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CSE 544 Relational Calculus

CSE 544 Relational Calculus. Lecture #2 January 11 th , 2011. Announcements. HW1 is posted First paper review due next week Friday: project groups are due. Announcements. Guest lecture on Tuesday, 1/18, by Bill Howe SQLShare : Smart Services for Ad Hoc Databases

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CSE 544 Relational Calculus

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  1. CSE 544Relational Calculus Lecture #2 January 11th, 2011 Dan Suciu -- 544, Winter 2011

  2. Announcements • HW1 is posted • First paper review due next week • Friday: project groups are due Dan Suciu -- 544, Winter 2011

  3. Announcements • Guest lecture on Tuesday, 1/18, by Bill HoweSQLShare: Smart Services for Ad Hoc Databases • Need to makeup one lecture this/next week. When ? • Friday 1/14, 12:30-2pm ? • Wednesday, 1/19, morning (9…12) ? • Friday, 1/21, late morning (11...1pm) ? Dan Suciu -- 544, Winter 2011

  4. Today’s Agenda Relational calculus: • Domain Relational Calculus • Non-recursive datalog (a reasonable abstraction of SQL) • Relational algebra They are equivalent and why we care Dan Suciu -- 544, Winter 2011

  5. Domain Relational Calculus Given: • A vocabulary: R1, …, Rk • An arity, ar(Ri), for each i=1,…,k • An infinite supply of variables x1, x2, x3, … • Constants: c1, c2, c3, ... Dan Suciu -- 544, Winter 2011

  6. Domain Relational Calculus Terms t and Formulas  are: t ::= x | a /* variable or constant */ • ::= R(t1, ..., tk) | ti = tj | ’ | ’ | ’ | x. | x. A query Q is any formula , usually written as: Q = { x |  } where x = (x1,x2,…) are the free variables in . Dan Suciu -- 544, Winter 2011

  7. Example: Querying a Graph R encodes a graph What do these queries return ? {x | y.R(x,y)} 1 4 2 3 {x | y.z.u.(R(x,y) R(y,z) R(z,u)) R= {(x,y) | z.(R(x,z) R(y,z)}

  8. Example: Querying a Graph R encodes a graph What do these queries return ? {x | y.R(x,y)} 1 4 Nodes that have at least one child: {1,2,3} 2 3 {x | y.z.u.(R(x,y) R(y,z) R(z,u)) R= Nodes that have a great-grand-child: {1,2} {(x,y) | z.(R(x,z) R(y,z)} Every child of x is a child of y:{(1,1),(2,2),(3,1),(3,3),(4,1), (4,2), (4,3)}

  9. Semantics of a Formula • A database instance (or a model) isD = (D, R1D, …, RkD) • D = a set, called domain, or universe • RiD D  D  ...  D, (ar(Ri) times) i = 1,...,k • A valuation is a function s : {x1, x2, ...} D • The semantics of a formula says when D ⊨ [s] holds The domain D is not part of the database,yet we need it to define the semantics. Where ? Dan Suciu -- 544, Winter 2011

  10. Semantics of D ⊨ [s] D ⊨ (R(t1, ..., tn)) [s] If (s(t1), ..., s(tn))  RD If s(t) = s(t’) D ⊨ (t= t’) [s] If D ⊨ [s] and D ⊨ (’) [s] D ⊨ (’) [s] If D ⊨ [s] or D ⊨ ’[s] D ⊨ (’) [s] If it is not the case D ⊨ [s] D ⊨ (’) [s] Dan Suciu -- 544, Winter 2011

  11. Semantics of D ⊨ [s] Notation: sa/x = the valuation s modified to map x to a D ⊨ (x.) [s] If for every constant a in D,D ⊨ [sa/x] If for some constant a in D,D ⊨ [sa/x] D ⊨ (x.) [s] Dan Suciu -- 544, Winter 2011

