believe&it&or&not&–& adding&belief&annota4ons&&...

Believe It or Not – Adding belief annota4ons to databases Wolfgang Ga*erbauer, Magda Balazinska, Nodira Khoussainova, and Dan Suciu

University of Washington h*p://db.cs.washington.edu/beliefDB/

August 25, VLDB 2009

http://db.cs.washington.edu/beliefDB/

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High-‐level overview

•  DBMS: manage consistent data •  ApplicaLons need inconsistent DM –  ScienLfic databases –  Internet community databases •  Community DBMS: manage inconsistent views

•  This work: Belief databases – manage data and curaLon –  grounded in modal and default logic –  implemented on top of relaLonal model

reason: disagreement !

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Agenda

• MoLvaLng example

•  Logic foundaLons

•  RelaLonal implementaLon

•  Discussion

Observa4ons

id uid species date locaLon comment

2 Alice Crow 06-‐14-‐08 Lake Placid found feathers

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Mo4va4ng applica4on

•  NatureMapping project (h*p://depts.washington.edu/natmap/) –  volunter contribute animal observaLons –  one person curates the database

Sigh4ngs (S)

sid uid species date locaLon

s2 Alice Crow 06-‐14-‐08 Lake Placid

Comments (C)

cid comment sid

c1 found feathers s2

problem: does not scale!

Alice

Bob

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1. Dis4nct database instances

S sid uid species date locaLon

s2 Alice Raven 06-‐14-‐08 Lake Placid



D1: Belief worlds: individually consistent, mutually possibly inconsistent

Alice

Bob

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1. Dis4nct database instances





BeliefSQL

insert into BELIEF ‘Alice’ SighLngs values (‘s2’,‘Alice’,‘Crow’,’06-‐14-‐08’,‘Lake Placid’)

select U2.name, S1.species, S2.species from Users as U,

BELIEF ‘Alice’ SighLngs as S1, BELIEF U.uid SighLngs as S2,

where S1.sid = S2.sid and S1.species <> S2.species

Q: Who believes something different than Alice and what?

A: {(‘Bob’, ‘Crow’, ‘Raven’)}

I: Alice believes that she saw a crow.

insert into BELIEF ‘Bob’ SighLngs values (‘s2’,‘Alice’,‘Raven’,’06-‐14-‐08’,‘Lake Placid’)

I: Bob believes that she actually saw a raven.

Alice

Bob

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2. Open world assump4on





Carol



s2 Alice Raven 06-‐14-‐08 Lake Placid −

−

D2: Model incomplete knowledge with exlicit negaLve beliefs

Adapted key constraints !

Alice

Bob

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Carol




−

I: Carol does not believe that Alice saw a crow nor a raven. insert into BELIEF ‘Carol’ not SighLngs values (‘s2’,‘Alice’,‘Crow’,’06-‐14-‐08’,‘Lake Placid’) insert into BELIEF ‘Carol’ not SighLngs values (‘s2’,‘Alice’,‘Raven’,’06-‐14-‐08’,‘Lake Placid’)

Alice

Bob

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Carol




−

Q: Who disagrees with a sighGng from ‘06-‐14-‐08’ that Alice believes?

A: {(‘Bob’, ‘Crow’), (‘Carol’, ‘Crow’)}

select U.name, S1.species from Users as U,

BELIEF ‘Alice’ SighLngs as S1, BELIEF U.uid not SighLngs as S2

where S1.sid = S2.sid and S1.uid = S2.uid and S1.species = S2.species and S1.date = ‘06-‐14-‐08’ and S2.date = ‘06-‐14-‐08’ and S1.locaLon = S2.locaLon

Bob Alice

Alice

Bob

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3. Higher-‐order beliefs

C cid comment sid

c1 purple-‐black feathers s2

C cid comment sid

c1 plain black feathers s2





D3: Beliefs about other user’s beliefs: allow discussion between users

Bob Alice

Alice

Bob

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C cid comment sid


C cid comment sid






I: According to Bob, Alice believes that the feathers of the sighted animal were plain black.

insert into BELIEF ‘Bob’ BELIEF ‘Alice’ Comments values (‘c1’, ‘plain black feathers’, ‘s2’)

Bob Alice

Alice

Bob

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C cid comment sid


C cid comment sid





s2 Alice Crow 06-‐14-‐08 Lake Placid select C1.cid, C1.comment from BELIEF ‘Bob’ BELIEF ‘Alice’ Comments as C1,

BELIEF ‘Bob’ not Comments as C2 where C1.cid = C2.cid and C1.comment = C2.comment and C1.sid = C2.sid

Q: Which comments does Alice believe according to Bob, which he does not believe himself?

A: {(‘c1’,‘plain-‐black feathers’)}

Bob Alice

Alice

Bob

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C cid comment sid


C cid comment sid





s2 Alice Crow 06-‐14-‐08 Lake Placid select U.name, C1.sid, C1.comment from Users as U,

BELIEF U.uid BELIEF ‘Alice’ Comments as C1, BELIEF U.uid not Comments as C2

where C1.cid = C2.cid and C1.comment = C2.comment and C1.sid = C2.sid

Q: Which comments does Alice believe according to somebody, which (s)he does not believe themself?

A: {(‘Bob’, ‘c1’, ‘plain-‐black feathers’)}

C cid comment sid

c1 purple-‐black feathers s2 Bob Alice

Alice

Bob C cid comment sid




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4. Message board assump4on





D4: Default assumpLon: models a trusLng ahtude & avoids repeated inserts

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4. Message board assump4on

Alice

Bob

Bob Alice C cid comment sid


C cid comment sid








Q: Which animal sighGngs does Alice believe according to Bob, which he does not believe himself?

select S1.sid, S1.species from BELIEF ‘Bob’ BELIEF ‘Alice’ SighLngs as S1,

BELIEF ‘Bob’ not SighLngs as S2 where S1.sid = S2.sid and S1.uid = S2.uid and S1.species = S2.species and S1.date = S2.date and S1.locaLon = S2.locaLon

A: {(‘s2’, ‘Crow’)}

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What we have seen so far

•  4 Design decisions for Belief Database model –  DisLnct belief worlds –  Open world assumpLon (OWA) –  Higher-‐order beliefs – Message board assumpLon

•  BeliefSQL –  SQL + ‘BELIEF’ (+ ‘not’)

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Agenda




•  Discussion

BobAlice Bob

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Logic founda4ons: Belief statements

Carol

Alice

Carol

insert into BELIEF ‘Alice’ S values (‘s2’, ‘Alice’, ‘Crow’,…)

i: ☐Alice S+(‘s2’,‘Alice’,‘Crow’,…)

belief statement

ϕ = ☐w ts

relaLonal tuple (t) sign (s)

modal operator & belief path (w)

Bob

Carol

Alice

Bob

…

…

…

…

…

…

Alice

ε

Belief database D = {ϕ1, …, ϕn}

Alice

Alice

S sid uid species …

s2 Alice Crow …

“annotaLon of ground tuple”

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Logic founda4ons: Entailment

Carol

Carol

Alice


s2 Alice Crow … Bob

Carol

Alice

Bob

…

…

…

…

…

ε

ϕ1=☐Alice S+(…‘Crow’,…)

Bob


s2 Alice Crow … Alice

Bob Alice

…

Alice select * from BELIEF ‘Bob’ BELIEF ‘Alice’ S

D ☐BobAlice S+(…‘Crow’,…) |=

BobAlice

Alice

One belief annotaLon:

More than one entailed belief:

D = {ϕ1}

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Logic founda4ons: Message board assump4on

If D ☐w ts |=Message board assumpLon

then D ☐uw ts |=and ☐uw ts consistent with D

ϕ : ☐u ϕ

☐u ϕ

Default logic

D Explicit beliefs (annotaLons)

D Entailed beliefs (extension)

D \ D Implicit beliefs (assumpLons)

non-‐monotonic reasoning !

belief database “contains” more than the explicit belief annotaLons !

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“Semi-‐formal” problem statement

i1: ϕ1

i2: ϕ2

... in: ϕn

Belief statements

Message board assump4on

ϕ : ☐u ϕ

☐u ϕ

INPUT OUTPUT

Adapted key constraints

D ϕ ? |=

D ☐w1…wd R+(x1,…xl) ? |=

Belief Conjunc4ve Queries (BCQ)

q(x) :− ☐w Ri+(xi)

q(x) :− ☐w1R1s1(x1), …, ☐wgRgsg(xg)

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Agenda




•  Discussion

{s11−,s12−,s22+,c22+}

{s11+,s21+,c11+,c21+} {s11+}

{s11+,s21+,c11+}

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Canonical Kripke structure

i1: ☐Alice s11+

i2: ☐Bob s11−

i3: ☐Bob s12−

i4: ☐Alice s21+

i5: ☐Alice c11+

i6: ☐Bob s22+

i7: ☐BobAlice c21+

i8: ☐Bob c22+

Belief statements*

Message board assump4on

ϕ : ☐i ϕ

☐i ϕ

Alice

Bob Bob Carol

Bob Carol

Alice

Carol

#3

#1

#0

#2

* Running example from the paper

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Rela4onal representa4on

Comments_INTERNAL

Ld cid comment sid

c1.1 c1 found feathers s2

c2.1 c2 plain black feathers s2

c2.2 c2 purple-‐black feathers s2

Sigh4ngs_INTERNAL

Ld sid uid species date locaLon

s1.1 s1 Carol Bald eagle 06-‐14-‐08 Lake Forest

s1.2 s1 Carol Fish eagle 06-‐14-‐08 Lake Forest

s2.1 s2 Alice Crow 06-‐14-‐08 Lake Placid

s2.2 s2 Alice Raven 06-‐14-‐08 Lake Placid

Sigh4ngs_V

wid Ld sid s e

#0 s1.1 s1 + y

#1 s1.1 s1 + n

#1 s2.1 s2 + y

#2 s1.1 s1 − y

#2 s1.2 s1 − y

#2 s2.2 s2 + y

#3 s1.1 s1 + n

#3 s2.1 s2 + n

Comments_V

wd Ld cid s e

#1 c1.1 c1 + y

#2 c2.2 c2 + y

#3 c1.1 c1 + n

#3 c2.1 c2 + y

E

wid1 uid wid2

#0 Alice #1

#0 Bob #2

#0 Carol #0

#1 Bob #2

#1 Carol #0

#2 Alice #3

#2 Carol #0

#3 Bob #2

#3 Carol #0

D

wid1 uid

#0 0

#1 1

#2 1

#3 2

S

wid1 uid

#1 #0

#2 #0

#3 #1

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Example Transla4on of a Belief CQ (BCQ)

BeliefSQL

select U.name, S1.species from Users as U,

BELIEF ‘Alice’ SighLngs as S1, BELIEF U.uid not SighLngs as S2

where S1.sid = S2.sid and S1.uid = S2.uid and S1.species = S2.species and S1.date = ‘06-‐14-‐08’ and S2.date = ‘06-‐14-‐08’ and S1.locaLon = S2.locaLon

Q: Who disagrees with a sighGng from ’06-‐14-‐08’ that Alice believes?

select E1.uid as uid1, V.Ld, V.sid, R.uid, R.species, R.date, R.locaLon, V.s into T2 from E as E1, SighLngs_V as V, SighLngs_STAR as R where E1.wid1=0 and V.wid=E1.wid2 and V.Ld=R.Ld and E1.uid='1';

select E1.uid as uid1, V.Ld, V.sid, R.uid, R.species, R.date, R.locaLon, V.s into T1 from E as E1, SighLngs_V as V, SighLngs_STAR as R where E1.wid1=0 and V.wid=E1.wid2 and V.Ld=R.Ld;

select T1.uid1, T2.species from T1 as T1, T2 as T2 where T1.sid=T2.sid and ((T1.s=0 and T1.uid=T2.uid and T1.species=T2.species

and T1.date='6-‐14-‐08' and T1.locaLon=T2.locaLon) or (T1.s=1 and (T1.uid<>T2.uid or T1.species<>T2.species or T1.date<>'6-‐14-‐08' or T1.locaLon<>T2.locaLon)))

and T2.s=1 and T2.date='6-‐14-‐08';

drop table T2; drop table T1;

Transla4on into SQL

q(x,y) :− ☐Alice S+(u,v,y,‘06-‐14-‐08’,z),

☐x S−(u,v,y,‘06-‐14-‐08’,z)

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Agenda




•  Discussion

27

Experiments

RelaLve overhead ρ := |R*| n ρ = O(mdmax) m … #users

dmax … maximum depth of belief annotaLon In theory: e.g. 100 users, max. depth 2

Experiments:

ρ → 10,000 ρ → 21 – 1,009

Size not limitaLon of semanLcs, but of relaLonal implementaLon!

Size

Time Depends on type of query (3 types in paper)

Experiments on 10,000 annotaLons (ρ =22.4):

Considerable speed-‐up to come!

Q1: ~0.1 s

Q2: ~0.4 s

Q3: ~4.5 s

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Inspira4ons and related work (excerpt) •  AnnotaLons –  Buneman et al. [ICDT 2001 / ICDT 2007] –  Bhagwat et al. [VLDBJ 2005], Geerts et al. [ICDE 2006] –  Srivastava & Velegrakis [SIGMOD 2007]

•  Modal logic –  Fagin et al. [1995] –  Calvanese et al. [IS 2008] –  Nguyen [LJ-‐IGPL 2008]

•  Uncertain / incomplete informaLon –  Sarma et al. [ICDE 2006] –  Green & Tannen [IEEE Data Eng. 2006] –  Dalvi & Suciu [PODS 2007]

•  Inconsistency / key violaLons –  Arenas et al. [PODS 1999] –  Fuxman et al. [SIGMOD 2005]

•  Peer-‐to-‐peer compuLng / collaboraLve data sharing –  Bernstein et al. [WebDB 2002] –  Ives et al. [SIGMOD record 2008]

29

Conclusions

•  Proposed BELIEF databases –  Goal: manage, curate inconsistent data

•  ImplementaLon –  Logical foundaLons –  RelaLonal translaLon

•  Current work – making it compact and fast

30

BACKUP

31

Rela4ve overhead of rela4onal representa4on

1E+1

1E+2

1E+3

1E+4

1E+1 1E+2 1E+3 1E+4

RelaLve overhe

d (|R|/n)

Number of annotaLons (n)

!"### !"### !"### !"$%% !"& !"!#

' ! $ ( ' ! $ (

$)*$+

$)*(+

$)*#+

$)*$+ $)*(+ $)*#+ $)*,+

!"#$%#&'()*)+,-'

./01#$'23'4,,25462,7'(,-'

-./0.1$+

-./0.1(+

!+

$+

!+ $+ (+

!+

$+

!+ $+ (+

6

3-7-2009 Example world insert: Before

U = {1,2}

Existing worlds:

1

1!2

2

2!1

2!1!2

Add world

w = 1!2!1

1E+1

1E+2

1E+3

1E+1 1E+2 1E+3 1E+4

Ov

erh

ed

(|R

|/n

)

Number of annotations (n)

Series1

Series2

!"### !"### !"### !"$%% !"& !"!#

' ! $ ( ' ! $ (

$)*$+

$)*(+

$)*#+

$)*$+ $)*(+ $)*#+ $)*,+

!"#$%#&'()*)+,-'

./01#$'23'4,,25462,7'(,-'

-./0.1$+

-./0.1(+

!+

$+

!+ $+ (+

!+

$+

!+ $+ (+

6

3-7-2009 Example world insert: Before

U = {1,2}

Existing worlds:

1

1!2

2

2!1

2!1!2

Add world

w = 1!2!1

1E+1

1E+2

1E+3

1E+1 1E+2 1E+3 1E+4

Overh

ed

(|R

|/n

)

Number of annotations (n)

Series1

Series2

Distribution of belief path depths (Pr[k=x])

Bound for relative overhead |R∗|n = O(mdmax)

Pr[d = {0, 1, 2}] uniform Zipf[0.3̇,0.3̇,0.3̇] 1,009 130

[0.8, 0.19, 0.01] 162 68[0.199, 0.8, 0.001] 26 21

Measured relative overhead |R∗|n for n = 10, 000 annotations, m = 100 users,

uniform or Zipf user participation, and 3 distributions of annotation depth:

Measured relative overhead |R∗|n for m = 100 users, uniform user participation,

and 2 distributions of annotation depth:

32

Query types and execu4on 4mes

1. Query for content : “What does Alice believe?”d ∈ {0, . . . , 4}:

q1,d(x, y) :− �wS+(x, , y, , ), with |w| ∈{ 0, . . . , 4}

2. Query for conflicts: “Which animal sightings does Bob believe that Alicebelieves, which he does not believe himself?”

q2(x, y) :− �2·1S+(x, z, y, u, v), �2S

−(x, z, y, u, v)

3. Query for users: Who disagrees with any of Alice’s beliefs of sightings atLake Placid?”

q3(x):−�xS−(y, z, u, v, ’a’), �1S+(y, z, u, v, ’a’)

Execution times and size of result sets for example queries executed over a beliefdatabase with 10,000 annotations and relative overhead 22.4.

q1,0 q1,1 q1,2 q1,3 q1,4 q2 q3

E(Time) [msec] 105 145 146 152 144 436 4473σ(Time) [msec] 120 168 153 162 162 186 661

Result size 1626 2816 2253 2061 1931 196 99

33

Belief Conjunc4ve Queries (BCQ)

q1(D) ={(’Bob’,’bald eagle’),(’Bob’,’crow’)}

q1 : “Who disagrees with any sighting from ’06-14-08’ that Alice believes?”

q1(x, y) :− �Alice S+(u, v, y, ’06-14-08’, z),�xS−(u, v, y, ’06-14-08’, z)

Conjunctive Queries (CQ) in Datalog form:

Belief Conjunctive Queries (BCQ) in ”Modal Datalog” form:

q(x̄) :− R1(x̄1), . . . , Rg(x̄g)

q(x̄) :− �w̄1Rs11 (x̄1), . . . ,�w̄gR

sgg (x̄g)

34

Revisi4ng the seman4cs / the user

BELIEF ’Alice’ (…,’eagle’,…)

-‐> ’Alice’ASSERTS (…,’eagle’,…)

BELIEF ’Bob’ BELIEF ’Alice’ (…,’black feathers’,…)

-‐> ’Bob’SUGGESTS that the ASSUMPTION (…,’black feathers’,…) has led ‘Alice’ to her original observaLon

Standard relaLonal model

Conflicts in belief worlds: OWA, keys, ML, DA

-‐> Structured discourse

(1) SQL

(2) BeliefSQL

(3) ?

believe&it&or&not&–& adding&belief&annota4ons&&...

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