relational algebra Tuple Relational Calculus - database management system

RELATIONAL ALGEBRA
AND TUPLE RELATIONAL
CALCULUS
9/27/2020 1

Relational Algebra
• Procedural Query Languages.
• Operation to be performed on exciting
relation to derive the result relation.
• Unary operation.
– Select
– Project
– Rename
• Binary operation.
– Cartesian product
– Union
– Set difference
• Additional Operation
– Intersection
– Join
– Division
– Assignment

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Division Operation (Cont.)
 Property
 Let q – r  s
 Then q is the largest relation satisfying q x s  r
 Definition in terms of the basic algebra operation
Let r(R) and s(S) be relations, and let S  R
r  s = R-S (r) –R-S ( (R-S (r) x s) – R-S,S(r))
To see why
 R-S,S(r) simply reorders attributes of r
 R-S(R-S (r) x s) – R-S,S(r)) gives those tuples t in
R-S (r) such that for some tuple u  s, tu  r.

9/27/2020 23
Division Operation – Example
Relations r, s: B
1
2
A B











1
2
3
1
1
1
3
4
6
1
2
r  s:
A


r
s

9/27/2020 24
Another Division Example
A B








a
a
a
a
a
a
a
a
C D








a
a
b
a
b
a
b
b
E
1
1
1
1
3
1
1
1
Relations r, s:
r  s:
D
a
b
E
1
1
A B


a
a
C


r
s

9/27/2020 25
Assignment Operation
 The assignment operation () provides a convenient way to express
complex queries.
 Write query as a sequential program consisting of
 a series of assignments
 followed by an expression whose value is displayed as a result
of the query.
 Assignment must always be made to a temporary relation variable.
 Example: Write r  s as
temp1  R-S (r)
temp2  R-S ((temp1 x s) – R-S,S (r))
result = temp1 – temp2
 The result to the right of the  is assigned to the relation variable on
the left of the .
 May use variable in subsequent expressions.

9/27/2020 26
Extended Relational-Algebra-
Operations
 Generalized Projection
 Outer Join
 Aggregate Functions

9/27/2020 27
Generalized Projection
 Extends the projection operation by allowing
arithmetic functions to be used in the projection
list.
 F1, F2, …, Fn(E)
 E is any relational-algebra expression
 Each of F1, F2, …, Fn are are arithmetic
expressions involving constants and attributes
in the schema of E.
 Given relation credit-info(customer-name,
limit, credit-balance), find how much more
each person can spend:
customer-name, limit – credit-balance (credit-info)

9/27/2020 28
Aggregate Functions and Operations
 Aggregation function takes a collection of values and
returns a single value as a result.
avg: average value
min: minimum value
max: maximum value
sum: sum of values
count: number of values
 Aggregate operation in relational algebra
G1, G2, …, Gn g F1( A1), F2( A2),…, Fn( An) (E)
 E is any relational-algebra expression
 G1, G2 …, Gn is a list of attributes on which to group
(can be empty)
 Each Fi is an aggregate function
 Each Ai is an attribute name

9/27/2020 29
Aggregate Operation –
Example
 Relation r: A B








C
7
7
3
10
g sum(c) (r) sum-C
27

9/27/2020 30
Aggregate Operation –
Example
 Relation account grouped by
branch-name:
branch-name g sum(balance) (account)
branch-name account-number balance
Perryridge
Perryridge
Brighton
Brighton
Redwood
A-102
A-201
A-217
A-215
A-222
400
900
750
750
700
branch-name balance
Perryridge
Brighton
Redwood
1300
1500
700

9/27/2020 31
Aggregate Functions (Cont.)
 Result of aggregation does not have a
name
 Can use rename operation to give it a
name
 For convenience, we permit renaming
as part of aggregate operation
branch-name g sum(balance) as sum-balance (account)

9/27/2020 32
Outer Join
 An extension of the join operation that avoids loss of
information.
 Computes the join and then adds tuples form one
relation that does not match tuples in the other
relation to the result of the join.
 Uses null values:
 null signifies that the value is unknown or does
not exist
 All comparisons involving null are (roughly
speaking) false by definition.
 Will study precise meaning of comparisons
with nulls later

9/27/2020 33
Outer Join – Example
 Relation loan
 Relation borrower
customer-name loan-number
Jones
Smith
Hayes
L-170
L-230
L-155
3000
4000
1700
loan-number amount
L-170
L-230
L-260
branch-name
Downtown
Redwood
Perryridge

9/27/2020 34
 Inner Join
loan Borrower
loan-number amount
L-170
L-230
3000
4000
customer-name
Jones
Smith
branch-name
Downtown
Redwood
Jones
Smith
null
loan-number amount
L-170
L-230
L-260
3000
4000
1700
customer-namebranch-name
Downtown
Redwood
Perryridge
 Left Outer Join
loan Borrower
loan Borrower

