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Compiling Probabilistic Logic Programs into Sentential Decision - - PowerPoint PPT Presentation

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Compiling Probabilistic Logic Programs into Sentential Decision Diagrams Jonas Vlasselaer, Joris Renkens, Guy Van den Broeck and Luc De Raedt KU Leuven, Belgium PLP 2014 July 17, 2014 Motivation Inference in Probabilistic Logic Programs

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Jonas Vlasselaer, Joris Renkens, Guy Van den Broeck and Luc De Raedt KU Leuven, Belgium PLP 2014 July 17, 2014

Compiling Probabilistic Logic Programs into Sentential Decision Diagrams

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Motivation

Inference in Probabilistic Logic Programs (PLPs) :

Probabilistic Logic Program

(Relevant to Query)

Grounded Program

(Break cycles)

Boolean Circuit

(Compilation)

Tractable Target Representation

(Weighted Model Counting)

Probability of Query

2

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Motivation

Inference in Probabilistic Logic Programs (PLPs)

Probabilistic Logic Program

(Relevant to Query)

Grounded Program

(Break cycles)

Boolean Circuit

(Compilation)

Tractable Target Representation

(Weighted Model Counting)

Probability of Query

2

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Motivation

Compiling Probabilistic Logic Programs Boolean Circuit

[De Raedt et al., …] [Fierens et al.]

OBDD d-DNNF

3

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Motivation

Compiling Probabilistic Logic Programs Boolean Circuit

[De Raedt et al., …] [Fierens et al.]

OBDD d-DNNF

3

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Motivation

Compiling Bayesian Networks Boolean Circuit

[Darwiche, …] [Darwiche, …] [Chavira et al., …]

OBDD SDD

d-DNNF

4

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Motivation

Compiling Bayesian Networks Boolean Circuit

[Darwiche, …] [Darwiche, …] [Chavira et al., …]

OBDD SDD

d-DNNF

4

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Motivation

Compiling Bayesian Networks Boolean Circuit

[Darwiche, …] [Darwiche, …] [Chavira et al., …]

OBDD SDD

d-DNNF

4

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Motivation

Compiling Probabilistic Logic Programs Boolean Circuit

[De Raedt et al., …] [Fierens et al.]

OBDD SDD

d-DNNF

5

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Outline

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 From PLPs to Boolean Circuits  Properties of Target Representations  Circuit Compilation  Experimental Results

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From PLPs to Boolean Circuits

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Probabilistic Logic Program (PLP)

0.4 :: friends(a,b). 0.5 :: friends(b,a). 0.8 :: friends(a,c). 0.9 :: friends(c,a). 0.2 :: friends(c,b). 0.1 :: friends(b,c). 0.1 :: stress(a). 0.5 :: stress(b). 0.9 :: stress(c). smokes(X) :- stress(X). smokes(X) :- friends(X,Y), smokes(Y).

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From PLPs to Boolean Circuits

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Grounded PLP (probabilistic facts are omitted)

smokes(a) :- stress(a). smokes(b) :- stress(b). smokes(c) :- stress(c). smokes(a) :- friends(a,b), smokes(b). smokes(a) :- friends(a,c), smokes(c). smokes(b) :- friends(b,a), smokes(a). smokes(b) :- friends(b,c), smokes(c). smokes(c) :- friends(c,a), smokes(a). smokes(c) :- friends(c,b), smokes(b).

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From PLPs to Boolean Circuits

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Grounded PLP (probabilistic facts are omitted)

smokes(a) :- stress(a). smokes(b) :- stress(b). smokes(c) :- stress(c). smokes(a) :- friends(a,b), smokes(b). smokes(a) :- friends(a,c), smokes(c). smokes(b) :- friends(b,a), smokes(a). smokes(b) :- friends(b,c), smokes(c). smokes(c) :- friends(c,a), smokes(a). smokes(c) :- friends(c,b), smokes(b). Cycle !

