2. Advanced Database System
8.1 Introduction
The algebra provides a set of explicit operators that can be used to tell the system how to construct some desired relation from certain given relations.
The calculus merely provides a notation for stating the definition of that desired relation in terms of those given relations.
The query “Get supplier numbers and cities for suppliers who supply part P2”
Algebra: 1. join supplier and shipment over S#.
2. restrict the result of that join to tuples for part P2.
3. project the result of that restriction over S# and City.
Calculus: Get S# and City for suppliers such that there exists a shipment
Sp with the same S# value and with P# value P2.
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3. Advanced Database System
8.1 Introduction (Cont.)
prescriptive (Algebra) vs. descriptive (Calculus)
procedural (Algebra) vs. nonprocedural (Calculus)
The algebra and the calculus are logically equivalent.
Relation calculus is based on a branch of mathematical logic called predicate calculus.
Tuple calculus: range variables over relations
Language called QUEL
Range of Sx Is S;
Retrieve (Sx.S#) Where Sx.City=“London”;
Domain calculus: range variables over domains
Language called Query-By-Example (QBE)
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4. Advanced Database System
8.2 Tuple Calculus
Syntax (See Page )
<relation exp> ::= Relation {<tuple exp commalist>}
| <relvar name>
| <relation op inv>
| <with exp>
| <introduced name>
| (<relation exp>)
<range var def> ::= Rangevar <range var name>
Ranges Over <relation exp commalist>;
<range attribute ref> ::= <range var name> . <attribute name>
[ As <attribute name> ]
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5. Advanced Database System
8.2 Tuple Calculus (Cont.)
<bool exp> ::= …all the usual possibilities, together with:
| <quantified bool exp>
<quantified bool exp> ::= <quantifier> <range var name>
( <bool exp> )
<quantifier> ::= Exists | Forall
<relation op inv> ::= <proto tuple> [ Where <bool exp> ]
<proto tuple> ::= …see the body of the text
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6. Advanced Database System
8.2 Tuple Calculus (Cont.)
Range variables
e.g. Rangevar Sx Ranges Over S;
Rangevar Spx Ranges Over Sp;
Rangevar Su Ranges Over
( Sx Where Sx.City = ‘London’ ),
( Sx Where Exists Spx ( Spx.S# = Sx.S# And
Spx.P# = P#(‘P1’) ) );
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7. Advanced Database System
8.2 Tuple Calculus (Cont.)
Free and bound variable references
Let V be a range variable
References to V in Not p
⇒ according as they are free or bound in p
References to V in p And q and p Or q
⇒ according as they are free or bound in p or q
References to V that are free in p are bound in
Exists V (p) and Forall V (p)
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8. Advanced Database System
8.2 Tuple Calculus (Cont.)
Examples
Sx.S#=S#(‘S1’)
Sx.S#=Spx.S#
Spx.P#≠Px.P#
⇒ Sx, Px, and Spx are free
Px.Weight < Weight(15.5) And Px.City=’Oslo’
Not (Sx.City=‘London’)
Sx.S#=Spx.S# And Spx.P#≠Px.P#
Px.Color=Color(‘Red’) Or Px.City=‘London’
Exists Spx (Spx.S#=Sx.S# And Spx.P#=P#(‘P2’))
Forall Px (Px.Color=Color(‘Red’))
⇒ Spx and Px are bound and Sx is free.
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9. Advanced Database System
8.2 Tuple Calculus (Cont.)
Quantifiers
Exists V ( p (V) )
⇒ False Or p(t1) Or ... Or p(tm)
Forall V ( p(V) )
⇒ True And p(t1) And ... And p(tm)
Free and bound variable references revisited
Exists x ( x > 3 ) ⇔ Exists y ( y > 3 )
Exists x ( x > 3 ) and x < 0 ⇔ Exists y ( y > 3 ) and x < 0
Exists y ( y > 3 ) and y < 0
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8.3 Examples
Exam 1: {Sx.S#, Sx.Status}
Where Sx.City=‘Paris’ And Sx.Status > 20
Exam 2: {Sx.S# As Sa, Sy.S# As Sb}
Where Sx.City=Sy.City And Sx.S# < Sy.S#
Exam 3: Sx
Where Exists Spx (Spx.S#=Sx.S# And Spx.P#=P#(‘P2’))
Exam 4: Sx.Sname
Where Exists Spx (Sx.S#=Spx.S# And
Exists Px (Px.P#=Spx.P# And
Px.Color=Color(‘Red’)))
Exam 5: Sx.Sname
Where Exists Spx ( Exists Spy (Sx.S#=Spx.S# And
(Spx.P#=Spy.P# And
Spy.S#=S#(‘S2’)))
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8.3 Examples (Cont.)
Exam 6: Sx.Sname
Where Forall Px ( Exists Spx (Spx.S#=Sx.S# And
Spx.P#=Px.P#))
Exam 7: Sx.Sname Where Not Exists Spx
(Spx.S#=Sx.S# And Spx.P#=P#(‘P2’))
Exam 8: Sx.S# Where Forall Spx (Spx.S#≠S#(‘S2’) Or
Exists Spy (Spy.S#=Sx.S# And
Spy.P#=Spx.P#))
If p Then q End If ⇔ (Not p) Or q
Exam 9: Rangevar Pu Ranges Over
(Px.P# Where Px.Weight > Weight(16.0)),
(Spx.P# Where Spx.S#=S#(‘S2’));
Pu.P#
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8.4 Calculus vs. Algebra
The algebra is at least as powerful as the calculus.
Codd‘s reduction algorithm:
by which an arbitrary expression of the calculus could be
reduced to a semantically equivalent expression of the algebra.
Example
Query: Get names and cities for suppliers who supply at least
one Athens project with at least 50 of every part.
A calculus expression:
{Sx.Sname, Sx.City} Where Exists Jx Forall Px Exists Spjx
(Jx.City=‘Athens’ And
Jx.J#=Spjx.J# And
Px.P#=Spjx.P# And
Sx.S#=Spjx.S# And
Spjx.Qty≧Qty ( 50 ) )
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13. 8.4 Calculus vs. Algebra (Cont.)
(See Page & Fig. 8.1)
Step 1:
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14. 8.4 Calculus vs. Algebra (Cont.)
Step 2: Cartesian product
5＊6＊2＊24 =1440 tuples
Step 3: Restriction
Jx.J#=Spjx.J# And Px.P#=Spjx.P# And Sx.S#=Spjx.S#
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15. 8.4 Calculus vs. Algebra (Cont.)
Step 4: Apply the quantifiers from right to left
Exists V ⇒ project the current intermediate result to eliminate all
attributes of relation r.
Forall V ⇒ divide the current intermediate result by the “restricted range”
relation associated with V.
Step 5: Projection
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16. 8.4 Calculus vs. Algebra (Cont.)
Why Codd defined precisely the eight algebraic operators?
1. Inherently implementable
2. A yardstick for measuring the expressive power of any given database language
A language is said to be relationally complete if it is at least as
powerful as the calculus.
Any given language L is complete, if it is sufficient to show that
L includes analogs of each of the eight algebraic operators.
e.g. SQL
Relational completeness does not necessarily imply any other
kind of completeness.
Advanced Database System

17. 8.5 Computational Capabilities
Exam 1: {Px.P#, Px.Weight＊454 As Gmwt}
Where Px.Weight＊454 > Weight( )
Exam 2: {Sx, ‘Supplier’ As Tag}
Exam 3: {Spx, Px.Weight＊Spx.Qty As Shipwt}
Where Px.P#=Spx.P#
Exam 4: {Px.P#, Sum(Spx Where Spx.P#=Px.P#, Qty) As Totqty}
Exam 5: Sum(Spx, Qty) As Grandtotal
Exam 6: {Sx.S#, Count(Spx Where Spx.S#=Sx.S#) As #_Of_Part}
Exam 7: Rangevar Py Ranges Over P;
Px.City Where Count (Py Where Py.City=Px.City
And Py.Color=(‘Red’)) > 5
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8.6 SQL Facilities
Exam 1: Select Px.Color, Px.City
From P As Px
Where Px.City <> ‘Paris’
And Px.Weight > Weight (10.0);
Exam 2: Select P.P#, P.Weight＊454 As Gmwt
From P;
Exam 3: 1. Select S.*, P.P#, P.Pname, P.Color, P.Weight
From S, P
Where S.City=P.City;
2. S Join P Using City;
3. S Natural Join P;
Exam 4: Select Distinct S.City As Scity, P.City As Pcity
From S Join SP Using S# Join P Using P#;
Exam5: Select A.S# As Sa, B.S# As Sb
From S As A, S As B
Where A.City=B.City
And A.S# < B.S#;
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19. Advanced Database System
8.6 SQL Facilities (Cont.)
Exam 6: Select Count(*) As N
From S;
Exam 7: Select Max(Sp.Qty) As Maxq, Min(Sp.Qty) As Minq
From Sp
Where Sp.P#=P#(‘P2’);
Exam 8: Select Sp.P#, Sum(Sp.Qty) As Totqty
Group By Sp.P#;
Exam 9: Select Sp.P#
Group By Sp.P#
Having Count(Sp.S#) > 1;
Exam 10: Select Distinct S.Sname
From S
Where S.S# In
(Select Sp.S#
Where Sp.P#=P#(‘P2’));
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20. Advanced Database System
8.6 SQL Facilities (Cont.)
Exam 11: Select Distinct S.Sname
From S
Where S.S# In
(Select Sp.S#
From Sp
Where Sp.P# In
(Select P.P#
From P
Where P.Color=Color(‘Red’)));
Exam 12: Select S.S#
Where S.Status <
(Select Max(S.Status)
From S);
Exam 13: Select Distinct S.Sname
From S
Where Exists
(Select *
From Sp
Where Sp.S#=S.S#
And Sp.P#=P#(‘P2’));
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21. Advanced Database System
8.6 SQL Facilities (Cont.)
Exam 14: Select Distinct S.Sname
From S
Where Not Exists
(Select *
From Sp
Where Sp.S#=S.S#
And Sp.P#=P#(‘P2’));
Exam 15: Select Distinct S.Sname
From P
And Sp.P#=P.P#));
Exam 16: Select P.P#
From P
Where P.Weight > P#(16.0)
Union
Select Sp.P#
From Sp
Where Sp.S#=S#(‘S2’)
Exam 17:
With T1 As
(Select P.P#, P.Weight＊454 As Gmwt
From P)
Select T1.P#, T1.Gmwt
From T1
Where T1.Gmwt > Weight( );
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22. Advanced Database System
8.7 Domain Calculus
The most immediately obvious difference between the domain and the tuple calculus is that the former supports an additional form of <bool exp> called a membership condition.
R { <pair commalist> }
e.g. Sx
Sx Where S {S# Sx}
Sx Where S {S# Sx, City ‘London’}
{Sx, Cityx} Where S {S# Sx, City Cityx}
And Sp {S# Sx, P# P#(‘P2’)}
{Sx, Px} Where S {S# Sx, City Cityx}
And P {P# Px, City Cityy}
And Cityx≠Cityy
Advanced Database System

23. 8.7 Domain Calculus (Cont.)
Exam 1: Sx Where Exists Statusx (Statusx > 20 And
S {S# Sx, Status Statusx, City ‘Paris’})
Exam 2: {Sx As Sa, Sy As Sb} Where Exists Cityz
(S {S# Sx, City Cityz} And
S {S# Sy, City Cityz} And
Sx < Sy)
Exam 3: Namex Where Exists Sx Exists Px
(S {S# Sx, Sname Namex} And
Sp {S# Sx, P# Px} And
P {P# Px, Color Color(‘Red’)})
Exam 4: Namex Where Exists Sx Exists Px
Sp {S# S#(‘S2’), P# Px})
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24. 8.7 Domain Calculus (Cont.)
Exam 5: Namex Where Exists Sx (S {S# Sx, Sname Namex} And
Forall Px (If P {P# Px}
Then Sp {S# Sx, P# Px}
End If))
Exam 6: Namex Where Exists Sx (S {S# Sx, Sname Namex} And
Not Sp {S# Sx, P# P#(‘P2’)})
Exam 7: Sx Where Forall Px (If Sp {S# S#(‘S2’), P# Px}
End If)
Exam 8: Px Where Exists Weightx
(P {P# Px, Weight Weightx} And
Weightx > Weight(16.0)) Or
Sp {S# S#(‘S2’), P# Px}
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25. Advanced Database System
8.8 Query-By-Example
Exam 1:
Exam 2:
Exam 3:
Exam 4: S S#
Sname
Status
City P.
> 20
Paris SP S# P#
Qty
UNQ. P. S S#
Status
City
P.AO(2).
P.DO(1).
Paris S S#
Status
City P.
Paris
> 20
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26. 8.8 Query-By-Example (Cont.)
Weight P.
>= 16.0
<= 19.0 P P#
Weight
Gmwt P.
_PW
P. _PW * 454 S S#
Sname
_SX P. SP S# P#
_SX P2 S S#
City
_SX
_CX P P#
City
_PX
_CX P.
_SX
_PX
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27. 8.8 Query-By-Example (Cont.)S S#
City
_SX
_CZ
_SY P.
_SX
_SY SP S# P#
Qty P2
_QX
P.SUM._QX SP S# P#
Qty
G.P.
_QY
P.SUM._QY SP S# P#
_SX
G.P.
Conditions
CNT._SX > 1
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28. 8.8 Query-By-Example (Cont.)
Weight
_PX
> 16.0 SP S# P# S2
_PY P.
_PX
_PY P P#
Pname
Color
Weight
City I. P7
24.0
Athens SP S# P#
Qty D.
> 300 P P#
Pname
Color
Weight
City P2
U.Yellow
_WT
U._WT+5
U.Oslo
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29. 8.8 Query-By-Example (Cont.)SP S#
Qty
_SX
U.5 S S#
City
_SX
London
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30. Advanced Database System
The End.
Advanced Database System