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 Informal: Defining Lifted Inference Exploit symmetries, Reason at first-order level, Reason about groups of objects, Scalable inference, High-level probabilistic reasoning, etc. A formal definition: Domain-lifted inference Inference runs in time polynomial in the number of objects in the domain. Polynomial in #people, #webpages, #cards Not polynomial in #predicates, #formulas, #logical variables Related to data complexity in databases [VdB 11, Jaeger 12]

3 Informal: Defining Lifted Inference Exploit symmetries, Reason at first-order level, Reason about groups of objects, Scalable inference, High-level probabilistic reasoning, etc. [Poole 03, etc.] A formal definition: Domain-lifted inference [VdB 11, Jaeger 12]

4 Informal: Defining Lifted Inference Exploit symmetries, Reason at first-order level, Reason about groups of objects, Scalable inference, High-level probabilistic reasoning, etc. [Poole 03, etc.] A formal definition: Domain-lifted inference Alternative in this tutorial: Lifted inference = Query Plan = FO Compilation [VdB 11, Jaeger 12]

5 Asymmetric WFOMC Rules Preprocess Q (omitted from this talk; see [Suciu 11]), then apply these rules (some have preconditions) WMC( Δ) = Z-WMC(Δ) Negation Normalization constant Z (easy to compute)

6 Asymmetric WFOMC Rules Preprocess Q (omitted from this talk; see [Suciu 11]), then apply these rules (some have preconditions) WMC( Δ) = Z-WMC(Δ) Negation Normalization constant Z (easy to compute) WMC(Δ 1 Δ 2 ) = WMC(Δ 1 ) * WMC(Δ 2 ) WMC(Δ 1 Δ 2 ) = Z -(Z 1 -WMC(Δ 1 ))*(Z 2 -WMC(Δ 2 )) Independent join / union

7 Asymmetric WFOMC Rules Preprocess Q (omitted from this talk; see [Suciu 11]), then apply these rules (some have preconditions) WMC( Δ) = Z-WMC(Δ) Negation Normalization constant Z (easy to compute) WMC(Δ 1 Δ 2 ) = WMC(Δ 1 ) * WMC(Δ 2 ) WMC(Δ 1 Δ 2 ) = Z -(Z 1 -WMC(Δ 1 ))*(Z 2 -WMC(Δ 2 )) Independent join / union WMC( z Δ) = Z Π C Domain (Z C WMC(Δ[C/z]) WMC( z Δ) = Π C Domain WMC(Δ[C/z] Independent project

8 Asymmetric WFOMC Rules Preprocess Q (omitted from this talk; see [Suciu 11]), then apply these rules (some have preconditions) WMC( Δ) = Z-WMC(Δ) Negation Normalization constant Z (easy to compute) WMC(Δ 1 Δ 2 ) = WMC(Δ 1 ) * WMC(Δ 2 ) WMC(Δ 1 Δ 2 ) = Z -(Z 1 -WMC(Δ 1 ))*(Z 2 -WMC(Δ 2 )) Independent join / union WMC( z Δ) = Z Π C Domain (Z C WMC(Δ[C/z]) WMC( z Δ) = Π C Domain WMC(Δ[C/z] Independent project WMC(Δ 1 Δ 2 ) = WMC(Δ 1 )+WMC(Δ 2 )-WMC(Δ 1 Δ 2 ) WMC(Δ 1 Δ 2 ) = WMC(Δ 1 )+WMC(Δ 2 )-WMC(Δ 1 Δ 2 ) Inclusion/ exclusion

9 Symmetric WFOMC Rules Simplification to independent project: If Δ[C 1 /x], Δ[C 2 /x], are independent WMC( z Δ) = Z (Z C1 -WMC(Δ[C 1 /z]) Domain WMC( z Δ) = WMC(Δ[C 1 /z]) Domain [VdB 11]

10 Symmetric WFOMC Rules Simplification to independent project: If Δ[C 1 /x], Δ[C 2 /x], are independent WMC( z Δ) = Z (Z C1 -WMC(Δ[C 1 /z]) Domain WMC( z Δ) = WMC(Δ[C 1 /z]) Domain A powerful new inference rule: atom counting Only possible with symmetric weights Intuition: Remove unary relations The workhorse of Symmetric WFOMC [VdB 11]

