Updating the Knowledge Compilation Map Simone Bova (TU Wien) - - PowerPoint PPT Presentation

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Updating the Knowledge Compilation Map

Simone Bova (TU Wien)

Dagstuhl Seminar on Recent Trends in Knowledge Compilation September 17–22 , 2017, Dagstuhl (Germany)

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Outline

Knowledge Compilation Map Updates Extensions

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Outline

Knowledge Compilation Map Updates Extensions

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Knowledge Compilation

Represent knowledge so to facilitate reasoning.

Example Represent DNFs by OBDDs to count models. (x ∧ y) ∨ (x ∧ z) ∨ (w ∧ y) ∨ (w ∧ z) ≡

⊥ ⊤ z x y w

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Knowledge Compilation

Represent knowledge so to facilitate reasoning.

Example Represent DNFs by OBDDs to count models. (x ∧ y) ∨ (x ∧ z) ∨ (w ∧ y) ∨ (w ∧ z) ≡

⊥ ⊤ z x y w

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Knowledge Compilation

Represent knowledge so to facilitate reasoning.

Example Represent DNFs by OBDDs to count models. (x ∧ y) ∨ (x ∧ z) ∨ (w ∧ y) ∨ (w ∧ z) ≡

16 12 = (8 + 16)/2 9 = (6 + 12)/2

⊥ ⊤ z x y w

8 = (0 + 16)/2 6 = (0 + 12)/2

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Knowledge Compilation

Represent propositional knowledge so to facilitate reasoning.

Find a class R of circuits expressing a class of Boolean functions

• Represent propositional knowledge. . .

such that certain logical tasks on the circuits in R are computationally feasible.

• . . . so to facilitate reasoning.

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Logical Tasks on Circuits

Given circuits C and C′ in a circuit class R. Counting: Count the models of C. Entailment: Decide if C entails C′. . . . . . . Negation: Find a circuit in R computing ¬C. Conjunction: Find a circuit in R computing C ∧ C′. Disjunction: Find a circuit in R computing C ∨ C′. . . . . . . A task is computationally feasible on R if it is polytime tractable wrt the input size ie, wrt the size of the circuits in R.

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Succinctness vs Tractability

Tradeoff succinctness/tractability in choosing a representation language: Truthtables: Tractable, but useless because too verbose. Circuits: Succinct, but useless because too hard. Darwiche and Marquis (2002) systematically investigate a hierarchy of representation languages wrt their succinctness/tractability tradeoffs:

• A. Darwiche and P. Marquis.

A Knowledge Compilation Map.

• J. Artif. Intell. Res., 17, 229-264, 2002.

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Representation Languages

NNF CNF PI FBDD IP DNNF DNF dDNNF OBDD MODS

Figure: Inclusions.

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Representation Languages

Negation Normal Forms (NNF) Boolean circuits having unbounded fanin AND and OR gates with negations pushed to the input gates. Decomposable NNFs (DNNF) NNFs where each AND gate has subcircuits using disjoint sets of variables. Deterministic DNNFs (dDNNF) DNNFs where each OR gate has pairwise inconsistent subcircuits. Prime Implicate Forms (PI) CNFs where entailed clauses are already entailed by a single clause in the CNF and no clause in the CNF is entailed by another. Models, MODS DNFs where each disjunct uses the same variables. Free Binary Decision Diagrams, FBDDs: Read-once branching programs.

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Determinism and Decomposability

x3 x4 ⊥ x1 x2 ⊥ x1 x4 ∨ ∨ ∨ ∨ ⊤ x3 x1 x2 x1 ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Counting via Determinism and Decomposability

x3 x4 ⊥ x1 x2 ⊥ x1 x4 + + + ⊤ x3 x1 x2 x1 + × × × × × × × ×

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Representation Languages

Negation Normal Forms (NNF) Boolean circuits having unbounded fanin AND and OR gates with negations pushed to the input gates. Decomposable NNFs (DNNF) NNFs where each AND gate has subcircuits using disjoint sets of variables. Deterministic DNNFs (dDNNF) DNNFs where each OR gate has pairwise inconsistent subcircuits. Prime Implicate Forms (PI) CNFs where entailed clauses are already entailed by a single clause in the CNF and no clause in the CNF is entailed by another. Models, MODS DNFs where each disjunct uses the same variables. Free Binary Decision Diagrams, FBDDs: Deterministic read-once branching programs.

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Succinctness Relation

Let S, T ⊆ NNF. S is (polysize) compilable into T (or T is at least as succinct as S) if there exists a polynomial p: N → N such that for all C ∈ S there exists D ∈ T equivalent to C such that size(D) ≤ p(size(C)). Write S T if S is compilable into T, and S T otherwise.

