Brief Tutorial on Probabilistic Databases

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1 Brief Tutorial on Probabilistic Databases Dan Suciu University of Washington Simons 206

2 About This Talk Probabilistic databases Tuple-independent Query evaluation Statistical relational models Representation, learning, inference in FO Reasoning/learning = lifted inference Sources: Book 20 [S.,Olteanu,Re,Koch] Upcoming F&T survey [van Den Broek,S] Simons 206 2

3 Background: Relational databases Smoker Friend X Y X Z Database D = Alice 2009 Alice 200 Bob 2009 Carol 200 Alice Alice Bob Carol Bob Carol Carol Bob Query: Q(z) = x (Smoker(x, 2009 ) Friend(x,z)) Q(D) = Z Bob Carol Constraint: Q = x (Smoker(x, 200 ) Friend(x, Bob )) Q(D) = true

4 Probabilistic Database Probabilistic database D: x y P Possible worlds semantics: p p 2 p 3 (-p )p 2 p 3 x y x a b xy a a b2 a b2 a2 b2 a2 b2 a2 y x y b x a b b2 a2 a b2 a b p a b2 p 2 a2 b2 p 3 Σ world P D (world) = P D (Q) = Σ world = Q P D (world) y x b2 a y x b a (-p )(-p 2 )(-p 3 ) y x y b2

5 Outline Model Counting Small Dichotomy Theorem Dichotomy Theorem Query Compilation Conclusions, Open Problems Simons 206 5

6 Model Counting Given propositional Boolean formula F, compute the number of models #F Example: F = (X X2) (X2 X3) (X3 X) #F = 4 [Valiant 79] #P-hard, even for 2CNF X X2 X3 F

7 Probability of a Formula Each variable X has a probability p(x); P(F) = probability that F=true, when each X is set to true independently Example: F = (X X2) (X2 X3) (X3 X) P(F) = (-p)*p2*p3 + p*(-p2)*p3 + p*p2*(-p3) + p*p2*p3 If p(x) = ½ for all X, then P(F) = #F / 2 n X X2 X3 F

8 Algorithms for Model Counting [Gomes, Sabharwal, Selman 2009] Based on full search DPLL: Shannon expansion. #F = #F[X=0] + #F[X=] Caching. Store #F, look it up later Components. If Vars(F) Vars(F2) = : #(F F2) = #F * #F2 Simons 206 8

Relational Representation (/2) Fix an FO sentence Q and a domain Δ Ground atom à Boolean variable Definition The lineage F Q,Δ is: F Q,Δ = Q if Q = ground atom F Q

9 Relational Representation (/2) Fix an FO sentence Q and a domain Δ Ground atom à Boolean variable Definition The lineage F Q,Δ is: F Q,Δ = Q if Q = ground atom F Q Q2,Δ = F Q,Δ F Q2,Δ same for, à, F x.q,δ = a Δ F Q[a/x],Δ F x.q,δ = a Δ F Q[a/x],Δ Q = x (Student(x) Person(x)) F Q,[n] = (Student() Person()) (Student(n) Person(n))

10 Relational Representation (2/2) For a database D, denote F Q,D = F Q,domain(D) where all tuples not in D are set to false F Q,Δ or F Q,D is called the lineage or the provenance or the grounding of Q

11 Weighted FO Model Counting Probabilities of ground atoms in D = probabilities of Boolean variables p(x) Fix Q. Given D, compute P(F Q,D ) x y P a b p a b2 p 2 a2 b2 p 3 Simple fact: P D (Q) = P(F Q,D ) Simons 206

12 This Talk Fix a query Q: What is the complexity of P D (Q) in the size of D? What is the best runtime of a DPLL-based algorithm on F Q,D in the size of D? Simons 206 2

13 Discussion: Correlations [Domingos&Richardson 06] MLN = popular FO framework for Machine Learning tasks Lise Getoor s talk today Smoker(x) Friends(x,y) àsmoker(y), weight = 2.3 Theorem [Jha,S ] One can construct effectively D s.t. P MLN, Δ (Q) = P D (Q Γ) = P D (Q Γ) / P D (Γ)

14 Outline Model Counting Small Dichotomy Theorem Dichotomy Theorem Query Compilation Conclusions, Open Problems Simons 206 4

