bud 2.50Sue’s Miller 3.00

JoeMenu := σbar=“Joe’s”(Sells):bar beer priceJoe’s Bud 2.50Joe’s Miller 2.75

SELECT *FROM SellsWHERE bar = ‘Joe’’s’

SELECT *FROM SellsWHERE bar = ‘Joe’’s’

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Example: ProjectionR1 := πA (R2)

Relation Sells:bar beer priceJoe’s Bud 2.50Joe’s Miller 2.75Sue’s Bud 2.50Sue’s Miller 3.00

Prices := πbeer,price(Sells):beer priceBud 2.50Miller 2.75Miller 3.00

SELECT beer, priceFROM SellsSELECT beer, priceFROM Sells

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Example: R3 := R1 Χ R2R1( A, B )

1 23 4

R2( B, C )5 67 89 10

R3( A, R1.B, R2.B, C )1 2 5 61 2 7 81 2 9 103 4 5 63 4 7 83 4 9 10

SELECT *FROM R1, R2SELECT *FROM R1, R2

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Example: Theta JoinR3 := R1 ⋈C R2

Sells( bar, beer, price ) Bars(name, addr)Joe’s Bud 2.50 Joe’s Maple St.Joe’s Miller 2.75 Sue’s River Rd.Sue’s Bud 2.50Sue’s Coors3.00

BarInfo := Sells ⋈Sells.bar = Bars.name Bars

BarInfo( bar, beer, price, name, addr )Joe’s Bud 2.50 Joe’s Maple St.Joe’s Miller 2.75 Joe’s Maple St.Sue’s Bud 2.50 Sue’s River Rd.Sue’s Coors3.00 Sue’s River Rd.

SELECT * FROM Sells, BarsWHERE Sells.bar=

Bars.name

SELECT * FROM Sells, BarsWHERE Sells.bar=

Bars.name

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Example: Natural JoinR3 := R1 ⋈ R2

Sells( bar, beer, price ) Bars( bar, addr )Joe’s Bud 2.50 Joe’s Maple St.Joe’s Miller 2.75 Sue’s River Rd.Sue’s Bud 2.50Sue’s Coors3.00

BarInfo( bar, beer, price, addr )Joe’s Bud 2.50 Maple St.Joe’s Milller 2.75 Maple St.Sue’s Bud 2.50 River Rd.Sue’s Coors3.00 River Rd.

BarInfo := Sells ⋈ Bars

SELECT S.bar, beer, price, addr FROM Sells S, Bars BWHERE S.bar=B.bar

SELECT S.bar, beer, price, addr FROM Sells S, Bars BWHERE S.bar=B.bar

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Renaming

The ρ operator gives a new schema to a relation.

R1 := ρR1(A1,…,An)(R2) makes R1 be a relation with attributes A1,…,An and the same tuples as R2.

Simplified notation: R1(A1,…,An) := R2.

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Example: Renaming

Bars( name, addr )Joe’s Maple St.Sue’s River Rd.

R( bar, addr )Joe’s Maple St.Sue’s River Rd.

R(bar, addr) := Bars

CREATE TABLE R AS SELECT name as bar, addr FROM Bars

CREATE TABLE R AS SELECT name as bar, addr FROM Bars

SELECT name as bar, addr FROM BarsSELECT name as bar, addr FROM Bars

OR

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Union, Intersection, and Difference

In Relational Algebra: R3 = R1 ∪ R2 (tuples in R1 or R2) R3 = R1 ∩ R2 (tuples in both R1 and R2) R3 = R1 — R2 (tuples in R1 and not in R2)

In SQL: (<subquery>) UNION (<subquery>)

• SELECT surname FROM instructors UNION

SELECT surname FROM students (<subquery>) INTERSECT (<subquery>)

• SELECT surname FROM instructors INTERSECT

SELECT surname FROM students (<subquery>) EXCEPT (<subquery>)

• SELECT surname FROM instructors EXCEPT

SELECT surname FROM students

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Building Complex Expressions Combine operators with parentheses and

precedence rules.

Example: the theta-join R3 := R1 ⋈C R2

can be written: R3 := σC (R1 Χ R2) Precedence of relational operators:

1. [σ, π, ρ] (highest).

2. [Χ, ⋈].

3. ∩.4. [∪, —]

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Expression Trees

Leaves are operands --- either variables standing for relations or particular, constant relations.

Interior nodes are operators, applied to their child or children.

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Query Tree: Find the names of all the bars that are either on Maple St. or sell Bud for

less than $3

Bars Sells

σaddr = “Maple St.” σprice<3 AND beer=“Bud”

πname

ρR(name)

πbar

∪

Bars(name, addr) Sells(bar, beer, price)

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Example: Self-Join

Using Sells(bar, beer, price), find the bars that sell two different beers at the same price.

Strategy: by renaming, define a copy of Sells, called S(bar, beer1, price). The natural join of Sells and S consists of quadruples (bar, beer, beer1, price) such that the bar sells both beers at this price.

