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CNF encodings of DNNFs and BDMCs Petr Kuera 1 Petr Savick 2 1 Charles University, Czech Republic 2 Institute of Computer Science, The Czech Academy of Sciences, Czech Republic KOCOON Workshop, Arras December 1619, 2019 Petr Kuera, Petr

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CNF encodings of DNNFs and BDMCs

Petr Kučera1 Petr Savický2

1Charles University, Czech Republic 2Institute of Computer Science, The Czech Academy of Sciences, Czech Republic

KOCOON Workshop, Arras December 16–19, 2019

Petr Kučera, Petr Savický CNF encodings of DNNFs and BDMCs KOCOON Workshop 2019 1 / 29

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Contents

1 CNF encodings and propagation strength 2 DNNF 3 Known encodings of DNNFs 4 Satisfying subtrees and separators 5 URC and PC encodings of DNNFs 6 Backdoor decomposable monotone circuits 7 Dual rail encoding 8 Encodings of BDMCs 9 Conclusion

Petr Kučera, Petr Savický CNF encodings of DNNFs and BDMCs KOCOON Workshop 2019 2 / 29

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CNF encoding

Consider a Boolean function f (x) on variables x (x1, . . . , xn). A formula ϕ(x, y) is a CNF encoding of a f (x) with auxiliary variables y (y1, . . . , yk) if f (x) (∃y)[ϕ(x, y)]. lit(x) — literals over variables x. For a partial assignment α ⊆ lit(x) and a clause C we denote ϕ ∧ α ⊢1 C the fact that C can be derived by unit propagation from ϕ ∧ α. ⊥ denotes the contradiction (empty clause).

Petr Kučera, Petr Savický CNF encodings of DNNFs and BDMCs KOCOON Workshop 2019 3 / 29

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Consistency

ϕ(x, y) ∧ α | ⊥ ⇔ ϕ(x, y) ∧ α ⊢1 ⊥ (1) CNF encoding ϕ(x, y) of function f (x) … …implements consistency checker by unit propagation (CC) if (1) holds for every α ⊆ lit(x). …is unit refutation complete (URC) if (1) holds for every α ⊆ lit(x ∪ y). URC formulas introduced by del Val (1994). The classifjcation of encodings follows Abío et al. (2016).

Petr Kučera, Petr Savický CNF encodings of DNNFs and BDMCs KOCOON Workshop 2019 4 / 29

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Propagating literals

ϕ(x, y) ∧ α | l ⇔ ϕ(x, y) ∧ α ⊢1 ⊥ or ϕ(x, y) ∧ α ⊢1 l (2) CNF encoding ϕ(x, y) of function f (x) … …implements domain consistency propagator by unit propagation (DC) if (2) holds for every α ⊆ lit(x) and every l ∈ lit(x). …is propagation complete (PC) if (2) holds for every α ⊆ lit(x ∪ y) and every l ∈ lit(x ∪ y). PC formulas introduced by Bordeaux and Marques-Silva (2012).

Petr Kučera, Petr Savický CNF encodings of DNNFs and BDMCs KOCOON Workshop 2019 5 / 29

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Decomposable Negation Normal Form (DNNF)

Negation normal form (NNF) D is a rooted DAG with

nodes V, root ρ ∈ V, edges E directed from root to leaves, inner nodes labeled with ∧ and ∨, leaves labeled with literals in lit(x).

var(v) — variables reachable from node v. Decomposable NNF (DNNF) — for every ∧ node v v1 ∧ · · · ∧ vk and 1 ≤ i < j ≤ k we have var(vi) ∩ var(vj) ∅. smooth DNNF — for every ∨ node v v1 ∨ · · · ∨ vk we have var(v) var(v1) · · · var(vk). DNNFs introduced by Darwiche (1999). Any DNNF can be made smooth in polynomial time (Darwiche, 2001).

Petr Kučera, Petr Savický CNF encodings of DNNFs and BDMCs KOCOON Workshop 2019 6 / 29

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CNF Encodings of DNNFs

CC encoding of a DNNF (Jung et al., 2008). DC encoding of a smooth DNNF (Abío et al., 2016; Jung et al., 2008). URC and PC encoding of a decision diagram (BDD, MDD) (Abío et al., 2016).

