Generating Random Logic Programs Using Constraint Programming

(1)

Generating Random Logic Programs Using Constraint Programming

Paulius Dilkas

¹

Vaishak Belle

^1,2

1University of Edinburgh, Edinburgh, UK

2Alan Turing Institute, London, UK

FATA Seminar

(2)

Probabilistic Logic Programming The Constraint Model Example Programs Experimental Results Summary

How Many Programs Are Used to Test Algorithms?

4 3 1

1

2

(3)

How Many Programs Are Used to Test Algorithms?

4 3 1

1

2

(4)

How Many Programs Are Used to Test Algorithms?

4 3

1

2

(5)

How Many Programs Are Used to Test Algorithms?

4 3 1

1

2

(6)

How Many Programs Are Used to Test Algorithms?

4 3 1

1

2

(7)

How Many Programs Are Used to Test Algorithms?

4 3 1

1 1

2

(8)

How Many Programs Are Used to Test Algorithms?

4 3 1

1

2

(9)

Outline

Probabilistic Logic Programming

The Constraint Model

Example Programs

Experimental Results

Summary

(10)

Probabilistic Logic Programs (ProbLog)

“Smokers” (Domingos et al. 2008; Fierens et al. 2015)

0.2 : :stress(P) : - person(P).

0.3 : :influences(P

1

, P

2

) : - friend(P

1

, P

2

).

0.1 : :cancer spont(P) : - person(P).

0.3 : :cancer smoke(P) : - person(P).

smokes(X) : - stress(X).

smokes(X) : - smokes(Y), influences(Y, X).

cancer(P) : - cancer spont(P).

cancer(P) : - smokes(P), cancer smoke(P).

person(mary ).

person(albert ).

friend(albert , mary ).

(11)

Applications

Moldovan et al. 2012 Dries et al. 2017

Cˆ orte-Real, Dutra, and Rocha 2017 De Maeyer et al. 2013

(12)

Probabilistic Inference: Reasoning by Hand

Let a ⊕ b := a + b − ab. Then

Pr[cancer(mary )] = Pr[cancer spont(mary )]

⊕ Pr[smokes(mary)]

× Pr[cancer smoke(mary)]

cancer(P) : - cancer spont(P).

cancer(P) : - smokes(P), cancer smoke(P).

(13)

Probabilistic Inference: Reasoning by Hand

Let a ⊕ b := a + b − ab. Then

Pr[cancer(mary )] = 0.1 ⊕ 0.3 × Pr[smokes(mary )]

0.1 : :cancer spont(P) : - person(P).

0.3 : :cancer smoke(P) : - person(P).

(14)

Probabilistic Inference: Reasoning by Hand

Let a ⊕ b := a + b − ab. Then

Pr[cancer(mary )] = 0.1 ⊕ 0.3 × Pr[smokes(mary )]

Pr[smokes(mary )] = Pr[stress(mary)]

⊕ Pr[smokes(albert)]

× Pr[influences(albert , mary)]

smokes(X) : - stress(X).

smokes(X) : - smokes(Y), influences(Y, X).

(15)

Probabilistic Inference: Reasoning by Hand

Let a ⊕ b := a + b − ab. Then

Pr[cancer(mary )] = 0.1 ⊕ 0.3 × Pr[smokes(mary )]

Pr[smokes(mary )] = 0.2 ⊕ 0.3 × Pr[ smokes(albert )]

0.2 : :stress(P) : - person(P).

0.3 : :influences(P

₁

, P

₂

) : - friend(P

₁

, P

₂

).

(16)

Probabilistic Inference: Reasoning by Hand

Let a ⊕ b := a + b − ab. Then

Pr[cancer(mary )] = 0.1 ⊕ 0.3 × Pr[smokes(mary )]

Pr[smokes(mary )] = 0.2 ⊕ 0.3 × Pr[ smokes(albert )]

Pr[smokes(albert )] = Pr[stress(albert )] = 0.2

0.2 : :stress(P) : - person(P).

smokes(X) : - stress(X).

(17)

Probabilistic Inference: Probabilities of Worlds

cancer(mary) = > Pr(world ) =

0.2 : :stress(P) : - person(P).

0.3 : :influences(P

₁

, P

₂

) : - friend(P

₁

, P

₂

).

0.1 : :cancer spont(P) : - person(P).

