Relational algebra in database management system

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.)
 Understanding Algebra & Calculus is key to
 understanding SQL, query processing!

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 Relational Algebra and SQL

Example Instances
sid sname rating age
22 dustin 7 45.0
31 lubber 8 55.5
58 rusty 10 35.0
28 yuppy 9 35.0
31 lubber 8 55.5
44 guppy 5 35.0
58 rusty 10 35.0
sid bid day
22 101 10/10/96
58 103 11/12/96
R1
S1
S2
 “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.

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”.)





Projection
sname rating
yuppy 9
lubber 8
guppy 5
rusty 10
sname rating
S
,
( )
2
age
35.0
55.5
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
S
8
2
( )
28 yuppy 9 35.0
58 rusty 10 35.0
sname rating
yuppy 9
rusty 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?
22 dustin 7 45.0
31 lubber 8 55.5
58 rusty 10 35.0
44 guppy 5 35.0
28 yuppy 9 35.0
31 lubber 8 55.5
58 rusty 10 35.0
S S
1 2

S S
1 2

22 dustin 7 45.0
S S
1 2


1
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 R
1 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:

1
Joins
 Condition Join: R C S = C (R  S)
S1 S1.sid < R1.sid R1
 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.
(sid) sname rating age (sid) bid day
22 dustin 7 45.0 58 103 11/12/96
31 lubber 8 55.5 58 103 11/12/96

1
Joins
 Equi-Join: A special case of condition join where
the condition c contains only equalities and ^.
S1 sid R1
 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 10/10/96
58 rusty 10 35.0 103 11/12/96

1
Find names of sailors who’ve reserved boat #103
 Solution 1:  
sname bid
serves Sailors
(( Re ) )
103

 Solution 2:  
( , Re )
Temp serves
bid
1
103

 ( , )
Temp Temp Sailors
2 1
 sname Temp
( )
2
 Solution 3:  
sname bid
serves Sailors
( (Re ))
103


1
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:
   
sname sid bid color red
Boats s Sailors
( ((
' '
) Re ) )

 
 A query optimizer can find this given the first solution!

1
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 green
Boats
  
 sname Tempboats serves Sailors
( Re )
 
 Can also define Tempboats using union! (How?)
 What happens if is replaced by in this query?
 

1
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 red
Boats serves


 sname Tempred Tempgreen Sailors
(( ) )
 
  
( , ((
' '
) Re ))
Tempgreen
sid color green
Boats serves



1
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.

1
Tuple Relational Calculus
 Query has the form: { T | p(T)}
 Answer includes all tuples T that
make the formula p(T) be true.
 Formula is 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.

2
TRC Formulas
 Atomic formula:
– R  Rel, or R.a op S.b, or R.a op constant
– op is one of
 Formula:
– an atomic formula, or
– , where p and q are formulas, or
– , where variable X is free in p(X), or
– , where variable X is free in p(X)
  
, , , , ,
  
p p q p q
, ,
X p X
( ( ))
X p X
( ( ))

2
Free and Bound Variables
 The use of quantifiers X and X in a
formula is said to bind X.
– A variable that is not bound is free.
 Let us revisit the definition of a query: {T|p(T)}
 There is an important restriction: the variable
T that appears to the left of `|’ must be the only
free variable in the formula p(...).

2
Find all sailors with a rating above 7
 {S | S  Sailors ^ S.rating > 7}
 Query is evaluated on an instance of Sailors
 Tuple variable S is instantiated to each tuple of this
instance in turn, and the condition “S.rating > 7” is
applied to each such tuple.
 Answer contains all instances of S (which are tuples
of Sailors) satisfying the condition.

2
Find sailors rated > 7 who’ve reserved boat #103
 {S | (S  Sailors) ^ (S.rating > 7) ^ ( R 
Reserves (R.sid = S.sid ^ R.bid = 103))}
 Note the use of  to find a tuple in Reserves
that `joins with’ the Sailors tuple under
consideration.
 R is bound, S is not

2
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
|  



















2
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.
 Relational calculus is non-operational, and users
define queries in terms of what they want, not in
terms of how to compute it. (Declarativeness.)
 Several ways of expressing a given query; a query
optimizer should choose the most efficient version.
 Algebra and safe calculus have same expressive power,
leading to the notion of relational completeness.

2
Nested Relations
 Attributes can be scalar (as before) or relation-
valued
 Definition is recursive
 Example:
create table Book (title: string,
author:string, copies: (publ: string,
pages: integer, date: integer))
 “copies” is a relation-valued field

2
Nested Relational Algebra
 A spectrum of algebras can be defined
 At one end:
– Relational algebra plus nest () and unnest ():
If B =
title author copies
Moby Dick Melville
Marmion Scott { }
publ pages date
Prentice Hall
McGraw Hill
613 1971
542 1942

2
Nesting and Unnesting
 … then  (B, copies) =
title author publ pages date
Moby Dick Melville
Moby Dick Melville
Marmion Scott
Prentice Hall
McGraw Hill
613 1971
542 1942
null null null
 Nulls must be supported in algebra
  ( (B, copies), copies (publ, pages, date)) = B
 ,  inverses iff S  N
– S is set of scalar fields
– N is set of non-scalar fields
– This is called PNF (partitioned normal form)

2
Extending Relational Operators
 At other end of spectrum:
– Selection allows set comparators and constants and use of select, project
inside the formula
– Projection allows arbitrary NF2 algebra expression in addition to simple field
names
– Union, difference: recursive definitions
– Cross product: usually just relational.
 Example: retrieve title, number of pages of all books by Melville:
 [title, [pages](copies)]([author=‘Melville’](B))

3
Nested Relations Summary
 An early step on the way to OODBMS
 No products, only prototypes, but:
– Many ideas from NF2 relations have survived
– Collection types in SQL3 (nesting, unnesting)
– Algebra ideas useful for Object Database QP
 Can provide a more natural model of data
 Are a straightforward extension of relations:
– many solutions are thus also straightforward
– formal foundation of relational model remains

Relational algebra in database management system

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