Faloutsos CMU - 15-415

CMU SCS CMU SCS

General Overview - rel. model Carnegie Mellon Univ. Dept. of Computer Science • history 15-415 - Database Applications • concepts • Formal query languages – relational algebra Lecture#6: Relational calculus – rel. tuple calculus – rel. domain calculus

Faloutsos CMU SCS 15-415 #2

CMU SCS CMU SCS

Overview - detailed Motivation

• rel. tuple calculus • Q: weakness of rel. algebra? – why? • A: procedural – details – describes the steps (ie., ‘how’) – examples – (still useful, for query optimization) – equivalence with rel. algebra – more examples; ‘safety’ of expressions • re. domain calculus + QBE

Faloutsos CMU SCS 15-415 #3 Faloutsos CMU SCS 15-415 #4

CMU SCS CMU SCS

Solution: rel. calculus Rel. tuple calculus (RTC)

– describes what we want • first order logic – two equivalent flavors: ‘tuple’ and ‘domain’ calculus {t | P(t)} – basis for SQL and QBE, resp. ‘Give me tuples ‘t’, satisfying predicate P - eg:

{t | t STUDENT}

Faloutsos CMU SCS 15-415 #5 Faloutsos CMU SCS 15-415 #6

1 Faloutsos CMU - 15-415

CMU SCS CMU SCS

Details Specifically

• symbols allowed: • Atom , , ,  t TABLE , , , , , , t.attr  const (, ),  t.attr  s.attr'

• quantifiers , 

Faloutsos CMU SCS 15-415 #7 Faloutsos CMU SCS 15-415 #8

CMU SCS CMU SCS

Specifically Specifically

• Formula: • Reminders: – atom – DeMorgan P1 P2  (P1 P2) – if P1, P2 are formulas, so are P1 P2;P1 P2... – implication: P1 P2  P1 P2 – if P(s) is a formula, so are s(P(s)) – double negation: s(P(s)) sTABLE (P(s))  sTABLE (P(s))

„every human is mortal : no human is immortal‟

Faloutsos CMU SCS 15-415 #9 Faloutsos CMU SCS 15-415 #10

CMU SCS CMU SCS

Reminder: our Mini-U db Examples

STUDENT CLASS • find all student records Ssn Name Address c-id c-name units 123 smith main str 15-413 s.e. 2 234 jones forbes ave 15-412 o.s. 2 {t | t STUDENT}

TAKES SSN c-id grade output 123 15-413 A of type „STUDENT‟ 234 15-413 B tuple

Faloutsos CMU SCS 15-415 #11 Faloutsos CMU SCS 15-415 #12

2 Faloutsos CMU - 15-415

CMU SCS CMU SCS

Examples Examples

• (selection) find student record with ssn=123 • (selection) find student record with ssn=123

{t | t STUDENT  t.ssn 123}

Faloutsos CMU SCS 15-415 #13 Faloutsos CMU SCS 15-415 #14

CMU SCS CMU SCS

Examples Examples

• (projection) find name of student with • (projection) find name of student with ssn=123 ssn=123 {t | t STUDENT  t.ssn 123} {t | s STUDENT(s.ssn 123  t.name  s.name)}

„t‟ has only one column

Faloutsos CMU SCS 15-415 #15 Faloutsos CMU SCS 15-415 #16

CMU SCS CMU SCS

‘Tracing’ Examples cont’d {t | s STUDENT(s.ssn 123  • (union) get records of both PT and FT t.name  s.name)} students

Name t aaaa …. STUDENT jones Ssn Name Address … s 123 smith main str zzzz 234 jones forbes ave

Faloutsos CMU SCS 15-415 #17 Faloutsos CMU SCS 15-415 #18

3 Faloutsos CMU - 15-415

CMU SCS CMU SCS

Examples cont’d Examples

• (union) get records of both PT and FT • difference: find students that are not staff students {t | t  FT _ STUDENT  t  PT _ STUDENT}

(assuming that STUDENT and STAFF are union-compatible)

Faloutsos CMU SCS 15-415 #19 Faloutsos CMU SCS 15-415 #20

CMU SCS CMU SCS

Examples Cartesian product

• difference: find students that are not staff • eg., dog-breeding: MALE x FEMALE • gives all possible couples {t | t STUDENT 

t STAFF} MALE FEMALE M.name F.name name x name = spike lassie spike lassie spike shiba spot shiba spot lassie spot shiba