0.3 : :cancer smoke(P) : - person(P).

smokes(X) : - stress(X).

smokes(X) : - smokes(Y), influences(Y, X).

cancer(P) : - cancer spont(P).

cancer(P) : - smokes(P), cancer smoke(P).

person(mary ).

person(albert ).

friend(albert , mary ).

(18)

Probabilistic Inference: Probabilities of Worlds

cancer(mary) = >

Pr(world ) = 0.2 × (1 − 0.3) × 0.1 × 0.3 0.2 : :stress(P) : - person(P).

0.3 : :influences(P

₁

, P

₂

) : - friend(P

₁

, P

₂

).

0.1 : :cancer spont(P) : - person(P).

0.3 : :cancer smoke(P) : - person(P).

smokes(X) : - stress(X).

smokes(X) : - smokes(Y), influences(Y, X).

cancer(P) : - cancer spont(P).

cancer(P) : - smokes(P), cancer smoke(P).

person(mary ).

person(albert ).

friend(albert , mary ).

(19)

Probabilistic Inference: Probabilities of Worlds

cancer(mary) = >

Pr(world ) = 0.2 × 0.3 × 0.1 × (1 − 0.3) 0.2 : :stress(P) : - person(P).

0.3 : :influences(P

₁

, P

₂

) : - friend(P

₁

, P

₂

).

0.1 : :cancer spont(P) : - person(P).

0.3 : :cancer smoke(P) : - person(P).

smokes(X) : - stress(X).

smokes(X) : - smokes(Y), influences(Y, X).

cancer(P) : - cancer spont(P).

cancer(P) : - smokes(P), cancer smoke(P).

person(mary ).

person(albert ).

friend(albert , mary ).

(20)

Probabilistic Inference: Probabilities of Worlds

cancer(mary) = >

Pr(world ) = (1 − 0.2) × (1 − 0.3) × 0.1 × 0.3 0.2 : :stress(P) : - person(P).

0.3 : :influences(P

₁

, P

₂

) : - friend(P

₁

, P

₂

).

0.1 : :cancer spont(P) : - person(P).

0.3 : :cancer smoke(P) : - person(P).

smokes(X) : - stress(X).

smokes(X) : - smokes(Y), influences(Y, X).

cancer(P) : - cancer spont(P).

cancer(P) : - smokes(P), cancer smoke(P).

person(mary ).

person(albert ).

friend(albert , mary ).

(21)

Probabilistic Inference: Probabilities of Worlds

cancer(mary) = ⊥

Pr(world ) = (1 − 0.2) × (1 − 0.3) × (1 − 0.1) × (1 − 0.3) 0.2 : :stress(P) : - person(P).

0.3 : :influences(P

₁

, P

₂

) : - friend(P

₁

, P

₂

).

0.1 : :cancer spont(P) : - person(P).

0.3 : :cancer smoke(P) : - person(P).

smokes(X) : - stress(X).

smokes(X) : - smokes(Y), influences(Y, X).

cancer(P) : - cancer spont(P).

cancer(P) : - smokes(P), cancer smoke(P).

person(mary ).

person(albert ).

friend(albert , mary ).

(22)

Inference Algorithms and Knowledge Compilation Maps

NNF negation normal form BDD binary decision diagrams

SDD sentential decision diagrams

k-Best only use the k most probable proofs

d-DNNF deterministic decomposable negation normal form

• for every pair α ∨ β, we have α ∧ β = ⊥

• for every pair α ∧ β, no atoms are shared between α and β

• examples:

77 (A ∨ C ) ∧ (A ∨ ¬B) 73 C ∧ (A ∨ ¬B)

37 B ∧ C ∧ [(B ∧ A) ∨ ¬B]

33 C ∧ [(B ∧ A) ∨ ¬B]

(23)

Inference Algorithms and Knowledge Compilation Maps

NNF negation normal form BDD binary decision diagrams

SDD sentential decision diagrams

k-Best only use the k most probable proofs

d-DNNF deterministic decomposable negation normal form

• for every pair α ∨ β, we have α ∧ β = ⊥

• for every pair α ∧ β, no atoms are shared between α and β

• examples:

77 (A ∨ C ) ∧ (A ∨ ¬B) 73 C ∧ (A ∨ ¬B)

37 B ∧ C ∧ [(B ∧ A) ∨ ¬B]

33 C ∧ [(B ∧ A) ∨ ¬B]

(24)