  • K. and Savický (2019b) (this talk)

URC and PC encoding of a smooth DNNF.

Petr Kučera, Petr Savický CNF encodings of DNNFs and BDMCs KOCOON Workshop 2019 7 / 29

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CNF Encodings of DNNFs

group clause condition N0 ρ ρ is the root of D N1 v → v1 ∨ · · · ∨ vk v v1 ∨ · · · ∨ vk N2 v → vi v v1 ∧ · · · ∧ vk, i 1, . . . , k N3 v → p1 ∨ · · · ∨ pk v has incoming edges from p1, . . . , pk N4 ¬l No leaf of D is associated with l ∈ lit(x) CC encoding of a DNNF Clauses N0–N2. DC encoding of a smooth DNNF Clauses N0–N4. The DC encoding is not unit refutation complete.

Petr Kučera, Petr Savický CNF encodings of DNNFs and BDMCs KOCOON Workshop 2019 8 / 29

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DC Encoding is not URC

In the smooth DNNF below, c ∧ d is contradictory, since c ≡ x1x2 ∨ x1x2 ≡ (x1 x2) d ≡ x1x2 ∨ x1x2 ≡ (x1 x2)

∨ ρ ∧ a ∨ c ∧ x1 x2 ∧ ¬x1 ¬x2 ∨ e ∧ x3 x4 ∧ ¬x3 ¬x4 ∧ b ∨ d ∧ ∧ ∨ f ∧ ∧

Petr Kučera, Petr Savický CNF encodings of DNNFs and BDMCs KOCOON Workshop 2019 9 / 29

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DC Encoding is not URC

In the smooth DNNF below, c ∧ d is contradictory, since c ≡ x1x2 ∨ x1x2 ≡ (x1 x2) d ≡ x1x2 ∨ x1x2 ≡ (x1 x2)

∨ ρ ∧ a ∨ c ∧ x1 x2 ∧ ¬x1 ¬x2 ∨ e ∧ x3 x4 ∧ ¬x3 ¬x4 ∧ b ∨ d ∧ ∧ ∨ f ∧ ∧

DC encoding contains unit clause ρ

Petr Kučera, Petr Savický CNF encodings of DNNFs and BDMCs KOCOON Workshop 2019 9 / 29

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DC Encoding is not URC

In the smooth DNNF below, c ∧ d is contradictory, since c ≡ x1x2 ∨ x1x2 ≡ (x1 x2) d ≡ x1x2 ∨ x1x2 ≡ (x1 x2)

∨ ρ ∧ a ∨ c ∧ x1 x2 ∧ ¬x1 ¬x2 ∨ e ∧ x3 x4 ∧ ¬x3 ¬x4 ∧ b ∨ d ∧ ∧ ∨ f ∧ ∧

a and b are derived using clauses in group N3: c → a and d → b

Petr Kučera, Petr Savický CNF encodings of DNNFs and BDMCs KOCOON Workshop 2019 9 / 29

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DC Encoding is not URC

In the smooth DNNF below, c ∧ d is contradictory, since c ≡ x1x2 ∨ x1x2 ≡ (x1 x2) d ≡ x1x2 ∨ x1x2 ≡ (x1 x2)

∨ ρ ∧ a ∨ c ∧ x1 x2 ∧ ¬x1 ¬x2 ∨ e ∧ x3 x4 ∧ ¬x3 ¬x4 ∧ b ∨ d ∧ ∧ ∨ f ∧ ∧

e and f are derived using clauses in group N2: a → e and b → f

Petr Kučera, Petr Savický CNF encodings of DNNFs and BDMCs KOCOON Workshop 2019 9 / 29

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DC Encoding is not URC

In the smooth DNNF below, c ∧ d is contradictory, since c ≡ x1x2 ∨ x1x2 ≡ (x1 x2) d ≡ x1x2 ∨ x1x2 ≡ (x1 x2)