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Knowledge Compilation in Practice

Want to count models of circuit C (offline) modulo a sequence of partial assignments g1, g2, g3, . . . (online): C|g1, C|g2, C|g3, . . . Ow, lookup in the KC map a maximally succinct language supporting counting mod assignments. Represent C by a form C∗ in the chosen language: C∗, C∗|g1, C∗|g2, C∗|g3, . . . High compilation cost is eventually amortized by low querying costs: t0 t1 · · · tj · · · · · · · · · ti · · · ti+1 · · · C|g1 · · · C|gj · · · · · · · · · C|gi · · · C|gi+1 · · · · · · · · · C∗ C∗|g1 C∗|g2 · · · C∗|gi C∗|gi+1 C∗|gi+2

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Outline

Knowledge Compilation Map Updates Extensions

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Succinctness Relation

Darwiche and Marquis (2002) summarize the status of the succinctness relation also appealing to previous work including

• Quine (1959),
• Chandra and Markowsky (1978),
• Bryant (1986),
• Wegener (1987),
• Gergov and Meinel (1994),
• Gogic, Kautz, Papadimitriou, and Selman (1995),
• Selman and Kautz (1996),
• Cadoli and Donini (1997), and
• Darwiche (1999).

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Succinctness Relation

MODS DNNF OBDD dDNNF FBDD DNF CNF NNF IP PI

Let S and T be languages in the diagram. If (S, T) in transitive closure of →, then S T. Else:

• If S T, then S

?

T (unknown).

• Else S T.

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Succinctness Relation

MODS DNNF OBDD dDNNF FBDD DNF CNF NNF IP PI MODS DNNF OBDD dDNNF FBDD DNF CNF NNF IP PI

Figure: S T means S T unknown. S → T means S T unless PH collapses.

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Succinctness Relation | 2002–2016

MODS DNNF OBDD dDNNF FBDD DNF CNF NNF IP PI

Let S and T be languages in the diagram. If (S, T) in transitive closure of →, then S T. Else:

• If S T, then S

?

T.

• If S → T, then S T unless PH collapses.
• Else S T.

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Succinctness | Updates | 1 | PI DNNF

MODS DNNF OBDD dDNNF FBDD DNF CNF NNF IP PI MODS DNNF OBDD dDNNF FBDD DNF CNF NNF IP PI

Figure: S → T means S T.

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Succinctness | Updates | 2 | DNNF dDNNF

MODS DNNF OBDD dDNNF FBDD DNF CNF NNF IP PI MODS DNNF OBDD dDNNF FBDD DNF CNF NNF IP PI

Figure: S → T means S T.

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Succinctness | Updates

MODS DNNF OBDD dDNNF FBDD DNF CNF NNF IP PI MODS DNNF OBDD dDNNF FBDD DNF CNF NNF IP PI

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Knowledge Compilation Meets Communication Complexity

Theorem (B, Capelli, Mengel, and Slivovsky) Let D be a DNNF (resp, dDNNF) computing a function F. The size of D is an upper bound on the size of a smallest rectangle cover (resp, disjoint rectange cover) of F.∗ A Z-rectangle is a function R(Z) of the form R ≡ S(X) ∧ T(Y) for functions S and T with X and Y forming a partition of Z. {Ri}i∈I is a rectangle cover of a Boolean function F(Z) if each Ri is a Z-rectangle and F ≡

i∈I Ri.

A rectangle cover {Ri}i∈I is disjoint if Ri ∧ Rj ≡ ⊥ (i = j ∈ I).

∗Balanced.

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Rectangle | R(x, y, z, w) ≡ S(x, y) ∧

× T(z, w)

R(x, y, z, w) ¯ x¯ y¯ z¯ w ¯ x¯ y¯ zw ¯ x¯ yz¯ w ¯ x¯ yzw ¯ xy¯ z¯ w ¯ xy¯ zw 1 ¯ xyz¯ w 1 ¯ xyzw x¯ y¯ z¯ w x¯ y¯ zw 1 x¯ yz¯ w 1 x¯ yzw xy¯ z¯ w xy¯ zw xyz¯ w xyzw ≡ ¯ z¯ w ¯ zw z¯ w zw ¯ x¯ y ¯ xy 1 1 x¯ y 1 1 xy

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Rectangle Cover | F ≡ R1 ∨ R2 ∨ R3

