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Lecture#5: Relational calculus

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1 Lecture#5: Relational calculus
Faloutsos - Pavlo CMU /615 Carnegie Mellon Univ. Dept. of Computer Science /615 – DB Applications C. Faloutsos & A. Pavlo Lecture#5: Relational calculus

2 General Overview - rel. model
Faloutsos - Pavlo CMU /615 General Overview - rel. model history concepts Formal query languages relational algebra rel. tuple calculus rel. domain calculus Faloutsos - Pavlo CMU SCS /615

3 Overview - detailed rel. tuple calculus rel. domain calculus + QBE
why? details examples equivalence with rel. algebra more examples; ‘safety’ of expressions rel. domain calculus + QBE Faloutsos - Pavlo CMU SCS /615

4 Motivation Q: weakness of rel. algebra? A: procedural
describes the steps (ie., ‘how’) (still useful, for query optimization) Faloutsos - Pavlo CMU SCS /615

5 Solution: rel. calculus
describes what we want two equivalent flavors: ‘tuple’ and ‘domain’ calculus basis for SQL and QBE, resp. Useful for proofs (see query optimization, later) Faloutsos - Pavlo CMU SCS /615

6 Rel. tuple calculus (RTC)
first order logic ‘Give me tuples ‘t’, satisfying predicate P - eg: Faloutsos - Pavlo CMU SCS /615

7 Details symbols allowed: quantifiers Faloutsos - Pavlo
CMU SCS /615

8 Specifically Atom Faloutsos - Pavlo CMU SCS /615

9 Specifically Formula: atom if P1, P2 are formulas, so are
if P(s) is a formula, so are Faloutsos - Pavlo CMU SCS /615

10 Specifically Reminders: DeMorgan implication: double negation:
‘every human is mortal : no human is immortal’ Faloutsos - Pavlo CMU SCS /615

11 Reminder: our Mini-U db
Faloutsos - Pavlo CMU SCS /615

12 Examples find all student records output tuple of type ‘STUDENT’
Faloutsos - Pavlo CMU SCS /615

13 (Goal: evidence that RTC = RA)
Full proof: complicated We’ll just show examples of the 5 RA fundamental operators, and how RTC can handle them (Quiz: which are the 5 fundamental op’s?) Faloutsos - Pavlo CMU SCS /615

14 FUNDAMENTAL Relational operators
Faloutsos - Pavlo CMU SCS /615 FUNDAMENTAL Relational operators selection projection cartesian product MALE x FEMALE set union set difference R - S R U S Faloutsos - Pavlo CMU SCS /615

15 Examples (selection) find student record with ssn=123
Faloutsos - Pavlo CMU SCS /615

16 Examples (selection) find student record with ssn=123 ✔ s selection
p projection X cartesian product U set union - set difference Examples (selection) find student record with ssn=123 Faloutsos - Pavlo CMU SCS /615

17 Examples (projection) find name of student with ssn=123
Faloutsos - Pavlo CMU SCS /615

18 Examples (projection) find name of student with ssn=123 ✔ ✔
s selection p projection X cartesian product U set union - set difference ✔ Examples (projection) find name of student with ssn=123 ‘t’ has only one column Faloutsos - Pavlo CMU SCS /615

19 ‘Tracing’ t s Faloutsos - Pavlo CMU SCS /615

20 Examples cont’d (union) get records of both PT and FT students ✔ ✔ ✔
s selection p projection X cartesian product U set union - set difference ✔ Examples cont’d ✔ (union) get records of both PT and FT students Faloutsos - Pavlo CMU SCS /615

21 Examples cont’d (union) get records of both PT and FT students
Faloutsos - Pavlo CMU SCS /615

22 Examples difference: find students that are not staff ✔ ✔ ✔ ✔
s selection p projection X cartesian product U set union - set difference ✔ Examples ✔ ✔ difference: find students that are not staff (assuming that STUDENT and STAFF are union-compatible) Faloutsos - Pavlo CMU SCS /615

23 Examples difference: find students that are not staff
Faloutsos - Pavlo CMU SCS /615

24 Cartesian product eg., dog-breeding: MALE x FEMALE
gives all possible couples = x Faloutsos - Pavlo CMU SCS /615

25 Cartesian product find all the pairs of (male, female) ✔ ✔ ✔ ✔ ✔
s selection p projection X cartesian product U set union - set difference ✔ Cartesian product ✔ ✔ ✔ find all the pairs of (male, female) Faloutsos - Pavlo CMU SCS /615

