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

Transcription

1 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: Symmetric Lifted Inference Complexity Part 8: Open-World Probabilistic Databases Part 9: Discussion & Conclusions

2 Question 2. Are lifted rules stronger than grounded? Alternative to lifting: 1. Ground the FO sentence 2. Do WMC on the propositional formula There is no reason why grounded inference should be weaker than lifted inference However, existing grounded algorithms are strictly weaker than lifted inference

3 Algorithms for Model Counting [Gomes 08] Based on full search DPLL: Shannon expansion. #F = #F[X=0] + #F[X=1] Caching. Store #F, look it up later Components. If Vars(F1) Vars(F2) = : #(F1 F2) = #F1 * #F2

4 Knowledge Compilation Definition (informal): represent the Boolean formula F in a circuit where WMC(F) is in PTIME in the size of the representation Why we care: The trace of any inference algorithm is a knowledge compilation Lower bounds on size(kc) give lower bounds on the algorithm s runtime

5 Knowledge Compilation Targets 0 0 Y X 1 0 U Z 0 1 FBDD: Decision-, sink-nodes OBDD: fixed variable order 1 0 X Y U 0 Z Decision-DNNF add: -nodes Children of have disjoint sets of variables 1 0 1

6 DPLL and Knowledge Compilation Fact: Trace of full-search DPLL KC: Basic DPLL decision trees DPLL + caching OBDD (fixed variable order) FBDD DPLL + caching + components decision-dnnf [Huang&Darwiche 2005]

7 Hard Queries H 0 = x y (R(x) S(x,y) T(y)) = non-hierarchical H k = hierarchical, has inversion, for k 1 Grounded Boolean formulas: F n (H 0 ) = i [n], j [n] (R i S ij T j ) Th. [Beame 14] Any FBDD for F n (H k ) has size 2 n-1 /n. Same holds for any non-hierarchical query. What about Decision-DNNFs?

8 Decision-DNNF to FBDD Optimal [Razgon] Theorem If F has a Decision-DNNF with N nodes, then F has an FBDD with at most N 1+log(N) nodes. Proof idea No-op No-op 0 1

9 Decision-DNNF to FBDD Optimal [Razgon] Theorem If F has a Decision-DNNF with N nodes, then F has an FBDD with at most N 1+log(N) nodes. Proof idea No-op No-op 0 1 Problem: 0 X 1

10 Decision-DNNF to FBDD Optimal [Razgon] Theorem If F has a Decision-DNNF with N nodes, then F has an FBDD with at most N 1+log(N) nodes. Proof idea No-op No-op 0 1 Problem: 0 X 1 Solution: copy the smaller child 0 X 1

11 Hard Queries Corollary Any Decision-DNNF for F n (H k ) has size 2 Ω( n) Same holds for any non-hierarchical query. Proof. N-node Decision-DNNF to N 1+log(N) nodes FBDD. N 1+log(N) > 2 n-1 /n, log(n) + log 2 (N) > n 1 log(n) log 2 (N) = Ω(n) log(n) = Ω( n)

12 Lifted v.s. Grounded Inference Lifted P(Q) Grounded P(F n (Q)) Non-hierarchical Q (e.g. H 0 ) #P-hard 2 Ω( n) What about hierarchical queries?

13 Inversion-Free Queries Definition An inversion in Q is a sequence of co-occurring vars: (x 0,y 0 ), (x 1,y 1 ),, (x k,y k ), such that: at(x 0 ) at(y 0 ), at(x 1 )=at(y 1 ),, at(x k-1 )=at(y k-1 ), at(x k ) at(y k ) For all i=1,..,k-1 there exists two atoms in Q of the form: S i (,x i-1,,y i-1, ) and S i (,x i,, y i, ) Inversion-free implies hierarchical, but converse fails Q=[R(x 0 ) S(x 0,y 0 )] [S(x 1,y 1 ) T(x 1 )] Inversion-free Inversion H 1 =[R(x 0 ) S(x 0,y 0 )] [S(x 1,y 1 ) T(y 1 )]

14 Easy Queries [Jha&S.11], [Beame 15] Theorem Let Q in FO un 1. If Q has inversion then OBDD for F n (Q) has size 2 n-1 /n 2. Else, F n (Q) has OBDD of width 2 #atoms(q) (size O(n)) Proof (part 2 only next slide)

15 Easy Queries [Jha&S.11], [Beame 15] [Beame&Liew 15] Extended to SDD. Thus, over FO un, OBDD SDD Theorem Let Q in FO un 1. If Q has inversion then OBDD for F n (Q) has size 2 n-1 /n 2. Else, F n (Q) has OBDD of width 2 #atoms(q) (size O(n)) Proof (part 2 only next slide)

16 Easy Queries [Bova 16] SDD more succint than OBDD (HWB) [Jha&S.11], [Beame 15] [Beame&Liew 15] Extended to SDD. Thus, over FO un, OBDD SDD Theorem Let Q in FO un 1. If Q has inversion then OBDD for F n (Q) has size 2 n-1 /n 2. Else, F n (Q) has OBDD of width 2 #atoms(q) (size O(n)) Proof (part 2 only next slide)