∨ ρ ∧ a ∨ c ∧ x1 x2 ∧ ¬x1 ¬x2 ∨ e ∧ x3 x4 ∧ ¬x3 ¬x4 ∧ b ∨ d ∧ ∧ ∨ f ∧ ∧

Unit propagation stops without deriving contradiction

Petr Kučera, Petr Savický CNF encodings of DNNFs and BDMCs KOCOON Workshop 2019 9 / 29

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Minimal satisfying subtrees

A minimal satisfying subtree T of a DNNF D is a rooted tree satisfying: Root ρ belongs to T. v v1 ∧ · · · ∧ vk is in D′ ⇒ all v1, . . . , vk belong to D′. v v1 ∨ · · · ∨ vk is in D′ ⇒ vi ∈ D for exactly one i ∈ {1, . . . , k}.

∨ ρ ∧ ∨ ∧ x1 x2 ∧ ¬x1 ¬x2 ∨ ∧ x3 x4 ∧ ¬x3 ¬x4 ∧ ∨ ∧ ∧ ∨ ∧ ∧

Petr Kučera, Petr Savický CNF encodings of DNNFs and BDMCs KOCOON Workshop 2019 10 / 29

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Properties of Minimal Satisfying Subtrees

Consider a smooth DNNF D representing a function f (x). Assume T is a minimal satisfying subtree of D. For every variable x ∈ x there is exactly one leaf in T labeled with literal l ∈ lit(x). The leaves of T determine a full assignment γT ⊆ lit(x).

Lemma

f (x) is consistent with a partial assignment α ⊆ lit(x), if and only if there is a minimal satisfying subtree T of D such that γT is consistent with α.

Petr Kučera, Petr Savický CNF encodings of DNNFs and BDMCs KOCOON Workshop 2019 11 / 29

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Minimal Satisfying Subtrees and Paths

Consider a smooth DNNF D representing a function f (x). Let Di be a subgraph of D induced by Hi {v | xi ∈ var(v)}. A subgraph T of D is a minimal satisfying subtree of D if and only if T ∩ Di is a path from the root ρ to a leaf for every i 1, . . . , n.

D1 ∨ ρ ∧ ∨ ∧ x1 x2 ∧ ¬x1 ¬x2 ∨ ∧ x3 x4 ∧ ¬x3 ¬x4 ∧ ∨ ∧ ∧ ∨ ∧ ∧

Petr Kučera, Petr Savický CNF encodings of DNNFs and BDMCs KOCOON Workshop 2019 12 / 29

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Covering With Separators

Consider a smooth DNNF D representing a function f (x).

Defjnition

Let Di be a subgraph of D induced by Hi {v | xi ∈ var(v)}. Set S ⊂ Hi is a separator in Di, if every path in Di from the root to a leaf contains precisely one node from S. D can be covered by separators, if for each i 1, . . . , n there is a collection of separators Si of separators in Di such that

  • S∈Si S Hi.

Not every DNNF can be covered by separators. Every DNNF can be modifjed in polynomial time into an equivalent DNNF which can be covered by separators. Separators can be based on levels of nodes, edges going across several levels can be subdivided by a node.

Petr Kučera, Petr Savický CNF encodings of DNNFs and BDMCs KOCOON Workshop 2019 13 / 29

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Covering With Separators

Consider a smooth DNNF D covered by separators Si, i 1, . . . , n. A subgraph T of D is a minimal satisfying subtree of D if and only if T contains exactly one node from each separator in S n

i1 Si. D1 ∨ ρ ∧ ∨ ∧ x1 x2 ∧ ¬x1 ¬x2 ∨ ∧ x3 x4 ∧ ¬x3 ¬x4 ∧ ∨ ∧ ∧ ∨ ∧ ∧

Petr Kučera, Petr Savický CNF encodings of DNNFs and BDMCs KOCOON Workshop 2019 14 / 29

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URC and PC Encodings

group clause condition N0 ρ ρ is the root of D N1 v → v1 ∨ · · · ∨ vk v v1 ∨ · · · ∨ vk N2 v → vi v v1 ∧ · · · ∧ vk, i 1, . . . , k N3 v → p1 ∨ · · · ∨ pk v has incoming edges from p1, . . . , pk N4 ¬l No leaf of D is associated with l ∈ lit(x) N5 at-most-one(S) S ∈ S N6 exactly-one(S) S ∈ S S n

i1 Si is a separator cover of D.