9/27/2020 35
 Right Outer Join
loan borrower
loan borrower
 Full Outer Join
loan-number amount
L-170
L-230
L-155
3000
4000
null
customer-name
Jones
Smith
Hayes
branch-name
Downtown
Redwood
null
loan-number amount
L-170
L-230
L-260
L-155
3000
4000
1700
null
customer-name
Jones
Smith
null
Hayes
branch-name
Downtown
Redwood
Perryridge
null
loan borrower

9/27/2020 36
Tuple Relational Calculus
 A nonprocedural query language, where each query
is of the form
{t | P (t) }
 It is the set of all tuples t such that
predicate P is true for t
 t is a tuple variable, t[A] denotes the value of tuple t
on attribute A
 t  r denotes that tuple t is in relation r
 P is a formula similar to that of the
predicate calculus

9/27/2020 37
Predicate Calculus Formula
1. Set of attributes and constants
2. Set of comparison operators: (e.g., , , , , , )
3. Set of connectives: and (), or (v)‚ not ()
4. Implication (): x  y, if x is true, then y is true
5. Set of quantifiers:
  t  r (Q(t))  ”there exists” a tuple in t in relation r
such that predicate Q(t) is true
 t r (Q(t)) Q is true “for all” tuples t in relation r
s[x] Ѳ u[y] : s, u are tuple variables. x,y are attributes
and Ѳ is comparison operator.
s[x] Ѳ c , where c is a constant.

9/27/2020 38
Banking Example
 branch (branch-name, branch-city, assets)
 customer (customer-name, customer-
street, customer-city)
 account (account-number, branch-name,
balance)
 loan (loan-number, branch-name, amount)
 depositor (customer-name, account-
number)
 borrower (customer-name, loan-number)

Example Queries
 Find all the tuples in loan
relation
 {t | t  loan}
 Find those tuples for which
loans of greater than 1200
 {t | t  loan  t [amount]  1200}
9/27/2020 39

9/27/2020 40
Example Queries
Find the loan number for each loan of an amount greater
than $1200
The set of all tuples t such that there exists a tuple s in relation
loan for which the values t and s for the loan-number attributes
are equal and the value of s for the amount attribute is greater
than $1200
Notice that a relation on schema [loan-number] is implicitly
defined by the query
{t |  s loan (t[loan-number] = s[loan-number]  s [amount]  1200)}

9/27/2020 41
Example Queries
 Find the names of all customers having a loan, an
account, or both at the bank
{t | s  borrower( t[customer-name] = s[customer-name])
 u  depositor( t[customer-name] = u[customer-name])
 Find the names of all customers who have a loan and
an account at the bank
{t | s  borrower( t[customer-name] = s[customer-name])
 u  depositor( t[customer-name] = u[customer-name])

9/27/2020 42
Safety of Expressions
 It is possible to write tuple calculus expressions that
generate infinite relations.
 For example, {t |  t r} results in an infinite relation if
the domain of any attribute of relation r is infinite
 To guard against the problem, we restrict the set of
allowable expressions to safe expressions.
 An expression {t | P(t)} in the tuple relational calculus is
safe if every component of t appears in one of the
relations, tuples, or constants that appear in P
 NOTE: this is more than just a syntax condition.
 E.g. { t | t[A]=5  true } is not safe --- it defines
an infinite set with attribute values that do not
appear in any relation or tuples or constants in P.

9/27/2020 43
Null Values
 It is possible for tuples to have a null value, denoted by null, for
some of their attributes
 null signifies an unknown value or that a value does not exist.
 The result of any arithmetic expression involving null is null.
 Aggregate functions simply ignore null values
 Is an arbitrary decision. Could have returned null as result
instead.
 We follow the semantics of SQL in its handling of null
values
 For duplicate elimination and grouping, null is treated like any
other value, and two nulls are assumed to be the same
 Alternative: assume each null is different from each other
 Both are arbitrary decisions, so we simply follow SQL

9/27/2020 44
Null Values
 Comparisons with null values return the special truth value
unknown
 If false was used instead of unknown, then not (A < 5)
would not be equivalent to A >= 5
 Three-valued logic using the truth value unknown:
 OR: (unknown or true) = true,
(unknown or false) = unknown
(unknown or unknown) = unknown
 AND: (true and unknown) = unknown,
(false and unknown) = false,
(unknown and unknown) = unknown
 NOT: (not unknown) = unknown
 In SQL “P is unknown” evaluates to true if predicate P
evaluates to unknown
 Result of select predicate is treated as false if it evaluates to
unknown

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