Inference Algorithms and Knowledge Compilation Maps

NNF negation normal form BDD binary decision diagrams

SDD sentential decision diagrams

k-Best only use the k most probable proofs

d-DNNF deterministic decomposable negation normal form

• for every pair α ∨ β, we have α ∧ β = ⊥

• for every pair α ∧ β, no atoms are shared between α and β

• examples:

77 (A ∨ C ) ∧ (A ∨ ¬B)

73 C ∧ (A ∨ ¬B)

37 B ∧ C ∧ [(B ∧ A) ∨ ¬B]

33 C ∧ [(B ∧ A) ∨ ¬B]

(25)

Inference Algorithms and Knowledge Compilation Maps

NNF negation normal form BDD binary decision diagrams

SDD sentential decision diagrams

k-Best only use the k most probable proofs

d-DNNF deterministic decomposable negation normal form

• for every pair α ∨ β, we have α ∧ β = ⊥

• for every pair α ∧ β, no atoms are shared between α and β

• examples:

77 (A ∨ C ) ∧ (A ∨ ¬B) 73 C ∧ (A ∨ ¬B)

37 B ∧ C ∧ [(B ∧ A) ∨ ¬B]

33 C ∧ [(B ∧ A) ∨ ¬B]

(26)

Inference Algorithms and Knowledge Compilation Maps

NNF negation normal form BDD binary decision diagrams

SDD sentential decision diagrams

k-Best only use the k most probable proofs

d-DNNF deterministic decomposable negation normal form

• for every pair α ∨ β, we have α ∧ β = ⊥

• for every pair α ∧ β, no atoms are shared between α and β

• examples:

77 (A ∨ C ) ∧ (A ∨ ¬B) 73 C ∧ (A ∨ ¬B)

37 B ∧ C ∧ [(B ∧ A) ∨ ¬B]

33 C ∧ [(B ∧ A) ∨ ¬B]

(27)

Inference Algorithms and Knowledge Compilation Maps

NNF negation normal form BDD binary decision diagrams

SDD sentential decision diagrams

k-Best only use the k most probable proofs

d-DNNF deterministic decomposable negation normal form

• for every pair α ∨ β, we have α ∧ β = ⊥

• for every pair α ∧ β, no atoms are shared between α and β

• examples:

77 (A ∨ C ) ∧ (A ∨ ¬B) 73 C ∧ (A ∨ ¬B)

37 B ∧ C ∧ [(B ∧ A) ∨ ¬B]

33 C ∧ [(B ∧ A) ∨ ¬B]

(28)

Example Diagrams for C ∧ (A ∨ ¬B)

∧

C ∨

A ¬B

Figure: NNF

A B C

1 0

Figure: BDD

∧

∨

∧ B

∧

∨

A ¬A

¬B C

Figure: d-DNNF

1 ¬A 3

¬B C B ⊥

A C

Figure: SDD

1 0 A

3 2 B

4 C

Figure: vtree

(29)

What Characterises a (Probabilistic) Logic Program?



 



 



0.2 : :stress(P) : - person(P).

0.3 : :influences(P

1

, P

2

) : - friend(P

1

, P

2

).

0.1 : :cancer spont(P) : - person(P).

0.3 : :cancer smoke(P) : - person(P).

smokes(X) : - stress(X).

smokes(X) : - smokes(Y), influences(Y, X).

cancer(P) : - cancer spont(P).

cancer(P) : - smokes(P), cancer smoke(P).

person(mary ).

person(albert).

friend(albert, mary ).

• predicates, arities

• variables

• constants

• probabilities

• length

• complexity

(30)

What Characterises a (Probabilistic) Logic Program?



 



 



0.2 : :stress(P) : - person(P).

0.3 : :influences(P

1

, P

2

) : - friend(P

1

, P

2

).

0.1 : :cancer spont(P) : - person(P).

0.3 : :cancer smoke(P) : - person(P).

smokes(X) : - stress(X).

smokes(X) : - smokes(Y), influences(Y, X).

cancer(P) : - cancer spont(P).

cancer(P) : - smokes(P), cancer smoke(P).

person(mary ).

person(albert).

friend(albert, mary ).

• predicates, arities

• variables

• constants

• probabilities

• length

• complexity

(31)

What Characterises a (Probabilistic) Logic Program?



 



 



0.2 : :stress(P) : - person(P).