11 WFOMC Inference: Example FO-Model Counting: w(r) = w( R) = 1 Apply inference rules backwards (step )

12 WFOMC Inference: Example FO-Model Counting: w(r) = w( R) = 1 Apply inference rules backwards (step ) 4. Δ = (Stress(Alice) Smokes(Alice)) Domain = {Alice}

13 WFOMC Inference: Example FO-Model Counting: w(r) = w( R) = 1 Apply inference rules backwards (step ) 4. Δ = (Stress(Alice) Smokes(Alice)) 3 models Domain = {Alice}

14 WFOMC Inference: Example FO-Model Counting: w(r) = w( R) = 1 Apply inference rules backwards (step ) 4. Δ = (Stress(Alice) Smokes(Alice)) Domain = {Alice} WMC( Stress(Alice) Smokes(Alice))) = = Z WMC(Stress(Alice)) WMC( Smokes(Alice)) = = 3 models

15 WFOMC Inference: Example FO-Model Counting: w(r) = w( R) = 1 Apply inference rules backwards (step ) 4. Δ = (Stress(Alice) Smokes(Alice)) Domain = {Alice} WMC( Stress(Alice) Smokes(Alice))) = = Z WMC(Stress(Alice)) WMC( Smokes(Alice)) = = 3 models 3. Δ = x, (Stress(x) Smokes(x)) Domain = {n people}

16 WFOMC Inference: Example FO-Model Counting: w(r) = w( R) = 1 Apply inference rules backwards (step ) 4. Δ = (Stress(Alice) Smokes(Alice)) Domain = {Alice} WMC( Stress(Alice) Smokes(Alice))) = = Z WMC(Stress(Alice)) WMC( Smokes(Alice)) = = 3 models 3. Δ = x, (Stress(x) Smokes(x)) Domain = {n people} 3 n models

17 WFOMC Inference: Example 3. Δ = x, (Stress(x) Smokes(x)) Domain = {n people} 3 n models

18 WFOMC Inference: Example 3. Δ = x, (Stress(x) Smokes(x)) Domain = {n people} 3 n models 2. Δ = y, (ParentOf(y) Female MotherOf(y)) D = {n people}

19 WFOMC Inference: Example 3. Δ = x, (Stress(x) Smokes(x)) Domain = {n people} 3 n models 2. Δ = y, (ParentOf(y) Female MotherOf(y)) D = {n people} If Female = true? Δ = y, (ParentOf(y) MotherOf(y)) 3 n models

20 WFOMC Inference: Example 3. Δ = x, (Stress(x) Smokes(x)) Domain = {n people} 3 n models 2. Δ = y, (ParentOf(y) Female MotherOf(y)) D = {n people} If Female = true? Δ = y, (ParentOf(y) MotherOf(y)) 3 n models If Female = false? Δ = true 4 n models

21 WFOMC Inference: Example 3. Δ = x, (Stress(x) Smokes(x)) Domain = {n people} 3 n models 2. Δ = y, (ParentOf(y) Female MotherOf(y)) D = {n people} If Female = true? Δ = y, (ParentOf(y) MotherOf(y)) 3 n models If Female = false? Δ = true 4 n models 3 n + 4 n models

22 WFOMC Inference: Example 3. Δ = x, (Stress(x) Smokes(x)) Domain = {n people} 3 n models 2. Δ = y, (ParentOf(y) Female MotherOf(y)) D = {n people} WMC(Δ ) = WMC( Female y, (ParentOf(y) MotherOf(y))) = 2 * 2 n * 2 n - (2 1) * (2 n * 2 n WMC( y, (ParentOf(y) MotherOf(y)))) = 2 * 4 n (4 n 3 n ) 3 n + 4 n models

23 WFOMC Inference: Example 3. Δ = x, (Stress(x) Smokes(x)) Domain = {n people} 3 n models 2. Δ = y, (ParentOf(y) Female MotherOf(y)) D = {n people} WMC(Δ ) = WMC( Female y, (ParentOf(y) MotherOf(y))) = 2 * 2 n * 2 n - (2 1) * (2 n * 2 n WMC( y, (ParentOf(y) MotherOf(y)))) = 2 * 4 n (4 n 3 n ) 3 n + 4 n models 1. Δ = x,y, (ParentOf(x,y) Female(x) MotherOf(x,y)) D = {n people}