  12. Semantics of D ⊨ [s] Notation: sa/x = the valuation s modified to map x to a D ⊨ (x.) [s] If for every constant a in D,D ⊨ [sa/x] If for some constant a in D,D ⊨ [sa/x] D ⊨ (x.) [s] Note: here we need the domain D Dan Suciu -- 544, Winter 2011

  13. Semantics of a Query The semantics of a queryQ = { x |  } is Q(D) = { s(x) | for all s such that D ⊨ [s] } Dan Suciu -- 544, Winter 2011

  14. The drinkers-bars-beers example Likes(drinker, beer) Frequents(drinker, bar) Serves(bar, beer) What does this query compute ? x: y. z. Frequents(x, y) Serves(y,z) Likes(x,z)

  15. The drinkers-bars-beers example Likes(drinker, beer) Frequents(drinker, bar) Serves(bar, beer) What does this query compute ? x: y. z. Frequents(x, y) Serves(y,z) Likes(x,z) Find drinkers that frequent some bar that serves some beer they like.

  16. The drinkers-bars-beers example Likes(drinker, beer) Frequents(drinker, bar) Serves(bar, beer) What does this query compute ? x: y. Frequents(x, y) (z. Serves(y,z)Likes(x,z))

  17. The drinkers-bars-beers example Likes(drinker, beer) Frequents(drinker, bar) Serves(bar, beer) What does this query compute ? x: y. Frequents(x, y) (z. Serves(y,z)Likes(x,z)) Find drinkers that frequent only bars that serves some beer they like.

  18. The drinkers-bars-beers example Likes(drinker, beer) Frequents(drinker, bar) Serves(bar, beer) What does this query compute ? x: y. Frequents(x, y)z.(Serves(y,z) Likes(x,z))

  19. The drinkers-bars-beers example Likes(drinker, beer) Frequents(drinker, bar) Serves(bar, beer) What does this query compute ? x: y. Frequents(x, y)z.(Serves(y,z) Likes(x,z)) Find drinkers that frequent some bar that serves only beers they like.

  20. The drinkers-bars-beers example Likes(drinker, beer) Frequents(drinker, bar) Serves(bar, beer) What does this query compute ? x:y.Frequents(x, y)z.(Serves(y,z) Likes(x,z)) Dan Suciu -- 544, Winter 2011

  21. The drinkers-bars-beers example Likes(drinker, beer) Frequents(drinker, bar) Serves(bar, beer) What does this query compute ? x:y.Frequents(x, y)z.(Serves(y,z) Likes(x,z)) Find drinkers that frequent only bars that serves only beer they like. Dan Suciu -- 544, Winter 2011

  22. Summary of Relational Calculus • Same as First Order Logic • See book for the two variants: • Domain relational calculus (what we discussed) • Tuple relational calculus • This is a powerful, concise language Dan Suciu -- 544, Winter 2011

  23. Datalog Rule Body Head Li = a literal, or atom, or subgoal; can be one of: • R(x1, x2, …) = relational atom • u = v, or u ≠ v, where u, v = variables or constants S = a new symbol x = (x1, x2, …) are head variables Q(x) :- L1, L2, …, Lk Example: Q(x) :- Likes(x,y),Likes(x,z),y≠z Dan Suciu -- 544, Winter 2011

  24. Semantics of Datalog Rule Q(x) :- L1, L2, …, Lk Q = {x | y1.y2… L1L2…Lk} y1, y2,… are all non-head variables The semantics is: Q(D) :- { s(x) | s = valuation s.t. D ⊨ L1[s],…,D ⊨ Lk[s] } Note: a datalog rule is also called a conjunctive query Dan Suciu -- 544, Winter 2011

  25. Non-Recursive Datalog S1(x1) :- body1 S2(x2) :- body2 S3(x3) :- body3 . . . Each symbol Sk may appearin bodyk+1, bodyk+2, …,but may not appearin body1,…,bodyk. Example: What does Q compute ? A(x,y) :- R(x,z),R(z,y) B(x,y) :- A(x,z),A(z,y) C(x,y) :- B(x,z),B(z,y) Q(x,y) :- C(x,z),C(z,y) Dan Suciu -- 544, Winter 2011