Faloutsos CMU SCS 15-415 #21 Faloutsos CMU SCS 15-415 #22

CMU SCS CMU SCS

Cartesian product ‘Proof’ of equivalence

• find all the pairs of (male, female) • rel. algebra <-> rel. tuple calculus

{t | m MALE  f  FEMALE t.m  name  m.name  t. f  name  f .name}

Faloutsos CMU SCS 15-415 #23 Faloutsos CMU SCS 15-415 #24

4 Faloutsos CMU - 15-415

CMU SCS CMU SCS

Overview - detailed More examples

• rel. tuple calculus • join: find names of students taking 15-415 – why? – details – examples – equivalence with rel. algebra – more examples; ‘safety’ of expressions • re. domain calculus + QBE

Faloutsos CMU SCS 15-415 #25 Faloutsos CMU SCS 15-415 #26

CMU SCS CMU SCS

Reminder: our Mini-U db More examples

STUDENT CLASS • join: find names of students taking 15-415 Ssn Name Address c-id c-name units 123 smith main str 15-413 s.e. 2 {t | s  STUDENT 234 jones forbes ave 15-412 o.s. 2  eTAKES ( s.ssn  e.ssn 

TAKES t.name  s.name SSN c-id grade e.c  id 15  415)} 123 15-413 A 234 15-413 B

Faloutsos CMU SCS 15-415 #27 Faloutsos CMU SCS 15-415 #28

CMU SCS CMU SCS

More examples More examples

• join: find names of students taking 15-415 • 3-way join: find names of students taking a 2-unit course {t | s  STUDENT  eTAKES ( s.ssn  e.ssn  join t.name  s.name projection e.c  id 15  415)} selection

Faloutsos CMU SCS 15-415 #29 Faloutsos CMU SCS 15-415 #30

5 Faloutsos CMU - 15-415

CMU SCS CMU SCS

Reminder: our Mini-U db More examples

STUDENT CLASS • 3-way join: find names of students taking a Ssn Name Address c-id c-name units 2-unit course 123 smith main str 15-413 s.e. 2 234 jones forbes ave 15-412 o.s. 2 {t | s  STUDENT  eTAKES c CLASS( s.ssn  e.ssn  join TAKES e.c  id  c.c  id  SSN c-id grade t.name  s.name 123 15-413 A projection 234 15-413 B c.units  2)} selection

Faloutsos CMU SCS 15-415 #31 Faloutsos CMU SCS 15-415 #32

CMU SCS CMU SCS

More examples Even more examples:

• 3-way join: find names of students taking a • self -joins: find Tom’s grandparent(s) 2-unit course - in rel. algebra??

PC p-id c-id Mary Tom  name(units2 (STUDENT TAKES  CLASS)) Peter Mary John Tom

Faloutsos CMU SCS 15-415 #33 Faloutsos CMU SCS 15-415 #34

CMU SCS CMU SCS

Even more examples: Hard examples: DIVISION

• self -joins: find Tom’s grandparent(s) • find suppliers that shipped all the ABOMB parts SHIPMENT {t | p  PC  q  PC s# p# ABOMB BAD_S s1 p1 p# s# ( p.c  id  q.p  id  s2 p1   p1 s1 p.p  id  t.p  id  s1 p2 p2 s3 p1 q.c  id "Tom")} s5 p3

Faloutsos CMU SCS 15-415 #35 Faloutsos CMU SCS 15-415 #36

6 Faloutsos CMU - 15-415

CMU SCS CMU SCS

Hard examples: DIVISION General pattern

• find suppliers that shipped all the ABOMB • three equivalent versions: parts – 1) if it’s bad, he shipped it {t | p(p ABOMB (P(t))} {t | p( p  ABOMB  ( – 2)either it was good, or he shipped it s  SHIPMENT ( {t | p(p ABOMB (P(t))} t.s# s.s#  – 3) there is no bad shipment that he missed s.p# p.p#)))} {t | p(p ABOMB (P(t))}

Faloutsos CMU SCS 15-415 #37 Faloutsos CMU SCS 15-415 #38

CMU SCS CMU SCS

a  b is the same as a  b More on division b • If a is true, b must be • find (SSNs of) students that take all the T F true for the courses that ssn=123 does (and maybe even implication to be true. more) T T F If a is true and b is a false, the implication find students ‘s’ so that evaluates to false. if 123 takes a course => so does ‘s’ F T T • If a is not true, we don’t care about b, the expression is always true. Faloutsos CMU SCS 15-415 #39 Faloutsos CMU SCS 15-415 #40