24 WFOMC Inference: Example 3. Δ = x, (Stress(x) Smokes(x)) Domain = {n people} 3 n models 2. Δ = y, (ParentOf(y) Female MotherOf(y)) D = {n people} WMC(Δ ) = WMC( Female y, (ParentOf(y) MotherOf(y))) = 2 * 2 n * 2 n - (2 1) * (2 n * 2 n WMC( y, (ParentOf(y) MotherOf(y)))) = 2 * 4 n (4 n 3 n ) 3 n + 4 n models 1. Δ = x,y, (ParentOf(x,y) Female(x) MotherOf(x,y)) D = {n people} (3 n + 4 n ) n models

25 Atom Counting: Example Δ = x,y, (Smokes(x) Friends(x,y) Smokes(y)) Domain = {n people}

26 Atom Counting: Example Δ = x,y, (Smokes(x) Friends(x,y) Smokes(y)) Domain = {n people} If we know precisely who smokes, and there are k smokers? Database: Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 Smokes(Dave) = 1 Smokes(Eve) = 0... Smokes k n-k Friends Smokes k n-k

27 Atom Counting: Example Δ = x,y, (Smokes(x) Friends(x,y) Smokes(y)) Domain = {n people} If we know precisely who smokes, and there are k smokers? Database: Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 Smokes(Dave) = 1 Smokes(Eve) = 0... Smokes k n-k Friends Smokes k n-k

28 Atom Counting: Example Δ = x,y, (Smokes(x) Friends(x,y) Smokes(y)) Domain = {n people} If we know precisely who smokes, and there are k smokers? Database: Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 Smokes(Dave) = 1 Smokes(Eve) = 0... Smokes k n-k Friends Smokes k n-k

29 Atom Counting: Example Δ = x,y, (Smokes(x) Friends(x,y) Smokes(y)) Domain = {n people} If we know precisely who smokes, and there are k smokers? Database: Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 Smokes(Dave) = 1 Smokes(Eve) = 0... Smokes k n-k Friends Smokes k n-k

30 Atom Counting: Example Δ = x,y, (Smokes(x) Friends(x,y) Smokes(y)) Domain = {n people} If we know precisely who smokes, and there are k smokers? Database: Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 Smokes(Dave) = 1 Smokes(Eve) = 0... Smokes k n-k Friends Smokes k n-k

31 Atom Counting: Example Δ = x,y, (Smokes(x) Friends(x,y) Smokes(y)) Domain = {n people} If we know precisely who smokes, and there are k smokers? Database: Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 Smokes(Dave) = 1 Smokes(Eve) = 0... Smokes k n-k Friends Smokes k n-k

32 Atom Counting: Example Δ = x,y, (Smokes(x) Friends(x,y) Smokes(y)) Domain = {n people} If we know precisely who smokes, and there are k smokers? Database: Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 Smokes(Dave) = 1 Smokes(Eve) = 0... Smokes k n-k Friends Smokes k n-k

33 Atom Counting: Example Δ = x,y, (Smokes(x) Friends(x,y) Smokes(y)) Domain = {n people} If we know precisely who smokes, and there are k smokers? Database: Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 Smokes(Dave) = 1 Smokes(Eve) = 0... Smokes k n-k Friends Smokes k n-k

34 Atom Counting: Example Δ = x,y, (Smokes(x) Friends(x,y) Smokes(y)) Domain = {n people} If we know precisely who smokes, and there are k smokers? Database: Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 Smokes(Dave) = 1 Smokes(Eve) = 0... Smokes k n-k Friends Smokes k n-k

35 Atom Counting: Example Δ = x,y, (Smokes(x) Friends(x,y) Smokes(y)) Domain = {n people} If we know precisely who smokes, and there are k smokers? Database: Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 Smokes(Dave) = 1 Smokes(Eve) = 0... models Smokes k n-k Friends Smokes k n-k

36 Atom Counting: Example Δ = x,y, (Smokes(x) Friends(x,y) Smokes(y)) Domain = {n people} If we know precisely who smokes, and there are k smokers? Database: Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 Smokes(Dave) = 1 Smokes(Eve) = 0... models Smokes k n-k Friends Smokes k n-k If we know that there are k smokers?