17 Q = [R(x) S(x,y)] [T(x ) S(x,y )]

18 Q = [R(x) S(x,y)] [T(x ) S(x,y )] n = 2 Π = R 1 T 1 S 11 S 12 R 2 T 2 S 21 S 22 x = 1 x = 2

19 C 1 = R(x) S(x,y) C 2 = T(x ) S(x,y ) = Q = [R(x) S(x,y)] [T(x ) S(x,y )] n = 2 Π = R 1 T 1 S 11 S 12 R 2 T 2 S 21 S 22 x = 1 x = 2

20 C 1 = R(x) S(x,y) C 2 = T(x ) S(x,y ) = Q = [R(x) S(x,y)] [T(x ) S(x,y )] F 2 (C 1 ) = (R 1 S 11 ) (R 1 S 12 ) ( R 2 S 21 ) (R 2 S 22 ) n = 2 Π = R 1 T 1 S 11 S 12 R 2 T 2 S 21 S 22 x = 1 x = 2

21 C 1 = R(x) S(x,y) C 2 = T(x ) S(x,y ) = Q = [R(x) S(x,y)] [T(x ) S(x,y )] F 2 (C 1 ) = (R 1 S 11 ) (R 1 S 12 ) ( R 2 S 21 ) (R 2 S 22 ) n = 2 Π = R 1 T 1 S 11 S 12 R 2 T 2 S 21 S 22 1 R 1 0 x = 1 x = 2 1 S 11 0 x = 1 S R S 22 S x =

22 C 1 = R(x) S(x,y) C 2 = T(x ) S(x,y ) = Q = [R(x) S(x,y)] [T(x ) S(x,y )] F 2 (C 1 ) = (R 1 S 11 ) (R 1 S 12 ) ( R 2 S 21 ) (R 2 S 22 ) n = 2 Π = R 1 T 1 S 11 S 12 R 2 T 2 S 21 S 22 1 R T 1 1 x = 1 x = 2 1 S 11 0 x = 1 0 S 11 1 R 2 S T 2 S Same variable order Π in both OBDDs! 22 1 S 22 S x = S 22 S OBDD for Q = C 1 C 2 has width = width1 width2

23 Lifted v.s. Grounded Inference Nonhierarchical Q (e.g. H 0 ) Inversion -free Q Lifted P(Q) #P-hard PTIME Grounded P(F n (Q)) 2 Ω( n) PTIME

24 Easy/Hard Queries Main result: a class of queries Q such that: Lifted inference: P((Q)) in PTIME Grounded inference: P(F n (Q)) exponential time Significance: limitation of DPLL-based algorithms for model counting

25 H k0 = x y R(x) S 1 (x,y) H k1 = x y S 1 (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) Clauses of H k

26 Clauses of H k H k0 = x y R(x) S 1 (x,y) H k1 = x y S 1 (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 1,, Z k ) = a Boolean function

27 Clauses of H k H k0 = x y R(x) S 1 (x,y) H k1 = x y S 1 (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 1,, Z k ) = a Boolean function Q = f(h k0, H k1,, H kk )

28 Clauses of H k H k0 = x y R(x) S 1 (x,y) H k1 = x y S 1 (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 1,, Z k ) = a Boolean function Q = f(h k0, H k1,, H kk ) Examples: f = Z 0 Z 1 Z k then f(h k0, H k1,, H kk ) = H k f = Z 0 Z 2 Z 0 Z 3 Z 1 Z 3 then f(h 30, H 31, H 31, H 33 ) = Q W

29 Easy/Hard Queries [Beame 14] Theorem For any Boolean function f(z 0, Z 1,, Z k ), denoting Q = f(h k0, H k1,, H kk ): Any FBDD for F n (Q) has size 2 Ω(n) Any Decision-DNNF has size 2 Ω( n). Consequence: Lifted inference computes compute P(Q W ) in PTIME Any DPLL-based algorithm takes time 2 Ω( n) Many other queries are like Q W

30 Lifted v.s. Grounded Inference Nonhierarchical Q (e.g. H 0 ) Inversion -free Q Q = f(h k0,,h kk ) Lifted P(Q) #P-hard PTIME PTIME or #P-hard Grounded P(F n (Q)) 2 Ω( n) PTIME 2 Ω( n)

31 Two Questions Question 1: Are the lifted rules complete? We know that they get stuck on some queries Should we add more rules? Complete for unate FO and for unate FO Question 2: Are lifted rules stronger than grounded? Lifted rules can also be grounded Any advantage over grounded inference? Strictly stronger than DPLL-based algorithms

32 #P-hard Möbius Über Alles FO un, FO un

33 #P-hard Möbius Über Alles FO un, FO un PTIME Q W Q 9

34 #P-hard Möbius Über Alles FO un, FO un PTIME Q W Q 9 Read Once Q U

35 #P-hard Möbius Über Alles FO un, FO un PTIME Q W Q 9 Poly-size OBDD,SDD = = inversion-free Q J Read Once Q U

36 #P-hard Möbius Über Alles FO un, FO un PTIME Q W Q 9 Poly-size FBDD, dec-dnnf Poly-size OBDD,SDD = = inversion-free Q J Q V Read Once Q U Open

37 Möbius Über Alles #P-hard Non-hierarchical hierarchical H 0 H 1 H 2 FO un, FO un 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