Probabilities in Databases and Logics I
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Transcript of Probabilities in Databases and Logics I
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Probabilities inDatabases and Logics
INilesh Dalvi and Dan SuciuUniversity of Washington
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Two Lectures•Today:
probabilistic database to model imprecisions
probabilistic logics
•Tomorrow:
probabilistic database to model incompletness
random graphs
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Motivation
Record reconciliation
Information extraction
Constraint violations
Schema matching
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name rating p
Monkey Love good .5
fair .2
fair .6
poor .9
Review
Queries:A(x,y) :- A(x,y) :- Review(x,y),Review(x,y), Movie(x,z), z Movie(x,z), z > 1991> 1991
Problem SettingTables
:
title year p
Twelve Monkeys 1995 .8
Monkey Love 1997 .4
Monkey Love 1935 .9
Monkey Love Pl 2005 .7
Answers:title rating p
Twelve Monkeys fair .53
Monkey Love good .42
Monkey Love Pl fair .15
Movie
Top k
Problem: complexity of
query evaluation
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Two Problems
Fix answer tuple (a,b)Given database I, compute Pr(Q(a,b))
Query evaluation problem
Fixed schema S, conjunctive query Q(x,y)
Fix k > 0Given database I, find k answer tuples with highest probabilities
Top-k answering problem
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Related Work: DB
Cavallo&Pitarelli:1987
Barbara,Garcia-Molina, Porter:1992
Lakshmanan,Leone,Ross&Subrahmanian:1997
Fuhr&Roellke:1997
Dalvi&S:2004
Widom:2005
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Related Work: Logic
Query reliability [Gradel,Gurevitch,Hirsch’98]
Degrees of belief [Bacchus,Grove,Halpern,Koller’96]
Probabilistic Logic [Nielson]
Probabilistic model checking [Kwiatkowska’02]
Probabilistic Relational Model[Taskar,Abbeel,Koller’02]
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Outline
Definitions
Query Evaluation
Top-k answering (joint with Chris Re)
Conclusions
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Pr : Inst Pr : Inst →→ [0,1], ∑ [0,1], ∑II Pr[I] = 1 Pr[I] = 1
Probabilistic Database
•Schema S, Domain D, Set of instances Inst
•DefinitionProbabilistic database is a probability distribution
If Pr[I] > 0 then I is called “possible world”
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Probabilistic Database
•Representation:
•Independent tuples:I-database DB over some schema Si
•Independent and disjoint tuples:ID-database DB over some schema Sid
Semantics:
DB “means” probability distribution Pr over schema S
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I-DatabasesMovie Score P
m42 good p1
m99 good p2
m76 poor p3
Pr[I1] + Pr[I2] + . . . + Pr[I8] = 1
Reviewsi(M,S,p)
Mov Scor
m76 poor
Mov Scor
m42 good
m76 poor
Mov Scor
m42 1995
m76 poor
Mov Scor
m42 good
Mov Scor
m99 good
Mov Scor
(1-p1)*(1-
p2)*(1-p
3)
Pr[I1]=
Mov Scor
m42 good
m99 good
p1*p
2*(1-
p3)
Pr[I4]=
Mov Scor
m42 good
m99 good
m76 poor
p1*p
2*p
3Pr[I8]=
Representation
Possible worlds semantics
Reviews(M,S)
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ID-DatabasesTimed Activit
yP
t walk p1
t run p2
t+1 walk p3
Pr[I1] + Pr[I2] + . . . + Pr[I6] = 1
Activitiesid
Time Act Time Act
t run
Time Act
t walk
t+1 walk
Time Act
t walk
Time Act
t+1 walk
Time Act
t run
t+1 walk(1-p1-
p2)*(1-p
3)
Pr[I1
]=p2*(1-p
3)
Pr[I3
]= p1*p
3
Pr[I5
]=
Activities
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ID subsumes I
Movied Scored P
m42 good p1
m99 good p2
m76 poor p3
Reviewsid
Movie Score P
m42 good p1
m99 good p2
m76 poor p3
Reviewsi
=
Note: Movie Score P
m42 good p1
m99 good p2
m76 poor p3
Reviewsid means alltuples aredisjoint
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Queries
id year Pm42 1995 0.95
m99 2002 0.65m76 2002 0.1m05 2005 0.7
mid rating p
m42 4 0.7
m42 5 0.45m99 5 0.82
m99 4 0.68
m05 5 0.79
Moviei Reviewi
Q(y) :- Movie(x,y), Q(y) :- Movie(x,y), Review(x,z), z >= 3Review(x,z), z >= 3
•Syntax: conjunctive queries over schema S
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Two Query Semantics
•Possible answer sets
Given set A:
Used for views
•Possible tuples
Given tuple t:
Used for query evaluation and top-k
Pr[{t | I Pr[{t | I ⊨⊨ Q(t)} = A]Q(t)} = A]
Pr[I Pr[I ⊨⊨ Q(t)]Q(t)]
ThisThistalktalk
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p1id year
m42 2004
m99 1901
m76 1902
p2id year
m99 1935
m05 1903
p4id year
m87 1934
m44 1904
p3id year
m76 1995
m99 1935
m05 2004
Q(y) :- Movie(x,y), Q(y) :- Movie(x,y), Review(x,z)Review(x,z)
top k
year p
1935 p2 + p3 = 0.6
2004 p1 + p3 = 0.5
1995 p3 = 0.2
. . . . . .
Query Semantics
Tupleprobabilities
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Summary on Data Model
Data Model:Semantics = possible worldsSyntax = I-databases or ID-databases
Queries:Syntax = unchanged (conjunctive queries)Semantics = tuple probabilities
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Outline
Definitions
Query evaluation
Top-k answering
Conclusions
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Problem Definition
•Fix schema S, query Q, answer tuple t
•Problem: given I/ID-database DB, compute Pr[I ⊨ Q(t)]•Conventions: For upper bounds (P or #P): probabilities are rationalsFor lower bounds (#P): probabilities are 1/2
Pr[Q(t)]Pr[Q(t)]notation:
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Query Evaluationon I-Databases
•Outline
Intuition
Extensional plans: PTIME case
Hard queries: #P-complete case
Dichotomy Theorem
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Intuition
Year p
1995
2002
p1 × (1 - (1 - q1) ×(1 - q2)×(1 - q3))
1 - (1 - ) × (1 - )
p2 × (1 - (1 - q4)×(1 - q5))p3 × q6
id year p
m42 1995 p1
m99 2002 p2
m76 2002 p3
m05 2005 p4
mid rate pm42 4 q1
m42 2 q2
m42 3 q3
m99 1 q4
m99 3 q5
m76 5 q6
Moviei Reviewi
Answer
Q(y) :- Q(y) :- Movie(x,y),Movie(x,y), Review(x,z)Review(x,z)
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Add Join ⋈ p = p1 * p2
Projection ∏ p = 1-(1-p1)(1-p2)...(1-pn)Selection σ p = p
Note: data complexity is PTIME
p
I-Extensional Plans
[Barbara92,Lakshmanan97]
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⋈
∏
Movie Review
CORRECTINCORRECT!
1995 m1 pq1
1995 m1 pq2
1995 m1 pq3
19951-(1-pq1)(1-pq2)
(1-pq3)
⋈
∏
∏
MovieReview
m11 - (1-q1)(1-
q2)(1-q3)
1995 m1p(1-(1-q1)(1-q2)
(1-q3))
m1 q1
m1 q2
m1 q3
1995 p
m1 q1
m1 q2
m1 q3
1995 p
Q(y) :- Q(y) :- Movie(x,y),Movie(x,y), Review(x,z)Review(x,z)
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QQbadbad :- R :- Rii(x), (x), S(x,y), TS(x,y), Tii(y)(y)
A pp1
p2
p3
p4
B pq1
q2
q3
q4
A B
Ri S Ti
TheoremTheorem: Data complexity is #P-: Data complexity is #P-completecomplete
#P-Complete Queries
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Proof:
A px1 1/2
x2 1/2x3 1/2x4 1/2
B py1 1/2
y2 1/2y3 1/2
A Bx2 y3
x1 y2
x4 y3
x3 y1
Ri S Ti
Reduction:x2y3 V x1y2 V x4y3 V x3y1
QQbadbad :- R :- Rii(x), (x), S(x,y), TS(x,y), Tii(y)(y)
TheoremTheorem [Provan&Ball83] Counting the [Provan&Ball83] Counting the number of satisfying assignments for number of satisfying assignments for bipartite DNF is #P-completebipartite DNF is #P-complete
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I-Dichotomy
Definition 1. For each Definition 1. For each variable x:variable x: goals(x) = set of goals that goals(x) = set of goals that contain xcontain x
Q = boolean conjunctive query
Definition 2. Q is hierarchical Definition 2. Q is hierarchical if forall x, y:if forall x, y: (a) goals(x) (a) goals(x) ∩∩ goals(y) = goals(y) = ∅∅, or, or (b) goals(x) (b) goals(x) ⊆⊆ goals(y), or goals(y), or (c) goals(y) (c) goals(y) ⊆⊆ goals(x) goals(x)
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Q :- Q :- R(x),S(x,y),T(x,y,z),KR(x),S(x,y),T(x,y,z),K(x,v)(x,v)
QQ :- R(x), :- R(x), S(x,y), T(y)S(x,y), T(y)
x yz
RRSS
TTv KK
x y
RR SS TT
“hierarchical”
“non-hierarchical”
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I-Dichotomy[Dalvi&S.’04]
Theorem Let Q = conjunctive query w/o self-joins.Then one of the following holds:
Q is in PTIMEQ is in PTIMEQ has a correct Q has a correct extensional planextensional planQ is hierarchicalQ is hierarchicalor:Q is #P-completeQ is #P-completeQ has subgoals Q has subgoals R(x,...),S(x,y,...),T(y,...)R(x,...),S(x,y,...),T(y,...)
Schema Si = {R1i, R2
i, . . ., Rmi}
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ProofLemma 1. Lemma 1. If Q is non-hierarchical, then If Q is non-hierarchical, then #P-complete#P-completeProof:
x y
RR SS TTz
KKv
Q :- RQ :- Rii(v,(v,xx), S), Sii((x,yx,y), ), TTii((yy,z), K,z), Kii(z)(z) rest is like for
Qbad
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ProofLemma 2. If Q is hierarchical, Lemma 2. If Q is hierarchical, then PTIMEthen PTIMEProof:
Case 1: has no Case 1: has no rootroot
Pr(Q) = Pr(Q1) Pr(Q2) Pr(Q3)
This is extensional join ⋈
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Proof
Case 2: has root Case 2: has root xx
x
Pr(Q) = 1 - (1-Pr(Q(a1/x))(1-Pr(Q(a2/x))...(1-Pr(Q(an/x)))
This is an extensional projection: ∏
Dom={a1, a2, . . ., an}
QED
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Query Evaluationon ID-Databases
ID-extensional plans
#P-complete queries
Dichotomoy Theorem
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Only difference: two kinds of projections:independent 1-(1-p1)...(1-pn)disjoint p1 + ... + pn
Extensional Plans for ID-DBs
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#P-Complete Queries
QQ22 :- R :- Rdd(x(xdd,y), ,y), SSdd(y(ydd,z),z)
QQ11 :- R :- Rii(x), S(x), Sii(x,y), (x,y), TTii(y)(y)
QQ33 :- R :- Rdd(x(xdd,y), ,y), SSdd(z(zdd,y),y)
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I-DB Dichotomy[Dalvi&S.’04]
Theorem Let Q = conjunctive query w/o self-joins.Then one of the following holds:Q is in PTIMEQ is in PTIME
Q has a correct Q has a correct extensional planextensional plan
or:Q is #P-completeQ is #P-completeQ has one of QQ has one of Q11, Q, Q22, Q, Q33 as as subqueriessubqueries
Schema Sid s.t. each table is either Ri or Rid
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Extensions
•Extensions of the dichotomoy theorem exists for:
Mixed schemas (some relations are deterministic)
Functional dependencies
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Summary on Query Evaluation
•Extensional plans: popular, efficient, BUT
“Equivalent” plans lead to different results
Some queries admit “correct” plans
•Some simple queries: #P-complete complexity
•Dichotomy theorem
•Future work: remove ‘no-self-join’ restriction
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Outline
Definitions
Query evaluation
Top-k answering (joint with Chris Re)
Conclusions
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Top-k Ranking Problem
•Fix schema S, query Q, number k > 0
•Problem: given I- or ID-database DB,find k answers t1,...,tk with highest probabilities
•Note: Checking Pr[Q(ti)] > Pr[Q(tj)] is PP-completeGoal: efficient polynomial time approximation
Pr[Q(tPr[Q(t11)] > Pr[Q(t)] > Pr[Q(t22)] > .... > )] > .... > Pr[Q(tPr[Q(tkk)] > ...)] > ...
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Probabilities of Boolean
ExpressionsWhat is the probability of e1⋀e2 ⋁ e1⋀e2 ⋁ e1⋀e3?
(1-p1)p2p3 + p1(1-p2)p3 + p1p2(1-p3) + p1p2p3
e1 e2 e3 Pr
0 0 0 (1-p1)(1-p2)(1-p3)
0 0 1 (1-p1)(1-p2)p3
0 1 0 (1-p1)p2(1-p3)
0 1 1 (1-p1)p2p3
1 0 0 p1(1-p2)(1-p3)
1 0 1 p1(1-p2)p3
1 1 0 p1p2(1-p3)
1 1 1 p1p2p3
Theorem #P-hard [Valiant]Theorem #P-hard [Valiant]
A p
e1 p1
e2 p2
e3 p3
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Monte Carlo Simulation
Better: PTAS
Pr( |p’-p| < Pr( |p’-p| < ε ε ) ) > 1-> 1-δδ
[Karp&Luby’83]
Algorithm:Algorithm:
radomly pick each eradomly pick each e11, e, e22, e, e33 = false = false or trueor true compute ecompute e11∧e∧e2 2 ∨ e∨ e11∧e∧e3 3 ∨ e∨ e22∧e∧e33: true or : true or false ?false ? repeatrepeat
Approximate probability p with Approximate probability p with frequency p’frequency p’
p’p’p’- p’- εε p’+ p’+ εε
pp
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Monte Carlo Simulation
N=0
0 1p
N=1
N=2
N=3
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The Multisimulation
Problem
Year P
1995 ??
2002 ??
1933 ??
1984 ??
Schedule simulation steps to find top-k
0 1
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Multisimulation
How to find the top k out of n ?
Example: looking for top k=2;
0 1
12
345
Which one simulate next ?
p5
p1p4
p2
p3
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Multisimulation
Critical region: (k’th left, k+1’th right)
0 1
k=2
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Multisimulation Algorithm
Case 1: pick a “double crosser” and simulate it
0 1
this this
k=2
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Multisimulation Algorithm
Case 2: pick both a “left” AND a “right” crosser
k=2
0 1
thi thiss
and and this this
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Multisimulation Algorithm
Case 3: pick a “max crosser” and simulate it
0 1
thi thiss
k=2
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Multisimulation Algorithm
End: when critical region is “empty”
0 1
k=2
To sort the top k, find the top k-1, etc
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Multisimulation Algorithm
Theorem Theorem (1) It runs in < 2 Optimal # steps(1) It runs in < 2 Optimal # steps (2) no other deterministic algorithm (2) no other deterministic algorithm does betterdoes better
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Experiments
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Summary on Top-k Answering
Simple algorithm, optimal (x2) w.r.t. a very powerful standard
Marriage of probabilistic and top-k answers make probabilistic databases practical
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Outline
Definitions
Query evaluation
Top-k answering
Conclusions
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Conclusions
Strong motivation from practical applicationsOpportunity to merge query and search technologies
Probabilistic DB’s are hard !Great opportunity for impactful theory work
Tomorrow: applications of random graphs to model incompleteness in databases
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Thank you !
Questions ?