[PDF] - Stratied Negation Negation wrapp ed inside a recursion mak PDF Document

SLIDE 1 Stratied Negation

Negation

wrapp ed inside a recursion mak es no sense.

Ev

en when negation and recursion are separated, there can b e am biguit y ab

ut

what the rules mean, and some

ne

meaning m ust b e selected.

Str

atie d ne gation is an additional restrain t

n

recursiv e rules (lik e safet y) that solv es b

th

problems: 1. It rules

ut

negation wrapp ed in recursion. 2. When negation is separate from recursion, it yields the in tuitiv el y correct meaning

f

rules.

Stratication

recen tly adopted in the SQL3 standard for recursiv e SQL. 1

SLIDE 2 Problem with Recursiv e Negation Consider: P(x) <- Q(x) AND NOT P(x)

Q

= EDB = f1; 2g.

Compute

IDB P iterativ el y?

✦

Initial ly , P = ;.

✦

Round 1: P = f1; 2g.

✦

Round 2: P = ;, etc., etc. 2

SLIDE 3 Strata In tuitiv ely : stratum

f

an IDB predicate = maxim um n um b er

f

negations y

u

can pass through

n

the w a y to an EDB predicate.

Must

not b e 1 in \stratied" rules.

Dene

str atum gr aph:

✦

No des = IDB predicates.

✦

Arc P ! Q if Q app ears in the b

dy
f

a rule with head P .

✦

Lab el that arc

if

Q is in a negated subgoal. Example P(x) <- Q(x) AND NOT P(x) P

3

SLIDE 4 Example Reach(x) <- Source(x) Reach(x) <- Reach(y) AND Arc(y,x) NoReach(x) <- Target(x) AND NOT Reach(x) Reach NoReach

4

SLIDE 5 Computing Strata Str atum

f

an IDB predicate A = maxim um n um b er

f
arcs
n

an y path from A in the stratum graph. Examples

F
r

rst example, stratum

f

P is 1.

F
r

second example, stratum

f

Reach is 0; stratum

f

NoReach is 1. Stratied Negation A Datalog program with recursion and negation is str atie d if ev ery IDB predicate has a nite stratum. Stratied Mo del If a Datalog program is stratied, w e can compute the relations for the IDB predicates lo w est- stratum-rst. 5

SLIDE 6 Example Reach(x) <- Source(x) Reach(x) <- Reach(y) AND Arc(y,x) NoReach(x) <- Target(x) AND NOT Reach(x)

EDB:

✦

Source = f1g.

✦

Arc = f(1; 2); (3; 4); (4; 3)g.

✦

Target = f2; 3g. 1 2 3 4 source target target

First

compute Reach = f1; 2g (stratum 0).

Next

compute NoReach = f3g. 6

SLIDE 7 Is the Stratied Solution \Ob vious"? Not really .

There

is another mo del that mak es the rules true no matter what v alues w e substitute for the v ariables.

✦

Reach = f1; 2; 3; 4g.

✦

NoReach = ;.

Remem

b er: the

nly

w a y to mak e a Datalog rule false is to nd v alues for the v ariables that mak e the b

dy

true and the head false.

✦

F

r

this mo del, the heads

f

the rules for Reach are true for all v alues, and in the rule for NoReach the subgoal NOT Reach(x) assures that the b

dy

cannot b e true. 7

SLIDE 8 SQL3 Recursion WITH stu that lo

ks

lik e Datalog rules an SQL query ab

ut

EDB, IDB

Rule

= [RECURSIVE] R (<argumen ts>) AS SQL query 8

SLIDE 9 Example Find Sally's cousins, using EDB Par(child, parent). WITH Sib(x,y) AS SELECT p1.child, p2,child FROM Par p1, Par p2 WHERE p1.parent = p2.parent AND p1.child <> p2.child, RECURSIVE Cousin(x,y) AS Sib UNION (SELECT p1.child, p2.child FROM Par p1, Par p2, Cousin WHERE p1.parent = Cousin.x AND p2.parent = Cousin.y ) SELECT y FROM Cousin WHERE x = 'Sally'; 9

SLIDE 10 Plan for Describing Legal SQL3 recursion 1. Dene \monotonicit y ," a prop ert y that generalizes \straticati

n."

2. Generalize stratum graph to apply to SQL queries instead

f

Datalog rules.

✦

(Non)monotonicit y replaces NOT in subgoals. 3. Dene seman tically correct SQL3 recursions in terms

f

stratum graph. Monotonicit y If relation P is a function

f

relation Q (and p erhaps

ther

things), w e sa y P is monotone in Q if adding tuples to Q cannot cause an y tuple

f

P to b e deleted. 10

SLIDE 11 Monotonicit y Example In addition to certain negations, an aggregation can cause nonmonotonicit y . Sells(bar , beer , price) SELECT AVG(price) FROM Sells WHERE bar = 'Joe''s Bar';

Adding

to Sells a tuple that giv es a new b eer Jo e sells will usually c hange the a v erage price

f

b eer at Jo e's.

Th

us, the former result, whic h migh t b e a single tuple lik e (2:78) b ecomes another single tuple lik e (2:81), and the

ld

tuple is lost. 11

SLIDE 12 Generalizing Stratum Graph to SQL

No

de for eac h relation dened b y a \rule."

No

de for eac h sub query in the \b

dy"
f

a rule.

Arc

P ! Q if a) P is \head"

f

a rule, and Q is a relation app earing in the FROM list

f

the rule (not in the FROM list

f

a sub query). b) P is head

f

a rule, and Q is a sub query directly used in that rule (not nested within some larger sub query). c) P is a sub query , and Q is a relation

r

sub query used directly within P .

Lab

el the arc

if

P is not monotone in Q.

Requiremen

t for legal SQL3 recursion: nite strata

nly

. 12

SLIDE 13 Example F

r

the Sib/Cousin example, there are three no des: Sib, Cousin, and S Q (the second term

f

the union in the rule for Cousin. Sib Cousin S Q

No

nonmonotonicit y , hence legal. 13

SLIDE 14 A Nonmonotonic Example Change the UNION to EXCEPT in the rule for Cousin. RECURSIVE Cousin(x,y) AS Sib EXCEPT (SELECT p1.child, p2.child FROM Par p1, Par p2, Cousin WHERE p1.parent = Cousin.x AND p2.parent = Cousin.y )

No

w, Adding to the result

f

the sub query can delete Cousin facts; i.e., Cousin is nonmonotone in S G. Sib Cousin S Q

Innite

n um b er

f

's in cycle, so illegal in SQL3. 14

SLIDE 15 Another Example: NOT Do esn't Mean Nonmonotone Lea v e Cousin as it w as, but negate

ne
f

the conditions in the where-clause. RECURSIVE Cousin(x,y) AS Sib UNION (SELECT p1.child, p2.child FROM Par p1, Par p2, Cousin WHERE p1.parent = Cousin.x AND NOT (p2.parent = Cousin.y) )

Y
u

migh t think that S G dep ends negativ ely

n

Cousin, but it do esn't.

✦

If I add a new tuple to Cousin, all the

ld

tuples still exist and yield whatev er tuples in S G they used to yield.

✦

In addition, the new Cousin tuple migh t com bine with

ld

p1 and p2 tuples to yield something new. 15