0.3 : :influences(P

1

, P

2

) : - friend(P

1

, P

2

).

0.1 : :cancer spont(P) : - person(P).

0.3 : :cancer smoke(P) : - person(P).

smokes(X) : - stress(X).

smokes(X) : - smokes(Y), influences(Y, X).

cancer(P) : - cancer spont(P).

cancer(P) : - smokes(P), cancer smoke(P).

person(mary ).

person(albert).

friend(albert, mary ).

• predicates, arities

• variables

• constants

• probabilities

• length

• complexity

(32)

What Characterises a (Probabilistic) Logic Program?



 



 



0.2 : :stress(P) : - person(P).

0.3 : :influences(P

1

, P

2

) : - friend(P

1

, P

2

).

0.1 : :cancer spont(P) : - person(P).

0.3 : :cancer smoke(P) : - person(P).

smokes(X) : - stress(X).

smokes(X) : - smokes(Y), influences(Y, X).

cancer(P) : - cancer spont(P).

cancer(P) : - smokes(P), cancer smoke(P).

person(mary).

person(albert ).

friend(albert , mary).

• predicates, arities

• variables

• constants

• probabilities

• length

• complexity

(33)

What Characterises a (Probabilistic) Logic Program?



 



 



0.2 : :stress(P) : - person(P).

0.3 : :influences(P

1

, P

2

) : - friend(P

1

, P

2

).

0.1 : :cancer spont(P) : - person(P).

0.3 : :cancer smoke(P) : - person(P).

smokes(X) : - stress(X).

smokes(X) : - smokes(Y), influences(Y, X).

cancer(P) : - cancer spont(P).

cancer(P) : - smokes(P), cancer smoke(P).

person(mary).

person(albert ).

friend(albert , mary).

• predicates, arities

• variables

• constants

• probabilities

• length

• complexity

(34)

What Characterises a (Probabilistic) Logic Program?



 



 



0.2 : :stress(P) : - person(P).

0.3 : :influences(P

1

, P

2

) : - friend(P

1

, P

2

).

0.1 : :cancer spont(P) : - person(P).

0.3 : :cancer smoke(P) : - person(P).

smokes(X) : - stress(X).

smokes(X) : - smokes(Y), influences(Y, X).

cancer(P) : - cancer spont(P).

cancer(P) : - smokes(P), cancer smoke(P).

person(mary).

person(albert ).

friend(albert , mary).

• predicates, arities

• variables

• constants

• probabilities

• length

• complexity

(35)

What Characterises a (Probabilistic) Logic Program?



 



 



0.2 : :stress(P) : - person(P).

0.3 : :influences(P

1

, P

2

) : - friend(P

1

, P

2

).

0.1 : :cancer spont(P) : - person(P).

0.3 : :cancer smoke(P) : - person(P).

smokes(X) : - stress(X).

smokes(X) : - smokes(Y), influences(Y, X).

cancer(P) : - cancer spont(P).

cancer(P) : - smokes(P), cancer smoke(P).

person(mary).

person(albert ).

friend(albert , mary).

• predicates, arities

• variables

• constants

• probabilities

• length

• complexity

(36)

Formulas As Trees

¬ p(X) ∨ (q(X) ∧ r(X))

∨

¬

p(X)

∧

q(X) r(X)

• s is sorted

• s

_i

6= i =⇒ v

_i

6= >

s : 0 0 0 1 2 2 6

v: ∨ ¬ ∧ p(X) q(X) r(X) >

c : 2 1 2 0 0 0 0

• s is a forest with T = 2 trees

• length L = 7

• number of nodes N := L − T + 1 = 6

• for i = 1, . . . , L − 1,

• if i < N, then s

i

< i

• else s

i

= i and v

i

= >

• c

_i

= 0 ⇐⇒ v

_i

= > or an atom

• c

_i

= 1 ⇐⇒ v

_i

= ¬

• c

_i

> 1 ⇐⇒ v

_i

∈ {∧, ∨}

(37)

Formulas As Trees

¬ p(X) ∨ (q(X) ∧ r(X))

∨

¬

p(X)

∧

q(X) r(X)

