General Overview - rel. model. Carnegie Mellon Univ. Dept. of Computer Science /615 DB Applications. Motivation. Overview - detailed

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1 Carnegie Mellon Univ. Dep of Computer Science /615 DB Applications C. Faloutsos & A. Pavlo Lecture#5: Relational calculus General Overview - rel. model history concepts Formal query languages relational algebra rel. tuple calculus rel. domain calculus Faloutsos - Pavlo /615 #2 Overview - detailed rel. tuple calculus why? details examples equivalence with rel. algebra more examples; safety of expressions rel. domain calculus + QBE Motivation Q: weakness of rel. algebra? A: procedural describes the steps (ie., how ) (still useful, for query optimization) Faloutsos - Pavlo /615 #3 Faloutsos - Pavlo /615 #4 1

2 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 optimizatio later) Rel. tuple calculus (RTC) first order logic { t P( t)} Give me tuples t, satisfying predicate P - eg: { t t STUDENT} Faloutsos - Pavlo /615 #5 Faloutsos - Pavlo /615 #6 Details Specifically symbols allowed:,,, >, <, =, (, ),,,, Atom t TABLE attr const attr s. attr' quantifiers, Faloutsos - Pavlo /615 #7 Faloutsos - Pavlo /615 #8 2

3 Specifically Formula: atom if P1, P2 are formulas, so are if P(s) is a formula, so are P1 P2; P1 P2... s( P( s)) s( P( s)) Specifically Reminders: DeMorgan P 1 P2 ( P1 P2) implication: P1 P2 P1 P2 double negation: s TABLE ( P( s)) s TABLE ( P( s)) every human is mortal : no human is immortal Faloutsos - Pavlo /615 #9 Faloutsos - Pavlo /615 #10 Reminder: our Mini-U db STUDENT Ssn Name Address 123 smith main str 234 jones forbes ave CLASS c-id c-name units s.e o.s. 2 find all student records { t t STUDENT} TAKES SSN c-id grade A B output tuple of type STUDENT Faloutsos - Pavlo /615 #11 Faloutsos - Pavlo /615 #12 3

4 (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?) FUNDAMENTAL Relational operators selection projection cartesian product set union set difference σ condition π att list MALE x FEMALE R U S R - S (R) (R) Faloutsos - Pavlo /615 #13 Faloutsos - Pavlo /615 #14 (selection) find student record with ssn=123 σ selection π projection X cartesian product U set union - set difference (selection) find student record with ssn=123 { t t STUDENT ssn = 123} Faloutsos - Pavlo /615 #15 Faloutsos - Pavlo /615 #16 4

5 (projection) find name of student with ssn=123 { t t STUDENT ssn = 123} (projection) find name of student with ssn=123 { t s STUDENT ( s. ssn = 123 name = s. name)} σ selection π projection X cartesian product U set union - set difference t has only one column Faloutsos - Pavlo /615 #17 Faloutsos - Pavlo /615 #18 Tracing { t s STUDENT ( s. ssn = 123 name = s. name)} cont d (union) get records of both PT and FT students σ selection π projection X cartesian product U set union - set difference t Name aaaa. jones zzzz s STUDENT Ssn Name Address 123 smith main str 234 jones forbes ave Faloutsos - Pavlo /615 #19 Faloutsos - Pavlo /615 #20 5

6 cont d (union) get records of both PT and FT students { t t FT _ STUDENT t PT _ STUDENT} difference: find students that are not staff σ selection π projection X cartesian product U set union - set difference (assuming that STUDENT and STAFF are union-compatible) Faloutsos - Pavlo /615 #21 Faloutsos - Pavlo /615 #22 difference: find students that are not staff { t t STUDENT t STAFF} Cartesian product eg., dog-breeding: MALE x FEMALE gives all possible couples MALE name spike spot FEMALE x name = lassie shiba M.name spike spike spot spot F.name lassie shiba lassie shiba Faloutsos - Pavlo /615 #23 Faloutsos - Pavlo /615 #24 6

7 Cartesian product find all the pairs of (male, female) { t m MALE f FEMALE m name = m. name f name = f. name} σ selection π projection X cartesian product U set union - set difference Proof of equivalence rel. algebra <-> rel. tuple calculus σ selection π projection X cartesian product U set union - set difference Faloutsos - Pavlo /615 #25 Faloutsos - Pavlo /615 #26 Overview - detailed rel. tuple calculus why? details examples equivalence with rel. algebra more examples; safety of expressions re. domain calculus + QBE More examples join: find names of students taking Faloutsos - Pavlo /615 #27 Faloutsos - Pavlo /615 #28 7

