[PPT] - Query Con tainmen t for Conjunctiv e Queries With Regular PowerPoint Presentation

SLIDE 1 Query Con tainmen t for Conjunctiv e Queries With Regular Expressions Daniela Florescu, Alon Levy , Dan Suciu Query Con tainmen t for Conjunctiv e Queries With Regular Expressions PODS'98 Dan Suciu A T&T Labs 1

SLIDE 2 Intr

duction

Semistructured Data. F eatures:

Do

es not t in to pre-existing, xed sc hema

Missing

attributes

A

ttributes

f

unkno wn cardinalities

Set
f

attributes unkno wn in adv ance

Irregular

nesting Query Languages for Semistructured Data. F eatures:

Lab

el v ariables

Regular

path expressions: recursiv e queries Query Con tainmen t for Conjunctiv e Queries With Regular Expressions PODS'98 Dan Suciu A T&T Labs 2

SLIDE 3 Motivation Curren t applications

f

semistructured data:

Data

in tegration [Tsimmis]

W

eb querying [W ebSQL, W ebOQL]

General-purp
se

[Lore, UnQL]

W

eb-site managemen t [Strudel] Query Con tainmen t needed in semistructured data for:

Chec

king in tegrit y constrain ts for W eb-sites [Strudel]

Query

rewriting to accomo date v ariet y

f

storage metho ds. Query Con tainmen t for Conjunctiv e Queries With Regular Expressions PODS'98 Dan Suciu A T&T Labs 3

SLIDE 4 Pr evious Work

n

Query Containment and Equivalenc e Non-recursiv e Conjunctiv e Queries:

decidable

for conjunctiv e queries [Chandra and Merlin 77]

w/

union [Sagiv and Y annak akis 81]

w/
rder

and inequalities [Klug 88]

w/

nested relations [Levy and Suciu 97] Results

n

Recursiv e Queries:

undecidable

for Datalog [Sh umeli 93]

recursiv

e v.s. nonrecursiv e [Chaudh uri and V ardi 92]

sev

eral results

n

deciding b

unde

dness

f

recursiv e queries

No

p

sitiv

e results for true recursiv e queries. Query Con tainmen t for Conjunctiv e Queries With Regular Expressions PODS'98 Dan Suciu A T&T Labs 4

SLIDE 5 Outline

Bac

kground (semistructured data and queries)

Denitions
Main

results

Con

tainmen t for simple regular expressions.

Con

tainmen t for arbitrary regular expressions.

Conclusions

Query Con tainmen t for Conjunctiv e Queries With Regular Expressions PODS'98 Dan Suciu A T&T Labs 5

SLIDE 6 Backgr

und:

Semistructur e d Data Denition Graph Database = Graph with lab eled edges

a b a c b d tup tup tup phone 5469 sue 2654 joe 1234 john name name phone name phone

Other p

ssible

v ariations: v alues attac hed to lea v es. Query Con tainmen t for Conjunctiv e Queries With Regular Expressions PODS'98 Dan Suciu A T&T Labs 6

SLIDE 7 Backgr

und:

Query L anguages for Semistructur e d Data Languages: LOREL, UnQL, W ebSQL, W ebOQL, StruQL. Denition An atomic condition X R Y where X ; Y = v ariables, R = regular path expression. Examples: X a Y simple lab el X a:(bjc:d) Y a follo w ed b y b

r

b y c:d X a

:b

+ Y regular path expression X L:(a:L)

Y

lab el v ariables Query Con tainmen t for Conjunctiv e Queries With Regular Expressions PODS'98 Dan Suciu A T&T Labs 7

SLIDE 8 Conjunctive Queries with R e gular Expr essions Denition Q : q (

X

) : Y 1 R 1 Z 1 ; : : : ; Y n R n Z n Meaning

n

database D B : a k

ary

relation, where k =j

X

j. Example: q (L; Y ) : X (a:L) Y ; Y L Z

d d c d u1 u4 c a b c b c b b b u5 u10 u3 u8 u14

DB =

u2 u9 u7 u6 u11 u12 u13 u15 a

Q(D B ) L Y b u5 c u8 Query Con tainmen t for Conjunctiv e Queries With Regular Expressions PODS'98 Dan Suciu A T&T Labs 8

SLIDE 9 The Containment and Equivalenc e Pr

blems

Con tainmen t Giv en Q1; Q2 c hec k if 8D B ; Q1(D B )

Q2(D

B ) Equiv alence Giv en Q1; Q2 c hec k if 8D B ; Q1(D B ) = Q2(D B ) Example: Q1 : q 1(X ; Z ) :

X

a + Z Q2 : q 2(X ; Z ) :

X

L + Z ; Y aZ ; X (a + j(a:b

))Z

Q1

Q2

Note Con tainmen t and Equiv alence Problem for regular expressions is PSP A CE-complete [Stokmey er and Mey er 1973]. Query Con tainmen t for Conjunctiv e Queries With Regular Expressions PODS'98 Dan Suciu A T&T Labs 9

SLIDE 10 Main R esults Theorem Con tainmen t (and equiv alence)

f

conjunctiv e queries with regular path expressions is decidable in exp

nen

tial space. Theorem Con tainmen t (and equiv alence)

f

conjunctiv e queries with simple regular path expressions is NP-complete. Query Con tainmen t for Conjunctiv e Queries With Regular Expressions PODS'98 Dan Suciu A T&T Labs 10

SLIDE 11 Simple R e gular Expr essions Denition R ::= r 1 :r 2 : : : r n where eac h r i is constan t

r
Examples

a:b:c a:

:c:
:a

:a:

:
:b

(= :a:

:b)

Query Con tainmen t for Conjunctiv e Queries With Regular Expressions PODS'98 Dan Suciu A T&T Labs 11

SLIDE 12 Containment

f

Conjunctive Queries with Simple R e gular Expr essions Con tainmen t test: Q

Q

i there exists a query mapping f : Q ! Q s.t.

X

aY mapp ed to similar condition X aY

X
Y

mapp ed to an y c hain X r Z 1 ; Z 1 r 1 Z 2 ; : : : ; Z n r n Y Example: X c Y ; Y ( :a)Z ; Y (b:d) U

X
Y

; Y a Z ; X

U

; U b W .

c b a * * * b d a

Hence:

NP-complete. Query Con tainmen t for Conjunctiv e Queries With Regular Expressions PODS'98 Dan Suciu A T&T Labs 12

SLIDE 13 Do es Not Work in Gener al Case One query mapping is not sucien t: Q1 : q 1(X ) :

X

a

U;

U b V ; U (a:b) V Q2 : q 2(X ) :

X

((a:a)

:b)

Y Q1

Q2,

but it is \witnessed" b y t w

query

mappings.

X X a* (a.a)*.b

?

a.b b

Need to consider sev eral query mappings. Query Con tainmen t for Conjunctiv e Queries With Regular Expressions PODS'98 Dan Suciu A T&T Labs 13

SLIDE 14 A nother Example q 1(X ) :

X

(a:)Y ; Y (b:c)Z ; Y cU q 2(X ) :

X

(a: :

:c)W

Need to map W either to Z

r

to U Query Con tainmen t for Conjunctiv e Queries With Regular Expressions PODS'98 Dan Suciu A T&T Labs 14

SLIDE 15 Canonic al Datab ases Denition Giv en Q1, a canonical database D B is

ne
btained

b y \expanding" the query graph. Example: Q1 : q 1(X ) : X a

U;

U b V ; U (a:b) V Q1 =

X a* b a.b

D B =

a b X a b a a a a

In general there are innitely man y canonical D B for Q1. Prop

sition

Q1

Q2

i

X

2 Q2(D B ) for all canonical databases. Query Con tainmen t for Conjunctiv e Queries With Regular Expressions PODS'98 Dan Suciu A T&T Labs 15

SLIDE 16 Canonic al Datab ases and Query Mappings

Eac

h mapping f : Q2 ! Q1 pro v es

X

2 Q(D B ) for a certain set S f

f

canonical D B 's

There

are

nly

exp

nen

tially man y mappins f : Q2 ! Q1.

Hence:

to c hec k Q1

Q2

suces to c hec k that S f S f = all canonical databases.

Ho

w to compute S f ? Query Con tainmen t for Conjunctiv e Queries With Regular Expressions PODS'98 Dan Suciu A T&T Labs 16

SLIDE 17 Enc

ding

Canonic al D B 's as Wor ds Example Giv en query Q1 : q 1(X ) : X a

U;

U b V ; U (a:b) V Ob ject Enco ding Canonical D B :

a b X a b a a a a

w D B = a:a:a:a:a$b$a:b All canonical D B 's W Q1 = a

$b$a:b

Query Mapping f : Q2 ! Q1 W f enco des S f

X X a*

f1

(a.a)*.b a.b b

W f 1 = (a:a)

$b$a:b

X X a*

f2

(a.a)*.b a.b b

W f 2 = a:(a:a)

$b$a:b

Query Con tainmen t for Conjunctiv e Queries With Regular Expressions PODS'98 Dan Suciu A T&T Labs 17

SLIDE 18 Containment T est Prop

sition

Q1

Q2

i S f W f

W

Q1 Complexit y

there

are exp

nen

tially man y query mappings f

eac

h W f is

f

p

lynomial

size Hence: Exp

nen

tial Space Query Con tainmen t for Conjunctiv e Queries With Regular Expressions PODS'98 Dan Suciu A T&T Labs 18

SLIDE 19 Conclusions

Dened

usefull class

f

recursiv e queries for whic h con tainmen t is NP-complete.

More

general case: Con tainmen t decidable EXPSP A CE.

Op

en questions: can it b e done in PSP A CE ?

\Reasoning

b y cases" required.

Note:

conjunctiv e queries with r.e. not closed under comp

sition.

T

study

con tainmen t

f

comp

sed

queries.

Query

rewriting. Query Con tainmen t for Conjunctiv e Queries With Regular Expressions PODS'98 Dan Suciu A T&T Labs 19