• s is sorted

• s

_i

6= i =⇒ v

_i

6= >

s : 0 0 0 1 2 2 6

v: ∨ ¬ ∧ p(X) q(X) r(X) >

c : 2 1 2 0 0 0 0

• s is a forest with T = 2 trees

• length L = 7

• number of nodes N := L − T + 1 = 6

• for i = 1, . . . , L − 1,

• if i < N, then s

i

< i

• else s

i

= i and v

i

= >

• c

_i

= 0 ⇐⇒ v

_i

= > or an atom

• c

_i

= 1 ⇐⇒ v

_i

= ¬

• c

_i

> 1 ⇐⇒ v

_i

∈ {∧, ∨}

(38)

Formulas As Trees

¬ p(X) ∨ (q(X) ∧ r(X))

∨

¬

p(X)

∧

q(X) r(X)

• s is sorted

• s

_i

6= i =⇒ v

_i

6= >

s : 0 0 0 1 2 2 6

v: ∨ ¬ ∧ p(X) q(X) r(X) >

c : 2 1 2 0 0 0 0

• s is a forest with T = 2 trees

• length L = 7

• number of nodes N := L − T + 1 = 6

• for i = 1, . . . , L − 1,

• if i < N, then s

i

< i

• else s

i

= i and v

i

= >

• c

_i

= 0 ⇐⇒ v

_i

= > or an atom

• c

_i

= 1 ⇐⇒ v

_i

= ¬

• c

_i

> 1 ⇐⇒ v

_i

∈ {∧, ∨}

(39)

Formulas As Trees

¬ p(X) ∨ (q(X) ∧ r(X))

∨

¬

p(X)

∧

q(X) r(X)

• s is sorted

• s

_i

6= i =⇒ v

_i

6= >

s : 0 0 0 1 2 2 6

v: ∨ ¬ ∧ p(X) q(X) r(X) >

c : 2 1 2 0 0 0 0

• s is a forest with T = 2 trees

• length L = 7

• number of nodes N := L − T + 1 = 6

• for i = 1, . . . , L − 1,

• if i < N, then s

i

< i

• else s

i

= i and v

i

= >

• c

_i

= 0 ⇐⇒ v

_i

= > or an atom

• c

_i

= 1 ⇐⇒ v

_i

= ¬

• c

_i

> 1 ⇐⇒ v

_i

∈ {∧, ∨}

(40)

Formulas As Trees

¬ p(X) ∨ (q(X) ∧ r(X))

∨

¬

p(X)

∧

q(X) r(X)

• s is sorted

• s

_i

6= i =⇒ v

_i

6= >

s : 0 0 0 1 2 2 6

v: ∨ ¬ ∧ p(X) q(X) r(X) >

c : 2 1 2 0 0 0 0

• s is a forest with T = 2 trees

• length L = 7

• number of nodes N := L − T + 1 = 6

• for i = 1, . . . , L − 1,

• if i < N, then s

i

< i

• else s

i

= i and v

i

= >

• c

_i

= 0 ⇐⇒ v

_i

= > or an atom

• c

_i

= 1 ⇐⇒ v

_i

= ¬

• c

_i

> 1 ⇐⇒ v

_i

∈ {∧, ∨}

(41)

Formulas As Trees

¬ p(X) ∨ (q(X) ∧ r(X))

∨

¬

p(X)

∧

q(X) r(X)

• s is sorted

• s

_i

6= i =⇒ v

_i

6= >

s : 0 0 0 1 2 2 6

v: ∨ ¬ ∧ p(X) q(X) r(X) >

c : 2 1 2 0 0 0 0

• s is a forest with T = 2 trees

• length L = 7

• number of nodes N := L − T + 1 = 6

• for i = 1, . . . , L − 1,

• if i < N, then s

i

< i

• else s

i

= i and v

i

= >

• c

_i

= 0 ⇐⇒ v

_i

= > or an atom

• c

_i

= 1 ⇐⇒ v

_i

= ¬

• c

_i

> 1 ⇐⇒ v

_i

∈ {∧, ∨}

(42)

Formulas As Trees

¬ p(X) ∨ (q(X) ∧ r(X))

∨

¬

p(X)

∧

q(X) r(X)

• s is sorted

• s

_i

6= i =⇒ v

_i

6= >

s : 0 0 0 1 2 2 6

v: ∨ ¬ ∧ p(X) q(X) r(X) >

c : 2 1 2 0 0 0 0

• s is a forest with T = 2 trees

• length L = 7

• number of nodes N := L − T + 1 = 6

• for i = 1, . . . , L − 1,

• if i < N, then s

i

< i

• else s

i

= i and v

i

= >

• c

_i

= 0 ⇐⇒ v

_i

= > or an atom

• c

_i

= 1 ⇐⇒ v

_i

= ¬

• c

_i

> 1 ⇐⇒ v

_i

∈ {∧, ∨}

(43)

