Draft ConnectingWidthandStructureinKnowledgeCompilation

Dr aft

Connecting Width and Structure in Knowledge Compilation

Antoine Amarilli¹, Mikaël Monet^1,3,Pierre Senellart^1,2,3 October 4th, 2018 (results from ICDT 2018 paper)

1LTCI, Télécom ParisTech, Université Paris-Saclay; Paris, France

2École normale supérieure, PSL University; Paris, France

3Inria; Paris, France

Dr aft

What is Knowledge Compilation?

•

^{You have atask}

→ Boolean SAT (is there a satisfying assignment?)

→ #SAT (model counting) (how many satisfying assignments?)

→ probabilistic evaluation

→ enumeration

•

^{Idea: compile}the input into a format that isdesignedto solve efficiently your task

1/20

Dr aft

What is Knowledge Compilation?

•

^{You have atask}

→ Boolean SAT (is there a satisfying assignment?)

→ #SAT (model counting) (how many satisfying assignments?)

→ probabilistic evaluation

→ enumeration

•

^{Idea: compile}the input into a format that isdesignedto solve efficiently your task

1/20

Dr aft

What is Knowledge Compilation?

•

^{You have atask}

→ Boolean SAT (is there a satisfying assignment?)

→ #SAT (model counting) (how many satisfying assignments?)

→ probabilistic evaluation

→ enumeration

•

^{Idea: compile}the input into a format that isdesignedto solve efficiently your task

1/20

Dr aft

What is Knowledge Compilation?

•

^{You have atask}

→ Boolean SAT (is there a satisfying assignment?)

→ #SAT (model counting) (how many satisfying assignments?)

→ probabilistic evaluation

→ enumeration

•

^{Idea: compile}the input into a format that isdesignedto solve efficiently your task

1/20

Dr aft

What is Knowledge Compilation?

•

^{You have atask}

→ Boolean SAT (is there a satisfying assignment?)

→ #SAT (model counting) (how many satisfying assignments?)

→ probabilistic evaluation

→ enumeration

•

^{Idea: compile}the input into a format that isdesignedto solve efficiently your task

1/20

Dr aft

Why would I do that?

•

Without knowledge compilation Input classC₁ Algo. 1 Result

Input classC₂ Algo. 2 Result Input classC₃ Algo. 3 Result

...

•

With knowledge compilation:

modularity!

Input class C₁ Input class C₂ Input class C₃

Compilation target

for your task Generic algo. Result Algo.1⁰

Algo. 2⁰ Algo.3⁰

2/20

Dr aft

Why would I do that?

•

Without knowledge compilation Input classC₁ Algo. 1 Result Input classC₂ Algo. 2 Result

Input classC₃ Algo. 3 Result ...

•

With knowledge compilation:

modularity!

Input class C₁ Input class C₂ Input class C₃

Compilation target

for your task Generic algo. Result Algo.1⁰

Algo. 2⁰ Algo.3⁰

2/20

Dr aft

Why would I do that?

•

Without knowledge compilation Input classC₁ Algo. 1 Result Input classC₂ Algo. 2 Result Input classC₃ Algo. 3 Result

...

•

With knowledge compilation:

modularity!

Input class C₁ Input class C₂ Input class C₃

Compilation target

for your task Generic algo. Result Algo.1⁰

Algo. 2⁰ Algo.3⁰

2/20

Dr aft

Why would I do that?

•

Without knowledge compilation Input classC₁ Algo. 1 Result Input classC₂ Algo. 2 Result Input classC₃ Algo. 3 Result

...

•

With knowledge compilation:

modularity!

Input class C₁ Input class C₂ Input class C₃

Compilation target

for your task Generic algo. Result Algo.1⁰

Algo. 2⁰ Algo.3⁰

2/20

Dr aft

Why would I do that?

•

Without knowledge compilation Input classC₁ Algo. 1 Result Input classC₂ Algo. 2 Result Input classC₃ Algo. 3 Result

...

•

With knowledge compilation:

modularity!

Input class C₁ Input class C₂ Input class C₃

Compilation target

for your task Generic algo. Result

Algo.1⁰ Algo. 2⁰ Algo.3⁰

2/20

Dr aft

Why would I do that?

•

Without knowledge compilation Input classC₁ Algo. 1 Result Input classC₂ Algo. 2 Result Input classC₃ Algo. 3 Result

...

•

With knowledge compilation:

modularity!

Input class C₁ Input class C₂ Input class C₃

Compilation target

for your task Generic algo. Result Algo.1⁰

Algo. 2⁰ Algo.3⁰

2/20

Dr aft

Why would I do that?