15 Background: Query Plans Q(z) = R(z,x), S(x,y),T(y,u), u=23 Query plan = expressions over the input relation Operators = selection, projection, join, union, difference Π z x σ u=23 y R(z,x) S(x,y) T(y,u)

16 Boolean query An Example Q() = R(x), S(x,y) = x y(r(x) S(x,y)) P D (Q) = - {- p*[ -(-q)*(-q2) ]} * {- p2*[ -(-q3)*(-q4)*(-q5) ] } One can compute P D (Q) in PTIME in the size of the database D R x a a2 a3 P p p2 p3 S x y P a b q a b2 q2 a2 b3 q3 a2 b4 q4 a2 b5 q5

17 Extensional Plans Modify each operator to compute output probabilities, assuming independent events Simons 206 7

18 Q() = R(x), S(x,y) P(Q) = [-p *(-(-q )*(-q 2 ))] *[- p 2 *(-(-q 3 )*(-q 4 )*(-q 5 ))] -(-p q )(-p q 2 )(-p 2 q 3 )(-p 2 q 4 )(-p 2 q 5 ) -{-p [-(-q )(-q 2 )]}* {-p 2 [-(-q 4 )(-q 5 ) (-q 6 )]} p q p q 2 Π Φ Wrong Π Φ Right p 2 q 3 p 2 q 4 p 2 q 5 x y P -(-q )(-q 2 ) -(-q 4 )(-q 5 ) (-q 6 ) a b q x P a b 2 q 2 Π x a p a 2 p 2 a 3 p 3 R(x) S(x,y) a 2 b 3 q 3 a 2 b 4 q 4 a 2 b 5 q 5 R(x) S(x,y) 8

19 Safe Queries Definition A plan for Q is safe if it computes the probabilities correctly. Q is safe if it has a safe plan. In AI, computing Q using a safe plan is called lifted inference Safe query = Liftable query If Q is safe then P D (Q) is in PTIME

20 Unsafe Queries R X P x p x2 p2 S X Y x y x y2 x2 y2 H 0 () = R(x), S(x,y), T(y) T Y y y2 P q q2 R P P S Wrong T Theorem. [Dalvi&S.2004] P D (H 0 ) is #P-hard However:. This plan computes an upper bound [VLDB 5] 2. Use samples on T [VLDB 6] Wolfgang Gatterbauer s talk today

21 Hierarchical Queries Fix Q; at(x) = set of atoms (=literals) containing the variable x Definition Q is hierarchical if forall variables x, y: at(x) at(y) or at(x) at(y) or at(x) at(y) = Hierarchical Q() = R(x,y), S(x,z) Non-hierarchical H 0 () = R(x), S(x,y), T(y) y R x S z x R S y T 2

22 The Small Dichotomy Theorem Non-repeating Conjunctive Query = = Conjunctive Query without self-joins = Simple conjunctive query [Dalvi&S.04] Theorem Let Q be a non-repeating CQ If Q is hierarchical, then P D (Q) is in PTIME. If Q is not hierarchical then P D (Q) is #P-hard. By duality, the same holds for a non-repeating clause

23 Summary so Far Complexity of P D (Q) PTIME Non-repeating CQ Non-repeating clause Hierarchical #P - hard Non-hierarchical Simons

24 Outline Model Counting Small Dichotomy Theorem Dichotomy Theorem Query Compilation Conclusions, Open Problems Simons

25 The Rules for Lifted Inference Preprocess Q (omitted from this talk; see book), then apply these rules (some have preconditions) P( Q) = P(Q) negation P(Q Q2) = P(Q)P(Q2) P(Q Q2) = ( P(Q))( P(Q2)) P( z Q) = Π a Domain ( P(Q[a/z]) P( z Q) = Π a Domain P(Q[a/z] Independent join / union Independent project P(Q Q2) = P(Q) + P(Q2) - P(Q Q2) P(Q Q2) = P(Q) + P(Q2) - P(Q Q2) Inclusion/ exclusion

26 FO un = Unate FO An FO sentence is unate if: Negations occur only on atoms Every relational symbol R either occurs only positively, or only negatively FO un = FO restricted to unate sentences Simons