26 ‘Proof’ of equivalence
rel. algebra <-> rel. tuple calculus ✔ s selection p projection X cartesian product U set union - set difference ✔ ✔ ✔ ✔ Faloutsos - Pavlo CMU SCS /615

27 Overview - detailed rel. tuple calculus re. domain calculus + QBE why?
details examples equivalence with rel. algebra more examples; ‘safety’ of expressions re. domain calculus + QBE Faloutsos - Pavlo CMU SCS /615

28 More examples join: find names of students taking 15-415
Faloutsos - Pavlo CMU SCS /615

29 Reminder: our Mini-U db
Faloutsos - Pavlo CMU SCS /615

30 More examples join: find names of students taking 15-415
Faloutsos - Pavlo CMU SCS /615

31 More examples join: find names of students taking 15-415 join
projection selection (Remember: ‘SPJ’ ) Faloutsos - Pavlo CMU SCS /615

32 More examples 3-way join: find names of students taking a 2-unit course Faloutsos - Pavlo CMU SCS /615

33 Reminder: our Mini-U db
Faloutsos - Pavlo CMU SCS /615

34 More examples 3-way join: find names of students taking a 2-unit course join projection selection Faloutsos - Pavlo CMU SCS /615

35 More examples 3-way join: find names of students taking a 2-unit course - in rel. algebra?? Faloutsos - Pavlo CMU SCS /615

36 More examples 3-way join: find names of students taking a 2-unit course - in rel. algebra?? Faloutsos - Pavlo CMU SCS /615

37 Even more examples: self -joins: find Tom’s grandparent(s)
Faloutsos - Pavlo CMU SCS /615

38 Even more examples: self -joins: find Tom’s grandparent(s)
Faloutsos - Pavlo CMU SCS /615

39 Hard examples: DIVISION
! Hard examples: DIVISION find suppliers that shipped all the ABOMB parts Faloutsos - Pavlo CMU SCS /615

40 Hard examples: DIVISION
! Hard examples: DIVISION find suppliers that shipped all the ABOMB parts Faloutsos - Pavlo CMU SCS /615

41 General pattern ! three equivalent versions:
1) if it’s bad, he shipped it 2)either it was good, or he shipped it 3) there is no bad shipment that he missed Faloutsos - Pavlo CMU SCS /615

42 General pattern ! three equivalent versions:
1) if it’s bad, he shipped it 2)either it was good, or he shipped it 3) there is no bad shipment that he missed a  b  a  b Faloutsos - Pavlo CMU SCS /615

43 General pattern ! three equivalent versions:
1) if it’s bad, he shipped it 2)either it was good, or he shipped it 3) there is no bad shipment that he missed Faloutsos - Pavlo CMU SCS /615

44 a  b is the same as a  b b T F T T F a F T T
If a is true, b must be true for the implication to be true. If a is true and b is false, the implication evaluates to false. If a is not true, we don’t care about b, the expression is always true. T F T T F a F T T Faloutsos - Pavlo CMU SCS /615

45 More on division find (SSNs of) students that take all the courses that ssn=123 does (and maybe even more) find students ‘s’ so that if 123 takes a course => so does ‘s’ Faloutsos - Pavlo CMU SCS /615

46 More on division find students that take all the courses that ssn=123 does (and maybe even more) Faloutsos - Pavlo CMU SCS /615

47 Safety of expressions FORBIDDEN: It has infinite output!!
Instead, always use Faloutsos - Pavlo CMU SCS /615

48 Overview - conclusions
rel. tuple calculus: DECLARATIVE dfn details equivalence to rel. algebra rel. domain calculus + QBE Faloutsos - Pavlo CMU SCS /615

49 General Overview relational model Formal query languages
relational algebra rel. tuple calculus rel. domain calculus Faloutsos - Pavlo CMU SCS /615

50 Rel. domain calculus (RDC)
Q: why? A: slightly easier than RTC, although equivalent - basis for QBE. idea: domain variables (w/ F.O.L.) - eg: ‘find STUDENT record with ssn=123’ Faloutsos - Pavlo CMU SCS /615

51 Rel. Dom. Calculus find STUDENT record with ssn=123’ Faloutsos - Pavlo
CMU /615 Rel. Dom. Calculus find STUDENT record with ssn=123’ solutions: {<a,b,c>| <a,b,c> in STUDENT} Faloutsos - Pavlo CMU SCS /615

52 Details Like R.T.C - symbols allowed: quantifiers Faloutsos - Pavlo
CMU SCS /615

53 Details but: domain (= column) variables, as opposed to tuple variables, eg: ssn address name Faloutsos - Pavlo CMU SCS /615

54 Reminder: our Mini-U db
Faloutsos - Pavlo CMU SCS /615

55 Examples find all student records RTC: Faloutsos - Pavlo
CMU SCS /615

56 Examples (selection) find student record with ssn=123
Faloutsos - Pavlo CMU SCS /615

57 (‘Proof’ of RDC = RA) Again, we show examples of the 5 fundamental operators, in RDC Faloutsos - Pavlo CMU SCS /615

58 Examples (selection) find student record with ssn=123 RTC:
Faloutsos - Pavlo CMU SCS /615

59 Examples (selection) find student record with ssn=123 or RTC:
Faloutsos - Pavlo CMU SCS /615

60 Examples (projection) find name of student with ssn=123
Faloutsos - Pavlo CMU SCS /615

61 Examples (projection) find name of student with ssn=123
need to ‘restrict’ “a” RTC: Faloutsos - Pavlo CMU SCS /615

62 Examples cont’d (union) get records of both PT and FT students RTC:
Faloutsos - Pavlo CMU SCS /615

63 Examples cont’d (union) get records of both PT and FT students
Faloutsos - Pavlo CMU SCS /615

64 Examples difference: find students that are not staff RTC:
Faloutsos - Pavlo CMU SCS /615

65 Examples difference: find students that are not staff
Faloutsos - Pavlo CMU SCS /615

66 Cartesian product eg., dog-breeding: MALE x FEMALE
gives all possible couples = x Faloutsos - Pavlo CMU SCS /615

67 Cartesian product find all the pairs of (male, female) - RTC:
Faloutsos - Pavlo CMU SCS /615

68 Cartesian product find all the pairs of (male, female) - RDC:
Faloutsos - Pavlo CMU SCS /615

69 Cartesian product find all the pairs of (male, female) - RDC:
Faloutsos - Pavlo CMU SCS /615

70 ‘Proof’ of equivalence
rel. algebra <-> rel. domain calculus <-> rel. tuple calculus Faloutsos - Pavlo CMU SCS /615

71 Overview - detailed rel. domain calculus why? details examples
equivalence with rel. algebra more examples; ‘safety’ of expressions Faloutsos - Pavlo CMU SCS /615

72 More examples join: find names of students taking 15-415
Faloutsos - Pavlo CMU SCS /615

73 Reminder: our Mini-U db
Faloutsos - Pavlo CMU SCS /615

74 More examples join: find names of students taking 15-415 - in RTC
Faloutsos - Pavlo CMU SCS /615

75 More examples join: find names of students taking 15-415 - in RDC
Faloutsos - Pavlo CMU SCS /615

76 Sneak preview of QBE: Faloutsos - Pavlo CMU SCS /615

77 Glimpse of QBE: very user friendly heavily based on RDC
very similar to MS Access interface and pgAdminIII Faloutsos - Pavlo CMU SCS /615

78 More examples 3-way join: find names of students taking a 2-unit course - in RTC: join projection selection Faloutsos - Pavlo CMU SCS /615

79 Reminder: our Mini-U db
_x .P _y 2 _x _y Faloutsos - Pavlo CMU SCS /615

80 More examples 3-way join: find names of students taking a 2-unit course Faloutsos - Pavlo CMU SCS /615

81 More examples 3-way join: find names of students taking a 2-unit course Faloutsos - Pavlo CMU SCS /615

82 Even more examples: self -joins: find Tom’s grandparent(s)
Faloutsos - Pavlo CMU SCS /615

83 Even more examples: self -joins: find Tom’s grandparent(s)
Faloutsos - Pavlo CMU SCS /615

84 Even more examples: self -joins: find Tom’s grandparent(s)
Faloutsos - Pavlo CMU SCS /615

85 Even more examples: self -joins: find Tom’s grandparent(s)
Faloutsos - Pavlo CMU SCS /615

86 Hard examples: DIVISION
find suppliers that shipped all the ABOMB parts Faloutsos - Pavlo CMU SCS /615

87 Hard examples: DIVISION
find suppliers that shipped all the ABOMB parts Faloutsos - Pavlo CMU SCS /615

88 Hard examples: DIVISION
find suppliers that shipped all the ABOMB parts Faloutsos - Pavlo CMU SCS /615

89 More on division find students that take all the courses that ssn=123 does (and maybe even more) Faloutsos - Pavlo CMU SCS /615

90 More on division find students that take all the courses that ssn=123 does (and maybe even more) Faloutsos - Pavlo CMU SCS /615

91 Safety of expressions similar to RTC FORBIDDEN: Faloutsos - Pavlo
CMU SCS /615

92 Overview - detailed rel. domain calculus + QBE dfn details
equivalence to rel. algebra Faloutsos - Pavlo CMU SCS /615

93 Fun Drill:Your turn … Schema: Movie(title, year, studioName)
ActsIn(movieTitle, starName) Star(name, gender, birthdate, salary) Faloutsos - Pavlo CMU SCS /615

94 Your turn … Queries to write in TRC:
Find all movies by Paramount studio … movies starring Kevin Bacon Find stars who have been in a film w/Kevin Bacon Stars within six degrees of Kevin Bacon* Stars connected to K. Bacon via any number of films** * Try two degrees for starters ** Good luck with this one! Faloutsos - Pavlo CMU SCS /615

95 Answers … {M | MMovie  M.studioName = ‘Paramount’}
Find all movies by Paramount studio {M | MMovie  M.studioName = ‘Paramount’} Faloutsos - Pavlo CMU SCS /615

96 Answers … Movies starring Kevin Bacon {M | MMovie  AActsIn(A.movieTitle = M.title  A.starName = ‘Bacon’))} Faloutsos - Pavlo CMU SCS /615

97 Answers … Stars who have been in a film w/Kevin Bacon {S | SStar 
AActsIn(A.starName = S.name  A2ActsIn(A2.movieTitle = A.movieTitle  A2.starName = ‘Bacon’))} S: name … movie star A: ‘Bacon’ movie star A2: Faloutsos - Pavlo CMU SCS /615

98 Answers … two Stars within six degrees of Kevin Bacon {S | SStar 
AActsIn(A.starName = S.name  A2ActsIn(A2.movieTitle = A.movieTitle  A3ActsIn(A3.starName = A2.starName  A4ActsIn(A4.movieTitle = A3.movieTitle  A4.starName = ‘Bacon’))} Faloutsos - Pavlo CMU SCS /615

99 Two degrees: S: A3: A4: name … movie star ‘Bacon’ movie star
Faloutsos - Pavlo CMU SCS /615

100 Two degrees: S: A: A2: A3: A4: name … movie star movie star ‘Bacon’
Faloutsos - Pavlo CMU SCS /615

101 Answers … Sorry … that was a trick question
Stars connected to K. Bacon via any number of films Sorry … that was a trick question Not expressible in relational calculus!! What about in relational algebra? No – RA, RTC, RDC are equivalent Faloutsos - Pavlo CMU SCS /615

102 Expressive Power Expressive Power (Theorem due to Codd):
Faloutsos - Pavlo Expressive Power CMU /615 Expressive Power (Theorem due to Codd): Every query that can be expressed in relational algebra can be expressed as a safe query in RDC / RTC; the converse is also true. Relational Completeness: Query language (e.g., SQL) can express every query that is expressible in relational algebra/calculus. (actually, SQL is more powerful, as we will see…) Faloutsos - Pavlo CMU SCS /615

103 Faloutsos - Pavlo CMU /615 Summary The relational model has rigorously defined query languages — simple and powerful. Relational algebra is more operational/procedural useful as internal representation for query evaluation plans Relational calculus is declarative users define queries in terms of what they want, not in terms of how to compute it. Faloutsos - Pavlo CMU SCS /615

104 Summary - cnt’d Several ways of expressing a given query
Faloutsos - Pavlo CMU /615 Summary - cnt’d Several ways of expressing a given query a query optimizer chooses best plan. Algebra and safe calculus: same expressive power => relational completeness. Faloutsos - Pavlo CMU SCS /615

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