•

Without knowledge compilation Input classC₁ Algo. 1 Result Input classC₂ Algo. 2 Result Input classC₃ Algo. 3 Result

...

•

With knowledge compilation:

modularity!

Input class C₁ Input class C₂ Input class C₃

Compilation target

for your task Generic algo. Result Algo.1⁰

Algo. 2⁰

Algo.3⁰

2/20

Dr aft

Why would I do that?

•

Without knowledge compilation Input classC₁ Algo. 1 Result Input classC₂ Algo. 2 Result Input classC₃ Algo. 3 Result

...

•

With knowledge compilation:

modularity!

Input class C₁ Input class C₂ Input class C₃

Compilation target

for your task Generic algo. Result Algo.1⁰

Algo. 2⁰ Algo.3⁰

2/20

Dr aft

Why would I do that?

•

Without knowledge compilation Input classC₁ Algo. 1 Result Input classC₂ Algo. 2 Result Input classC₃ Algo. 3 Result

...

•

With knowledge compilation:modularity!

Input class C₁ Input class C₂ Input class C₃

Compilation target

for your task Generic algo. Result Algo.1⁰

Algo. 2⁰ Algo.3⁰

2/20

Dr aft

Studying the compilation targets

•

Tradeoffs between:

→ Complexity of compilation (concisenessof thecompilation target)

→ Complexity of solving the task

Input Compilation C_target

Result Complexity

Evaluation SAT

#SAT

→ When can we convert from one target to another?

•

We are interested in#SATandprobability evaluation

3/20

Dr aft

Studying the compilation targets

•

Tradeoffs between:

→ Complexity of compilation (concisenessof thecompilation target)

→ Complexity of solving the task

Input Compilation C_target

Result Complexity

Truth table

Evaluation SAT

#SAT

→ When can we convert from one target to another?

•

We are interested in#SATandprobability evaluation

3/20

Dr aft

Studying the compilation targets

•

Tradeoffs between:

→ Complexity of compilation (concisenessof thecompilation target)

→ Complexity of solving the task

Input Compilation C_target

Result Complexity

Truth table Evaluation

SAT

#SAT

→ When can we convert from one target to another?

•

We are interested in#SATandprobability evaluation

3/20

Dr aft

Studying the compilation targets

•

Tradeoffs between:

→ Complexity of compilation (concisenessof thecompilation target)

→ Complexity of solving the task

Input Compilation C_target

Result Complexity

Truth table O(1) Evaluation

SAT

#SAT

→ When can we convert from one target to another?

•

We are interested in#SATandprobability evaluation

3/20

Dr aft

Studying the compilation targets

•

Tradeoffs between:

→ Complexity of compilation (concisenessof thecompilation target)

→ Complexity of solving the task

Input Compilation C_target

Result Complexity

Truth table O(1) Evaluation SAT

#SAT

→ When can we convert from one target to another?

•

We are interested in#SATandprobability evaluation

3/20

Dr aft

Studying the compilation targets

•

Tradeoffs between:

→ Complexity of compilation (concisenessof thecompilation target)

→ Complexity of solving the task

Input Compilation C_target

Result Complexity

Truth table O(1) O(n)

Evaluation SAT

#SAT

→ When can we convert from one target to another?

•

We are interested in#SATandprobability evaluation

3/20

Dr aft

Studying the compilation targets

•

Tradeoffs between:

→ Complexity of compilation (concisenessof thecompilation target)

→ Complexity of solving the task

Input Compilation C_target

Result Complexity

Truth table O(1) O(n)

Evaluation SAT

#SAT

→ When can we convert from one target to another?

•

We are interested in#SATandprobability evaluation

3/20

Dr aft

Studying the compilation targets

•

Tradeoffs between:

→ Complexity of compilation (concisenessof thecompilation target)

→ Complexity of solving the task

Input Compilation C_target

Result Complexity

Truth table O(1) O(n) O(n)

Evaluation SAT

#SAT

→ When can we convert from one target to another?

•

We are interested in#SATandprobability evaluation

3/20

Dr aft

Studying the compilation targets

•

Tradeoffs between:

→ Complexity of compilation (concisenessof thecompilation target)

→ Complexity of solving the task

Input Compilation C_target

Result Complexity

DNF Evaluation

SAT

#SAT

→ When can we convert from one target to another?

•

We are interested in#SATandprobability evaluation

3/20

Dr aft

Studying the compilation targets

•

Tradeoffs between:

→ Complexity of compilation (concisenessof thecompilation target)

→ Complexity of solving the task

Input Compilation C_target

Result Complexity

DNF O(n) Evaluation SAT

#SAT

→ When can we convert from one target to another?

