Draft Connecting Width and Structure in Knowledge Compilation - - PowerPoint PPT Presentation
Draft Connecting Width and Structure in Knowledge Compilation Antoine Amarilli 1 , Mikal Monet 1 , 3 , Pierre Senellart 2 , 3 March 28th, 2018 1 LTCI, Tlcom ParisTech, Universit Paris-Saclay; Paris, France 2 cole normale suprieure,
Connecting Width and Structure in Knowledge Compilation
Antoine Amarilli1, Mikaël Monet1,3, Pierre Senellart2,3 March 28th, 2018 1LTCI, Télécom ParisTech, Université Paris-Saclay; Paris, France 2École normale supérieure, PSL Reasearch University; Paris, France 3Inria Paris; Paris, France What is Knowledge Compilation?
What is Knowledge Compilation?
What is Knowledge Compilation?
What is Knowledge Compilation?
What is Knowledge Compilation?
efficiently your task
1/20 Why would I do that?
Input class C1 Result
Why would I do that?
Input class C1 Result
Input class C2 Result
Why would I do that?
Input class C1 Result
Input class C2 Result
Input class C3 Result
...
2/20 Why would I do that?
Input class C1 Result
Input class C2 Result
Input class C3 Result
...
Why would I do that?
Input class C1 Result
Input class C2 Result
Input class C3 Result
...
Input class C1 Input class C2 Input class C3 Compilation target for your task Result Generic algo.
2/20 Why would I do that?
Input class C1 Result
Input class C2 Result
Input class C3 Result
...
Input class C1 Input class C2 Input class C3 Compilation target for your task Result Generic algo.
Why would I do that?
Input class C1 Result
Input class C2 Result
Input class C3 Result
...
Input class C1 Input class C2 Input class C3 Compilation target for your task Result Generic algo.
Why would I do that?
Input class C1 Result
Input class C2 Result
Input class C3 Result
...
Input class C1 Input class C2 Input class C3 Compilation target for your task Result Generic algo.
Why would I do that?
Input class C1 Result
Input class C2 Result
Input class C3 Result
...
Input class C1 Input class C2 Input class C3 Compilation target for your task Result Generic algo.
Studying the compilation targets
Input Ctarget Compilation Result Complexity
3/20 Studying the compilation targets
Input Ctarget Compilation Result Complexity Truth table
3/20 Studying the compilation targets
Input Ctarget Compilation Result Complexity Truth table Evaluation
3/20 Studying the compilation targets
Input Ctarget Compilation Result Complexity Truth table O(1) Evaluation
3/20 Studying the compilation targets
Input Ctarget Compilation Result Complexity Truth table O(1) Evaluation SAT
3/20 Studying the compilation targets
Input Ctarget Compilation Result Complexity Truth table O(1) O(n) Evaluation SAT
3/20 Studying the compilation targets
Input Ctarget Compilation Result Complexity Truth table O(1) O(n) Evaluation SAT #SAT
3/20 Studying the compilation targets
Input Ctarget Compilation Result Complexity Truth table O(1) O(n) O(n) Evaluation SAT #SAT
3/20 Studying the compilation targets
Input Ctarget Compilation Result Complexity DNF Evaluation SAT #SAT
3/20 Studying the compilation targets
Input Ctarget Compilation Result Complexity DNF O(n) Evaluation SAT #SAT
3/20 Studying the compilation targets
Input Ctarget Compilation Result Complexity DNF O(n) O(n) Evaluation SAT #SAT
3/20 Studying the compilation targets
Input Ctarget Compilation Result Complexity DNF O(n) O(n) #P hard Evaluation SAT #SAT
3/20 Studying the compilation targets
Input Ctarget Compilation Result Complexity Boolean circuit Evaluation SAT #SAT
3/20 Studying the compilation targets
Input Ctarget Compilation Result Complexity Boolean circuit O(n) Evaluation SAT #SAT
3/20 Studying the compilation targets
Input Ctarget Compilation Result Complexity Boolean circuit O(n) NP hard Evaluation SAT #SAT
3/20 Studying the compilation targets
Input Ctarget Compilation Result Complexity Boolean circuit O(n) NP hard #P hard Evaluation SAT #SAT
3/20 Studying the compilation targets
Input Ctarget Compilation Result Complexity Boolean circuit O(n) NP hard #P hard Evaluation SAT #SAT → When can we convert from one target to another?
3/20 Studying the compilation targets
Input Ctarget Compilation Result Complexity Boolean circuit O(n) NP hard #P hard Evaluation SAT #SAT → When can we convert from one target to another?