F(x, y, z, w) = ¯ z¯ w ¯ zw z¯ w zw ¯ x¯ y 1 1 ¯ xy 1 1 1 x¯ y 1 1 1 xy 1 1 ≡ R1 = ¯ z¯ w ¯ zw z¯ w zw ¯ x¯ y 1 1 ¯ xy 1 1 x¯ y xy ∨ R2 = ¯ z¯ w ¯ zw z¯ w zw ¯ x¯ y ¯ xy 1 1 x¯ y 1 1 xy ∨ R3 = ¯ z¯ w ¯ zw z¯ w zw ¯ x¯ y ¯ xy x¯ y 1 1 xy 1 1

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Disjoint Rectangle Cover | F ≡ R1

+

∨ R′

2 +

∨ R′′

2 +

∨ R3

F(x, y, z, w) = ¯ z¯ w ¯ zw z¯ w zw ¯ x¯ y 1 1 ¯ xy 1 1 1 x¯ y 1 1 1 xy 1 1 ≡ R1 = ¯ z¯ w ¯ zw z¯ w zw ¯ x¯ y 1 1 ¯ xy 1 1 x¯ y xy

+

∨ R′

2

= ¯ z¯ w ¯ zw z¯ w zw ¯ x¯ y ¯ xy 1 x¯ y xy

+

∨ R′′

2

= ¯ z¯ w ¯ zw z¯ w zw ¯ x¯ y ¯ xy x¯ y 1 xy

+

∨ R3 = ¯ z¯ w ¯ zw z¯ w zw ¯ x¯ y ¯ xy x¯ y 1 1 xy 1 1

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Knowledge Compilation Meets Communication Complexity

Theorem (B, Capelli, Mengel, and Slivovsky) Let D be a DNNF (resp, dDNNF) computing a function F. The size of D is an upper bound on the size of a smallest rectangle cover (resp, disjoint rectange cover) of F.† Details? Ask 2F + 2S.

†Balanced.

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Succinctness | Updates | 1 | PI DNNF

Fn =

xy∈En(x ∨ y) where En are edges of size n member of expander family.

Theorem (Jukna and Schnitger; B, Capelli, Mengel, and Slivovsky) A rectangle cover of the function Fn has size at least 2Ω(n).‡ Corollary The DNNF size of Fn is 2Ω(n). Proposition The PI size of Fn is O(n2). Corollary PI DNNF.

‡Balanced.

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Succinctness | Updates | 2 | DNNF dDNNF

Sn(X) = Rn(X) ∨ Cn(X) where Rn (resp, Cn) is the sum modulo 2 of the sums modulo 3 of the rows (resp, columns) of X = (xij)i,j∈[n]. Theorem (Sauerhoff) A disjoint rectangle cover of the function Sn has size 2Ω(n).§ Corollary The dDNNF size of Sn is 2Ω(n). Proposition The DNNF size of Sn is O(n2). Corollary DNNF dDNNF.

§Balanced.

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Succinctness | Updates

MODS DNNF OBDD dDNNF FBDD DNF CNF NNF IP PI MODS DNNF OBDD dDNNF FBDD DNF CNF NNF IP PI

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Transformations | 2002–2016

∃X ∃x ∧c ∧n ∨c ∨n ¬ DNNF Yes Yes No No Yes Yes No dDNNF No No No No No No ? FBDD No No No No No No Yes OBDD No Yes No No No No Yes Reads: ? — Unknown. No — Does not support unless P = NP. No — Does not support. Yes — Supports.

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Transformations | Updates

∃X ∃x ∧c ∧n ∨c ∨n ¬ DNNF Yes Yes No No Yes Yes No dDNNF No No No No No No ? FBDD No No No No No No Yes OBDD No Yes No No No No Yes Reads: ? — Unknown. No — Does not support unless P = NP. No — Does not support. Yes — Supports.

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Transformations | Updates | 1

∃X ∃x ∧c ∧n ∨c ∨n ¬ DNNF Yes Yes No No Yes Yes No dDNNF No No No No No No ? FBDD No No No No No No Yes OBDD No Yes No No No No Yes S∗

n = (x ∧

Rn

• small OBDD

) ∨ (¬x ∧ Cn

• small OBDD

)

• small FBDD

∃xS∗

n ≡

Sn

• large dDNNF [BCMS16]

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Transformations | Updates | 2

∃X ∃x ∧c ∧n ∨c ∨n ¬ DNNF Yes Yes No No Yes Yes No dDNNF No No No No No No ? FBDD No No No No No No Yes OBDD No Yes No No No No Yes G = (V, E) expander graph of degree 3. G is c-edge-colorable for c ∈ {3, 4} by Vizing theorem, ie, E = c

i=1 Ei such that {x, y} ∩ {z, w} = ∅ for all {xy, zw} ⊆ Ei.

Then: F =

c

• i=1
• xy∈Ei

(x ∨ y)

• small OBDD
• large DNNF [BCMS16]

Still No (conditional) for ∧2 and ∧3.