•

We are interested in#SATandprobability evaluation

3/20

Dr aft

Studying the compilation targets

•

Tradeoffs between:

→ Complexity of compilation (concisenessof thecompilation target)

→ Complexity of solving the task

Input Compilation C_target

Result Complexity

DNF O(n)

O(n)

Evaluation SAT

#SAT

→ When can we convert from one target to another?

•

We are interested in#SATandprobability evaluation

3/20

Dr aft

Studying the compilation targets

•

Tradeoffs between:

→ Complexity of compilation (concisenessof thecompilation target)

→ Complexity of solving the task

Input Compilation C_target

Result Complexity

DNF O(n)

O(n)

#P-hard

Evaluation SAT

#SAT

→ When can we convert from one target to another?

•

We are interested in#SATandprobability evaluation

3/20

Dr aft

Studying the compilation targets

•

Tradeoffs between:

→ Complexity of compilation (concisenessof thecompilation target)

→ Complexity of solving the task

Input Compilation C_target

Result Complexity

Boolean circuit Evaluation SAT

#SAT

→ When can we convert from one target to another?

•

We are interested in#SATandprobability evaluation

3/20

Dr aft

Studying the compilation targets

•

Tradeoffs between:

→ Complexity of compilation (concisenessof thecompilation target)

→ Complexity of solving the task

Input Compilation C_target

Result Complexity

Boolean circuit O(n) Evaluation SAT

#SAT

→ When can we convert from one target to another?

•

We are interested in#SATandprobability evaluation

3/20

Dr aft

Studying the compilation targets

•

Tradeoffs between:

→ Complexity of compilation (concisenessof thecompilation target)

→ Complexity of solving the task

Input Compilation C_target

Result Complexity

Boolean circuit O(n) NP-hard

Evaluation SAT

#SAT

→ When can we convert from one target to another?

•

We are interested in#SATandprobability evaluation

3/20

Dr aft

Studying the compilation targets

•

Tradeoffs between:

→ Complexity of compilation (concisenessof thecompilation target)

→ Complexity of solving the task

Input Compilation C_target

Result Complexity

Boolean circuit O(n) NP-hard

#P-hard

Evaluation SAT

#SAT

→ When can we convert from one target to another?

•

We are interested in#SATandprobability evaluation

3/20

Dr aft

Studying the compilation targets

•

Tradeoffs between:

→ Complexity of compilation (concisenessof thecompilation target)

→ Complexity of solving the task

Input Compilation C_target

Result Complexity

Boolean circuit O(n) NP-hard

#P-hard

Evaluation SAT

#SAT

→ When can we convert from one target to another?

•

We are interested in#SATandprobability evaluation

3/20

Dr aft

Studying the compilation targets

•

Tradeoffs between:

→ Complexity of compilation (concisenessof thecompilation target)

→ Complexity of solving the task

Input Compilation C_target

Result Complexity

Boolean circuit O(n) NP-hard

#P-hard

Evaluation SAT

#SAT

→ When can we convert from one target to another?

•

We are interested in#SATandprobability evaluation

3/20

Dr aft

Target classes in knowledge compilation

For #SAT and probabilistic evaluation, two main restrictions on compilation targets:

Width-based:

•

^{Boundedpathwidth/treewidth}Boolean circuits, CNFs, DNFs, etc.

→ message passingalgorithm for #SAT and probabilistic evaluation

• Links withBayesian networks Semantics-based:

•

Ordered Binary Decision Diagrams(OBDDs)/Deterministic Structured Decomposable Negation Normal Forms(d-SDNNFs)

→ #SAT and probabilistic evaluation are easy because these classes have strongsemantic constraints

• Used to understand#SAT solvers

Question: what are the links beetween the two?

4/20

Dr aft

Target classes in knowledge compilation

For #SAT and probabilistic evaluation, two main restrictions on compilation targets:

Width-based:

•

^{Boundedpathwidth/treewidth}Boolean circuits, CNFs, DNFs, etc.

→ message passingalgorithm for #SAT and probabilistic evaluation

• Links withBayesian networks Semantics-based:

•

Ordered Binary Decision Diagrams(OBDDs)/Deterministic Structured Decomposable Negation Normal Forms(d-SDNNFs)

→ #SAT and probabilistic evaluation are easy because these classes have strongsemantic constraints

• Used to understand#SAT solvers

Question: what are the links beetween the two?

4/20

Dr aft

Target classes in knowledge compilation

For #SAT and probabilistic evaluation, two main restrictions on compilation targets:

Width-based:

•

^{Boundedpathwidth/treewidth}Boolean circuits, CNFs, DNFs, etc.