Target classes in knowledge compilation
For #SAT and probabilistic evaluation, two main restrictions on compilation targets:
4/20 Target classes in knowledge compilation
For #SAT and probabilistic evaluation, two main restrictions on compilation targets: Width-based:
Target classes in knowledge compilation
For #SAT and probabilistic evaluation, two main restrictions on compilation targets: Width-based:
Target classes in knowledge compilation
For #SAT and probabilistic evaluation, two main restrictions on compilation targets: Width-based:
Target classes in knowledge compilation
For #SAT and probabilistic evaluation, two main restrictions on compilation targets: Width-based:
Semantics-based:
Structured Decomposable Negation Normal Forms (d-SDNNFs)
4/20 Target classes in knowledge compilation
For #SAT and probabilistic evaluation, two main restrictions on compilation targets: Width-based:
Semantics-based:
Structured Decomposable Negation Normal Forms (d-SDNNFs)
→ #SAT and probabilistic evaluation are easy because these classes have strong semantic constraints 4/20 Target classes in knowledge compilation
For #SAT and probabilistic evaluation, two main restrictions on compilation targets: Width-based:
Semantics-based:
Structured Decomposable Negation Normal Forms (d-SDNNFs)
→ #SAT and probabilistic evaluation are easy because these classes have strong semantic constraints Target classes in knowledge compilation
For #SAT and probabilistic evaluation, two main restrictions on compilation targets: Width-based:
Semantics-based:
Structured Decomposable Negation Normal Forms (d-SDNNFs)
→ #SAT and probabilistic evaluation are easy because these classes have strong semantic constraintsQuestion: what are the links beetween the two?
4/20 Plan
d-SDNNF O(|C| × exp(k))
5/20 Plan
d-SDNNF O(|C| × exp(k))
Plan
d-SDNNF O(|C| × exp(k))
Then
DNF/CNF ϕ of pathwidth k OBDD O(|ϕ| × exp(k))
5/20 Plan
d-SDNNF O(|C| × exp(k))
Then
DNF/CNF ϕ of pathwidth k OBDD O(|ϕ| × exp(k))
Plan
d-SDNNF O(|C| × exp(k))
Then
DNF/CNF ϕ of pathwidth k OBDD O(|ϕ| × exp(k))
Then
Treewidth and d-SDNNFs
6/20 Bounded treewidth Boolean circuits
∧ ∧ x ¬ ∨ t ¬ ∨ ∧ z y
7/20 Bounded treewidth Boolean circuits
∧ ∧ x ¬ ∨ t ¬ ∨ ∧ z y Treewidth of C = that of the underlying graph
7/20 Bounded treewidth Boolean circuits
∧ ∧ x ¬ ∨ t ¬ ∨ ∧ z y Treewidth of C = that of the underlying graph We can do message passing: Theorem (Lauritzen & Spielgelhalter, 1988) Fix k ∈ N. Given a Boolean circuit C of treewidth k, we can compute its probability in time O(f(k) × |C|), where f is singly exponential
7/20 d-SDNNF
∨ ∧ x ∧ y ¬ z ∧ ¬ x ∧ y z
8/20 d-SDNNF
∨ ∧ x ∧ y ¬ z ∧ ¬ x ∧ y z
d-SDNNF
∨ ∧ x ∧ y ¬ z ∧ ¬ x ∧ y z
independent (no variable has a path to two different inputs of the same ∧-gate)
8/20 d-SDNNF
∨ ∧ x ∧ y ¬ z ∧ ¬ x ∧ y z
independent (no variable has a path to two different inputs of the same ∧-gate)
→ SAT can be solved efficiently 8/20 d-SDNNF
∨ ∧ x ∧ y ¬ z ∧ ¬ x ∧ y z
independent (no variable has a path to two different inputs of the same ∧-gate)
→ SAT can be solved efficientlymutually exclusive
8/20 d-SDNNF
∨ ∧ x ∧ y ¬ z ∧ ¬ x ∧ y z
independent (no variable has a path to two different inputs of the same ∧-gate)
→ SAT can be solved efficientlymutually exclusive
→ #SAT and probability evaluation 8/20 d-SDNNF
∨ ∧ x ∧ y ¬ z ∧ ¬ x ∧ y z
y x
independent (no variable has a path to two different inputs of the same ∧-gate)
→ SAT can be solved efficientlymutually exclusive
→ #SAT and probability evaluationstructures the ∧-gates
8/20 d-SDNNF
∨ ∧ x ∧ y ¬ z ∧ ¬ x ∧ y z
y x
independent (no variable has a path to two different inputs of the same ∧-gate)
→ SAT can be solved efficientlymutually exclusive
→ #SAT and probability evaluationstructures the ∧-gates
→ Enumeration 8/20 Treewidth and d-SDNNFs: Upper bound