27 Dichotomy Theorem [Dalvi&S 2] Theorem For any Q in * FO un (or * FO un ) If rules succeed, then P D (Q) in PTIME in D If rules fail, then P D (Q) is #P hard in D Note: Unions of Conjunctive queries (UCQ) is essentially * FO un

28 Example: Liftable Query Q J () = S(x,y ), R(y ), S(x 2,y 2 ), T(y 2 ) = [S(x,y ),R(y )] [S(x 2,y 2 ),T(y 2 )] Q Q 2 P(Q J ) = P(Q ) + P(Q 2 ) - P(Q Q 2 ) PTIME PTIME (have (have seen seen before) already) Q Q 2 = y [S(x,y),R(y) S(x 2,y)),T(y)] y = y = y2 P(Q Q 2 ) = = Π b Domain ( P[S(x,b),R(b) S(x 2,b)),T(b)]) = Π b Domain ( P[S(x,b)] * P[R(b) T(b)]) = etc Runtime = O(n 2 ). Simons

29 Example: Liftable Query Q J = x y x 2 y 2 (S(x,y ) R(y ) S(x 2,y 2 ) T(y 2 )) = [ x y S(x,y ) R(y )] [ x 2 y 2 S(x 2,y 2 ) T(y 2 )] Q Q 2 P(Q J ) = P(Q ) + P(Q 2 ) - P(Q Q 2 ) PTIME PTIME (have (have seen seen before) already) y = y = y2 Q Q 2 = y [( x S(x,y) R(y)) ( x 2 S(x 2,y)) T(y)] = y [ x S(x,y) (R(y) T(y))] P(Q Q 2 ) = Π b Domain P[ x.s(x,b) (R(b) T(b))] = etc Runtime = O(n 2 ). Simons

30 Unliftable Queries H k H 0 = R(x) S(x,y) T(y) Will drop to reduce clutter H = [R(x 0 ) S(x 0,y 0 )] [S(x,y ) T(y )] H 2 = [R(x 0 ) S (x 0,y 0 )] [S (x,y ) S 2 (x,y )] [S 2 (x 2,y 2 ) T(y 2 )] H 3 = [R(x 0 ) S (x 0,y 0 )] [S (x,y ) S 2 (x,y )] [S 2 (x 2,y 2 ) S 3 (x 2,y 2 )] [S 3 (x 3,y 3 ) T(y 3 )]... Every H k, k is hierarchical Theorem. [Dalvi&S 2] Every query H k is #P-hard Simons

31 A Closer Look at H k If we drop any one clause à in PTIME H 3 = [R(x 0 ) S (x 0,y 0 )] [S (x,y ) S 2 (x,y )] [S 2 (x 2,y 2 ) S 3 (x 2,y 2 )] [S 3 (x 3,y 3 ) T(y 3 )] Independent join

32 Summary so Far Complexity of P D (Q) Non-repeating CQ Non-repeating clauses * FO un * FO un PTIME Hierarchical Rules succeed #P - hard Non-hierarchical Rules fail Simons

33 Outline Model Counting Small Dichotomy Theorem Dichotomy Theorem Query Compilation Conclusions, Open Problems Simons

34 Lifted v.s. Grounded Inference To compute P D (Q): compute the lineage F Q,D use DPLL-based algorithm for P(F Q,D ) For which queries Q can this be in PTIME? [Huang&Darwiche 2005] The trace of a DPLL-based algorithm is decision-dnnf DL

35 Decision-DNNF Def [Darwiche] A Decision-DNNF is a rooted DAG where: Internal nodes are decision or Sink nodes are 0 or Children of have disjoint sets of variables Every root-to-sink path reads each variable at most once 0 X Y 0 0 U 0 Z

36 Notations H k0 = x y R(x) S (x,y) H k = x y S (x,y) S 2 (x,y) H k2 = x y S 2 (x,y) S 3 (x,y) H kk = x y S k (x,y) T(y) f(z 0, Z,, Z k ) = a Boolean function Q = f(h k0, H k,, H kk ) Example: f = Z 0 Z Z k then f(h k0, H k,, H kk ) = H k 36