→ message passingalgorithm for #SAT and probabilistic evaluation

• Links withBayesian networks Semantics-based:

•

Ordered Binary Decision Diagrams(OBDDs)/Deterministic Structured Decomposable Negation Normal Forms(d-SDNNFs)

→ #SAT and probabilistic evaluation are easy because these classes have strongsemantic constraints

• Used to understand#SAT solvers

Question: what are the links beetween the two?

4/20

Dr aft

Target classes in knowledge compilation

For #SAT and probabilistic evaluation, two main restrictions on compilation targets:

Width-based:

•

^{Boundedpathwidth/treewidth}Boolean circuits, CNFs, DNFs, etc.

→ message passingalgorithm for #SAT and probabilistic evaluation

• Links withBayesian networks

Semantics-based:

•

Ordered Binary Decision Diagrams(OBDDs)/Deterministic Structured Decomposable Negation Normal Forms(d-SDNNFs)

→ #SAT and probabilistic evaluation are easy because these classes have strongsemantic constraints

• Used to understand#SAT solvers

Question: what are the links beetween the two?

4/20

Dr aft

Target classes in knowledge compilation

For #SAT and probabilistic evaluation, two main restrictions on compilation targets:

Width-based:

•

^{Boundedpathwidth/treewidth}Boolean circuits, CNFs, DNFs, etc.

→ message passingalgorithm for #SAT and probabilistic evaluation

• Links withBayesian networks Semantics-based:

•

Ordered Binary Decision Diagrams(OBDDs)/Deterministic Structured Decomposable Negation Normal Forms(d-SDNNFs)

→ #SAT and probabilistic evaluation are easy because these classes have strongsemantic constraints

• Used to understand#SAT solvers

Question: what are the links beetween the two?

4/20

Dr aft

Target classes in knowledge compilation

For #SAT and probabilistic evaluation, two main restrictions on compilation targets:

Width-based:

•

^{Boundedpathwidth/treewidth}Boolean circuits, CNFs, DNFs, etc.

→ message passingalgorithm for #SAT and probabilistic evaluation

• Links withBayesian networks Semantics-based:

•

Ordered Binary Decision Diagrams(OBDDs)/Deterministic Structured Decomposable Negation Normal Forms(d-SDNNFs)

→ #SAT and probabilistic evaluation are easy because these classes have strongsemantic constraints

• Used to understand#SAT solvers

Question: what are the links beetween the two?

4/20

Dr aft

Target classes in knowledge compilation

For #SAT and probabilistic evaluation, two main restrictions on compilation targets:

Width-based:

•

^{Boundedpathwidth/treewidth}Boolean circuits, CNFs, DNFs, etc.

→ message passingalgorithm for #SAT and probabilistic evaluation

• Links withBayesian networks Semantics-based:

•

Ordered Binary Decision Diagrams(OBDDs)/Deterministic Structured Decomposable Negation Normal Forms(d-SDNNFs)

→ #SAT and probabilistic evaluation are easy because these classes have strongsemantic constraints

• Used to understand#SAT solvers

Question: what are the links beetween the two?

4/20

Dr aft

Target classes in knowledge compilation

For #SAT and probabilistic evaluation, two main restrictions on compilation targets:

Width-based:

•

^{Boundedpathwidth/treewidth}Boolean circuits, CNFs, DNFs, etc.

→ message passingalgorithm for #SAT and probabilistic evaluation

• Links withBayesian networks Semantics-based:

•

Ordered Binary Decision Diagrams(OBDDs)/Deterministic Structured Decomposable Negation Normal Forms(d-SDNNFs)

→ #SAT and probabilistic evaluation are easy because these classes have strongsemantic constraints

• Used to understand#SAT solvers

Question: what are the links beetween the two?