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Transformations | Updates | 3

∃X ∃x ∧c ∧n ∨c ∨n ¬ DNNF Yes Yes No No Yes Yes No dDNNF No No No No No No ? FBDD No No No No No No Yes OBDD No Yes No No No No Yes Sn = Rn

• small OBDD

∨ Cn

• small OBDD
• large dDNNF [BCMS16]

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Transformations | Updates | 4

∃X ∃x ∧c ∧n ∨c ∨n ¬ DNNF Yes Yes No No Yes Yes No dDNNF No No No No No No ? FBDD No No No No No No Yes OBDD No Yes No No No No Yes F

• CNF with large DNNF [BCMS16]

≡ ¬ ¬F

• small DNF hence small DNNF

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Transformations | Questions

∃X ∃x ∧c ∧n ∨c ∨n ¬ DNNF Yes Yes No No Yes Yes No dDNNF No No No No No No ? FBDD No No No No No No Yes OBDD No Yes No No No No Yes Question Does dDNNF have polysize (resp, polytime) negation? If not, then dDNNF and dDNNF¬ are separated.

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Understanding (Semantic) Determinism

Let F(X) = 1 iff the n × n Boolean matrix X has a full row or a full column.¶ Both F and ¬F have large decDNNF size (since F has large FBDD size). F/dDNNF ¬F/dDNNF ¬/dDNNF IP/dDNNF DNF/dDNNF small small ?

• →

small large No

• →

large small No

• large

large ?

• ¶Lineage of ∃x∀yE(x, y) ∨ ∃y∀xE(x, y) over Kn.

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Queries | 2002-2017

≡ dDNNF ? FBDD ? Observation (Wegener) Equivalence of FBDDs is not coNP-hard unless NP=RP.

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Outline

Knowledge Compilation Map Updates Extensions

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Traces of Counting Algorithms

Two techniques for (CNF) model counting:

• exhaustive DPLL with caching and components (practical);
• dynamic-programming based on syntactic structure (theoretical).

Traces of such counting algorithms are respectively: Decision DNNFs (decDNNF) FBDDs with decomposable ANDs; Structured dDNNFs (sdDNNF) dDNNFs with ANDs “respecting a vtree”. Lower bounds on traces’ size imply lower bounds on algorithms’ runtime.

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Structured Determinism

A structured deterministic NNF. . . . . . with its variable tree. x3 x4 ⊥ x1 x2 ⊥ x1 x4 ∨ ∨ ∨ ∨ ⊤ x3 x1 x2 x1 ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ x1 x2 x3 x4

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Traces of Counting Algorithms

SDD FBDD OBDD sdDNNF decDNNF

Figure: Inclusions.

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Sentential Decision Diagrams (SDD)

An SDD respecting vtree T with left/right subtrees Tl(X) and Tr(Y) has form

m

• i=1

(Pi ∧ Si) where:

• the Pi’s are SDDs respecting Tl(X) and distinguishing cases on X;
• the Si’s are SDDs respecting Tr(Y).

Literals are SDDs respecting any vtree containing their variable. ⊥ and ⊤ are SDDs respecting any vtree.

Exhaustively and disjointly.

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Counting Techniques | Separations

SDD FBDD decDNNF OBDD sdDNNF

Let S and T be languages in the diagram. If (S, T) in transitive closure of →, then S T. Else:

• If S T, then S

?

T.

• Else S T.

Unstructured/decision simulates structured/determinism?

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Literature

• P. Beame, J. Li, S. Roy, and D. Suciu.

Lower Bounds for Exact Model Counting and Applications in Probabilistic Databases. In Proc. of UAI 2013, pp. 52–61. AUAI, 2013.

• S. Bova.

SDDs are Exponentially More Succinct than OBDDs. In Proc. of AAAI 2016, pp. 929–935. AAAI, 2016.

• S. Bova, F. Capelli, S. Mengel, and F. Slivovsky.

On Compiling CNFs into Structured Deterministic DNNFs. In Proc. of SAT 2015, pp. 199–214. Springer, 2015.

• S. Bova, F. Capelli, S. Mengel, and F. Slivovsky.

Knowledge Compilation Meets Communication Complexity. In Proc. of IJCAI 2016, pp. 1008–1014. IJCAI/AAAI, 2016.

• F. Capelli.

Understanding the Complexity of ♯SAT Using Knowledge Compilation. In Proc. of LICS 2017, pp. 1–10. LICS, 2017.

• A. Darwiche and P. Marquis.

A Knowledge Compilation Map.

• J. Artif. Intell. Res., 17, 229-264, 2002.

KNOWLEDGE COMPILATION MAP UPDATES EXTENSIONS

Thank you for your attention!