CNF encodings of DNNFs and BDMCs Petr Kuera 1 Petr Savick 2 1 Charles - PowerPoint PPT Presentation

CNF encodings of DNNFs and BDMCs Petr Kučera 1 Petr Savický 2 1 Charles University, Czech Republic 2 Institute 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

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

Petr Kučera, Petr Savický CNF encoding CNF encodings of DNNFs and BDMCs KOCOON Workshop 2019 3 / 29 Consider a Boolean function f ( x ) on variables x � ( x 1 , . . . , x n ) . A formula ϕ ( x , y ) is a CNF encoding of a f ( x ) with auxiliary variables y � ( y 1 , . . . , y k ) 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).

Consistency (1) …implements consistency checker by unit propagation (CC) …is unit refutation complete (URC) 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 ϕ ( x , y ) ∧ α | � ⊥ ⇔ ϕ ( x , y ) ∧ α ⊢ 1 ⊥ CNF encoding ϕ ( x , y ) of function f ( x ) … if (1) holds for every α ⊆ lit ( x ) . if (1) holds for every α ⊆ lit ( x ∪ y ) .

Propagating literals (2) …implements domain consistency propagator by unit propagation (DC) …is propagation complete (PC) PC formulas introduced by Bordeaux and Marques-Silva (2012). Petr Kučera, Petr Savický CNF encodings of DNNFs and BDMCs KOCOON Workshop 2019 5 / 29 ϕ ( x , y ) ∧ α | � l ⇔ ϕ ( x , y ) ∧ α ⊢ 1 ⊥ or ϕ ( x , y ) ∧ α ⊢ 1 l CNF encoding ϕ ( x , y ) of function f ( x ) … if (2) holds for every α ⊆ lit ( x ) and every l ∈ lit ( x ) . if (2) holds for every α ⊆ lit ( x ∪ y ) and every l ∈ lit ( x ∪ y ) .

Decomposable Negation Normal Form (DNNF) DNNFs introduced by Darwiche (1999). KOCOON Workshop 2019 CNF encodings of DNNFs and BDMCs Petr Kučera, Petr Savický 2001). Any DNNF can be made smooth in polynomial time (Darwiche, 6 / 29 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 � v 1 ∧ · · · ∧ v k and 1 ≤ i < j ≤ k we have var ( v i ) ∩ var ( v j ) � ∅ . smooth DNNF — for every ∨ node v � v 1 ∨ · · · ∨ v k we have var ( v ) � var ( v 1 ) � · · · � var ( v k ) .

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

CNF Encodings of DNNFs group KOCOON Workshop 2019 CNF encodings of DNNFs and BDMCs Petr Kučera, Petr Savický The DC encoding is not unit refutation complete. DC encoding of a smooth DNNF Clauses N0–N4. CC encoding of a DNNF Clauses N0–N2. N4 N3 N2 N1 N0 condition clause 8 / 29 ρ is the root of D ρ v → v 1 ∨ · · · ∨ v k v � v 1 ∨ · · · ∨ v k v → v i v � v 1 ∧ · · · ∧ v k , i � 1 , . . . , k v → p 1 ∨ · · · ∨ p k v has incoming edges from p 1 , . . . , p k ¬ l No leaf of D is associated with l ∈ lit ( x )

DC Encoding is not URC Petr Kučera, Petr Savický KOCOON Workshop 2019 CNF encodings of DNNFs and BDMCs 9 / 29 In the smooth DNNF below, c ∧ d is contradictory, since c ≡ x 1 x 2 ∨ x 1 x 2 ≡ ( x 1 � x 2 ) d ≡ x 1 x 2 ∨ x 1 x 2 ≡ ( x 1 � x 2 ) ρ ∨ a b ∧ ∧ f c d e ∨ ∨ ∨ ∨ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ x 1 x 2 ¬ x 1 ¬ x 2 x 3 x 4 ¬ x 3 ¬ x 4

DC Encoding is not URC DC encoding contains KOCOON Workshop 2019 CNF encodings of DNNFs and BDMCs Petr Kučera, Petr Savický 9 / 29 In the smooth DNNF below, c ∧ d is contradictory, since c ≡ x 1 x 2 ∨ x 1 x 2 ≡ ( x 1 � x 2 ) d ≡ x 1 x 2 ∨ x 1 x 2 ≡ ( x 1 � x 2 ) ρ ∨ unit clause ρ a b ∧ ∧ f c d e ∨ ∨ ∨ ∨ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ x 1 x 2 ¬ x 1 ¬ x 2 x 3 x 4 ¬ x 3 ¬ x 4