37 Atom Counting: Example Δ = x,y, (Smokes(x) Friends(x,y) Smokes(y)) Domain = {n people} If we know precisely who smokes, and there are k smokers? Database: Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 Smokes(Dave) = 1 Smokes(Eve) = 0... models Smokes k n-k Friends Smokes k n-k If we know that there are k smokers? models

38 Atom Counting: Example Δ = x,y, (Smokes(x) Friends(x,y) Smokes(y)) Domain = {n people} If we know precisely who smokes, and there are k smokers? Database: Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 Smokes(Dave) = 1 Smokes(Eve) = 0... models Smokes k n-k Friends Smokes k n-k If we know that there are k smokers? models In total

39 Atom Counting: Example Δ = x,y, (Smokes(x) Friends(x,y) Smokes(y)) Domain = {n people} If we know precisely who smokes, and there are k smokers? Database: Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 Smokes(Dave) = 1 Smokes(Eve) = 0... models Smokes k n-k Friends Smokes k n-k If we know that there are k smokers? models In total models

40 Augment Rules with Logical Rewritings [Suciu 11]

41 Augment Rules with Logical Rewritings 1. Remove constants (shattering) Δ = x (Friend(Alice, x) Friend(x, Bob)) [Suciu 11]

42 Augment Rules with Logical Rewritings 1. Remove constants (shattering) Δ = x (Friend(Alice, x) Friend(x, Bob)) F 1 (x) = Friend(Alice,x) F 2 (x) = Friend(x,Bob) F 3 = Friend(Alice, Alice) F 4 = Friend(Alice,Bob) F 5 = Friend(Bob,Bob) Δ = x (F 1 (x) F 2 (x)) (F 3 F 4 ) (F 4 F 5 ) [Suciu 11]

43 Augment Rules with Logical Rewritings 1. Remove constants (shattering) Δ = x (Friend(Alice, x) Friend(x, Bob)) F 1 (x) = Friend(Alice,x) F 2 (x) = Friend(x,Bob) F 3 = Friend(Alice, Alice) F 4 = Friend(Alice,Bob) F 5 = Friend(Bob,Bob) Δ = x (F 1 (x) F 2 (x)) (F 3 F 4 ) (F 4 F 5 ) 2. Rank variables (= occur in the same order in each atom) Δ = (Friend(x,y) Enemy(x,y)) (Friend(x,y) Enemy(y,x)) Wrong order [Suciu 11]

44 Augment Rules with Logical Rewritings 1. Remove constants (shattering) Δ = x (Friend(Alice, x) Friend(x, Bob)) F 1 (x) = Friend(Alice,x) F 2 (x) = Friend(x,Bob) F 3 = Friend(Alice, Alice) F 4 = Friend(Alice,Bob) F 5 = Friend(Bob,Bob) Δ = x (F 1 (x) F 2 (x)) (F 3 F 4 ) (F 4 F 5 ) 2. Rank variables (= occur in the same order in each atom) Δ = (Friend(x,y) Enemy(x,y)) (Friend(x,y) Enemy(y,x)) Wrong order F 1 (u,v) = Friend(u,v),u<v F 2 (u) = Friend(u,u) F 3 (u,v) = Friend(v,u),v<u E 1 (u,v) = Friend(u,v),u<v E 2 (u) = Friend(u,u) E 3 (u,v) = Friend(v,u),v<u Δ = (F 1 (x,y) E 1 (x,y)) (F 1 (x,y) E 3 (x,y)) (F 2 (x) E 2 (x)) (F 3 (x,y) E 3 (x,y)) (F 3 (x,y) E 1 (x,y)) [Suciu 11]

45 Augment Rules with Logical Rewritings 3. Perform Resolution [Gribkoff 14] Δ = x y (R(x) S(x,y)) x y (S(x,y) T(y)) Rules stuck Resolution on S(x,y): x y (R(x) T(y)) Add resolvent: Δ = x y (R(x) S(x,y)) x y (S(x,y) T(y)) x y (R(x) T(y)) Now apply I/E!

46 Augment Rules with Logical Rewritings 4. Skolemization [VdB 14] Δ = p, c, Card(p,c) Inference rules assume one type of quantifier! Mix / in encodings of MLNs with quantifiers and probabilistic programs Datalog smokes(x) :- friends(x,y), smokes(y). FOL Δ = x, Smokes(x) y, Friends(x,y), Smokes(y). Skolemization Input: Mix / Output: Only BUT: cannot introduce Skolem constants or functions! p, Card(p,S(p))