  26. Non-Recursive Datalog S1(x1) :- body1 S2(x2) :- body2 S3(x3) :- body3 . . . Each symbol Sk may appearin bodyk+1, bodyk+2, …,but may not appearin body1,…,bodyk. Example: What does Q compute ? A(x,y) :- R(x,z),R(z,y) B(x,y) :- A(x,z),A(z,y) C(x,y) :- B(x,z),B(z,y) Q(x,y) :- C(x,z),C(z,y) A: all pairs (x,y) connected bya path of length 16. Dan Suciu -- 544, Winter 2011

  27. Comparison Is non-recursive datalog equivalentto the Relational Calculus ? Dan Suciu -- 544, Winter 2011

  28. Comparison Is non-recursive datalog equivalentto the Relational Calculus ? What about writing this query in non-recursive datalog ? Q = {x | y. Frequents(x,y) Serves(y,’Bud’) z. Frequents(x,z) Serves(z,’Miller’) } Dan Suciu -- 544, Winter 2011

  29. Comparison Is non-recursive datalog equivalentto the Relational Calculus ? What about writing this query in non-recursive datalog ? Q = {x | y. Frequents(x,y) Serves(y,’Bud’) z. Frequents(x,z) Serves(z,’Miller’) } Q(x) :- Frequents(x,y), Serves(y,’Bud’) Q(x) :- Frequents(x,z), Serves(z,’Miller’) Dan Suciu -- 544, Winter 2011

  30. Comparison Is non-recursive datalog equivalentto the Relational Calculus ? Now what about writing this query in non-recursive datalog ? Q = {x | y. Frequents(x,y) z.Likes(x,z)Serves(y,z) Dan Suciu -- 544, Winter 2011

  31. Comparison Is non-recursive datalog equivalentto the Relational Calculus ? Now what about writing this query in non-recursive datalog ? Q = {x | y. Frequents(x,y) z.Likes(x,z)Serves(y,z) Impossible ! Proof on next slides. Dan Suciu -- 544, Winter 2011

  32. Monotone Queries • Given two database instancesD = (D, R1, …, Rk) and D’ = (D’, R’1, …, R’k)we write D⊆ D’ if D⊆D’, R1⊆R’1,…, Rk⊆R’k. • In other words, D’ is obtained by adding tuples to D. • We say that Q is monotoneif whenever D⊆ D’, Q(D)⊆ Q(D’). Dan Suciu -- 544, Winter 2011

  33. Monotone Queries Fact. Every datalog rule is monotone. Why ? Fact. Every datalog program is monotone. Why ? Fact. This query is not monotone:Q = {x | y. Likes(x,y) z.Frequents(x,z)Serves(z,y) Why ? Dan Suciu -- 544, Winter 2011

  34. Monotone Queries Fact. Every datalog rule is monotone. Why ? Recall the semantics: Q(D) = { s(x) | D ⊨ L1[s],…,D ⊨ Lk[s] } If D ⊨ Li[s] then D’ ⊨ Li[s] hence Q(D)⊆ Q(D’). Fact. Every datalog program is monotone. Why ? Fact. This query is not monotone:Q = {x | y. Likes(x,y) z.Frequents(x,z)Serves(z,y) Why ? Dan Suciu -- 544, Winter 2011

  35. Monotone Queries Fact. Every datalog rule is monotone. Why ? Recall the semantics: Q(D) = { s(x) | D ⊨ L1[s],…,D ⊨ Lk[s] } If D ⊨ Li[s] then D’ ⊨ Li[s] hence Q(D)⊆ Q(D’). Fact. Every datalog program is monotone. Why ? By induction on the number of rules Fact. This query is not monotone:Q = {x | y. Likes(x,y) z.Frequents(x,z)Serves(z,y) Why ? Dan Suciu -- 544, Winter 2011