Formulas As Trees

¬ p(X) ∨ (q(X) ∧ r(X))

∨

¬

p(X)

∧

q(X) r(X)

• s is sorted

• s

_i

6= i =⇒ v

_i

6= >

s : 0 0 0 1 2 2 6

v: ∨ ¬ ∧ p(X) q(X) r(X) >

c : 2 1 2 0 0 0 0

• s is a forest with T = 2 trees

• length L = 7

• number of nodes N := L − T + 1 = 6

• for i = 1, . . . , L − 1,

• if i < N, then s

i

< i

• else s

i

= i and v

i

= >

• c

_i

= 0 ⇐⇒ v

_i

= > or an atom

• c

_i

= 1 ⇐⇒ v

_i

= ¬

• c

_i

> 1 ⇐⇒ v

_i

∈ {∧, ∨}

(44)

Variable Symmetry Breaking

Let {W, X, Y} be the set of variables. Then

smokes(X) : - smokes(Y), influences(Y, X)

is equivalent to

smokes(Y) : - smokes(X), influences(X, Y) and to

smokes(W) : - smokes(X), influences(X, W)

(45)

Variable Symmetry Breaking

X Y Y X

Occurrences (channeling)

W 7→ ∅ X 7→ {0, 3}

Y 7→ {1, 2}

Introductions 1 + min occurrences(v ) or 0

W 7→ 0 X 7→ 1 Y 7→ 2

sorted!

(46)

Variable Symmetry Breaking

Y X X Y

Occurrences (channeling)

W 7→ ∅ X 7→ {1, 2}

Y 7→ {0, 3}

Introductions 1 + min occurrences(v ) or 0

W 7→ 0 X 7→ 2 Y 7→ 1

not sorted!

(47)

Variable Symmetry Breaking

W X X W

Occurrences (channeling)

W 7→ {0, 3}

X 7→ {1, 2}

Y 7→ ∅

Introductions 1 + min occurrences(v ) or 0

W 7→ 1 X 7→ 2 Y 7→ 0

not sorted!

(48)

Predicate Dependency Graph

friend influences

stress

cancer spont

cancer smoke

person cancer

smokes

(49)

Stratification and Negative Cycles

0.1 : : friend(X, Y) : - \+ smokes(Y).

• Stratum 1: predicates with no negation

• Stratum 2: new predicates s.t. negative literals refer to predicates in Stratum 1

friend influences

stress

cancer spont

cancer smoke

person cancer

smokes

(50)

Stratification and Negative Cycles

0.1 : : friend(X, Y) : - \+ smokes(Y).

• Stratum 1: predicates with no negation

• Stratum 2: new predicates s.t. negative literals refer to predicates in Stratum 1

friend influences

stress

cancer spont

cancer smoke

person cancer

smokes

(51)

Stratification and Negative Cycles

0.1 : : friend(X, Y) : - \+ smokes(Y).

• Stratum 1: predicates with no negation

• Stratum 2: new predicates s.t. negative literals refer to predicates in Stratum 1

friend influences

stress

cancer spont

cancer smoke

person cancer

smokes

(52)

Predicate Independence: friend ⊥ ⊥ stress

friend influences

stress

cancer spont

cancer smoke

person cancer

smokes

(53)

Predicate Independence: friend ⊥ ⊥ stress

friend influences

stress

cancer spont

cancer smoke

person cancer

smokes

(54)

Predicate Independence: friend ⊥ ⊥ stress

friend influences

stress

cancer spont

cancer smoke

person cancer

smokes

(55)

Predicate Independence: friend ⊥ ⊥ stress

friend influences

stress

cancer spont

cancer smoke

person cancer

smokes

(56)

Predicate Independence: friend ⊥ ⊥ stress

friend influences

stress

cancer spont

cancer smoke

person cancer

smokes

(57)

Predicate Independence: friend ⊥ ⊥ stress

friend influences

stress

cancer spont

cancer smoke

person cancer

smokes

(58)

One-Liners

Setup

• predicate p/1

• variable X

• no constants

• 1 clause

• 4 nodes

• no negative cycles

(All) Programs

• p(X).