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The Tree

Sells Sells

ρS(bar, beer1, price)

⋈

πbar

σbeer != beer1

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Schemas for Results

Union, intersection, and difference: the schemas of the two operands must be the same, so use that schema for the result.

Selection: schema of the result is the same as the schema of the operand.

Projection: list of attributes tells us the schema.

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Schemas for Results --- (2)

Product: schema is the attributes of both relations. Use R.A, etc., to distinguish two

attributes named A. Theta-join: same as product. Natural join: union of the attributes

of the two relations. Renaming: the operator tells the

schema.

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Relational Algebra on Bags A bag (or multiset ) is like a set, but

an element may appear more than once.

Example: {1,2,1,3} is a bag. Example: {1,2,3} is also a bag that

happens to be a set. SQL is actually a bag language. Some operations, like projection, are

more efficient on bags than sets.

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Example: SelectionR( A, B )

1 25 61 2

A B1 21 2

SELECT *FROM RWHERE A+B < 5

SELECT *FROM RWHERE A+B < 5

σA+B < 5 (R) = A B1 2

SELECT DISTINCT *FROM RWHERE A+B < 5

SELECT DISTINCT *FROM RWHERE A+B < 5

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Example: ProjectionR( A, B )

1 25 61 2

A151

SELECT DISTINCT AFROM R

SELECT DISTINCT AFROM R

πA (R) = A15

SELECT AFROM RSELECT AFROM R

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Example: Theta-JoinR( A, B ) S(B, C ) 1 2 3 4 5 6 7 8 1 2

A R.B S.B C1 2 3 41 2 7 85 6 7 81 2 3 41 2 7 8

A R.B S.B C1 2 3 41 2 7 85 6 7 8

SELECT A, R.B, S.B, CFROM R, S WHERE R.B < S.B

SELECT A, R.B, S.B, CFROM R, S WHERE R.B < S.B

R ⋈ R.B<S.B S

SELECT DISTINCT A, R.B, S.B, CFROM R, S WHERE R.B < S.B

SELECT DISTINCT A, R.B, S.B, CFROM R, S WHERE R.B < S.B

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Union, Intersection, Difference

In SQL UNION, INTERSECTION, EXCEPT return distinct tuples (no duplicates)

Use UNION ALL, INTERSECTION ALL, EXCEPT ALL to return duplicates (if any)

Bag Union: {1,2,1} ∪ {1,1,2,3,1} = {1,1,1,1,1,2,2,3}

Bag Intersection: {1,2,1,1} ∩ {1,2,1,3} = {1,1,2}

Bag Difference: {1,2,1,1} – {1,2,3} = {1,1}

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SQL Semantics

Select-From-Where Statements

Multirelation QueriesSubqueries

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Example

Using Beers(name, manf), what beers are made by Anheuser-Busch?

SELECT name

FROM Beers

WHERE manf = ’Anheuser-Busch’;

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Result of Query

nameBudBud LiteMichelob . . .

The answer is a relation with a single attribute,name, and tuples with the name of each beerby Anheuser-Busch, such as Bud.

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Meaning of Single-Relation Query

Begin with the relation in the FROM clause.

Apply the selection indicated by the WHERE clause.

Apply the extended projection indicated by the SELECT clause.

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Operational Semantics

Check ifAnheuser-Busch

name manf

Bud Anheuser-Busch Include t.name in the result, if so

Tuple-variable tloops over alltuples

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Operational Semantics --- General

Think of a tuple variable visiting each tuple of the relation mentioned in FROM.

Check if the “current” tuple satisfies the WHERE clause.

If so, compute the attributes or expressions of the SELECT clause using the components of this tuple.

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* In SELECT clauses When there is one relation in the FROM

clause, * in the SELECT clause stands for “all attributes of this relation.”

Example: Using Beers(name, manf):SELECT *

FROM Beers

WHERE manf = ’Anheuser-Busch’;

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Result of Query:

name manfBud Anheuser-BuschBud Lite Anheuser-BuschMichelob Anheuser-Busch . . . . . .

Now, the result has each of the attributesof Beers.

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Renaming Attributes

If you want the result to have different attribute names, use “AS <new name>” to rename an attribute.

Example: Using Beers(name, manf):SELECT name AS beer, manf

FROM Beers

WHERE manf = ’Anheuser-Busch’

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Result of Query:

beer manfBud Anheuser-BuschBud Lite Anheuser-BuschMichelob Anheuser-Busch . . . . . .

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Expressions in SELECT Clauses

Any expression that makes sense can appear as an element of a SELECT clause.

Example: Using Sells(bar, beer, price):SELECT bar, beer,

price*114 AS priceInYen

FROM Sells;

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Result of Query

bar beerpriceInYen

Joe’s Bud 285Sue’s Miller 342 … … …

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Example: Constants as Expressions

Using Likes(drinker, beer):

SELECT drinker,

’likes Bud’ AS whoLikesBud

FROM Likes

WHERE beer = ’Bud’;

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Result of Query

drinker whoLikesBudSally likes BudFred likes Bud … …

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Example: Information Integration

We often build “data warehouses” from the data at many “sources.”

Suppose each bar has its own relation Menu(beer, price) .

To contribute to Sells(bar, beer, price) we need to query each bar and insert the name of the bar.

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Information Integration --- (2)

For instance, at Joe’s Bar we can issue the query: SELECT ’Joe’’s Bar’as bar, beer, price FROM Menu; bar beer price

Joe’s BarBud 2.50Joe’s BarBud Light 2.75 … …

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Complex Conditions in WHERE Clause

Boolean operators AND, OR, NOT. Comparisons =, <>, <, >, <=, >=.

And many other operators that produce boolean-valued results.

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Example: Complex Condition

Using Sells(bar, beer, price), find the price Joe’s Bar charges for Bud:

SELECT price

FROM Sells

WHERE bar = ’Joe’’s Bar’ AND

beer = ’Bud’;

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Patterns

A condition can compare a string to a pattern by: <Attribute> LIKE <pattern> or

<Attribute> NOT LIKE <pattern> Pattern is a quoted string with %

= “any string”; _ = “any character.”

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Example: LIKE

Using Drinkers(name, addr, phone) find the drinkers with exchange 555:

SELECT name

FROM Drinkers

WHERE phone LIKE ’%555-_ _ _ _’;

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NULL Values

Tuples in SQL relations can have NULL as a value for one or more components.

Meaning depends on context. Two common cases: Missing value : e.g., we know Joe’s Bar has

some address, but we don’t know what it is. Inapplicable : e.g., the value of attribute

spouse for an unmarried person.

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Comparing NULL’s to Values

The logic of conditions in SQL is really 3-valued logic: TRUE, FALSE, UNKNOWN.

Comparing any value (including NULL itself) with NULL yields UNKNOWN.

A tuple is in a query answer iff the WHERE clause is TRUE (not FALSE or UNKNOWN).

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Three-Valued Logic To understand how AND, OR, and NOT

work in 3-valued logic, think of TRUE = 1, FALSE = 0, and UNKNOWN = ½.

AND = MIN; OR = MAX, NOT(x) = 1-x. Example:TRUE AND (FALSE OR NOT(UNKNOWN))

= MIN(1, MAX(0, (1 - ½ ))) =MIN(1, MAX(0, ½ )) = MIN(1, ½ ) = ½.

Three-valued Logic AND = MIN; OR = MAX, NOT(x) = 1-x

OR TRUE and UNKNOWN = UNKNOWN FALSE and UNKNOWN = FALSE TRUE or UNKNOWN = TRUE FALSE or UNKNOWN = UNKNOWN not UNKNOWN = UNKNOWN Example:TRUE AND (FALSE OR NOT(UNKNOWN)) = TRUE AND (FALSE OR UNKNOWN)= TRUE AND UNKNOWN = UNKNOWN

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Surprising Example

From the following Sells relation:bar beer priceJoe’s BarBud NULL

SELECT barFROM SellsWHERE price < 2.00 OR price >= 2.00; UNKNOWN UNKNOWN

UNKNOWN

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Reason: 2-Valued Laws != 3-Valued Laws

Some common laws, like commutativity of AND, hold in 3-valued logic.

But not others, e.g., the law of the excluded middle : p OR NOT p = TRUE. When p = UNKNOWN, the left side is

MAX( ½, (1 – ½ )) = ½ != 1. i.e., UNKNOWN or not UNKNOWN =

UNKNOWN

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Multirelation Queries

Interesting queries often combine data from more than one relation.

We can address several relations in one query by listing them all in the FROM clause.

Distinguish attributes of the same name by “<relation>.<attribute>” .

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Example: Joining Two Relations

Using relations Likes(drinker, beer) and Frequents(drinker, bar), find the beers liked by at least one person who frequents Joe’s Bar.SELECT beerFROM Likes, FrequentsWHERE bar = ’Joe’’s Bar’ AND

Frequents.drinker = Likes.drinker;

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Formal Semantics

Almost the same as for single-relation queries:

1. Start with the product of all the relations in the FROM clause.

2. Apply the selection condition from the WHERE clause.

3. Project onto the list of attributes and expressions in the SELECT clause.

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Operational Semantics

Imagine one tuple-variable for each relation in the FROM clause. These tuple-variables visit each

combination of tuples, one from each relation.

If the tuple-variables are pointing to tuples that satisfy the WHERE clause, send these tuples to the SELECT clause.

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Example

drinker bar drinker beer

tv1 tv2Sally Bud

Sally Joe’s

Likes Frequents

to outputcheck theseare equal

checkfor Joe

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Explicit Tuple-Variables

Sometimes, a query needs to use two copies of the same relation.

Distinguish copies by following the relation name by the name of a tuple-variable, in the FROM clause.

It’s always an option to rename relations this way, even when not essential.

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Example: Self-Join

From Beers(name, manf), find all pairs of beers by the same manufacturer. Do not produce pairs like (Bud, Bud). Produce pairs in alphabetic order, e.g.

(Bud, Miller), not (Miller, Bud).

SELECT b1.name, b2.nameFROM Beers b1, Beers b2WHERE b1.manf = b2.manf AND

b1.name < b2.name;