Datalog. Lecture #16 Autumn, Fall, 2001, LRX

Transcription

1 Dataog Lecture #16 Autumn, 2001 Fa, 2001, LRX #16 Dataog HUST,Wuhan,China 477

2 Logica Query Languages Motivation:» 1. Logica rues extend more naturay to recursive queries than does reationa agebra. Used in SQL3 recursion.» 2. Logica rues form the basis for many information-integration systems and appications. Fa, 2001, LRX #16 Dataog HUST,Wuhan,China 478

3 Dataog Exampe Likes(drinker, beer) Ses(bar, beer, price) Frequents(drinker, bar) Happy(d) <- Frequents(d,bar) AND Likes(d,beer) AND Ses(bar,beer,p) Fa, 2001, LRX #16 Dataog HUST,Wuhan,China 479

4 Dataog Exampe (II) Above is a rue. Left side = head. Right side = body = AND of subgoas. Head and subgoas are atoms.» Atom = predicate and arguments.» Predicate = reation name or arithmetic predicate, e.g. <.» Arguments are variabes or constants.» Subgoas (not head) may optionay be negated by NOT. Fa, 2001, LRX #16 Dataog HUST,Wuhan,China 480

5 Meaning of Rues Head is true of its arguments if there exist vaues for oca variabes (those in body, not in head) that make a of the subgoas true. If no negation or arithmetic comparisons, just natura join the subgoas and project onto the head variabes. Fa, 2001, LRX #16 Dataog HUST,Wuhan,China 481

6 Exampe Above rue equivaent to Happy(d) = π drinker (Frequents Likes Ses) Fa, 2001, LRX #16 Dataog HUST,Wuhan,China 482

7 Genera Evauation of Rues In principe, consider a possibe assignments of vaues to variabes. If body becomes true, add the head to the constructed reation. Fa, 2001, LRX #16 Dataog HUST,Wuhan,China 483

8 Exampe S(x,y) <- R(x,z) AND R(z,y) AND NOT R(x,y) R = A B Fa, 2001, LRX #16 Dataog HUST,Wuhan,China 484

9 Exampe (II) Ony assignments that make first subgoa true:» 1. x 1, z 2.» 2. x 2, z 3. In case (1), y 3 makes second subgoa true. Since (1, 3) is not in R, the third subgoa is aso true.» Thus, add (x, y) = (1, 3) to reation S. Fa, 2001, LRX #16 Dataog HUST,Wuhan,China 485

10 Exampe (III) In case (2), no vaue of y makes the second subgoa true. Thus, S = A B 1 3 Fa, 2001, LRX #16 Dataog HUST,Wuhan,China 486

11 Safety A rue can make no sense if variabes appear in funny ways. Exampes» S(x) <- R(y)» S(x) <- NOT R(x)» S(x) <- R(y) AND x < y In each of these cases, the resut is infinite, even if the reation R is finite. Fa, 2001, LRX #16 Dataog HUST,Wuhan,China 487

12 Safety (II) To make sense as a database operation, we need to require three things of a variabe x. If x appears in either» 1. The head,» 2. A negated subgoa, or» 3. An arithmetic comparison, then x must aso appear in a nonnegated, "ordinary" (reationa) subgoa of the body. We insist that rues be safe, henceforth. Fa, 2001, LRX #16 Dataog HUST,Wuhan,China 488

13 A Tupe-Based Interpretation for Rues If a rue is safe, we can evauate ike an SQL rue.» 1. Consider tupe variabes for each reationa, positive subgoa that range over their reations.» 2. For each assignment of tupes to each of these subgoas, determine the impied assignment of vaues to variabes.» 3. If the assignment is a) Consistent and b) Makes arithmetic and negated subgoas true, then add the head tupe from this assignment to the resut reation. Fa, 2001, LRX #16 Dataog HUST,Wuhan,China 489

14 Exampe S(x,y) <- R(x,z) AND R(z,y) AND NOT R(x,y) R = A B Fa, 2001, LRX #16 Dataog HUST,Wuhan,China 490

15 Exampe (II) Four assignments of tupes to subgoas: R(x, z) R(z, y) (1, 2) (1, 2) (1, 2) (2, 3) (2, 3) (1, 2) (2, 3) (2, 3) Ony the second gives a consistent vaue to z. That assignment aso makes NOT R(x,y) true. Thus, (1, 3) is the ony tupe for the head. Fa, 2001, LRX #16 Dataog HUST,Wuhan,China 491

16 Dataog Programs A coection of rues is a Dataog program. Predicates/reations divide into two casses:» EDB = extensiona database = reation stored in DB.» IDB = intensiona database = reation defined by one or more rues. A predicate must be IDB or EDB, not both.» Thus, an IDB predicate can appear in the body or head of a rue; EDB ony in the body. Fa, 2001, LRX #16 Dataog HUST,Wuhan,China 492

17 Exampe Convert the foowing SQL (Find the manufacturers of the beers Joe ses):» Beers(name, manf) Ses(bar, beer, price) SELECT manf FROM Beers WHERE name IN( SELECT beer FROM Ses WHERE bar = 'Joe''s Bar' ); Fa, 2001, LRX #16 Dataog HUST,Wuhan,China 493

18 to a Dataog program.» JoeSes(b) <- Ses('Joe''s Bar', b, p) Answer(m) <- JoeSes(b) AND Beers(b,m) Note: Beers, Ses = EDB; JoeSes, Answer = IDB. Fa, 2001, LRX #16 Dataog HUST,Wuhan,China 494

19 Expressive Power of Dataog Nonrecursive Dataog = reationa agebra. Dataog simuates SQL seect-from-where without aggregation and grouping. Recursive Dataog expresses queries that cannot be expressed in SQL. But none of these anguages have fu expressive power (Turing competeness). Fa, 2001, LRX #16 Dataog HUST,Wuhan,China 495

20 Reationa Agebra to Dataog Text has constructions for each of the operators of R.A.» Ony hard part: seections with OR's and NOT's. Simuate a R.A. expression in Dataog by creating an IDB predicate for each interior node and using the construction for the operator at that node. Fa, 2001, LRX #16 Dataog HUST,Wuhan,China 496

21 Exampe Find the bars that se two different beers at the same price. Fa, 2001, LRX #16 Dataog HUST,Wuhan,China 497

22 Exampe(II) π bar σbeer¹beer1 ρ (bar, beer1, price) Ses Ses Fa, 2001, LRX #16 Dataog HUST,Wuhan,China 498

23 Dataog to Reationa Agebra Genera rue is compex; the foowing often works for singe rues:» Probems not handed: constant arguments and variabes appearing twice in the same atom.» Can you provide the necessary fixes? 1. Use ρ to create for each reationa subgoa a reation whose schema is the variabes of that subgoa. Fa, 2001, LRX #16 Dataog HUST,Wuhan,China 499

24 Dataog to Reationa Agebra (II) 2. Hande negated subgoas by finding an expression for the finite set of a possibe vaues for each of its variabes (π a suitabe coumn) and take their product. Then subtract. 3. Natura join the reations from (1), (2). 4. Get the effect of arithmetic comparisons with σ. Fa, 2001, LRX #16 Dataog HUST,Wuhan,China 500

25 Dataog to Reationa Agebra (III) 5. Project onto head with π. Severa rues for same predicate: use. Fa, 2001, LRX #16 Dataog HUST,Wuhan,China 501

26 Exampe S(x,y) <- R(x,z) AND R(z,y) AND NOT R(x,y) S1(x,y,z) := ρ R1 (x,z) (R) ρ R2(z,y) (R); S2(x,y) := π x (S1) X y(s1); S3(x,y) := S2 - ρ (R3(x.y) (R); S(x,y) := π x,y (S1(x,y,z) S3(x,y)); Fa, 2001, LRX #16 Dataog HUST,Wuhan,China 502

27 Recursion IDB predicate P depends on predicate Q if there is a rue with P in the head and Q in a subgoa. Draw a graph: nodes = IDB predicates, arc P Q means P depends on Q. Cyces iff recursive. Fa, 2001, LRX #16 Dataog HUST,Wuhan,China 503

28 Recursive Exampe Sib(x,y) <- Par(x,p) AND Par(y,p) AND x <> y Cousin(x,y) <- Sib(x,y) Cousin(x,y) <- Par(x,xp) AND Par(y,yp) AND Cousin(xp,yp) Fa, 2001, LRX #16 Dataog HUST,Wuhan,China 504

29 Iterative Fixed-Point Evauates Recursive Rues Start IDB = 0 Appy rues to IDB, EDB yes Change to IDB? no done Fa, 2001, LRX #16 Dataog HUST,Wuhan,China 505

30 Exampe EDB Par= a d b c e f g h j k i Fa, 2001, LRX #16 Dataog HUST,Wuhan,China 506

31 Exampe (II) Note, because of symmetry, Sib and Cousin facts appear in pairs, so we sha mention ony (x; y) when both (x; y) and (y; x) are meant. Fa, 2001, LRX #16 Dataog HUST,Wuhan,China 507

32 Exampe (III) Fa, 2001, LRX Sib Initia 0 0 Round 1 add: Round 2 add: Round 3 add: Round 4 add: (b,c), (c,e) (g,h), (j,k) Cousin 0 (b,c), (c,e) (g,h), (j,k) (f,g), (f,h). (g,i),(h,i), (i,k) (k,k),(i,k) #16 Dataog HUST,Wuhan,China 508