8 Reminder: our Mini-U db STUDENT Ssn Name Address 123 smith main str 234 jones forbes ave TAKES SSN c-id grade A B CLASS c-id c-name units s.e o.s. 2 More examples join: find names of students taking { t s STUDENT e TAKES ( s. ssn = e. ssn name = s. name e. c id = )} Faloutsos - Pavlo /615 #29 Faloutsos - Pavlo /615 #30 More examples join: find names of students taking { t s STUDENT e TAKES ( s. ssn = e. ssn name = s. name e. c id = )} projection selection join More examples 3-way join: find names of students taking a 2-unit course (Remember: SPJ ) Faloutsos - Pavlo /615 #31 Faloutsos - Pavlo /615 #32 8

9 Reminder: our Mini-U db STUDENT Ssn Name Address 123 smith main str 234 jones forbes ave TAKES SSN c-id grade A B CLASS c-id c-name units s.e o.s. 2 Faloutsos - Pavlo /615 #33 More examples 3-way join: find names of students taking a 2-unit course { t s STUDENT e TAKES c CLASS ( s. ssn = e. ssn e. c id = c. c id name = s. name c. units = 2)} projection selection join Faloutsos - Pavlo /615 #34 More examples 3-way join: find names of students taking a 2-unit course - in rel. algebra?? More examples 3-way join: find names of students taking a 2-unit course - in rel. algebra?? π name ( σ 2( STUDENT TAKES CLASS )) units = Faloutsos - Pavlo /615 #35 Faloutsos - Pavlo /615 #36 9

10 Even more examples: self -joins: find Tom s grandparent(s) Even more examples: self -joins: find Tom s grandparent(s) PC p-id Mary Peter John c-id Tom Mary Tom PC p-id Mary Peter John c-id Tom Mary Tom { t p PC q PC ( p. c id = q. p id p. p id = p id q. c id = " Tom")} Faloutsos - Pavlo /615 #37 Faloutsos - Pavlo /615 #38 Hard examples: DIVISION! Hard examples: DIVISION! find suppliers that shipped all the ABOMB parts SHIPMENT s# p# s1 p1 s2 p1 s1 p2 s3 p1 s5 p3 ABOMB p# p1 p2 = BAD_S s# s1 find suppliers that shipped all the ABOMB parts {t p(p ABOMB ( s SHIPMENT ( s# = s.s# s.p# = p.p#)))} Faloutsos - Pavlo /615 #39 Faloutsos - Pavlo /615 #40 10

11 General pattern! General pattern! three equivalent versions: 1) if it s bad, he shipped it { t p( p ABOMB ( P( t))} 2)either it was good, or he shipped it { t p( p ABOMB ( P( t))} 3) there is no bad shipment that he missed { t p( p ABOMB ( P( t))} three equivalent versions: 1) if it s bad, he shipped it { t p( p ABOMB ( P( t))} 2)either it was good, or he shipped it {t p(p ABOMB (P(t))} 3) there is no bad shipment that he missed { t p( p ABOMB ( P( t))} a b a b Faloutsos - Pavlo /615 #41 Faloutsos - Pavlo /615 #42 General pattern three equivalent versions: 1) if it s bad, he shipped it { t p( p ABOMB ( P( t))} 2)either it was good, or he shipped it {t p(p ABOMB (P(t))} 3) there is no bad shipment that he missed { t p( p ABOMB ( P( t))} Faloutsos - Pavlo /615 #43! x(p(x)) x( P(x)) a T F a b is the same as a b T T T b F T F 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. Faloutsos - Pavlo /615 #44 11

12 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 More on division find students that take all the courses that ssn=123 does (and maybe even more) { o t(( t TAKES ssn = 123) t1 TAKES( )} t1. c id = c id t1. ssn = o. ssn) Faloutsos - Pavlo /615 #45 Faloutsos - Pavlo /615 #46 FORBIDDEN: Safety of expressions It has infinite output!! Instead, always use { t t STUDENT} { t... t SOME TABLE} Overview - conclusions rel. tuple calculus: DECLARATIVE dfn details equivalence to rel. algebra rel. domain calculus + QBE Faloutsos - Pavlo /615 #47 Faloutsos - Pavlo /615 #48 12