47 Skolemization: Example Δ = p, c, Card(p,c) [VdB 14]

48 Skolemization: Example Δ = p, c, Card(p,c) Skolemization Δ = p, c, Card(p,c) S(p) [VdB 14]

49 Skolemization: Example Δ = p, c, Card(p,c) Skolemization Δ = p, c, Card(p,c) S(p) w(s) = 1 and w( S) = -1 Skolem predicate [VdB 14]

50 Skolemization: Example Δ = p, c, Card(p,c) Skolemization Δ = p, c, Card(p,c) S(p) Consider one position p: c, Card(p,c) = true w(s) = 1 and w( S) = -1 Skolem predicate c, Card(p,c) = false [VdB 14]

51 Skolemization: Example Δ = p, c, Card(p,c) Skolemization Δ = p, c, Card(p,c) S(p) Consider one position p: c, Card(p,c) = true w(s) = 1 and w( S) = -1 Skolem predicate c, Card(p,c) = false S(p) = true Also model of Δ, weight * 1 [VdB 14]

52 Skolemization: Example Δ = p, c, Card(p,c) Skolemization Δ = p, c, Card(p,c) S(p) Consider one position p: c, Card(p,c) = true w(s) = 1 and w( S) = -1 Skolem predicate S(p) = true Also model of Δ, weight * 1 c, Card(p,c) = false S(p) = true No model of Δ, weight * 1 S(p) = false No model of Δ, weight * -1 [VdB 14] Extra models Cancel out

53 First-Order Knowledge Compilation Markov Logic 3.14 Smokes(x) Friends(x,y) Smokes(y) [Vdb 11, 13]

54 First-Order Knowledge Compilation Markov Logic 3.14 Smokes(x) Friends(x,y) Smokes(y) Weight Function w(smokes)=1 w( Smokes )=1 w(friends )=1 w( Friends )=1 w(f)=exp(3.14) w( F)=1 FOL Sentence x,y, F(x,y) [ Smokes(x) Friends(x,y) Smokes(y) ] [Vdb 11, 13]

55 First-Order Knowledge Compilation Markov Logic 3.14 Smokes(x) Friends(x,y) Smokes(y) Weight Function w(smokes)=1 w( Smokes )=1 w(friends )=1 w( Friends )=1 w(f)=exp(3.14) w( F)=1 FOL Sentence x,y, F(x,y) [ Smokes(x) Friends(x,y) Smokes(y) ] Compile? First-Order d-dnnf Circuit [Vdb 11, 13]

56 First-Order Knowledge Compilation Markov Logic 3.14 Smokes(x) Friends(x,y) Smokes(y) Weight Function w(smokes)=1 w( Smokes )=1 w(friends )=1 w( Friends )=1 w(f)=exp(3.14) w( F)=1 FOL Sentence x,y, F(x,y) [ Smokes(x) Friends(x,y) Smokes(y) ] Compile? First-Order d-dnnf Circuit Domain Alice Bob Charlie Z = WFOMC = [Vdb 11, 13]

57 First-Order Knowledge Compilation Markov Logic 3.14 Smokes(x) Friends(x,y) Smokes(y) Weight Function w(smokes)=1 w( Smokes )=1 w(friends )=1 w( Friends )=1 w(f)=exp(3.14) w( F)=1 FOL Sentence x,y, F(x,y) [ Smokes(x) Friends(x,y) Smokes(y) ] Compile? First-Order d-dnnf Circuit Domain Alice Bob Charlie Z = WFOMC = Evaluation in time polynomial in domain size [Vdb 11, 13]

58 First-Order Knowledge Compilation Markov Logic 3.14 Smokes(x) Friends(x,y) Smokes(y) Weight Function w(smokes)=1 w( Smokes )=1 w(friends )=1 w( Friends )=1 w(f)=exp(3.14) w( F)=1 FOL Sentence x,y, F(x,y) [ Smokes(x) Friends(x,y) Smokes(y) ] Compile? First-Order d-dnnf Circuit Domain Alice Bob Charlie Z = WFOMC = Evaluation in time polynomial in domain size Domain-lifted! [Vdb 11, 13]