URC encoding of a smooth DNNF Clauses N0–N3, N5. PC encoding of a smooth DNNF Clauses N0–N3, N6.

Petr Kučera, Petr Savický CNF encodings of DNNFs and BDMCs KOCOON Workshop 2019 15 / 29

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Backdoor Decomposable Monotone Circuits

C-BDMC is a DNNF circuit in which the leaves are associated with CNF encodings from a base class C.

∨ ρ ∧ a (x1 ∨ ¬x2)(¬x1 ∨ x2) c (x3 ∨ ¬x4)(¬x3 ∨ x4) e ∧ b (x1 ∨ x2)(¬x1 ∨ ¬x2) d (x3 ∨ x4)(¬x3 ∨ ¬x4) f

ℓ leaves labeled with CNF encodings ϕi(xi, yi) ∈ C, i 1, . . . , ℓ. Decomposability and smoothness defjned with respect to the input variables x ℓ

i1 xi.

Auxiliary variables yi are local to each encoding.

Petr Kučera, Petr Savický CNF encodings of DNNFs and BDMCs KOCOON Workshop 2019 16 / 29

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Related concepts

DNNFs form a special case of C-BDMCs provided C be a base class which contains single literals as encodings. C-backdoor trees (Samer and Szeider, 2008) form a special case

  • f C-BDMCs.

URC encodings and their disjunctions (Bordeaux, Janota, et al., 2012) form a special case of URC-BDMCs.

  • K. and Savický (2019a) (this talk)

C-BDMCs are equally succint with C encodings for C being any of the classes of CC, DC, URC, or PC encodings.

Petr Kučera, Petr Savický CNF encodings of DNNFs and BDMCs KOCOON Workshop 2019 17 / 29

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Suitable base classes

URC encodings

co-NP complete to check if a formula is URC (Čepek, K., and Vlček, 2012; Gwynne and Kullmann, 2013). Renamable Horn formulas are URC. q-Horn formulas (Boros, Crama, and Hammer, 1990) are not URC in general, but have polynomial size URC encodings (K. and Savický, 2020).

PC encodings

co-NP complete to check if a formula is PC (Babka et al., 2013). prime 2-CNFs are always PC.

CC and DC encodings

Equally succint (Bessiere et al., 2009). Recognition complete for the second level of polynomial hierarchy (K. and Savický, unpublished).

Petr Kučera, Petr Savický CNF encodings of DNNFs and BDMCs KOCOON Workshop 2019 18 / 29

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Dual Rail Encoding

Consider a CNF ϕ(x). We associate a meta-variable l with every l ∈ lit(x) ∪ {⊥}. Denote z {l | l ∈ lit(x) ∪ {⊥}}. The dual-rail encoding of ϕ(x) is the formula DR(ϕ, z)

  • C∈ϕ
  • l∈C
  • e∈C\{l}

¬e → l

  • ∧
  • x∈x

(x ∧ ¬x → ⊥). For α ⊆ lit(x), denote α {l | l ∈ α}. For every α ⊆ lit(x) and every l ∈ lit(x) ∪ {⊥} we have ϕ ∧ α ⊢1 l ⇐⇒ DR(ϕ, z) ∧ α ⊢1 l.

Petr Kučera, Petr Savický CNF encodings of DNNFs and BDMCs KOCOON Workshop 2019 19 / 29

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Extended Dual Rail Encoding

DR+(ϕ, z) DR(ϕ, z) ∧

  • l∈lit(x)

(⊥ → l) ∧

  • x∈x

(x ∨ ¬x) . If ϕ is PC, then DR+(ϕ, z) is PC as well. If ϕ is URC, then DR+(ϕ, z) is URC as well, moreover for every α ⊆ lit(z) assuming DR+(ϕ, z) ∧ α ⊥, we have that

If DR+(ϕ, meta(x)) ∧ α 1 ⊥, then DR+(ϕ, meta(x)) ∧ α ∧ ¬⊥ is satisfjable. If DR+(ϕ, meta(x)) ∧ α 1 ¬⊥, then DR+(ϕ, meta(x)) ∧ α ∧ ⊥ is satisfjable.