  36. Monotone Queries Fact. Every datalog rule is monotone. Why ? Recall the semantics: Q(D) = { s(x) | D ⊨ L1[s],…,D ⊨ Lk[s] } If D ⊨ Li[s] then D’ ⊨ Li[s] hence Q(D)⊆ Q(D’). Fact. Every datalog program is monotone. Why ? By induction on the number of rules Fact. This query is not monotone:Q = {x | y. Likes(x,y) z.Frequents(x,z)Serves(z,y) Why ? By adding a tuple to Frequents we may remove some answer Dan Suciu -- 544, Winter 2011

  37. Monotone Queries Recall the relational calculus: t ::= x | a /* variable or constant */ • ::= R(t1, ..., tk) | ti = tj | ’ | ’ | ’ | x. | x. What fragment of the relational calculus is monotone ? Dan Suciu -- 544, Winter 2011

  38. Monotone Queries Recall the relational calculus: t ::= x | a /* variable or constant */ • ::= R(t1, ..., tk) | ti = tj | ’ | ’ | ’ | x. | x. What fragment of the relational calculus is monotone ? t ::= x | a /* variable or constant */ • ::= R(t1, ..., tk) | ti = tj | ’ | ’ | x. Dan Suciu -- 544, Winter 2011

  39. Non-Recursive Datalog Li = may also be R(x1, x2, …) • R(x1, x2, …) = relational atom • u = v, or u ≠ v, where u, v = variables or constants Q(x) :- L1, L2, …, Lk Dan Suciu -- 544, Winter 2011

  40. Non-Recursive Datalog Example: Q = {x | y. Likes(x,y) z.Frequents(x,z)Serves(z,y) Dan Suciu -- 544, Winter 2011

  41. Non-Recursive Datalog Example: Q = {x | y. Likes(x,y) z.Frequents(x,z)Serves(z,y) FS(x,y) :- Frequents(x,z),Serves(z,y) Q(x) :- Likes(x,y), FS(x,y) Dan Suciu -- 544, Winter 2011

  42. Rel. Calculus  non-rec. Datalog Theorem. Every query in the Relational Calculuscan be translated to non-recursive Datalog Dan Suciu -- 544, Winter 2011

  43. Rel. Calculus  non-rec. Datalog Proof First remove  by using De Morgan: x. = x. Then, inductively, for each subformulacreate a datalog rule F(x):-bodythat computes Q = { x |  } Dan Suciu -- 544, Winter 2011

  44. Rel. Calculus  non-rec. Datalog R(t1, ..., tk) F(x) :- R(t1, ..., tk) Dan Suciu -- 544, Winter 2011

  45. Rel. Calculus  non-rec. Datalog F1(x1) :- … /* for 1*/ F2(x2) :- … /* for 2*/ 12 ? ? ? Dan Suciu -- 544, Winter 2011

  46. Rel. Calculus  non-rec. Datalog F1(x1) :- … /* for 1*/ F2(x2) :- … /* for 2*/ 12 F(x) :- F1(x1),F2(x2) Dan Suciu -- 544, Winter 2011

  47. Rel. Calculus  non-rec. Datalog F1(x1) :- … /* for 1*/ F2(x2) :- … /* for 2*/ 1 2 ? ? ? Dan Suciu -- 544, Winter 2011

  48. Rel. Calculus  non-rec. Datalog F1(x1) :- … /* for 1*/ F2(x2) :- … /* for 2*/ 1 2 F(x) :- F1(x1) F(x) :- F2(x2) Dan Suciu -- 544, Winter 2011

  49. Rel. Calculus  non-rec. Datalog F1(x) :- … /* for 1*/ x.1 ? ? ? Dan Suciu -- 544, Winter 2011

  50. Rel. Calculus  non-rec. Datalog F1(x) :- … /* for 1*/ x.1 F(y) :- F1(x) where y = x – {x} Dan Suciu -- 544, Winter 2011

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