59 Negation Normal Form [Darwiche 01]

60 Decomposable NNF Decomposable [Darwiche 01]

61 Deterministic Decomposable NNF Deterministic [Darwiche 01]

62 Deterministic Decomposable NNF Weighted Model Counting [Darwiche 01]

63 Deterministic Decomposable NNF Weighted Model Counting and much more! [Darwiche 01]

64 First-Order NNF [VdB 13]

65 First-Order Decomposability Decomposable [VdB 13]

66 First-Order Decomposability Decomposable [VdB 13]

67 First-Order Determinism Deterministic [VdB 13]

68 First-Order NNF = Query Plan [VdB 13]

69 Deterministic Decomposable FO NNF Weighted Model Counting [VdB 13]

70 Deterministic Decomposable FO NNF Weighted Model Counting Pr(belgian) x Pr(likes) + Pr( belgian) [VdB 13]

71 Deterministic Decomposable FO NNF Weighted Model Counting People Pr(belgian) x Pr(likes) + Pr( belgian) ( ) [VdB 13]

72 Symmetric WFOMC on FO NNF Complexity polynomial in domain size! Polynomial in NNF size for bounded depth. [VdB 13]

73 How to do first-order knowledge compilation?

74 Deterministic Decomposable FO NNF Δ = x,y People, (Smokes(x) Friends(x,y) Smokes(y)) [VdB 13]

75 Deterministic Decomposable FO NNF Δ = x,y People, (Smokes(x) Friends(x,y) Smokes(y)) [VdB 13]

76 Deterministic Decomposable FO NNF Δ = x,y People, (Smokes(x) Friends(x,y) Smokes(y)) [VdB 13]

77 Deterministic Decomposable FO NNF Δ = x,y People, (Smokes(x) Friends(x,y) Smokes(y)) [VdB 13]

78 Deterministic Decomposable FO NNF Δ = x,y People, (Smokes(x) Friends(x,y) Smokes(y)) Deterministic [VdB 13]

79 Deterministic Decomposable FO NNF Δ = x,y People, (Smokes(x) Friends(x,y) Smokes(y)) [VdB 13]

80 Compilation Rules Standard rules Shannon decomposition (DPLL) Detect decomposability Etc. FO Shannon decomposition: Δ [VdB 13]

81 Playing Cards Revisited... Let us automate this: Relational model p, c, Card(p,c) c, p, Card(p,c) p, c, c, Card(p,c) Card(p,c ) c = c Lifted probabilistic inference algorithm

82 Why not do propositional WMC? Reduce to propositional model counting: [VdB 15]

83 Why not do propositional WMC? Reduce to propositional model counting: Δ = Card(A,p 1 ) v v Card(2,p 1 ) Card(A,p 2 ) v v Card(2,p 2 ) Card(A,p 1 ) v v Card(A,p 52 ) Card(K,p 1 ) v v Card(K,p 52 ) Card(A,p 1 ) v Card(A,p 2 ) Card(A,p 1 ) v Card(A,p 3 ) [VdB 15]

84 Why not do propositional WMC? Reduce to propositional model counting: Δ = Card(A,p 1 ) v v Card(2,p 1 ) Card(A,p 2 ) v v Card(2,p 2 ) Card(A,p 1 ) v v Card(A,p 52 ) Card(K,p 1 ) v v Card(K,p 52 ) Card(A,p 1 ) v Card(A,p 2 ) Card(A,p 1 ) v Card(A,p 3 ) What will happen? [VdB 15]

85 3 Deck of Cards Graphically A 2 K [VdB 15]

86 3 Deck of Cards Graphically A 2 K Card(K,p 52 ) [VdB 15]

87 3 Deck of Cards Graphically A 2 K One model/perfect matching [VdB 15]

88 3 Deck of Cards Graphically A 2 K [VdB 15]

89 3 Deck of Cards Graphically A 2 K Card(K,p 52 ) [VdB 15]

90 3 Deck of Cards Graphically A 2 K Card(K,p 52 ) Model counting: How many perfect matchings? [VdB 15]

91 3 Deck of Cards Graphically A 2 K What if I set w(card(k,p 52 )) = 0? [VdB 15]

92 3 Deck of Cards Graphically A 2 K What if I set w(card(k,p 52 )) = 0? [VdB 15]

93 3 Deck of Cards Graphically A 2 K What if I set can set any asymmetric weight function? [VdB 15]

94 Observations Asymmetric weight function can remove edge Encode any bigraph Counting models = perfect matchings Problem is #P-complete! All non-lifted WMC solvers efficiently handle asymmetric weights No solver does cards problem efficiently! Later: Power of lifted vs. ground inference and complexities [VdB 15]