4/20

Dr aft

Plan

•

^{CircuitCoftreewidth6k O(|C| ×exp(k)) d-SDNNF}

•

^+'matching lower bound Then

•

^{DNF/CNFϕofpathwidth6k O(|ϕ| ×exp(k)) OBDD (not us)}

•

+ matching lower bound Then

•

Application to provenance and probabilistic databases

5/20

Dr aft

Plan

•

^{CircuitCoftreewidth6k O(|C| ×exp(k)) d-SDNNF}

•

^+'matching lower bound

Then

•

^{DNF/CNFϕofpathwidth6k O(|ϕ| ×exp(k)) OBDD (not us)}

•

+ matching lower bound Then

•

Application to provenance and probabilistic databases

5/20

Dr aft

Plan

•

^{CircuitCoftreewidth6k O(|C| ×exp(k)) d-SDNNF}

•

^+'matching lower bound Then

•

^{DNF/CNFϕofpathwidth6k O(|ϕ| ×exp(k)) OBDD (not us)}

•

+ matching lower bound Then

•

Application to provenance and probabilistic databases

5/20

Dr aft

Plan

•

^{CircuitCoftreewidth6k O(|C| ×exp(k)) d-SDNNF}

•

^+'matching lower bound Then

•

^{DNF/CNFϕofpathwidth6k O(|ϕ| ×exp(k)) OBDD (not us)}

•

+ matching lower bound

Then

•

Application to provenance and probabilistic databases

5/20

Dr aft

Plan

•

^{CircuitCoftreewidth6k O(|C| ×exp(k)) d-SDNNF}

•

^+'matching lower bound Then

•

^{DNF/CNFϕofpathwidth6k O(|ϕ| ×exp(k)) OBDD (not us)}

•

+ matching lower bound Then

•

Application to provenance and probabilistic databases

5/20

Dr aft

Treewidth and d-SDNNFs

6/20

Dr aft

Bounded treewidth Boolean circuits

∧

∧

x ¬ ∨

t ¬

∨

∧ z y

TreewidthofC= that of the underlying graph

We can domessage passing:

Theorem (Lauritzen & Spielgelhalter, 1988)

Fixk∈N. Given a Boolean circuitCoftreewidth6k, we can compute itsprobabilityin timeO(f(k)× |C|), wheref is singly exponential

7/20

Dr aft

Bounded treewidth Boolean circuits

∧

∧

x ¬ ∨

t ¬

∨

∧ z y

TreewidthofC= that of the underlying graph

We can domessage passing:

Theorem (Lauritzen & Spielgelhalter, 1988)

Fixk∈N. Given a Boolean circuitCoftreewidth6k, we can compute itsprobabilityin timeO(f(k)× |C|), wheref is singly exponential

7/20

Dr aft

Bounded treewidth Boolean circuits

∧

∧

x ¬ ∨

t ¬

∨

∧ z y

TreewidthofC= that of the underlying graph

We can domessage passing:

Theorem (Lauritzen & Spielgelhalter, 1988)

Fixk∈N. Given a Boolean circuitCoftreewidth6k, we can compute itsprobabilityin timeO(f(k)× |C|), wheref is singly exponential

7/20

Dr aft

d-SDNNF

∨

∧

x ∧

y ¬

z

∧

∧

¬

x ∧

y z

•

• y z x

•

Negation Normal Form: negations only applied to the leaves

•

Decomposable: inputs of∧-gates are independent(no variable has a path to two different inputs of the same

∧-gate)

→ SATcan be solved efficiently

•

Deterministic: inputs of∨-gates are mutually exclusive

→ #SATandprobability evaluation

•

^Structured: there is avtreethat structures the∧-gates

→ Enumeration

8/20

Dr aft

d-SDNNF

∨

∧

x ∧

y ¬

z

∧

∧

¬

x ∧

y z

•

• y z x

•

Negation Normal Form: negations only applied to the leaves

•

Decomposable: inputs of∧-gates are independent(no variable has a path to two different inputs of the same

∧-gate)

→ SATcan be solved efficiently

•

Deterministic: inputs of∨-gates are mutually exclusive

→ #SATandprobability evaluation

•

^Structured: there is avtreethat structures the∧-gates

→ Enumeration

8/20

Dr aft

d-SDNNF

∨

∧

x ∧

y ¬

z

∧

∧

¬

x ∧

y z

•

• y z x

•

Negation Normal Form: negations only applied to the leaves

•

Decomposable: inputs of∧-gates are independent(no variable has a path to two different inputs of the same

∧-gate)

→ SATcan be solved efficiently

•

Deterministic: inputs of∨-gates are mutually exclusive

→ #SATandprobability evaluation

•

^Structured: there is avtreethat structures the∧-gates

→ Enumeration

8/20

Dr aft

d-SDNNF

∨

∧

x ∧

y ¬

z

∧

∧

¬

x ∧

y z

•

• y z x

•

Negation Normal Form: negations only applied to the leaves

•

Decomposable: inputs of∧-gates are independent(no variable has a path to two different inputs of the same

∧-gate)

→ SATcan be solved efficiently

•

Deterministic: inputs of∨-gates are mutually exclusive

→ #SATandprobability evaluation

•

^Structured: there is avtreethat structures the∧-gates

→ Enumeration

8/20

Dr aft

d-SDNNF

∨

∧

x ∧

y ¬

z

∧

∧

¬

x ∧

y z

•

• y z x

•

Negation Normal Form: negations only applied to the leaves

•

Decomposable: inputs of∧-gates are independent(no variable has a path to two different inputs of the same