Theorem (Bova & Szeider, 2017) Let C be a Boolean circuit on m variables of treewidth k. There exists a d-SDNNF equivalent to C of size O(m × g(k)), where g is doubly exponential
9/20 Treewidth and d-SDNNFs: Upper bound
Theorem (Bova & Szeider, 2017) Let C be a Boolean circuit on m variables of treewidth k. There exists a d-SDNNF equivalent to C of size O(m × g(k)), where g is doubly exponential Drawbacks: non constructive
9/20 Treewidth and d-SDNNFs: Upper bound
Theorem (Bova & Szeider, 2017) Let C be a Boolean circuit on m variables of treewidth k. There exists a d-SDNNF equivalent to C of size O(m × g(k)), where g is doubly exponential Drawbacks: non constructive Theorem (This paper) Let C be a Boolean circuit of treewidth k. We can compute a d-SDNNF equivalent to C in time O(|C| × f(k)), where f is singly exponential
9/20 Treewidth and d-SDNNFs: Upper bound
Theorem (Bova & Szeider, 2017) Let C be a Boolean circuit on m variables of treewidth k. There exists a d-SDNNF equivalent to C of size O(m × g(k)), where g is doubly exponential Drawbacks: non constructive Theorem (This paper) Let C be a Boolean circuit of treewidth k. We can compute a d-SDNNF equivalent to C in time O(|C| × f(k)), where f is singly exponential Applications: recapturing message passing, and enumeration of satisfying valuations
9/20 Construction sketch
10/20 Construction sketch
10/20 Construction sketch
10/20 Construction sketch
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10/20 Construction sketch
10/20 Construction sketch
10/20 Construction sketch
10/20 Construction sketch
10/20 Construction sketch
10/20 Construction sketch
10/20 Construction sketch
10/20 Construction sketch
10/20 Construction sketch
10/20 Construction sketch
10/20 Construction sketch
10/20 Construction sketch
10/20 Construction sketch
10/20 Treewidth and d-SDNNFs: Lower bound
DNFs and CNFs
11/20 Treewidth and d-SDNNFs: Lower bound
DNFs and CNFs
Treewidth and d-SDNNFs: Lower bound
DNFs and CNFs
Treewidth and d-SDNNFs: Lower bound
DNFs and CNFs
Treewidth and d-SDNNFs: Lower bound
DNFs and CNFs
Theorem Let ϕ be a monotone DNF of treewidth k, let a := arity(ϕ) and d := degree(ϕ). Then any d-SDNNF for ϕ has size 2
Treewidth and d-SDNNFs: Lower bound
DNFs and CNFs
Theorem Let ϕ be a monotone DNF of treewidth k, let a := arity(ϕ) and d := degree(ϕ). Then any d-SDNNF for ϕ has size 2
Treewidth and d-SDNNFs: Lower bound
DNFs and CNFs
Theorem Let ϕ be a monotone DNF of treewidth k, let a := arity(ϕ) and d := degree(ϕ). Then any d-SDNNF for ϕ has size 2
Pathwidth and OBDDs
12/20 Ordered Binary Decision Diagrams (OBDDs)
nodes labeled by variables
13/20 Ordered Binary Decision Diagrams (OBDDs)
nodes labeled by variables
to get the value of the Boolean function
13/20 Ordered Binary Decision Diagrams (OBDDs)
nodes labeled by variables
to get the value of the Boolean function
v = X1 X2 X3 X4 such that each root-to-sink path is compatible with v
13/20 Ordered Binary Decision Diagrams (OBDDs)
nodes labeled by variables
to get the value of the Boolean function
v = X1 X2 X3 X4 such that each root-to-sink path is compatible with v
Ordered Binary Decision Diagrams (OBDDs)
nodes labeled by variables
to get the value of the Boolean function
v = X1 X2 X3 X4 such that each root-to-sink path is compatible with v
Prπ(•) = π(X3) × Prπ(•) +(1 − π(X3)) × Prπ(•)
13/20 Ordered Binary Decision Diagrams (OBDDs)
nodes labeled by variables
to get the value of the Boolean function
v = X1 X2 X3 X4 such that each root-to-sink path is compatible with v
Prπ(•) = π(X3) × Prπ(•) +(1 − π(X3)) × Prπ(•)
nodes that are labeled by the same variable
13/20 Pathwidth and OBDDs: Upper and lower bounds
Upper bound: Theorem (Bova & Slivovsky, 2017) Let ϕ be a CNF or DNF of pathwidth k. We can compile ϕ into an OBDD
Pathwidth and OBDDs: Upper and lower bounds
Upper bound: Theorem (Bova & Slivovsky, 2017) Let ϕ be a CNF or DNF of pathwidth k. We can compile ϕ into an OBDD
Lower bound: Theorem (This paper) Let ϕ be a monotone CNF or DNF of pathwidth k, and let a := arity(ϕ) and d := degree(ϕ). Then any OBDD for ϕ has width 2
Pathwidth and OBDDs: Upper and lower bounds
Upper bound: Theorem (Bova & Slivovsky, 2017) Let ϕ be a CNF or DNF of pathwidth k. We can compile ϕ into an OBDD
Lower bound: Theorem (This paper) Let ϕ be a monotone CNF or DNF of pathwidth k, and let a := arity(ϕ) and d := degree(ϕ). Then any OBDD for ϕ has width 2
Pathwidth and OBDDs: Upper and lower bounds
Upper bound: Theorem (Bova & Slivovsky, 2017) Let ϕ be a CNF or DNF of pathwidth k. We can compile ϕ into an OBDD
Lower bound: Theorem (This paper) Let ϕ be a monotone CNF or DNF of pathwidth k, and let a := arity(ϕ) and d := degree(ϕ). Then any OBDD for ϕ has width 2
smallest width of an OBDD for ϕ is 2Θ(pathwidth(ϕ))
14/20 Application to provenance
15/20 Provenance: definition
Definition The provenance Prov(q, I) of query q on relational instance I is the Boolean function with facts of I as variables and such that for any valuation ν : I → {0, 1}, Prov(q, I) evaluates to ⊤ under ν iff {F ∈ I|ν(F) = 1} | = q
16/20 Example: Provenance
∃x y z (R(x, y) ∧ S(y, z)) ∨ (S(x, y) ∧ R(y, z)) R b c c a c d S a b d b
17/20 Example: Provenance
∃x y z (R(x, y) ∧ S(y, z)) ∨ (S(x, y) ∧ R(y, z)) R b c c a c d S a b d b b d c a S R R S R
17/20 Example: Provenance
∃x y z (R(x, y) ∧ S(y, z)) ∨ (S(x, y) ∧ R(y, z)) b d c a S R R S R
17/20 Example: Provenance
∃x y z (R(x, y) ∧ S(y, z)) ∨ (S(x, y) ∧ R(y, z)) b d c a S R R S R Prov(q, I) = [S(a, b) ∧ (R(b, c) ∨ R(c, a))]
17/20 Example: Provenance
∃x y z (R(x, y) ∧ S(y, z)) ∨ (S(x, y) ∧ R(y, z)) b d c a S R R S R Prov(q, I) = [S(a, b) ∧ (R(b, c) ∨ R(c, a))] ∨ [S(d, b) ∧ (R(b, c) ∨ R(c, d))]
17/20 Example: Provenance
∃x y z (R(x, y) ∧ S(y, z)) ∨ (S(x, y) ∧ R(y, z)) b d c a S R R S R R(b, c) R(c, a) R(c, d) ∨ ∨ S(a, b) S(d, b) ∧ ∧ ∨ Prov(q, I) = [S(a, b) ∧ (R(b, c) ∨ R(c, a))] ∨ [S(d, b) ∧ (R(b, c) ∨ R(c, d))]
17/20 Treewidth of instances
treewidth of the database
18/20 Treewidth of instances
treewidth of the database
same treewidth as the instance
18/20 Treewidth of instances
treewidth of the database
same treewidth as the instance
the lineage have size exponential in the treewidth of the database!
18/20 Intricate queries
Theorem (This paper) There is a constant d ∈ N such that the following is true. Let σ be an arity-2 signature, and Q a connected UCQ= which is intricate on σ. For any instance I on σ of treewidth k, any d-SDNNF representing the lineage of Q on I has size 2Ω(k1/d)
19/20 Conclusion
restrictions in knowledge compilation:
20/20 Conclusion
restrictions in knowledge compilation:
→ Recapture message passing on bounded treewidth circuits by compiling them to d-SDNNF 20/20 Conclusion
restrictions in knowledge compilation:
→ Recapture message passing on bounded treewidth circuits by compiling them to d-SDNNF → Compilation is singly exponential in the treewidth of the circuit and cannot be avoided 20/20 Conclusion
restrictions in knowledge compilation:
→ Recapture message passing on bounded treewidth circuits by compiling them to d-SDNNF → Compilation is singly exponential in the treewidth of the circuit and cannot be avoided → The width of an OBDD and the pathwidth of a DNF/CNF are within a constant of each other 20/20 Conclusion
restrictions in knowledge compilation:
→ Recapture message passing on bounded treewidth circuits by compiling them to d-SDNNF → Compilation is singly exponential in the treewidth of the circuit and cannot be avoided → The width of an OBDD and the pathwidth of a DNF/CNF are within a constant of each otherThanks for your attention!
20/20