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From PLPs to Boolean Circuits

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Cycle – Free Grounded PLP

smokes(a) :- stress(a). … smokes(a) :- friends(a,b), smokes-a(b). smokes(a) :- friends(a,c), smokes-a(c). … smokes-a(b) :- stress(b). smokes-a(c) :- stress(c). smokes-a(b) :- friends(b,c), smokes-ab(c). smokes-a(c) :- friends(c,b), smokes-ac(b). … smokes-ab(c) :- stress(c). smokes-ac(b) :- stress(b). …

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From PLPs to Boolean Circuits

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Boolean Circuit

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From PLPs to Boolean Circuits

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Boolean Circuit

smokes-a(b)

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From PLPs to Boolean Circuits

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Boolean Circuit (propositional formula) Compile Tractable Target Representation (equivalent formula)

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From PLPs to Boolean Circuits

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Boolean Circuit (propositional formula) Compile Tractable Target Representation (equivalent formula) HOW ? WHICH language ?

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Properties of Target Representations

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 Three key properties for target languages:

 Succinctness  Class of tractable transformations  Class of tractable queries

A Knowledge Compilation Map [Darwiche et al.]

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Properties of Target Representations

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 Tractable Queries

 Weighted Model Counting linear in compiled circuit

 Tractable Transformations

 Apply-operator (˄, ˅, ¬) simplifies compilation process  Variable reordering (minimization)

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Properties of Target Representations

13

 Tractable Queries

 Weighted Model Counting linear in compiled circuit

 Tractable Transformations

 Apply-operator (˄, ˅, ¬) simplifies compilation process  Variable reordering (minimization)

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Properties of Target Representations

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 Succinctness

 Size of smallest compiled circuit

d-DNNF < SDD ≤ OBDD

 Upper bounds on the size of the compiled circuit

 OBDD : based on pathwidth  d-DNNF and SDD : based on treewidth

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Circuit Compilation

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 Compiling into OBDD

 Historically used as target representation  Supports bottom-up compilation approach  Supports minimization of the circuit size

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Circuit Compilation

15

 Compiling into OBDD

 Historically used as target representation  Supports bottom-up compilation approach  Supports minimization of the circuit size

smokes(a) :- friends(a,b), smokes-a(b). smokes-a(b) :- stress(b). smokes-a(b) :- friends(b,c), stress(c).

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Circuit Compilation

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 Compiling into OBDD

 Historically used as target representation  Supports bottom-up compilation approach  Supports minimization of the circuit size

smokes(a) :- friends(a,b), smokes-a(b). smokes-a(b) :- stress(b). smokes-a(b) :- friends(b,c), stress(c).

  • Variable true

Variable false

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Circuit Compilation

15

 Compiling into OBDD

 Historically used as target representation  Supports bottom-up compilation approach  Supports minimization of the circuit size

smokes(a) :- friends(a,b), smokes-a(b). smokes-a(b) :- stress(b). smokes-a(b) :- friends(b,c), stress(c).

Variable true Variable false

˅

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Circuit Compilation

15

 Compiling into OBDD

 Historically used as target representation  Supports bottom-up compilation approach  Supports minimization of the circuit size

smokes(a) :- friends(a,b), smokes-a(b). smokes-a(b) :- stress(b). smokes-a(b) :- friends(b,c), stress(c).

Variable true Variable false

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Circuit Compilation

15

 Compiling into OBDD

 Historically used as target representation  Supports bottom-up compilation approach  Supports minimization of the circuit size

smokes(a) :- friends(a,b), smokes-a(b). smokes-a(b) :- stress(b). smokes-a(b) :- friends(b,c), stress(c).

Variable true Variable false

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Circuit Compilation

15

 Compiling into OBDD

 Historically used as target representation  Supports bottom-up compilation approach  Supports minimization of the circuit size

smokes(a) :- friends(a,b), smokes-a(b). smokes-a(b) :- stress(b). smokes-a(b) :- friends(b,c), stress(c).