∧-gate)

→ SATcan be solved efficiently

•

Deterministic: inputs of∨-gates are mutually exclusive

→ #SATandprobability evaluation

•

^Structured: there is avtreethat structures the∧-gates

→ Enumeration

8/20

Dr aft

d-SDNNF

∨

∧

x ∧

y ¬

z

∧

∧

¬

x ∧

y z

•

• y z x

•

Negation Normal Form: negations only applied to the leaves

•

Decomposable: inputs of∧-gates are independent(no variable has a path to two different inputs of the same

∧-gate)

→ SATcan be solved efficiently

•

Deterministic: inputs of∨-gates are mutually exclusive

→ #SATandprobability evaluation

•

^Structured: there is avtreethat structures the∧-gates

→ Enumeration

8/20

Dr aft

d-SDNNF

∨

∧

x ∧

y ¬

z

∧

∧

¬

x ∧

y z

•

• y z x

•

Negation Normal Form: negations only applied to the leaves

•

Decomposable: inputs of∧-gates are independent(no variable has a path to two different inputs of the same

∧-gate)

→ SATcan be solved efficiently

•

Deterministic: inputs of∨-gates are mutually exclusive

→ #SATandprobability evaluation

•

^Structured: there is avtreethat structures the∧-gates

→ Enumeration

8/20

Dr aft

d-SDNNF

∨

∧

x ∧

y ¬

z

∧

∧

¬

x ∧

y z

•

• y z x

•

Negation Normal Form: negations only applied to the leaves

•

Decomposable: inputs of∧-gates are independent(no variable has a path to two different inputs of the same

∧-gate)

→ SATcan be solved efficiently

•

Deterministic: inputs of∨-gates are mutually exclusive

→ #SATandprobability evaluation

•

^Structured: there is avtreethat structures the∧-gates

→ Enumeration

8/20

Dr aft

Treewidth and d-SDNNFs: Upper bound

Theorem (Bova & Szeider, 2017)

LetCbe a Boolean circuit onmvariables oftreewidth6k.

There existsad-SDNNFequivalent toCof sizeO(m×g(k)), wheregisdoublyexponential

Drawbacks: non constructive Theorem (Our contribution)

LetCbe a Boolean circuit oftreewidth6k.

We can computead-SDNNFequivalent toCin timeO(|C| ×f(k)), wheref issinglyexponential

Applications: recapturing message passing, and enumeration of satisfying valuations

9/20

Dr aft

Treewidth and d-SDNNFs: Upper bound

Theorem (Bova & Szeider, 2017)

LetCbe a Boolean circuit onmvariables oftreewidth6k.

There existsad-SDNNFequivalent toCof sizeO(m×g(k)), wheregisdoublyexponential

Drawbacks: non constructive

Theorem (Our contribution)

LetCbe a Boolean circuit oftreewidth6k.

We can computead-SDNNFequivalent toCin timeO(|C| ×f(k)), wheref issinglyexponential

Applications: recapturing message passing, and enumeration of satisfying valuations

9/20

Dr aft

Treewidth and d-SDNNFs: Upper bound

Theorem (Bova & Szeider, 2017)

LetCbe a Boolean circuit onmvariables oftreewidth6k.

There existsad-SDNNFequivalent toCof sizeO(m×g(k)), wheregisdoublyexponential

Drawbacks: non constructive Theorem (Our contribution)

LetCbe a Boolean circuit oftreewidth6k.

We can computead-SDNNFequivalent toCin timeO(|C| ×f(k)), wheref issinglyexponential

Applications: recapturing message passing, and enumeration of satisfying valuations

9/20

Dr aft

Treewidth and d-SDNNFs: Upper bound

Theorem (Bova & Szeider, 2017)

LetCbe a Boolean circuit onmvariables oftreewidth6k.

There existsad-SDNNFequivalent toCof sizeO(m×g(k)), wheregisdoublyexponential

Drawbacks: non constructive Theorem (Our contribution)

LetCbe a Boolean circuit oftreewidth6k.

