Logical Query Languages. Motivation: 1. Logical rules extend more naturally to. recursive queries than does relational algebra.

Transcription

1 Logical Query Languages Motivation: 1. Logical rules extend more naturally to recursive queries than does relational algebra. Used in SQL3 recursion. 2. Logical rules form the basis for many information-integration systems and applications. 1

2 Datalog Example Likesèdrinker, beerè Sellsèbar, beer, priceè Frequentsèdrinker, barè Happyèdè é- Frequentsèd,barè AND Likesèd,beerè AND Sellsèbar,beer,pè Above isarule. Left side = head. Right side = body = AND of subgoals. Head and subgoals are atoms. Atom = predicate and arguments. Predicate = relation name or arithmetic predicate, e.g. é. Arguments are variables or constants. Subgoals ènot headè may optionally be negated by NOT. 2

3 Meaning of Rules Head is true of its arguments if there exist values for local variables èthose in body, not in headè that make all of the subgoals true. If no negation or arithmetic comparisons, just natural join the subgoals and project onto the head variables. Example Above rule equivalent to Happyèdè = ç drinker èfrequents.è Likes.è Sellsè 3

4 Evaluation of Rules Two, dual, approaches: 1. Variable-based: Consider all possible assignments of values to variables. If all subgoals are true, add the head to the result relation. 2. Tuple-based: Consider all assignments of tuples to subgoals that make each subgoal true. If the variables are assigned consistent values, add the head to the result. Example: Variable-Based Assignment R = Sèx,yè é- Rèx,zè AND Rèz,yè AND NOT Rèx,yè A B

5 Only assignments that make rst subgoal true: 1. x! 1, z! x! 2, z! 3. In case è1è, y! 3 makes second subgoal true. Since è1; 3è is not in R, the third subgoal is also true. Thus, add èx; yè =è1; 3è to relation S. In case è2è, no value of y makes the second subgoal true. Thus, S = A B 1 3 5

6 Example: Tuple-Based Assignment Trick: start with the positive ènot negatedè, relational ènot arithmeticè subgoals only. R = Sèx,yè é- Rèx,zè AND Rèz,yè AND NOT Rèx,yè A B Four assignments of tuples to subgoals: Rèx; zè Rèz; yè è1; 2è è1; 2è è1; 2è è2; 3è è2; 3è è1; 2è è2; 3è è2; 3è Only the second gives a consistent value to z. That assignment also makes NOT Rèx,yè true. Thus, è1; 3è is the only tuple for the head. 6

7 Safety A rule can make no sense if variables appear in funny ways. Examples Sèxè é- Rèyè Sèxè é- NOT Rèxè Sèxè é- Rèyè AND x é y In each of these cases, the result is innite, even if the relation R is nite. To make sense as a database operation, we need to require three things of a variable x è= denition of safetyè. If x appears in either 1. The head, 2. A negated subgoal, or 3. An arithmetic comparison, then x must also appear in a nonnegated, ëordinary" èrelationalè subgoal of the body. We insist that rules be safe, henceforth. 7

8 Datalog Programs A collection of rules is a Datalog program. Predicatesèrelations divide into two classes: EDB = extensional database = relation stored in DB. IDB = intensional database = relation dened by one or more rules. A predicate must be IDB or EDB, not both. Thus, an IDB predicate can appear in the body or head of a rule; EDB only in the body. 8

9 Example Convert the following SQL èfind the manufacturers of the beers Joe sellsè: Beersèname, manfè Sellsèbar, beer, priceè SELECT manf FROM Beers WHERE name INè SELECT beer FROM Sells WHERE bar = 'Joe''s Bar' è; to a Datalog program. JoeSellsèbè é- Sellsè'Joe''s Bar', b, pè Answerèmè é- JoeSellsèbè AND Beersèb,mè Note: Beers, Sells = EDB; JoeSells, Answer = IDB. 9

10 Expressive Power of Datalog Nonrecursive Datalog = relational algebra. Datalog simulates SQL select-from-where without aggregation and grouping. Recursive Datalog expresses queries that cannot be expressed in SQL. But none of these languages have full expressive power èturing completenessè. 10

11 Relational Algebra to Datalog Text has constructions for each of the operators of R.A. Only hard part: selections with OR's and NOT's. Simulate a R.A. expression in Datalog by creating an IDB predicate for each interior node and using the constuction for the operator at that node. 11

12 Example: Find the bars that sell two dierent beers at the same price. ç bar ç beer6=beer1.è ç Sèbar;beer1;priceè Sells Sells R1èbar,beer1,beer,priceè é- Sellsèbar,beer1,priceè AND Sellsèbar,beer,priceè; R2èbar,beer1,beer,priceè é- R1èbar,beer1,beer,priceè AND beer1 éé beer; Answerèbarè é- R2èbar,beer1,beer,priceè; 12

13 Datalog to Relational Algebra General rule is complex; the following often works for single rules: Problems not handled: constant arguments and variables appearing twice in the same atom. Can you provide the necessary xes? 1. Use ç to create for each relational subgoal a relation whose schema is the variables of that subgoal. 2. Handle negated subgoals by nding an expression for the nite set of all possible values for each of its variables èç a suitable columnè and take their product. Then subtract. 3. Natural join the relations from è1è, è2è. 4. Get the eect of arithmetic comparisons with ç. 5. Project onto head with ç. Several rules for same predicate: use ë. 13

14 Example Sèx,yè é- Rèx,zè AND Rèz,yè AND NOT Rèx,yè S1èx,y,zè := ç R1èx;zèèRè.è ç R2èz;yèèRè; S2èx,yè := ç x ès1è ç y ès1è; S3èx,yè := S2, ç R3èx;yèèRè; Sèx,yè := ç x;y ès1èx,y,zè.è S3èx,yèè; 14

15 Recursion IDB predicate P depends on predicate Q if there is a rule with P in the head and Q in a subgoal. Draw a graph: nodes = IDB predicates, arc P! Q means P depends on Q. Cycles i recursive. Recursive Example Sibèx,yè é- Parèx,pè AND Parèy,pè ANDxééy Cousinèx,yè é- Sibèx,yè Cousinèx,yè é- Parèx,xpè AND Parèy,ypè AND Cousinèxp,ypè 15

16 Iterative Fixed-Point Evaluates Recursive Rules Start IDB = ; Apply rules to IDB, EDB yes Change to IDB? no done 16

17 Example EDB Par = a d b c e f g h j k i Note, because of symmetry, Sib and Cousin facts appear in pairs, so we shall mention only èx; yè when both èx; yè and èy; xè are meant. 17

18 Sib Initial ; ; Round 1 èb; cè; èc; eè ; add: èg; hè; èj; kè Round 2 add: Round 3 add: Round 4 add: Cousin èb; cè; èc; eè èg; hè; èj; kè èf; gè; èf; hè èg; iè; èh; iè èi; kè èk; kè èi; jè 18