Variable true Variable false

˄

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Circuit Compilation

15

 Compiling into OBDD

 Historically used as target representation  Supports bottom-up compilation approach  Supports minimization of the circuit size

smokes(a) :- friends(a,b), smokes-a(b). smokes-a(b) :- stress(b). smokes-a(b) :- friends(b,c), stress(c).

Variable true Variable false

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Circuit Compilation

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 Compiling into d-DNNF

 Requires intermediate encoding in CNF  Auxiliary variables needed (for every rule with >1 literal in body)  Compile CNF into d-DNNF

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Circuit Compilation

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 Compiling into d-DNNF

 Requires intermediate encoding in CNF  Auxiliary variables needed (for every rule with >1 literal in body)  Compile CNF into d-DNNF

Conjunctions (˄) are decomposable Disjunctions (˅) are deterministic

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Circuit Compilation

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 Compiling into d-DNNF

 Requires intermediate encoding in CNF  Auxiliary variables needed (for every rule with >1 literal in body)  Compile CNF into d-DNNF

Conjunctions (˄) are decomposable Disjunctions (˅) are deterministic

“Real” d-DNNF is bigger

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Circuit Compilation

Boolean Circuit

[De Raedt et al., …] [Fierens et al.]

OBDD d-DNNF

17

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Circuit Compilation

Boolean Circuit

[De Raedt et al., …] [Fierens et al.]

OBDD d-DNNF

More succinct Tighter upper bounds Requires CNF representation Auxiliary variables needed

17

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Circuit Compilation

Boolean Circuit

[De Raedt et al., …] [Fierens et al.]

OBDD SDD

d-DNNF

18

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Circuit Compilation

Boolean Circuit

[De Raedt et al., …] [Fierens et al.]

OBDD SDD

d-DNNF Tighter upper bounds on their size

18

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Circuit Compilation

Boolean Circuit

[De Raedt et al., …] [Fierens et al.]

OBDD SDD

d-DNNF Apply operator (bottom-up compilation) Support for minimization

18

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Circuit Compilation

Boolean Circuit

[De Raedt et al., …] [Fierens et al.]

OBDD SDD

d-DNNF SDD combine desirable properties of OBDD and d-DNNF

18

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Circuit Compilation

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 Compiling into SDD

CNF compilation Bottom-up compilation (apply-operator) Minimization

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Circuit Compilation

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 Compiling into SDD

CNF compilation Bottom-up compilation (apply-operator) Minimization

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Circuit Compilation

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 Compiling into SDD

CNF compilation Bottom-up compilation (apply-operator) Minimization

Disjunctions (decisions) Conjunctions

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Experimental Results ‘Smokers example’

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 Four different approaches:

 c2d : compile CNF into d-DNNF  cnf : compile CNF into SDD  sdd_m : bottom-up compilation into SDD with minimization  sdd_f : bottom-up compilation into SDD with fixed variable ordering

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Experimental Results ‘Smokers example’

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 Four different approaches:

 c2d : compile CNF into d-DNNF  cnf : compile CNF into SDD  sdd_m : bottom-up compilation into SDD with minimization  sdd_f : bottom-up compilation into SDD with fixed variable ordering

Compilation Time

(until out of memory)

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Experimental Results ‘Smokers example’

22

 Four different approaches:

 c2d : compile CNF into d-DNNF  cnf : compile CNF into SDD  sdd_m : bottom-up compilation into SDD with minimization  sdd_f : bottom-up compilation into SDD with fixed variable ordering

Circuit Size

(until out of memory)

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Conclusions

Boolean Circuit

[De Raedt et al., …] [Vlasselaer et al.] [Fierens et al.]

OBDD SDD

d-DNNF

23

http://dtai.cs.kuleuven.be/problog/ http://reasoning.cs.ucla.edu/