We can computead-SDNNFequivalent toCin timeO(|C| ×f(k)), wheref issinglyexponential

Applications: recapturing message passing, and enumeration of satisfying valuations

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Construction sketch

V₁

L

x ^V 2 V3

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Construction sketch

V1

L

V

2

x

V3 V1 V₁

L

x ^V 2 V3

V1

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Dr aft

Construction sketch

V1

L

V

2

x

V3 V1 V₁

L

x ^V 2 V3

V1

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Construction sketch

V1

L

V

2

V1 V

2

x

V3 V1 V₁

L

x ^V 2 V3

V1

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Dr aft

Construction sketch

V1

L

V

2

V1 V

2 V1 V

2

x

V3 V1 V₁

L

x ^V 2 V3

V1

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Dr aft

Construction sketch

V1

L

V

2

V1 V

2 V1 V

2 V1 V

2

x

V3 V1 V₁

L

x ^V 2 V3

V1

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Dr aft

Construction sketch

V1

L

V

2

V1 V

2 V1 V

2 V1 V

2

V1 V3 V1 V3 V1 V3

x

V3 V1 V₁

L

x ^V 2 V3

V1

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Dr aft

Construction sketch

V1

L

V

2

V1 V

2 V1 V

2 V1 V

2

V1 V3 V1 V3 V1 V3 V1 L

x

V1 L x

V1 L x

x

V3 V1 V₁

L

x ^V 2 V3

V1

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Dr aft

Construction sketch

V1

L

V

2

V1 V

2 V1 V

2 V1 V

2

V1 V3 V1 V3 V1 V3 V1 L

x

V1 L x

V1 L x

x

V3 V1 V₁

L

x ^V 2 V3

V1

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Dr aft

Construction sketch

V1

L

V

2

V1 V

2 V1 V

2 V1 V

2

V1 V3 V1 V3 V1 V3 V1 L

x

V1 L x

V1 L x

x

V3 V1 V₁

L

x ^V 2 V3

V1

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Dr aft

Construction sketch

V1

L

V

2

V1 V

2 V1 V

2 V1 V

2

V1 V3 V1 V3 V1 V3 V1 L

x

V1 L x

V1 L x

x

V3 V1 V₁

L

x ^V 2 V3

V1

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Dr aft

Construction sketch

V1

L

V

2

V1 V

2 V1 V

2 V1 V

2

V1 V3 V1 V3 V1 V3 V1 L

x

V1 L x

V1 L x

x

V3 V1 V₁

L

x ^V 2 V3

V1

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Dr aft

Construction sketch

V1

L

V

2

V1 V

2 V1 V

2 V1 V

2

V1 V3 V1 V3 V1 V3 V1 L

x

V1 L x

V1 L x

x

V3 V1 V₁

L

x ^V 2 V3

V1

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Construction sketch

V1

L

V

2

V1 V

2 V1 V

2 V1 V

2

V1 V3 V1 V3 V1 V3 V1 L

x

V1 L x

V1 L x

x

V3 V1 V₁

L

x ^V 2 V3

V1

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Dr aft

Construction sketch

V1

L

V

2

V1 V

2 V1 V

2 V1 V

2

V1 V3 V1 V3 V1 V3 V1 L

x

V1 L x

V1 L x

x

V3 V1 V₁

L

x ^V 2 V3

V1

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Construction sketch

V1

L

V

2

V1 V

2 V1 V

2 V1 V

2

V1 V3 V1 V3 V1 V3 V1 L

x

V1 L x

V1 L x

x

V3 V1 V₁

L

x ^V 2 V3

V1

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Dr aft

Construction sketch

V1

L

V

2

V1 V

2 V1 V

2 V1 V

2

V1 V3 V1 V3 V1 V3 V1 L

x

V1 L x

V1 L x

x

V3 V1 V₁

L

x ^V 2 V3

V1

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Construction sketch

V1

L

V

2

V1 V

2 V1 V

2 V1 V

2

V1 V3 V1 V3 V1 V3 V1 L

x

V1 L x

V1 L x

x

V3 V1 V₁

L

x ^V 2 V3

V1

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Dr aft

Treewidth and d-SDNNFs: Lower bound

•

Already applies to very restricted Boolean circuits:monotone DNFs and CNFs

•

Treewidth of a DNF/CNF: that of itsGaifman graph

•

Arity: size of the largest clause

•

Degree: maximal number of clauses to which a variable belongs Theorem

Letϕbe amonotone DNFoftreewidthk, leta:=arity(ϕ)and d:=degree(ϕ). Then anyd-SDNNFforϕhas size>2

j k 3×a3×d2

k

−1

•

^ForCNFs, the bound even works for (non-deterministic)SDNNF

•

The bound isgeneric: it applies toanymonotone DNF/CNF

11/20

Dr aft

Treewidth and d-SDNNFs: Lower bound

•

Already applies to very restricted Boolean circuits:monotone DNFs and CNFs

•

Treewidth of a DNF/CNF: that of itsGaifman graph

•

Arity: size of the largest clause

•

Degree: maximal number of clauses to which a variable belongs Theorem

Letϕbe amonotone DNFoftreewidthk, leta:=arity(ϕ)and d:=degree(ϕ). Then anyd-SDNNFforϕhas size>2