CMU SCS CMU SCS

More on division Safety of expressions

• find students that take all the courses that • FORBIDDEN: {t | t STUDENT} ssn=123 does (and maybe even more) {o | t((t TAKES  t.ssn 123)  It has infinite output!! t1TAKES ( • Instead, always use t1.c  id  t.c  id  t1.ssn  o.ssn) {t |....t SOME TABLE} )}

Faloutsos CMU SCS 15-415 #41 Faloutsos CMU SCS 15-415 #42

7 Faloutsos CMU - 15-415

CMU SCS CMU SCS

Overview - conclusions General Overview

• rel. tuple calculus: DECLARATIVE • relational model – dfn • Formal query languages – details – relational algebra – equivalence to rel. algebra – rel. tuple calculus • rel. domain calculus + QBE – rel. domain calculus

Faloutsos CMU SCS 15-415 #43 Faloutsos CMU SCS 15-415 #44

CMU SCS CMU SCS

Overview - detailed Rel. domain calculus (RDC)

• rel. tuple calculus • Q: why? – dfn • A: slightly easier than RTC, although – details equivalent - basis for QBE. – equivalence to rel. algebra • idea: domain variables (w/ F.O.L.) - eg: • rel. domain calculus + QBE • ‘find STUDENT record with ssn=123’

Faloutsos CMU SCS 15-415 #45 Faloutsos CMU SCS 15-415 #46

CMU SCS CMU SCS

Rel. Dom. Calculus Details • find STUDENT record with ssn=123’ • Like R.T.C - symbols allowed:

, , ,  , , , , , , { s,n,a | s,n,a STUDENT  s 123} (, ), 

• quantifiers , 

Faloutsos CMU SCS 15-415 #47 Faloutsos CMU SCS 15-415 #48

8 Faloutsos CMU - 15-415

CMU SCS CMU SCS

Details Reminder: our Mini-U db

• but: domain (= column) variables, as STUDENT CLASS opposed to tuple variables, eg: Ssn Name Address c-id c-name units 123 smith main str 15-413 s.e. 2 234 jones forbes ave 15-412 o.s. 2  s,n,a STUDENT TAKES SSN c-id grade 123 15-413 A ssn 234 15-413 B name address

Faloutsos CMU SCS 15-415 #49 Faloutsos CMU SCS 15-415 #50

CMU SCS CMU SCS

Examples Examples

• find all student records • (selection) find student record with ssn=123

{ s,n,a | s,n,a STUDENT}

RTC: {t | t STUDENT}

Faloutsos CMU SCS 15-415 #51 Faloutsos CMU SCS 15-415 #52

CMU SCS CMU SCS

Examples Examples

• (selection) find student record with ssn=123 • (projection) find name of student with ssn=123 {123,n,a |123,n,a STUDENT} { n | 123,n,a STUDENT } or { s,n,a | s,n,a STUDENT  s 123}

RTC: {t | t STUDENT  t.ssn 123}

Faloutsos CMU SCS 15-415 #53 Faloutsos CMU SCS 15-415 #54

9 Faloutsos CMU - 15-415

CMU SCS CMU SCS

Examples Examples cont’d

• (projection) find name of student with • (union) get records of both PT and FT ssn=123 students { n | a(123,n,a STUDENT)}

need to „restrict‟ “a”

RTC: {t | t  FT _ STUDENT  RTC: {t | s STUDENT(s.ssn 123  t.name  s.name)} t  PT _ STUDENT}

Faloutsos CMU SCS 15-415 #55 Faloutsos CMU SCS 15-415 #56

CMU SCS CMU SCS

Examples cont’d Examples

• (union) get records of both PT and FT • difference: find students that are not staff students { s,n,a | s,n,a FT _ STUDENT   s,n,a PT _ STUDENT} RTC: {t | t STUDENT  t STAFF}

Faloutsos CMU SCS 15-415 #57 Faloutsos CMU SCS 15-415 #58

CMU SCS CMU SCS

Examples Cartesian product

• difference: find students that are not staff • eg., dog-breeding: MALE x FEMALE • gives all possible couples { s,n,a | s,n,a STUDENT 

 s,n,a STAFF} MALE FEMALE M.name F.name name x name = spike lassie spike lassie spike shiba spot shiba spot lassie spot shiba

Faloutsos CMU SCS 15-415 #59 Faloutsos CMU SCS 15-415 #60

10 Faloutsos CMU - 15-415

CMU SCS CMU SCS

Cartesian product Cartesian product

• find all the pairs of (male, female) - RTC: • find all the pairs of (male, female) - RDC:

{t | m MALE  f  FEMALE { m, f | m MALE  t.m  name  m.name   f FEMALE} t. f  name  f .name}

Faloutsos CMU SCS 15-415 #61 Faloutsos CMU SCS 15-415 #62

CMU SCS CMU SCS

‘Proof’ of equivalence Overview - detailed

• rel. algebra <-> rel. domain calculus • rel. domain calculus <-> rel. tuple calculus – why? – details – examples – equivalence with rel. algebra – more examples; ‘safety’ of expressions

Faloutsos CMU SCS 15-415 #63 Faloutsos CMU SCS 15-415 #64

CMU SCS CMU SCS

More examples Reminder: our Mini-U db

• join: find names of students taking 15-415 STUDENT CLASS Ssn Name Address c-id c-name units 123 smith main str 15-413 s.e. 2 234 jones forbes ave 15-412 o.s. 2

TAKES SSN c-id grade 123 15-413 A 234 15-413 B

Faloutsos CMU SCS 15-415 #65 Faloutsos CMU SCS 15-415 #66

11 Faloutsos CMU - 15-415

CMU SCS CMU SCS

More examples More examples

• join: find names of students taking 15-415 - • join: find names of students taking 15-415 - in RTC in RDC {t | s  STUDENT  eTAKES ( s.ssn  e.ssn  { n | s a g( s,n,a STUDENT t.name  s.name   s,15  415, g TAKES)} e.c  id 15  415)}

Faloutsos CMU SCS 15-415 #67 Faloutsos CMU SCS 15-415 #68

CMU SCS CMU SCS

Sneak preview of QBE: Sneak preview of QBE:

{ n | s a g( s,n,a STUDENT • very user friendly   s,15  415, g TAKES)} • heavily based on RDC • very similar to MS Access interface

STUDENT TAKES Ssn Name Address SSN c-id grade _x P. _x 15-415

Faloutsos CMU SCS 15-415 #69 Faloutsos CMU SCS 15-415 #70

CMU SCS CMU SCS

More examples Reminder: our Mini-U db _x .P _y 2 • 3-way join: find names of students taking a STUDENT CLASS 2-unit course - in RTC: Ssn Name Address c-id c-name units 123 smith main str 15-413 s.e. 2 {t | s  STUDENT  eTAKES 234 jones forbes ave 15-412 o.s. 2 c CLASS( s.ssn  e.ssn  join e.c  id  c.c  id  TAKES SSN c-id grade t.name  s.name projection 123 15-413 A c.units  2)} 234 15-413 B selection _x _y

Faloutsos CMU SCS 15-415 #71 Faloutsos CMU SCS 15-415 #72

12 Faloutsos CMU - 15-415

CMU SCS CMU SCS

More examples More examples

• 3-way join: find names of students taking a • 3-way join: find names of students taking a 2-unit course 2-unit course { n |...... { n | s,a,c, g,cn(  s,n,a STUDENT   s,n,a STUDENT   s,c, g TAKES   s,c, g TAKES   c,cn,2 CLASS}  c,cn,2 CLASS )}

Faloutsos CMU SCS 15-415 #73 Faloutsos CMU SCS 15-415 #74

CMU SCS CMU SCS

Even more examples: Even more examples:

• self -joins: find Tom’s grandparent(s) • self -joins: find Tom’s grandparent(s)

{t | p  PC  q  PC PC p-id c-id ( p.c  id  q.p  id  Mary Tom Peter Mary p.p  id  t.p  id  John Tom q.c  id "Tom")}

Faloutsos CMU SCS 15-415 #75 Faloutsos CMU SCS 15-415 #76

CMU SCS CMU SCS

Even more examples: Even more examples:

• self -joins: find Tom’s grandparent(s) • self -joins: find Tom’s grandparent(s)

{t | p  PC  q  PC { g | p( g, p PC  { g | p( g, p PC  ( p.c  id  q.p  id   p,"Tom"PC)} p.p  id  t.p  id   p,"Tom"PC)} q.c  id "Tom")}

Faloutsos CMU SCS 15-415 #77 Faloutsos CMU SCS 15-415 #78

13 Faloutsos CMU - 15-415

CMU SCS CMU SCS

Hard examples: DIVISION Hard examples: DIVISION

• find suppliers that shipped all the ABOMB • find suppliers that shipped all the ABOMB parts parts SHIPMENT s# p# ABOMB BAD_S {t | p( p  ABOMB  ( s1 p1 p# s# s  SHIPMENT ( s2 p1   p1 s1 t.s# s.s#  s1 p2 p2 s3 p1 s.p# p.p#)))} s5 p3