DR+(ϕ, meta(x)) is complete with respect to deriving literals on

⊥.

Petr Kučera, Petr Savický CNF encodings of DNNFs and BDMCs KOCOON Workshop 2019 20 / 29

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Variables in the Encodings of BDMCs

Assume a BDMC D with nodes V and ℓ leaves associated with formulas ϕ(xi, yi). Variables x — the input variables x n

i1 xi.

Variables zi {li | l ∈ lit(x) ∪ {⊥}} — the meta-variables used in DR+(ϕi, zi). Variables z ℓ

i1 zi.

Variables v — the inner nodes of D (labeled with ∧ or ∨). For a leaf v associated with formula ϕi(xi, yi), we identify v with literal ¬⊥i.

Petr Kučera, Petr Savický CNF encodings of DNNFs and BDMCs KOCOON Workshop 2019 21 / 29

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Clauses in the Encodings of BDMCs

group clause condition E1 DR+(ϕi, zi) i 1, . . . , ℓ E2 l → li l ∈ lit(xi), i 1, . . . , ℓ E3

  • i:l∈lit(xi)li
  • → l

l ∈ lit(x) CC encoding of a CC-BDMC CC encoding of the circuit part (N0–N2), clauses E1, E2. DC encoding of a smooth DC-BDMC DC encoding of the circuit part (N0–N3), clauses E1–E3. URC encoding of a smooth URC-BDMC covered by separators URC encoding of the circuit part (N0–N3, N5), clauses E1, E2. PC encoding of a smooth PC-BDMC covered by separators PC encoding of the circuit part (N0–N3, N6), clauses E1–E3.

Petr Kučera, Petr Savický CNF encodings of DNNFs and BDMCs KOCOON Workshop 2019 22 / 29

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Conclusion

Main results: Polynomial construction of URC and PC encodings of smooth DNNFs. Polynomial construction of CC, DC, URC, and PC encodings of BDMCs with the respective base classes. C-BDMCs are equally succint with C encodings for C being any of the classes of CC, DC, URC, or PC encodings.

Petr Kučera, Petr Savický CNF encodings of DNNFs and BDMCs KOCOON Workshop 2019 23 / 29

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Encoding Cardinality Constraints

We use cardinality constraints at-most-one(S) and exactly-one(S)

  • n a given set of variables S.

As a default we use canonical representations which use Θ(|S|2) clauses. We show that using sequential (or ladder) encoding of the at-most-one(S) (which has linear number of clauses) preserves unit refutation completeness. Conjunction of the sequential encoding of at-most-one(S) with a single at-least-one(S) clause is not a PC encoding of exactly-one(S). We propose another linear size encoding of the exactly-one(S) which is PC and using it preserves propagation completeness.

Petr Kučera, Petr Savický CNF encodings of DNNFs and BDMCs KOCOON Workshop 2019 24 / 29

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References I

Abío, Ignasi et al. (2016). “On CNF Encodings of Decision Diagrams”. In: Integration of AI and OR Techniques in Constraint Programming.

  • Ed. by Claude-Guy Quimper. Cham: Springer International

Publishing, pp. 1–17. isbn: 978-3-319-33954-2. Babka, Martin et al. (2013). “Complexity issues related to propagation completeness”. In: Artifjcial Intelligence 203.0, pp. 19–34. issn: 0004-3702. doi: http://dx.doi.org/10.1016/j.artint.2013.07.006. Bessiere, Christian et al. (2009). “Circuit Complexity and Decompositions of Global Constraints”. In: Proceedings of the Twenty-First International Joint Conference on Artifjcial Intelligence (IJCAI-09), pp. 412–418.