j k 3×a3×d2

k

−1

•

^ForCNFs, the bound even works for (non-deterministic)SDNNF

•

The bound isgeneric: it applies toanymonotone DNF/CNF

11/20

Dr aft

Treewidth and d-SDNNFs: Lower bound

•

Already applies to very restricted Boolean circuits:monotone DNFs and CNFs

•

Treewidth of a DNF/CNF: that of itsGaifman graph

•

Arity: size of the largest clause

•

Degree: maximal number of clauses to which a variable belongs Theorem

Letϕbe amonotone DNFoftreewidthk, leta:=arity(ϕ)and d:=degree(ϕ). Then anyd-SDNNFforϕhas size>2

j k 3×a3×d2

k

−1

•

^ForCNFs, the bound even works for (non-deterministic)SDNNF

•

The bound isgeneric: it applies toanymonotone DNF/CNF

11/20

Dr aft

Treewidth and d-SDNNFs: Lower bound

•

Already applies to very restricted Boolean circuits:monotone DNFs and CNFs

•

Treewidth of a DNF/CNF: that of itsGaifman graph

•

Arity: size of the largest clause

•

Degree: maximal number of clauses to which a variable belongs

Theorem

Letϕbe amonotone DNFoftreewidthk, leta:=arity(ϕ)and d:=degree(ϕ). Then anyd-SDNNFforϕhas size>2

j k 3×a3×d2

k

−1

•

^ForCNFs, the bound even works for (non-deterministic)SDNNF

•

The bound isgeneric: it applies toanymonotone DNF/CNF

11/20

Dr aft

Treewidth and d-SDNNFs: Lower bound

•

Already applies to very restricted Boolean circuits:monotone DNFs and CNFs

•

Treewidth of a DNF/CNF: that of itsGaifman graph

•

Arity: size of the largest clause

•

Degree: maximal number of clauses to which a variable belongs Theorem

Letϕbe amonotone DNFoftreewidthk, leta:=arity(ϕ)and d:=degree(ϕ). Then anyd-SDNNFforϕhas size>2

j k 3×a3×d2

k

−1

•

^ForCNFs, the bound even works for (non-deterministic)SDNNF

•

The bound isgeneric: it applies toanymonotone DNF/CNF

11/20

Dr aft

Treewidth and d-SDNNFs: Lower bound

•

Already applies to very restricted Boolean circuits:monotone DNFs and CNFs

•

Treewidth of a DNF/CNF: that of itsGaifman graph

•

Arity: size of the largest clause

•

Degree: maximal number of clauses to which a variable belongs Theorem

Letϕbe amonotone DNFoftreewidthk, leta:=arity(ϕ)and d:=degree(ϕ). Then anyd-SDNNFforϕhas size>2

j k 3×a3×d2

k

−1

•

^ForCNFs, the bound even works for (non-deterministic)SDNNF

•

The bound isgeneric: it applies toanymonotone DNF/CNF

11/20

Dr aft

Treewidth and d-SDNNFs: Lower bound

•

Already applies to very restricted Boolean circuits:monotone DNFs and CNFs

•

Treewidth of a DNF/CNF: that of itsGaifman graph

•

Arity: size of the largest clause

•

Degree: maximal number of clauses to which a variable belongs Theorem

Letϕbe amonotone DNFoftreewidthk, leta:=arity(ϕ)and d:=degree(ϕ). Then anyd-SDNNFforϕhas size>2

j k 3×a3×d2

k

−1

•

^ForCNFs, the bound even works for (non-deterministic)SDNNF

•

The bound isgeneric: it applies toanymonotone DNF/CNF

11/20

Dr aft

Pathwidth and OBDDs

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Dr aft

Ordered Binary Decision Diagrams (OBDDs)

•

^DAGwith sink nodes{>,⊥}and internal nodes labeled by variables

•

^Semantics:follow the path of an assignment to get the value of the Boolean function

•

^{There is a}total orderon the variables v=X₁ X₂ X₃X₄such that each root-to-sink path is compatible withv

•

Compute probabilitybottom-up Prπ(•) =π(X₃)×Prπ(•)

+(1−π(X₃))×Prπ(•)

•

^Widthof the OBDD'largest number of nodes that are labeled by the same variable

x

₁

x

₄

x

₃

x

₃

T T

x

₂

x

₂

0 1

0

0 0

0

0 1

1

1

1 1

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