37 Easy/Hard Queries [Beame 4] Theorem Let Q = f(h k0, H k,, H kk ) where f(z 0, Z,, Z k ) is a monotone Boolean function. Any Decision-DNNF for F Q,[n] has size 2 Ω( n). P D (Q) is in PTIME iff μ Q (0, ) = 0 μ = Möbius function of the implicates lattice of Q Consequence: Any DPLL-based algorithm takes time 2 Ω( n), even if the query is in PTIME! 37

38 Cancellations Q W = (H 30 H 32 ) (H 30 H 33 ) (H 3 H 33 ) H 30 = x y R(x) S (x,y) H 3 = x y S (x,y) S 2 (x,y) H 32 = x y S 2 (x,y) S 3 (x,y) H 33 = x y S 3 (x,y) T(y) P(Q W ) = P(H 30 H 32 ) + P(H 30 H 33 ) + P(H 3 H 33 ) + P(H 30 H 32 H 33 ) P(H 30 H 3 H 33 ) P(H 30 H 3 H 32 H 33 ) + P(H 30 H 3 H 32 H 33 ) = H 3 (hard!) Also = H 3 P(Q W ) is in PTIME

39 The CNF Lattice Q W = (H 30 H 32 ) (H 30 H 33 ) (H 3 H 33 ) Definition. The DNF lattice of Q = Q Q 2 is: Elements are prime implicants Order is implication H 30 H 32 H 30 H 33 H 3 H 33 H 30 H 32 H 33 H 30 H 3 H 33 H 30 H 3 H 32 H 33 Nodes in PTIME, Nodes #P hard.

40 The Möbius Function Def. The Möbius function: μ(, ) = μ(u, ) = - Σ u < v μ(v, ) Möbius Inversion Formula: P(Q) = - Σ Qi < μ(qi, ) P(Qi) Simons

41 The Möbius Function Def. The Möbius function: μ(, ) = μ(u, ) = - Σ u < v μ(v, ) Möbius Inversion Formula: P(Q) = - Σ Qi < μ(qi, ) P(Qi) Simons 206 4

42 The Möbius Function Def. The Möbius function: μ(, ) = μ(u, ) = - Σ u < v μ(v, ) Möbius Inversion Formula: P(Q) = - Σ Qi < μ(qi, ) P(Qi) Simons

43 The Möbius Function Def. The Möbius function: μ(, ) = μ(u, ) = - Σ u < v μ(v, ) Möbius Inversion Formula: P(Q) = - Σ Qi < μ(qi, ) P(Qi) Simons

44 The Möbius Function Def. The Möbius function: μ(, ) = μ(u, ) = - Σ u < v μ(v, ) Möbius Inversion Formula: P(Q) = - Σ Qi < μ(qi, ) P(Qi) Simons

45 Def. The Möbius function: The Möbius Function μ(, ) = μ(u, ) = - Σ u < v μ(v, ) Möbius Inversion Formula: P(Q) = - Σ Qi < μ(qi, ) P(Qi) Simons

46 Summary nr CQ nr clause * FO un f(h k0,,h kk ) * FO un PTIME Hierarchical Rules succeed μ(0,) = 0 #P - hard DPLL Non-hierarchical Rules fail μ(0,) 0 2 Ω( n) Simons

47 Möbius Über Alles Non-hierarchical H 0 * FO un or * FO un ( UCQ) #P-hard hierarchical H H 2 PTIME Q W Q 9 H 3 Poly-size FBDD, dec-dnnf Poly-size OBDD,SDD = = inversion-free Q J Q V Read Once Q U Open

48 Outline Model Counting Small Dichotomy Theorem Dichotomy Theorem Query Compilation Conclusions, Open Problems Simons

49 Summary Query evaluation on probabilistic databases = weighted model counting Each query Q defines a different WMC problem Dichotomy: depending on Q, WMC is in PTIME or #P-hard Using a DPLL-based algorithm on the grounded Q is suboptimal Simons

50 Discussion: Extensions Open problems: extend the dichotomy theorem to: Mixed probabilistic/deterministic relations Functional dependencies Interpreted predicates: <, Open problem: complexity of MAP Simons

51 Discussion: Symmetric Relations A relation R is symmetric if all ground tuples have the same probability [van den Broeck 4] For every Q in FO 2, P(Q) is in PTIME on symmetric databases. [Beame 5] Hardness results. In general the complexity is open Simons 206 5

52 Discussion: Negation [Fink&Olteanu 4] Restrict FO to non-repeating exrpessions Theorem Hierarchical expressions are in PTIME, non-hierarchical are #P-hard. [Gribkoff,S.,v.d.Broeck 4] * FO or * FO Need resolution compute some queries with negation Open problem: completeness/dichotomy? Dan Olteanu s talk next Simons

53 Simons

54 Thank You! Simons

55 BACKUP Simons

56 Weighted Model Counting Each variable X has a weight w(x); Weight of a model = X=true w(x) WMC(F) = sum of weights of models of F Example: F = (X X2) (X2 X3) (X3 X) WMC(F) = w2*w3 + w*w3 + w*w2 + w*w2*w3 Set w(x) = : then WMC(F) = #F X X2 X3 F

57 Probability of a Formula Each variable X has a probability p(x); P(F) = probability that F=true, when each X is set to true independently Example: F = (X X2) (X2 X3) (X3 X) P(F) = (-p)*p2*p3 + p*(-p2)*p3 + p*p2*(-p3) + p*p2*p3 Set w(x) = p(x)/(-p(x)) Then P(F) = WMC(F) / Z, where Z = Π X (+w(x)) X X2 X3 F

58 Discussion: Dichotomy for #SAT [Creignou&Hemann 96] consider model counting #F where the formula F is given by generalized clauses Dichotomy Theorem: #F is in PTIME when all clauses are affine, #P-hard otherwise Not helpful in our context: If Q is a UCQ, then the clauses of F Q,D are of the form X Y Z and are not affine; But P(F Q,D ) is not always #P-hard, because Q restricts the struture of the clauses Simons

59 Warm-up: Weights Replace probabilities with weights: R: x y w S: y w a b w b v a2 b w 2 b2 v 2 P D (world) = Weight(world)/Z Z = Σ world Weight(world ) a3 b2 w 3 b3 v 3 Weight of a possible world: R: x y y Weight( ) = w w 2 v 2 a b b2 a2 b S: Z = (+v ) (+v 2 ) (+v 3 ) (+w ) (+w 2 ) (+w 3 )

60 Markov Logic Networks Replace probabilities with weights: R: S: x y w y w Add soft constraints: R(x,y) S(y) w 4 a b w a2 b w 2 a3 b2 w 3 b v b2 v 2 b3 v 3

61 Markov Logic Networks Replace probabilities with weights: R: S: x y w y w Add soft constraints: R(x,y) S(y) w 4 a b w a2 b w 2 a3 b2 w 3 b v b2 v 2 b3 v 3 Weight of a possible world: R: S: x y y Weight( ) = w w 3 v w 4 w 4 w 4 w 4 w 4 a b b a3 b2

62 Markov Logic Networks Replace probabilities with weights: Add soft constraints: R: S: R(x,y) S(y) w 4 x y w y w a b w b v P MLN (world) = Weight(world)/Z a2 b w 2 a3 b2 w 3 b2 v 2 b3 v 3 Z = Σ world Weight(world ) Weight of a possible world: R: S: x y y Weight( ) = w w 3 v w 4 w 4 w 4 w 4 w 4 a b b a3 b2

Markov Logic Networks Replace probabilities with weights: Add soft constraints: R: S: R(x,y) S(y) w 4 x y w y w a b w b v P MLN (world) = Weight(world)/Z a2 b w 2 a3 b2 w 3 b2 v 2 Z = Σ

63 Markov Logic Networks Replace probabilities with weights: Add soft constraints: R: S: R(x,y) S(y) w 4 x y w y w a b w b v P MLN (world) = Weight(world)/Z a2 b w 2 a3 b2 w 3 b2 v 2 Z = Σ b3 v world Weight(world ) 3 Z = (+v) (+v2) (+v3) Weight of a possible world: R: x y y Weight( ) = w w 3 v w 4 w 4 w 4 w 4 w 4 a b b a3 b2 S: (+w ) (+w 2 ) (+w 3 ) Z is #P-hard to compute

64 Discussion Weights v.s. probabilities Soft constraints with probabilities may be inconsistent Soft constraints with weights ( 0, ) always consistent Weight values have no semantics! Learned from training data Inconsistent: S(x): p=0.5 S(x) R(x): p=0.9 Consistent: S(x): w=5 S(x) R(x): w=9 Simons

65 MLN s to Tuple-Independent PDB Replace probabilities with weights: Soft constraint: R: S: R(x,y) S(y) w 4 x y w y w Replace with hard constraint: a b w a2 b w 2 a3 b2 w 3 b v b2 v 2 b3 v 3

66 MLN s to Tuple-Independent PDB Replace probabilities with weights: R: x y w a b w a2 b w 2 a3 b2 w 3 S: y w b v b2 v 2 b3 v 3 Soft constraint: R(x,y) S(y) w 4 Replace with hard constraint: Γ x y A(x,y) (R(x,y) S(y)) New relation A: x y w a b w 4 a b2 w 4 a b3 w 4 a b w 4

67 MLN s to Tuple-Independent PDB Replace probabilities with weights: R: x y w a b w a2 b w 2 a3 b2 w 3 R: x S: y w b v b2 v 2 b3 v 3 Soft constraint: a b b a b Weight( ) = w w 3 v w 4 w 4 w 4 w 4 w 4 a3 y b2 S: y A: x a a2 a2 a3 y b2 b b2 b R(x,y) S(y) w 4 Replace with hard constraint: Γ x y A(x,y) (R(x,y) S(y)) New relation A: x y w a b w 4 a b2 w 4 a b3 w 4 a b w 4

68 MLN s to Tuple-Independent PDB Replace probabilities with weights: R: x y w a b w a2 b w 2 a3 b2 w 3 R: x y S: y w b v b2 v 2 b3 v 3 Soft constraint: a b b a b Weight( ) = w w 3 v w 4 w 4 w 4 w 4 w 4 a3 b2 S: y A: x a y b2 a2 b Theorem: P MLN (Q) = a2 Pb2 D (Q Γ) a3 b R(x,y) S(y) w 4 Replace with hard constraint: Γ x y A(x,y) (R(x,y) S(y)) New relation A: x y w a b w 4 a b2 w 4 a b3 w 4 a b w 4

69 Replace with Improved Translation A clause remains a clause! New weight is w 4 - Probability may be < 0!!! That s OK Theorem: P MLN (Q) = P D (Q Γ) Soft constraint: R(x,y) S(y) w 4 Replace with hard constraint: Γ x y A(x,y) (R(x,y) S(x)) New relation A: x y w a b w 4 - a b2 w 4 - a b3 w 4 - a b w 4 -

70 [VLDB 206] SlimShot = SafePlans + Sample Accuracy = f(number of Samples) Lower is better SlimShot VLDB 206 SlimShot needs N 200 for Accuracy=0%

71 SlimShot Dataset: Smokers Runtime SlimShot Runtime = f(n), where N=0000 Runtime = f(precision)

72 Dataset: Smokers Dataset: Drinkers Scalability SlimShot

73 Duality The dual of a query Q is the formula obtained by the following transformations: / à / / à / Q and its dual have the same complexity Query: H 0 () = R(x), S(x,y), T(y) Dual query: H 0 = x y (R(x) S(x,y) T(y)) x y (R(x) S(x,y) T(y)) Simons

74 S.R Example: Liftable Clause Q = x y S(x,y) R(y) = y ( x S(x,y) R(y)) P(Q) = Π b Domain P( x S(x,b) R(b)) Indep. P(Q) = Π b Domain [ P( x S(x,b)) ( P(R(b)))] Indep. or: P(X Y) = = P( X Y) = P(X) (-P(Y)) P(Q) = Π b Domain [ ( Π a Domain (-P(S(a,b)))) ( P(R(b)))] Runtime = O(n 2 ). Lookup Lookup the the probabilities probabilitiesin in D Indep. Simons

Outline. Part 1: Motivation Part 2: Probabilistic Databases Part 3: Weighted Model Counting Part 4: Lifted Inference for WFOMC

Outline. Part 1: Motivation Part 2: Probabilistic Databases Part 3: Weighted Model Counting Part 4: Lifted Inference for WFOMC Outline Part 1: Motivation Part 2: Probabilistic Databases Part 3: Weighted Model Counting Part 4: Lifted Inference for WFOMC Part 5: Completeness of Lifted Inference Part 6: Query Compilation Part 7: