[PDF] - Used in SQL recursion. 2. Logical rules form the basis PDF Document

SLIDE 1 Logical Query Languages Motiv atio n: 1. Logical rules extend more naturally to recursiv e queries than do es relational algebra.

✦

Used in SQL recursion. 2. Logical rules form the basis for man y information-in tegration systems and applications. 1

SLIDE 2 Datalog Example Likes(drinker , beer ) Sells(bar , beer , price) Frequents(drinker , bar) Happy(d) <- Frequents(d,bar) AND Likes(d,beer) AND Sells(bar,beer,p )

Ab
v

e is a rule.

Left

side = he ad.

Righ

t side = b

dy

= AND

f

sub go als.

Head

and subgoals are atoms.

✦

A tom = pr e dic ate and argumen ts.

✦

Predicate = relation name

r

arithmetic predicate, e.g. <.

✦

Argumen ts are v ariables

r

constan ts.

Subgoals

(not head) ma y

ptionally

b e negated b y NOT. 2

SLIDE 3 Meaning

f

Rules Head is true

f

its argumen ts if there exist v alues for lo c al v ariables (those in b

dy

, not in head) that mak e all

f

the subgoals true.

If

no negation

r

arithmetic comparisons, just natural join the subgoals and pro ject

n

to the head v ariables. Example Ab

v

e rule equiv alen t to Happy(d) =

dr

ink er (Frequents . / Likes . / Sells) 3

SLIDE 4 Ev aluation

f

Rules Tw

,

dual, approac hes: 1. V ariable-b ase d : Consider all p

ssible

assignmen ts

f

v alues to v ariables. If all subgoals are true, add the head to the result relation. 2. T uple-b ase d : Consider all assignmen ts

f

tuples to subgoals that mak e eac h subgoal true. If the v ariables are assigned consisten t v alues, add the head to the result. Example: V ariable-Based Assignmen t S(x,y) <- R(x,z) AND R(z,y) AND NOT R(x,y) R = A B 1 2 2 3 4

SLIDE 5

Only

assignmen ts that mak e rst subgoal true: 1. x ! 1, z ! 2. 2. x ! 2, z ! 3.

In

case (1), y ! 3 mak es second subgoal true. Since (1; 3) is not in R , the third subgoal is also true.

✦

Th us, add (x; y ) = (1; 3) to relation S .

In

case (2), no v alue

f

y mak es the second subgoal true. Th us, S = A B 1 3 5

SLIDE 6 Example: T uple-Based Assignmen t T ric k: start with the p

sitiv

e (not negated), relational (not arithmetic) subgoals

nly

. S(x,y) <- R(x,z) AND R(z,y) AND NOT R(x,y) R = A B 1 2 2 3

F
ur

assignmen ts

f

tuples to subgoals: R (x; z ) R (z ; y ) (1; 2) (1; 2) (1; 2) (2; 3) (2; 3) (1; 2) (2; 3) (2; 3)

Only

the second giv es a consisten t v alue to z .

That

assignmen t also mak es NOT R(x,y) true.

Th

us, (1; 3) is the

nly

tuple for the head. 6

SLIDE 7 Safet y A rule can mak e no sense if v ariables app ear in funn y w a ys. Examples

S(x)

<- R(y)

S(x)

<- NOT R(x)

S(x)

<- R(y) AND x < y In eac h

f

these cases, the result is innite, ev en if the relation R is nite.

T
mak

e sense as a database

p

eration, w e need to require three things

f

a v ariable x (= denition

f

safety). If x app ears in either 1. The head, 2. A negated subgoal,

r

3. An arithmetic comparison, then x m ust also app ear in a nonnegated, \ordinary" (relational) subgoal

f

the b

dy

.

W

e insist that rules b e safe, henceforth. 7

SLIDE 8 Datalog Programs

A

collect i

n
f

rules is a Datalo g pr

gr

am.

Predicates/relati
ns

divide in to t w

classes:

✦

EDB = extensional datab ase = relation stored in DB.

✦

IDB = intensional datab ase = relation dened b y

ne
r

more rules.

A

predicate m ust b e IDB

r

EDB, not b

th.

✦

Th us, an IDB predicate can app ear in the b

dy
r

head

f

a rule; EDB

nly

in the b

dy

. 8

SLIDE 9 Example Con v ert the follo wing SQL (Find the man ufacturers

f

the b eers Jo e sells): Beers(name , manf) Sells(bar , beer , price) SELECT manf FROM Beers WHERE name IN( SELECT beer FROM Sells WHERE bar = 'Joe''s Bar' ); to a Datalog program. JoeSells(b) <- Sells('Joe''s Bar', b, p) Answer(m) <- JoeSells(b) AND Beers(b,m)

Note:

Beers, Sells = EDB; JoeSells, Answer = IDB. 9

SLIDE 10 Expressiv e P

w

er

f

Datalog

Nonrecursiv

e Datalog = (classical ) relational algebra.

✦

See discussion in text.

Datalog

sim ulates SQL select-from-where without aggregation and grouping.

Recursiv

e Datalog expresses queries that cannot b e expressed in SQL.

But

none

f

these languages ha v e full expressiv e p

w

er (T uring c

mpleteness).

10

SLIDE 11 Recursion

IDB

predicate P dep ends

n

predicate Q if there is a rule with P in the head and Q in a subgoal.

Dra

w a graph: no des = IDB predicates, arc P ! Q means P dep ends

n

Q.

Cycles

i recursiv e. Recursiv e Example 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) 11

SLIDE 12 Iterativ e Fixed-P

in

t Ev aluates Recursiv e Rules Change to IDB? Start IDB = ; Apply rules to IDB, EDB y es no done 12

SLIDE 13 Example EDB Par = a d b c e f g j k i h

Note,

b ecause

f

symmetry , Sib and Cousin facts app ear in pairs, so w e shall men tion

nly

(x; y ) when b

th

(x; y ) and (y ; x) are mean t. 13

SLIDE 14 Sib Cousin Initial ; ; Round 1 (b; c); (c; e) ; add: (g ; h); (j; k ) Round 2 (b; c); (c; e) add: (g ; h); (j; k ) Round 3 (f ; g ); (f ; h) add: (g ; i); (h; i) (i; k ) Round 4 (k ; k ) add: (i; j ) 14

SLIDE 15 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 (the str atie d mo del). 15

SLIDE 16 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. 16

SLIDE 17 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

17

SLIDE 18 Example Whic h target no des cannot b e reac hed from an y source no de? Reach(x) <- Source(x) Reach(x) <- Reach(y) AND Arc(y,x) NoReach(x) <- Target(x) AND NOT Reach(x) Reach NoReach

18

SLIDE 19 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 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. 19

SLIDE 20 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. 20

SLIDE 21 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. 21