Faloutsos CMU SCS 15-415 #79 Faloutsos CMU SCS 15-415 #80

CMU SCS CMU SCS

Hard examples: DIVISION More on division

• find suppliers that shipped all the ABOMB • find students that take all the courses that parts ssn=123 does (and maybe even more) { s | p( p  ABOMB {o | t((t TAKES  t.ssn 123)   s, p SHIPMENT)} t1TAKES ( t1.c  id  t.c  id  t1.ssn  o.ssn) )}

Faloutsos CMU SCS 15-415 #81 Faloutsos CMU SCS 15-415 #82

CMU SCS CMU SCS

More on division Safety of expressions

• find students that take all the courses that • similar to RTC ssn=123 does (and maybe even more) • FORBIDDEN:

{ s,n,a | s,n,a STUDENT} { s | c(g(123,c, g TAKES)  g'( s,c, g')TAKES))}

Faloutsos CMU SCS 15-415 #83 Faloutsos CMU SCS 15-415 #84

14 Faloutsos CMU - 15-415

CMU SCS CMU SCS Fun Drill:Your turn … Overview - detailed • Schema: Movie(title, year, studioName) • rel. domain calculus + QBE ActsIn(movieTitle, starName) – dfn Star(name, gender, birthdate, salary) – details – equivalence to rel. algebra

Faloutsos CMU SCS 15-415 #85 Faloutsos CMU SCS 15-415 #86

CMU SCS CMU SCS Your turn … Answers … • Queries to write in TRC: • Find all movies by Paramount studio – 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* {M | MMovie  – Stars connected to K. Bacon via any number of films** M.studioName = ‘Paramount’}

* Try two degrees for starters ** Good luck with this one!

Faloutsos CMU SCS 15-415 #87 Faloutsos CMU SCS 15-415 #88

CMU SCS CMU SCS Answers … Answers … • Stars who have been in a film w/Kevin • Movies starring Kevin Bacon Bacon {S | S Star  {M | MMovie  AActsIn(A.starName = S.name  AActsIn(A.movieTitle = M.title  A2ActsIn(A2.movieTitle = A.movieTitle  A2.starName = ‘Bacon’))} A.starName = ‘Bacon’))} S: name …

A: movie star

‘Bacon’ Faloutsos CMU SCS 15-415 #89 Faloutsos A2: movieCMU SCSstar 15- 415 #90

15 Faloutsos CMU - 15-415

CMU SCS CMU SCS Answers … Two degrees: two • Stars within six degrees of Kevin Bacon S: name …

{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  A3: movie star A4.starName = ‘Bacon’))} A4: movie star ‘Bacon’

Faloutsos CMU SCS 15-415 #91 Faloutsos CMU SCS 15-415 #92

CMU SCS CMU SCS Two degrees: Answers …

S: name … • Stars connected to K. Bacon via any number of films A: movie star • Sorry … that was a trick question A2: movie star – Not expressible in relational calculus!!

A3: movie star • What about in relational algebra? A4: movie star – We will be able to answer this question shortly … ‘Bacon’

Faloutsos CMU SCS 15-415 #93 Faloutsos CMU SCS 15-415 #94

CMU SCS CMU SCS Expressive Power Question: • Expressive Power (Theorem due to Codd): • Can we express previous query (‘any # – Every query that can be expressed in relational algebra can be expressed as a safe query in DRC / steps’) in relational algebra? TRC; the converse is also true. • A: If we could, then by Codd’s theorem we • Relational Completeness: could also express it in relational calculus. Query language (e.g., SQL) can express every query However, we know the latter is not possible, that is expressible in relational algebra/calculus. so the answer is no. (actually, SQL is more powerful, as we will see…)

Faloutsos CMU SCS 15-415 #95 Faloutsos CMU SCS 15-415 #96

16 Faloutsos CMU - 15-415

CMU SCS CMU SCS

Summary Summary - cnt’d • The relational model has rigorously defined query • Several ways of expressing a given query languages — simple and powerful. – a query optimizer should choose the most efficient • Relational algebra is more operational/procedural version. – useful as internal representation for query evaluation • Algebra and safe calculus have same expressive plans power • Relational calculus is declarative – leads to the notion of relational completeness. – users define queries in terms of what they want, not in terms of how to compute it.

Faloutsos CMU SCS 15-415 #97 Faloutsos CMU SCS 15-415 #98

17