95 Playing Cards Revisited... Let us automate this: Relational model p, c, Card(p,c) c, p, Card(p,c) p, c, c, Card(p,c) Card(p,c ) c = c Lifted probabilistic inference algorithm

96 Playing Cards Revisited p, c, Card(p,c) c, p, Card(p,c) p, c, c, Card(p,c) Card(p,c ) c = c [VdB 15]

97 Playing Cards Revisited p, c, Card(p,c) c, p, Card(p,c) p, c, c, Card(p,c) Card(p,c ) c = c Skolemization [VdB 15]

98 Playing Cards Revisited p, c, Card(p,c) c, p, Card(p,c) p, c, c, Card(p,c) Card(p,c ) c = c Skolemization p, c, Card(p,c) S 1 (p) c, p, Card(p,c) S 2 (c) p, c, c, Card(p,c) Card(p,c ) c = c [VdB 15]

99 Playing Cards Revisited p, c, Card(p,c) c, p, Card(p,c) p, c, c, Card(p,c) Card(p,c ) c = c Skolemization p, c, Card(p,c) S 1 (p) c, p, Card(p,c) S 2 (c) p, c, c, Card(p,c) Card(p,c ) c = c w(s 1 ) = 1 and w( S 1 ) = -1 w(s 2 ) = 1 and w( S 2 ) = -1 [VdB 15]

100 Playing Cards Revisited p, c, Card(p,c) c, p, Card(p,c) p, c, c, Card(p,c) Card(p,c ) c = c Skolemization p, c, Card(p,c) S 1 (p) c, p, Card(p,c) S 2 (c) p, c, c, Card(p,c) Card(p,c ) c = c w(s 1 ) = 1 and w( S 1 ) = -1 w(s 2 ) = 1 and w( S 2 ) = -1 Atom counting [VdB 15]

101 Playing Cards Revisited p, c, Card(p,c) c, p, Card(p,c) p, c, c, Card(p,c) Card(p,c ) c = c Skolemization p, c, Card(p,c) S 1 (p) c, p, Card(p,c) S 2 (c) p, c, c, Card(p,c) Card(p,c ) c = c w(s 1 ) = 1 and w( S 1 ) = -1 w(s 2 ) = 1 and w( S 2 ) = -1 Atom counting p, c, c, Card(p,c) Card(p,c ) c = c [VdB 15]

102 Playing Cards Revisited p, c, Card(p,c) c, p, Card(p,c) p, c, c, Card(p,c) Card(p,c ) c = c Skolemization p, c, Card(p,c) S 1 (p) c, p, Card(p,c) S 2 (c) p, c, c, Card(p,c) Card(p,c ) c = c w(s 1 ) = 1 and w( S 1 ) = -1 w(s 2 ) = 1 and w( S 2 ) = -1 Atom counting p, c, c, Card(p,c) Card(p,c ) c = c -Rule [VdB 15]

103 Playing Cards Revisited p, c, Card(p,c) c, p, Card(p,c) p, c, c, Card(p,c) Card(p,c ) c = c Skolemization p, c, Card(p,c) S 1 (p) c, p, Card(p,c) S 2 (c) p, c, c, Card(p,c) Card(p,c ) c = c w(s 1 ) = 1 and w( S 1 ) = -1 w(s 2 ) = 1 and w( S 2 ) = -1 Atom counting p, c, c, Card(p,c) Card(p,c ) c = c -Rule c, c, Card(c) Card(c ) c = c [VdB 15]

104 Playing Cards Revisited p, c, Card(p,c) c, p, Card(p,c) p, c, c, Card(p,c) Card(p,c ) c = c Skolemization p, c, Card(p,c) S 1 (p) c, p, Card(p,c) S 2 (c) p, c, c, Card(p,c) Card(p,c ) c = c w(s 1 ) = 1 and w( S 1 ) = -1 w(s 2 ) = 1 and w( S 2 ) = -1 Atom counting p, c, c, Card(p,c) Card(p,c ) c = c -Rule c, c, Card(c) Card(c ) c = c [VdB 15]

105 Playing Cards Revisited... Let us automate this: Lifted probabilistic inference algorithm Computed in time polynomial in n [VdB 15]

106 Summary Lifted Inference By definition: PTIME data complexity Also: FO compilation = Query Plan However: only works for liftable queries Preprocessing based on logical rewriting The rules: Deceptively simple: the only surprising rules are I/E and atom counting Rules are captured by a query plan or first-order NNF circuit