13 General Overview relational model Formal query languages relational algebra rel. tuple calculus rel. domain calculus 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 /615 #49 Faloutsos - Pavlo /615 #50 Rel. Dom. Calculus find STUDENT record with ssn=123 { < s, a >< s, a > STUDENT s = 123} Details Like R.T.C - symbols allowed:,,, >, <, =,,,, (, ), quantifiers, Faloutsos - Pavlo /615 #51 Faloutsos - Pavlo /615 #52 13

14 Details but: domain (= column) variables, as opposed to tuple variables, eg: ssn < s, a > STUDENT name address Reminder: our Mini-U db STUDENT Ssn Name Address 123 smith main str 234 jones forbes ave TAKES SSN c-id grade A B CLASS c-id c-name units s.e o.s. 2 Faloutsos - Pavlo /615 #53 Faloutsos - Pavlo /615 #54 find all student records (selection) find student record with ssn=123 { < s, a >< s, a > STUDENT } RTC: { t t STUDENT} Faloutsos - Pavlo /615 #55 Faloutsos - Pavlo /615 #56 14

15 ( Proof of RDC = RA) Agai we show examples of the 5 fundamental operators, in RDC (selection) find student record with ssn=123 RTC: { t t STUDENT ssn = 123} Faloutsos - Pavlo /615 #57 Faloutsos - Pavlo /615 #58 (selection) find student record with ssn=123 { < 123, n, a >< 123, a > STUDENT } or { < s, a >< s, a > STUDENT s = 123} (projection) find name of student with ssn=123 { < n > < 123, a > STUDENT } RTC: { t t STUDENT ssn = 123} Faloutsos - Pavlo /615 #59 Faloutsos - Pavlo /615 #60 15

16 (projection) find name of student with ssn=123 { < n > a( < 123, a > STUDENT )} cont d (union) get records of both PT and FT students need to restrict a RTC: { t s STUDENT ( s. ssn = 123 name = s. name)} RTC: { t t FT _ STUDENT t PT _ STUDENT} Faloutsos - Pavlo /615 #61 Faloutsos - Pavlo /615 #62 cont d (union) get records of both PT and FT students difference: find students that are not staff { < s, a > < s, a > FT _ STUDENT < s, a > PT _ STUDENT} RTC: { t t STUDENT t STAFF} Faloutsos - Pavlo /615 #63 Faloutsos - Pavlo /615 #64 16

17 difference: find students that are not staff { < s, a > < s, a > STUDENT < s, a > STAFF} Cartesian product eg., dog-breeding: MALE x FEMALE gives all possible couples MALE name spike spot FEMALE x name = lassie shiba M.name spike spike spot spot F.name lassie shiba lassie shiba Faloutsos - Pavlo /615 #65 Faloutsos - Pavlo /615 #66 Cartesian product find all the pairs of (male, female) - RTC: Cartesian product find all the pairs of (male, female) - RDC: { t m MALE f FEMALE m name = m. name f name = f. name} Faloutsos - Pavlo /615 #67 Faloutsos - Pavlo /615 #68 17

18 Cartesian product find all the pairs of (male, female) - RDC: Proof of equivalence rel. algebra <-> rel. domain calculus <-> rel. tuple calculus {< m, f > < m > MALE < f > FEMALE } Faloutsos - Pavlo /615 #69 Faloutsos - Pavlo /615 #70 Overview - detailed rel. domain calculus why? details examples equivalence with rel. algebra more examples; safety of expressions More examples join: find names of students taking Faloutsos - Pavlo /615 #71 Faloutsos - Pavlo /615 #72 18

19 Reminder: our Mini-U db STUDENT Ssn Name Address 123 smith main str 234 jones forbes ave TAKES SSN c-id grade A B CLASS c-id c-name units s.e o.s. 2 More examples join: find names of students taking in RTC { t s STUDENT e TAKES ( s. ssn = e. ssn name = s. name e. c id = )} Faloutsos - Pavlo /615 #73 Faloutsos - Pavlo /615 #74 More examples join: find names of students taking in RDC Sneak preview of QBE: { < n > s a g( < s, a > STUDENT < s,15 415, g > TAKES)} { < n > s a g( < s, a > STUDENT < s,15 415, g > TAKES)} STUDENT Ssn Name Address _x P. TAKES SSN c-id grade _x Faloutsos - Pavlo /615 #75 Faloutsos - Pavlo /615 #76 19

20 STUDENT Ssn Name Address _x P. Glimpse of QBE: very user friendly heavily based on RDC very similar to MS Access interface and pgadminiii TAKES SSN c-id grade _x Faloutsos - Pavlo /615 #77 More examples 3-way join: find names of students taking a 2-unit course - in RTC: { t s STUDENT e TAKES c CLASS ( s. ssn = e. ssn e. c id = c. c id name = s. name c. units = 2)} projection selection join Faloutsos - Pavlo /615 #78 Reminder: our Mini-U db _x.p STUDENT Ssn Name Address 123 smith main str 234 jones forbes ave TAKES SSN c-id grade A B _x _y _y 2 CLASS c-id c-name units s.e o.s. 2 Faloutsos - Pavlo /615 #79 More examples 3-way join: find names of students taking a 2-unit course { < n >... < s, a > STUDENT < s, c, g > TAKES < c, c2 > CLASS} Faloutsos - Pavlo /615 #80 20

21 More examples 3-way join: find names of students taking a 2-unit course { < n > s, a, c, g, cn( < s, a > STUDENT < s, c, g > TAKES < c, c2 > CLASS )} Faloutsos - Pavlo /615 #81 Even more examples: self -joins: find Tom s grandparent(s) PC p-id c-id PC p-id c-id Mary Tom Mary Tom Peter Mary Peter Mary John Tom John Tom Faloutsos - Pavlo /615 #82 Even more examples: self -joins: find Tom s grandparent(s) Even more examples: self -joins: find Tom s grandparent(s) { t p PC q PC ( p. c id = q. p id p. p id = p id q. c id = " Tom")} { t p PC q PC ( p. c id = q. p id p. p id = p id q. c id = " Tom")} { < g > p( < g, p > PC < p," Tom" > PC)} Faloutsos - Pavlo /615 #83 Faloutsos - Pavlo /615 #84 21

22 Even more examples: self -joins: find Tom s grandparent(s) { < g > p( < g, p > PC < p," Tom" > PC)} Hard examples: DIVISION find suppliers that shipped all the ABOMB parts SHIPMENT s# p# s1 p1 s2 p1 s1 p2 s3 p1 s5 p3 ABOMB p# p1 p2 = BAD_S s# s1 Faloutsos - Pavlo /615 #85 Faloutsos - Pavlo /615 #86 Hard examples: DIVISION find suppliers that shipped all the ABOMB parts Hard examples: DIVISION find suppliers that shipped all the ABOMB parts {t p(p ABOMB ( s SHIPMENT ( s# = s.s# s.p# = p.p# )))} {t p(p ABOMB ( s SHIPMENT ( s# = s.s# s.p# = p.p# )))} {< s > p(< p > ABOMB < s, p > SHIPMENT )} Faloutsos - Pavlo /615 #87 Faloutsos - Pavlo /615 #88 22

23 More on division find students that take all the courses that ssn=123 does (and maybe even more) { o t(( t TAKES ssn = 123) t1 TAKES( )} t1. c id = c id t1. ssn = o. ssn) More on division find students that take all the courses that ssn=123 does (and maybe even more) { < s > c( g( < 123, c, g > TAKES ) g'( < s, c, g' > ) TAKES ))} Faloutsos - Pavlo /615 #89 Faloutsos - Pavlo /615 #90 similar to RTC FORBIDDEN: Safety of expressions {< s,a > < s,a > STUDENT } Overview - detailed rel. domain calculus + QBE dfn details equivalence to rel. algebra Faloutsos - Pavlo /615 #91 Faloutsos - Pavlo /615 #92 23

24 Fun Drill:Your turn Your turn Schema: Movie(title, year, studioname) ActsIn(movieTitle, starname) Star(name, gender, birthdate, salary) 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 /615 #93 Faloutsos - Pavlo /615 #94 Answers Answers Find all movies by Paramount studio Movies starring Kevin Bacon {M M Movie M.studioName = Paramount } {M M Movie A ActsIn(A.movieTitle = M.title A.starName = Bacon ))} Faloutsos - Pavlo /615 #95 Faloutsos - Pavlo /615 #96 24

25 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 A: A2: movie movie Faloutsos - Pavlo /615 #97 star star Bacon 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 /615 #98 Two degrees: Two degrees: S: name S: name A: movie star A2: movie star A3: movie star A3: movie star A4: movie star Bacon A4: movie star Bacon Faloutsos - Pavlo /615 #99 Faloutsos - Pavlo /615 #100 25

26 Answers Expressive Power 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 /615 #101 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 /615 #102 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 i 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 /615 #103 Faloutsos - Pavlo /615 #104 26

Database Applications (15-415)

Database Applications (15-415) Relational Calculus Lecture 6, January 26, 2016 Mohammad Hammoud Today Last Session: Relational Algebra Today s Session: Relational calculus Relational tuple calculus Announcements: