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# 8.1 Introduction

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1. 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. Advanced Database System

2. 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) Advanced Database System

3. 8.2 Tuple Calculus • Syntax (See Page 215-216) • <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> ] Advanced Database System

4. 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 Advanced Database System

5. 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’) ) ); Advanced Database System

6. 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) Advanced Database System

7. 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’ ⇒ Sx, Px, and Spx are free 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. Advanced Database System

8. 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 Advanced Database System

9. 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’))) Advanced Database System

10. 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# Advanced Database System

11. 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 ) ) Advanced Database System

12. 8.4 Calculus vs. Algebra (Cont.) (See Page 226-228 & Fig. 8.1) Step 1: Advanced Database System

13. 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# Advanced Database System

14. 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 Advanced Database System

15. 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

16. 8.5 Computational Capabilities • Exam 1: {Px.P#, Px.Weight＊454 As Gmwt} Where Px.Weight＊454 > Weight(10000.0) • 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 Advanced Database System

17. 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#; Advanced Database System

18. 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 From Sp Group By Sp.P#; • Exam 9: Select Sp.P# From Sp Group By Sp.P# Having Count(Sp.S#) > 1; • Exam 10: Select Distinct S.Sname From S Where S.S# In • (Select Sp.S# • From Sp • Where Sp.P#=P#(‘P2’)); Advanced Database System

19. 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# • From 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’)); Advanced Database System

20. 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 S Where Not Exists (Select * From P Where Not Exists (Select * From Sp Where Sp.S#=S.S# 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(10000.0); Advanced Database System

21. 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

22. 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 (S {S# Sx, Sname Namex} And Sp {S# Sx, P# Px} And Sp {S# S#(‘S2’), P# Px}) Advanced Database System

23. 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} Then Sp {S# Sx, 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} 8.7 Domain Calculus (Cont.) Advanced Database System

24. 8.8 Query-By-Example • Exam 1: • Exam 2: • Exam 3: • Exam 4: Advanced Database System

25. 8.8 Query-By-Example (Cont.) • Exam 5: • Exam 6: • Exam 7: • Exam 8: Advanced Database System

26. 8.8 Query-By-Example (Cont.) • Exam 9: • Exam 10: • Exam 11: • Exam 12: Advanced Database System

27. 8.8 Query-By-Example (Cont.) • Exam 13: • Exam 14: • Exam 15: • Exam 16: Advanced Database System

28. 8.8 Query-By-Example (Cont.) • Exam 17: Advanced Database System

29. The End. Advanced Database System