DC Encoding is not URC using clauses in group N3: KOCOON Workshop 2019 CNF encodings of DNNFs and BDMCs Petr Kučera, Petr Savický 9 / 29 In the smooth DNNF below, c ∧ d is contradictory, since c ≡ x 1 x 2 ∨ x 1 x 2 ≡ ( x 1 � x 2 ) d ≡ x 1 x 2 ∨ x 1 x 2 ≡ ( x 1 � x 2 ) ρ ∨ a and b are derived a b c → a and d → b ∧ ∧ f c d e ∨ ∨ ∨ ∨ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ x 1 x 2 ¬ x 1 ¬ x 2 x 3 x 4 ¬ x 3 ¬ x 4

DC Encoding is not URC using clauses in group N2: KOCOON Workshop 2019 CNF encodings of DNNFs and BDMCs Petr Kučera, Petr Savický 9 / 29 In the smooth DNNF below, c ∧ d is contradictory, since c ≡ x 1 x 2 ∨ x 1 x 2 ≡ ( x 1 � x 2 ) d ≡ x 1 x 2 ∨ x 1 x 2 ≡ ( x 1 � x 2 ) ρ ∨ a b ∧ ∧ e and f are derived f c d e a → e and b → f ∨ ∨ ∨ ∨ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ x 1 x 2 ¬ x 1 ¬ x 2 x 3 x 4 ¬ x 3 ¬ x 4

DC Encoding is not URC Unit propagation stops KOCOON Workshop 2019 CNF encodings of DNNFs and BDMCs Petr Kučera, Petr Savický without deriving contradiction 9 / 29 In the smooth DNNF below, c ∧ d is contradictory, since c ≡ x 1 x 2 ∨ x 1 x 2 ≡ ( x 1 � x 2 ) d ≡ x 1 x 2 ∨ x 1 x 2 ≡ ( x 1 � x 2 ) ρ ∨ a b ∧ ∧ f c d e ∨ ∨ ∨ ∨ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ x 1 x 2 ¬ x 1 ¬ x 2 x 3 x 4 ¬ x 3 ¬ x 4

Minimal satisfying subtrees Petr Kučera, Petr Savický KOCOON Workshop 2019 CNF encodings of DNNFs and BDMCs 10 / 29 A minimal satisfying subtree T of a DNNF D is a rooted tree satisfying: Root ρ belongs to T . v � v 1 ∧ · · · ∧ v k is in D ′ ⇒ all v 1 , . . . , v k belong to D ′ . v � v 1 ∨ · · · ∨ v k is in D ′ ⇒ v i ∈ D for exactly one i ∈ { 1 , . . . , k } . ρ ∨ ∧ ∧ ∨ ∨ ∨ ∨ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ x 1 x 2 ¬ x 1 ¬ x 2 x 3 x 4 ¬ x 3 ¬ x 4

Lemma Properties of Minimal Satisfying Subtrees Petr Kučera, Petr Savický CNF encodings of DNNFs and BDMCs KOCOON Workshop 2019 11 / 29 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 γ T ⊆ lit ( x ) . 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 γ T is consistent with α .

Minimal Satisfying Subtrees and Paths Petr Kučera, Petr Savický KOCOON Workshop 2019 CNF encodings of DNNFs and BDMCs 12 / 29 Consider a smooth DNNF D representing a function f ( x ) . Let D i be a subgraph of D induced by H i � { v | x i ∈ var ( v )} . A subgraph T of D is a minimal satisfying subtree of D if and only if T ∩ D i is a path from the root ρ to a leaf for every i � 1 , . . . , n . ρ D 1 ∨ ∧ ∧ ∨ ∨ ∨ ∨ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ x 1 x 2 ¬ x 1 ¬ x 2 x 3 x 4 ¬ x 3 ¬ x 4

Covering With Separators Not every DNNF can be covered by separators. KOCOON Workshop 2019 CNF encodings of DNNFs and BDMCs Petr Kučera, Petr Savický several levels can be subdivided by a node. Separators can be based on levels of nodes, edges going across DNNF which can be covered by separators. Every DNNF can be modifjed in polynomial time into an equivalent 13 / 29 Defjnition Consider a smooth DNNF D representing a function f ( x ) . Let D i be a subgraph of D induced by H i � { v | x i ∈ var ( v )} . Set S ⊂ H i is a separator in D i , if every path in D i 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 S i of separators in D i such that S ∈S i S � H i .

Covering With Separators Petr Kučera, Petr Savický KOCOON Workshop 2019 CNF encodings of DNNFs and BDMCs 14 / 29 Consider a smooth DNNF D covered by separators S i , i � 1 , . . . , n . if T contains exactly one node from each separator in S � � n A subgraph T of D is a minimal satisfying subtree of D if and only i � 1 S i . ρ D 1 ∨ ∧ ∧ ∨ ∨ ∨ ∨ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ x 1 x 2 ¬ x 1 ¬ x 2 x 3 x 4 ¬ x 3 ¬ x 4