Petr Kučera, Petr Savický CNF encodings of DNNFs and BDMCs KOCOON Workshop 2019 25 / 29

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References II

Bordeaux, Lucas, Mikoláš Janota, et al. (2012). “On Unit-refutation Complete Formulae with Existentially Quantifjed Variables”. In: Proceedings of the Thirteenth International Conference on Principles

  • f Knowledge Representation and Reasoning. KR’12. Rome, Italy:

AAAI Press, pp. 75–84. isbn: 978-1-57735-560-1. Bordeaux, Lucas and Joao Marques-Silva (2012). “Knowledge Compilation with Empowerment”. In: SOFSEM 2012: Theory and Practice of Computer Science. Ed. by Mária Bieliková et al.

  • Vol. 7147. Lecture Notes in Computer Science. Springer Berlin /

Heidelberg, pp. 612–624. isbn: 978-3-642-27659-0. Boros, E., Y. Crama, and P. L. Hammer (Sept. 1990). “Polynomial-time inference of all valid implications for Horn and related formulae”. In: Annals of Mathematics and Artifjcial Intelligence 1.1, pp. 21–32.

Petr Kučera, Petr Savický CNF encodings of DNNFs and BDMCs KOCOON Workshop 2019 26 / 29

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References III

Čepek, Ondřej, K., and Václav Vlček (2012). “Properties of SLUR Formulae”. In: SOFSEM 2012: Theory and Practice of Computer

  • Science. Ed. by Mária Bieliková et al. Vol. 7147. LNCS. Springer

Berlin Heidelberg, pp. 177–189. isbn: 978-3-642-27659-0. doi: 10.1007/978-3-642-27660-6{\_}15. Darwiche, Adnan (1999). “Compiling Knowledge into Decomposable Negation Normal Form”. In: Proceedings of the 16th International Joint Conference on Artifjcal Intelligence - Volume 1. IJCAI’99. San Francisco, CA, USA: Morgan Kaufmann Publishers Inc.,

  • pp. 284–289.

– (2001). “On the Tractable Counting of Theory Models and its Application to Truth Maintenance and Belief Revision”. In: Journal of Applied Non-Classical Logics 11.1-2, pp. 11–34. doi: 10.3166/jancl.11.11-34.

Petr Kučera, Petr Savický CNF encodings of DNNFs and BDMCs KOCOON Workshop 2019 27 / 29

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References IV

del Val, Alvaro (1994). “Tractable Databases: How to Make Propositional Unit Resolution Complete through Compilation”. In: Knowledge Representation and Reasoning, pp. 551–561. Gwynne, Matthew and Oliver Kullmann (2013). “Generalising and Unifying SLUR and Unit-Refutation Completeness”. In: SOFSEM 2013: Theory and Practice of Computer Science. Ed. by Peter van Emde Boas et al. Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 220–232. isbn: 978-3-642-35843-2. Jung, Jean C. et al. (2008). “Two Encodings of DNNF Theories”. In: ECAI’08 Workshop on Inference methods based on Graphical Structures of Knowledge.

  • K. and Petr Savický (2019a). “Backdoor Decomposable Monotone

Circuits and their Propagation Complete Encodings”. In: CoRR abs/1811.09435.

Petr Kučera, Petr Savický CNF encodings of DNNFs and BDMCs KOCOON Workshop 2019 28 / 29

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References V

  • K. and Petr Savický (2019b). “Propagation complete encodings of

smooth DNNF theories”. In: arXiv preprint arXiv:1909.06673. – (2020). “On the size of CNF formulas with high propagation strength”. To appear at ISAIM 2020. Samer, Marko and Stefan Szeider (2008). “Backdoor Trees”. In: Proceedings of the Twenty-Third AAAI Conference on Artifjcial Intelligence, AAAI 2008, Chicago, Illinois, USA, July 13-17, 2008.

  • Ed. by Dieter Fox and Carla P. Gomes. AAAI Press, pp. 363–368.

isbn: 978-1-57735-368-3.

Petr Kučera, Petr Savický CNF encodings of DNNFs and BDMCs KOCOON Workshop 2019 29 / 29