Recursion, Stratification in SQL3 and Datalog

Transcription

1 Recursion, Stratification in SQL3 and Dataog Lecture #17 Autumn, 2001 Fa, 2001, LRX #17 Recursion, Stratification in SQL3 and Dataog HUST,Wuhan,China 509

2 Stratified Negation Negation wrapped inside a recursion makes no sense. Even when negation and recursion are separated, there can be ambiguity about what the rues mean, and some one meaning must be seected. Fa, 2001, LRX #17 Recursion, Stratification in SQL3 and Dataog HUST,Wuhan,China 510

3 Stratified Negation (II) Stratified negation is an additiona restraint on recursive rues (ike safety) that soves both probems:» 1. It rues out negation wrapped in recursion.» 2. When negation is separate from recursion, it yieds the intuitivey correct meaning of rues. Stratification recenty adopted in the SQL3 standard for recursive SQL. Fa, 2001, LRX #17 Recursion, Stratification in SQL3 and Dataog HUST,Wuhan,China 511

4 Probem with Recursive Negation Consider: P(x) <- Q(x) AND NOT P(x) Q = EDB = {1, 2}. Compute IDB P iterativey?» Initiay, P = φ.» Round 1: P = {1, 2}.» Round 2: P = φ, etc., etc. Fa, 2001, LRX #17 Recursion, Stratification in SQL3 and Dataog HUST,Wuhan,China 512

5 Strata Intuitivey: stratum of an IDB predicate = maximum number of negations you can pass through on the way to an EDB predicate. Must not be in "stratified" rues. Define stratum graph:» Nodes = IDB predicates.» Arc P Q if Q appears in the body of a rue with head P.» Labe that arc -- if Q is in a negated subgoa. Fa, 2001, LRX #17 Recursion, Stratification in SQL3 and Dataog HUST,Wuhan,China 513

6 Exampe P(x) <- Q(x) AND NOT P(x) -- P Fa, 2001, LRX #17 Recursion, Stratification in SQL3 and Dataog HUST,Wuhan,China 514

7 Exampe Reach(x) <- Source(x) Reach(x) <- Reach(y) AND Arc(y,x) NoReach(x) <- Target(x) AND NOT Reach(x) NoReach -- Reach Fa, 2001, LRX #17 Recursion, Stratification in SQL3 and Dataog HUST,Wuhan,China 515

8 Computing Strata Stratum of an IDB predicate A = maximum number of -- arcs on any path from A in the stratum graph. Fa, 2001, LRX #17 Recursion, Stratification in SQL3 and Dataog HUST,Wuhan,China 516

9 Exampes» For first exampe, stratum of P is 1.» For second exampe, stratum of Reach is 0; stratum of NoReach is 1. Fa, 2001, LRX #17 Recursion, Stratification in SQL3 and Dataog HUST,Wuhan,China 517

10 Stratified Negation A Dataog program with recursion and negation is stratified if every IDB predicate has a finite stratum. Fa, 2001, LRX #17 Recursion, Stratification in SQL3 and Dataog HUST,Wuhan,China 518

11 Stratified Mode If a Dataog program is stratified, we can compute the reations for the IDB predicates owest-stratum-first. Fa, 2001, LRX #17 Recursion, Stratification in SQL3 and Dataog HUST,Wuhan,China 519

12 Exampe Reach(x) <- Source(x) Reach(x) <- Reach(y) AND Arc(y,x) NoReach(x) <- Target(x) AND NOT Reach(x) EDB:» Source = {1}.» Arc = {(1, 2), (3, 4),; (4, 3)}.» Target = {2, 3}. Fa, 2001, LRX #17 Recursion, Stratification in SQL3 and Dataog HUST,Wuhan,China 520

13 Exampe (II) Source Target Target First compute Reach = {1, 2} (stratum 0). Next compute NoReach = {3}. Fa, 2001, LRX #17 Recursion, Stratification in SQL3 and Dataog HUST,Wuhan,China 521

14 Is the Stratified Soution "Obvious"? Not reay. There is another mode that makes the rues true no matter what vaues we substitute for the variabes.» Reach = {1, 2, 3, 4}.» NoReach = φ. Fa, 2001, LRX #17 Recursion, Stratification in SQL3 and Dataog HUST,Wuhan,China 522

15 Is the Stratified Soution "Obvious"? (II) Remember: the ony way to make a Dataog rue fase is to find vaues for the variabes that make the body true and the head fase. For this mode, the heads of the rues for Reach are true for a vaues, and in the rue for NoReach the subgoa NOT Reach(x) assures that the body cannot be true. Fa, 2001, LRX #17 Recursion, Stratification in SQL3 and Dataog HUST,Wuhan,China 523

16 SQL3 Recursion WITH stuff that ooks ike Dataog rues an SQL query about EDB, IDB Rue = [RECURSIVE] R(<arguments>) AS SQL query Fa, 2001, LRX #17 Recursion, Stratification in SQL3 and Dataog HUST,Wuhan,China 524

17 Exampe Find Say's cousins, using EDB Par(chid, parent). Fa, 2001, LRX #17 Recursion, Stratification in SQL3 and Dataog HUST,Wuhan,China 525

18 Exampe (II) WITH Sib(x,y) AS SELECT p1.chid, p2,chid FROM Par p1, Par p2 WHERE p1.parent = p2.parent AND p1.chid <> p2.chid, RECURSIVE Cousin(x,y) AS Sib UNION (SELECT p1.chid, p2.chid FROM Par p1, Par p2, Cousin WHERE p1.parent = Cousin.x AND p2.parent = Cousin.y ) SELECT y FROM Cousin WHERE x = 'Say'; Fa, 2001, LRX #17 Recursion, Stratification in SQL3 and Dataog HUST,Wuhan,China 526

19 Pan for Describing Lega SQL3 recursion 1. Define "monotonicity," a property that generaizes "stratification." 2. Generaize stratum graph to appy to SQL queries instead of Dataog rues.»(non)monotonicity repaces NOT in subgoas. 3. Define semanticay correct SQL3 recursions in terms of stratum graph. Fa, 2001, LRX #17 Recursion, Stratification in SQL3 and Dataog HUST,Wuhan,China 527

20 Monotonicity If reation P is a function of reation Q (and perhaps other things), we say P is monotone in Q if adding tupes to Q cannot cause any tupe of P to be deeted. Fa, 2001, LRX #17 Recursion, Stratification in SQL3 and Dataog HUST,Wuhan,China 528

21 Monotonicity Exampe In addition to certain negations, an aggregation can cause nonmonotonicity. Ses(bar, beer, price) SELECT AVG(price) FROM Ses WHERE bar = 'Joe''s Bar'; Fa, 2001, LRX #17 Recursion, Stratification in SQL3 and Dataog HUST,Wuhan,China 529

22 Monotonicity Exampe (II) Adding to Ses a tupe that gives a new beer Joe ses wi usuay change the average price of beer at Joe's. Thus, the former resut, which might be a singe tupe ike (2:78) becomes another singe tupe ike (2:81), and the od tupe is ost. Fa, 2001, LRX #17 Recursion, Stratification in SQL3 and Dataog HUST,Wuhan,China 530

23 Generaizing Stratum Graph to SQL Node for each reation defined by a "rue." Node for each subquery in the "body" of a rue. Fa, 2001, LRX #17 Recursion, Stratification in SQL3 and Dataog HUST,Wuhan,China 531

24 Generaizing Stratum Graph to SQL (II) Arc P Q if»a) P is "head" of a rue, and Q is a reation appearing in the FROM ist of the rue (not in the FROM ist of a subquery).»b) P is head of a rue, and Q is a subquery directy used in that rue (not nested within some arger subquery).»c) P is a subquery, and Q is a reation or subquery used directy within P. Fa, 2001, LRX #17 Recursion, Stratification in SQL3 and Dataog HUST,Wuhan,China 532

25 Generaizing Stratum Graph to SQL (III) Labe the arc -- if P is not monotone in Q. Requirement for ega SQL3 recursion: finite strata ony. Fa, 2001, LRX #17 Recursion, Stratification in SQL3 and Dataog HUST,Wuhan,China 533

26 Exampe For the Sib/Cousin exampe, there are three nodes: Sib, Cousin, and SQ (the second term of the union in the rue for Cousin. Sib Sib Sib No nonmonotonicity, hence ega. Fa, 2001, LRX #17 Recursion, Stratification in SQL3 and Dataog HUST,Wuhan,China 534

27 A Nonmonotonic Exampe Change the UNION to EXCEPT in the rue for Cousin. RECURSIVE Cousin(x,y) AS Sib EXCEPT (SELECT p1.chid, p2.chid FROM Par p1, Par p2, Cousin WHERE p1.parent = Cousin.x AND p2.parent = Cousin.y ) Fa, 2001, LRX #17 Recursion, Stratification in SQL3 and Dataog HUST,Wuhan,China 535

28 A Nonmonotonic Exampe (II) Now, Adding to the resut of the subquery can deete Cousin facts; i.e., Cousin is nonmonotone in SG. Sib Infinite number of --'s in cyce, so iega in SQL3. -- Sib Sib Fa, 2001, LRX #17 Recursion, Stratification in SQL3 and Dataog HUST,Wuhan,China 536

29 Another Exampe: NOT Doesn't Mean Nonmonotone Fa, 2001, LRX Leave Cousin as it was, but negate one of the conditions in the where-cause. RECURSIVE Cousin(x,y) AS Sib UNION (SELECT p1.chid, p2.chid FROM Par p1, Par p2, Cousin WHERE p1.parent = Cousin.x AND NOT (p2.parent = Cousin.y) ) #17 Recursion, Stratification in SQL3 and Dataog HUST,Wuhan,China 537

30 Another Exampe: NOT Doesn't Mean Nonmonotone (II) You might think that SG depends negativey on Cousin, but it doesn't.»if I add a new tupe to Cousin, a the od tupes sti exist and yied whatever tupes in SG they used to yied.»in addition, the new Cousin tupe might combine with od p1 and p2 tupes to yied something new. Fa, 2001, LRX #17 Recursion, Stratification in SQL3 and Dataog HUST,Wuhan,China 538