• 0.7 : : p(X) : - p(X).

• 0.8 : : p(X) : - p(X); p(X).

• 0.7 : : p(X) : - p(X), p(X).

• 0.1 : : p(X) : - p(X); p(X); p(X).

• 0.8 : : p(X) : - p(X), p(X), p(X).

(59)

Symmetry Breaking in Action

Setup

• predicate p/3

• variables: X, Y, Z

• no constants

• 1 clause

• 1 node

• no cycles at all

(All) Programs

• 0.8 : : p(Z, Z, Z).

• p(Y, Y, Z).

• p(Y, Z, Y).

• p(Y, Z, Z).

• 0.1 : : p(X, Y, Z).

(60)

A Larger Example

Setup

• predicates: p/1, q/2, r/3

• variables: X, Y, Z

• constants: a, b, c

• 5 clauses

• 5 nodes

• no negative cycles

A Random Program

p(b) : - \+(q(a, b), q(X, Y), q(Z, X)).

0.4 : :q(X, X) : - \+ r(Y, Z, a).

q(X, a) : - r(Y, Y, Z).

q(X, a) : - r(Y, b, Z).

r(Y, b, Z).

(61)

Examples of Predicate Independence

Setup

• predicates: p/1, q/1, r/1

• no variables

• constant a

• 3 clauses

• 3 nodes

• no negative cycles

• p ⊥ ⊥ q

A Few Random Programs

• 0.5 : :p(a) : - p(a); p(a).

0.2 : :q(a) : - q(a), q(a).

0.4 : :r(a) : - \+ q(a).

• p(a) : - p(a).

0.5 : :q(a) : - r(a); q(a).

r(a) : - r(a); r(a).

• p(a) : - p(a); p(a).

0.6 : :q(a) : - q(a).

0.7 : :r(a) : - \+ q(a).

(62)

Scalability

0 2000 4000

1 2 4 8

Value

Sea rch No des

Variable

The number of predicates Maximum arity

The number of variables

The number of constants

The number of additional clauses

The maximum number of nodes

(63)

What Programs Should We Generate?

• each program is divided into:

• rules

• e.g., 0.2 : : stress(P) : - person(P).

• facts

• e.g., friend(albert, mary ).

• predicates, variables, nodes: 2, 4, 8

• maximum arity: 1, 2, 3

• all possible numbers of pairs of independent predicates

• 10 programs per configuration

• fully restarting the constraint solver

• probabilities sampled from {0.1, 0.2, . . . , 0.9}

• query: random unlisted fact

(64)

Rules

0.2 : : stress(P) : - person(P).

• no constants, no empty bodies

• one rule per predicate

• all rules are probabilistic

(65)

Facts

friend(albert , mary).

• proportion probabilistic: 25%, 50%, 75%

• constants: 100, 200, 400

• number of facts: 10

³

, 10

⁴

, 10

⁵

• but only up to 75% of all possible facts

(66)

Properties of Programs vs. Inference Algorithms

Facts (×10

³

) Prop. independent

0 25 50 75 100 0 0.25 0.5 0.75 1 0

10 20 30

Mean inference time (s) Algorithm

BDD

d-DNNF

K-Best

NNF

SDD

SDDX

(67)

Properties of Programs vs. Inference Algorithms

Maximum arity Probabilistic

2 3 0.25 0.5 0.75 20

40 60

Inference time (s)

(68)

How Encodings Compare Across Instances

2

^-6

2

^-4

2

^-2

2

⁰

2

²

2

⁴

2

⁶

2

^-6

2

^-4

2

^-2

2

⁰

2

²

2

⁴

2

⁶

SDD inference time (s)

SDD X in fere n ce time (s)

0 25 50 75 100

Independent (%)

(69)

The Ratio of Listed Facts to Possible Facts

0 5 10 15

0.0 0.2 0.4 0.6

Proportion of listed facts

Inference time (s)

Algorithm

BDD

d-DNNF

K-Best

NNF

SDD

SDDX

(70)

Summary

• The model can generate (approximately) realistic instances of reasonable size

• The main performance bottleneck can be addressed by generating programs with a simpler structure

• Open questions and future work

• Can we ensure uniform sampling?

• Why do all of the algorithms behave so similarly?

• Why does independence have no effect on inference time?