Compilation of CNF-formulas: upper and lower bounds Florent Capelli - PowerPoint PPT Presentation

Compilation of CNF-formulas: upper and lower bounds Florent Capelli joint work with Simone Bova, Stefan Mengel, Friedrich Slivovsky Birkbeck College, University of London Dagstuhl Seminar: “SAT and Interaction” September 22, 2016. Florent Capelli joint work with Simone Bova, Stefan Mengel, Friedrich Slivovsky (Birkbeck College, University of London) Compilation of CNF-formulas: upper and lower bounds 1 / 23

Outline of this talk Focus on compilation of CNF-formulas into DNNF Lower bounds : does not work in general: Theorem If all CNF-formulas can be compiled into polynomial size DNNF then PH collapses. Conditional result: no information on which CNF are hard In this talk: unconditional exponential lower bounds Connection with communication complexity Upper bounds : Algorithms based on structural restrictions of the input Generalize and unify several approaches used for #SAT Florent Capelli joint work with Simone Bova, Stefan Mengel, Friedrich Slivovsky (Birkbeck College, University of London) Compilation of CNF-formulas: upper and lower bounds 2 / 23

Previously in Stefan’s talk... DNNF are a restricted form of boolean circuits: input are literals ∨ and ∧ gates (no internal negation!) ∧ -gate are decomposable : input subcircuits have disjoint variables Florent Capelli joint work with Simone Bova, Stefan Mengel, Friedrich Slivovsky (Birkbeck College, University of London) Compilation of CNF-formulas: upper and lower bounds 3 / 23

Previously in Stefan’s talk... DNNF are a restricted form of boolean circuits: input are literals ∨ and ∧ gates (no internal negation!) ∧ -gate are decomposable : input subcircuits have disjoint variables More restrictive conditions: deterministic DNNF : ∨ -gates verify α ∨ β such that α ∧ β ≡ ⊥ decision DNNF : ∨ -gates are of the form ( x ∧ α ) ∨ ( ¬ x ∧ β ) structured DNNF : analog of OBDD for DNNF. The “order” is now a tree. Florent Capelli joint work with Simone Bova, Stefan Mengel, Friedrich Slivovsky (Birkbeck College, University of London) Compilation of CNF-formulas: upper and lower bounds 3 / 23

Previously in Stefan’s talk... DNNF are a restricted form of boolean circuits: input are literals ∨ and ∧ gates (no internal negation!) ∧ -gate are decomposable : input subcircuits have disjoint variables More restrictive conditions: deterministic DNNF : ∨ -gates verify α ∨ β such that α ∧ β ≡ ⊥ decision DNNF : ∨ -gates are of the form ( x ∧ α ) ∨ ( ¬ x ∧ β ) structured DNNF : analog of OBDD for DNNF. The “order” is now a tree. What can we do in PTIME? Satisfiability , “smallest” solution, enumeration (polynomial delay), existential projection... Determinism: (weighted) model counting Florent Capelli joint work with Simone Bova, Stefan Mengel, Friedrich Slivovsky (Birkbeck College, University of London) Compilation of CNF-formulas: upper and lower bounds 3 / 23

Connection with communication complexity Strategy of the proof: DNNF: “small” communication complexity + CNF with high communication complexity = lower bound! Florent Capelli joint work with Simone Bova, Stefan Mengel, Friedrich Slivovsky (Birkbeck College, University of London) Compilation of CNF-formulas: upper and lower bounds 4 / 23

Connection with communication complexity Strategy of the proof: DNNF: “small” communication complexity + CNF with high communication complexity = lower bound! Rectangle : notion from communication complexity X = X 1 ⊎ X 2 r boolean function is a ( X 1 , X 2 )-rectangle if r ≡ r 1 ∧ r 2 with r 1 boolean function over X 1 , r 2 boolean function over X 2 Florent Capelli joint work with Simone Bova, Stefan Mengel, Friedrich Slivovsky (Birkbeck College, University of London) Compilation of CNF-formulas: upper and lower bounds 4 / 23

Connection with communication complexity Strategy of the proof: DNNF: “small” communication complexity + CNF with high communication complexity = lower bound! Rectangle : notion from communication complexity X = X 1 ⊎ X 2 r boolean function is a ( X 1 , X 2 )-rectangle if r ≡ r 1 ∧ r 2 with r 1 boolean function over X 1 , r 2 boolean function over X 2 Rectangle cover R of f boolean function over X is a set of rectangles over X s.t.: � sat( f ) = sat( r ) r ∈ R Not necessarily the same underlying partition Florent Capelli joint work with Simone Bova, Stefan Mengel, Friedrich Slivovsky (Birkbeck College, University of London) Compilation of CNF-formulas: upper and lower bounds 4 / 23

Balanced rectangle Size of a rectangle cover is not a relevant complexity measure Any f over X has a rectangle cover of size 2: f ≡ ( f [ x �→ 0] ∧ ¬ x ) ∨ ( f [ x �→ 1] ∧ x ) disjunction of two ( X \ { x } , { x } )-rectangles Florent Capelli joint work with Simone Bova, Stefan Mengel, Friedrich Slivovsky (Birkbeck College, University of London) Compilation of CNF-formulas: upper and lower bounds 5 / 23

Balanced rectangle Size of a rectangle cover is not a relevant complexity measure Any f over X has a rectangle cover of size 2: f ≡ ( f [ x �→ 0] ∧ ¬ x ) ∨ ( f [ x �→ 1] ∧ x ) disjunction of two ( X \ { x } , { x } )-rectangles ( X 1 , X 2 ) balanced partition if | X | / 3 ≤ | X 1 | ≤ 2 | X | / 3 What is the size of the best balanced rectangle cover of f ? Florent Capelli joint work with Simone Bova, Stefan Mengel, Friedrich Slivovsky (Birkbeck College, University of London) Compilation of CNF-formulas: upper and lower bounds 5 / 23

DNNF have small complexity DNNF have small balanced rectangle cover: Theorem (Bova, C., Mengel, Slivovsky) For any DNNF D, there exists a balanced rectangle cover of D of size at most | D | . Intuition (proof is more technical): Find a ∧ -gate v in D such that | X | / 3 ≤ | var( D v ) | ≤ 2 | X | / 3 D v is a rectangle because of decomposability and it is balanced Factorize D ≡ D v ∨ D ′ and apply induction on D ′ Florent Capelli joint work with Simone Bova, Stefan Mengel, Friedrich Slivovsky (Birkbeck College, University of London) Compilation of CNF-formulas: upper and lower bounds 6 / 23

Proving lower bounds [Jukna, Schnigter]: infinite family of 3-CNF having no balanced rectangle cover smaller than 2 Ω( m + n ) ( m clauses, n variables) Thus no DNNF of size smaller than 2 Ω( m + n ) Florent Capelli joint work with Simone Bova, Stefan Mengel, Friedrich Slivovsky (Birkbeck College, University of London) Compilation of CNF-formulas: upper and lower bounds 7 / 23

Proving lower bounds [Jukna, Schnigter]: infinite family of 3-CNF having no balanced rectangle cover smaller than 2 Ω( m + n ) ( m clauses, n variables) Thus no DNNF of size smaller than 2 Ω( m + n ) Improvement: Theorem (Bova, C., Mengel, Slivovsky) There exists a family of (expander) graph G n = ( V n , E n ) such that: F n = � ( i , j ) ∈ E n ( x i ∨ x j ) has no balanced rectangle cover smaller than 2 Ω( | E n | + | V n | ) . Florent Capelli joint work with Simone Bova, Stefan Mengel, Friedrich Slivovsky (Birkbeck College, University of London) Compilation of CNF-formulas: upper and lower bounds 7 / 23

Generalizing the approach and applications Approach can be generalized to other target language: deterministic DNNF : D covered by | D | balanced rectangles having disjoint models . Application: bound [Sauerhoff] separates deterministic DNNF from DNNF. structured DNNF : D covered by | D | rectangles having the same underlying partition . Application: separate FBDD from structured DNNF. Lower bounds on possible transformations too: Negation, conjunction of DNNF can lead to an exponential blow-up Florent Capelli joint work with Simone Bova, Stefan Mengel, Friedrich Slivovsky (Birkbeck College, University of London) Compilation of CNF-formulas: upper and lower bounds 8 / 23

Tractable classes Goal : find large classes of family of CNF that can be efficiently compiled. Target language here: deterministic DNNF as it still supports model counting in PTIME. A class of CNF C is tractable if there exists a polynomial p such that every F ∈ C can be compiled into a d-DNNF of size p (size( F )). Natural idea : use tractable classes for #SAT Structure based algorithms : algorithms using the structure of the input formula. How to model this structure? Florent Capelli joint work with Simone Bova, Stefan Mengel, Friedrich Slivovsky (Birkbeck College, University of London) Compilation of CNF-formulas: upper and lower bounds 9 / 23

Incidence graph By associating a graph to a formula: x 7 x 6 C 4 x 5 C 3 x 4 C 2 x 3 C 1 x 2 x 1 Figure: ( x 1 ∨ x 2 ∨ x 3 ) ∧ ( x 3 ∨ x 4 ∨ ¬ x 5 ) ∧ ( x 1 ∨ x 5 ∨ x 6 ) ∧ ( x 1 ∨ ¬ x 3 ∨ x 5 ∨ ¬ x 7 )

Incidence graph By associating a graph to a formula: x 7 x 6 C 4 x 5 C 3 x 4 C 2 x 3 C 1 x 2 x 1 Figure: ( x 1 ∨ x 2 ∨ x 3 ) ∧ ( x 3 ∨ x 4 ∨ ¬ x 5 ) ∧ ( x 1 ∨ x 5 ∨ x 6 ) ∧ ( x 1 ∨ ¬ x 3 ∨ x 5 ∨ ¬ x 7 )

Hypergraph Or a hypergraph x 5 x 6 x 4 x 7 x 1 x 2 x 3 Figure: ( x 1 ∨ x 2 ∨ x 3 ) ∧ ( x 3 ∨ x 4 ∨ ¬ x 5 ) ∧ ( x 1 ∨ x 5 ∨ x 6 ) ∧ ( x 1 ∨ ¬ x 3 ∨ x 5 ∨ ¬ x 7 )

Hypergraph Or a hypergraph x 5 x 5 x 6 x 6 x 4 x 4 x 7 x 7 x 1 x 1 x 2 x 2 x 3 x 3 Figure: ( x 1 ∨ x 2 ∨ x 3 ) ∧ ( x 3 ∨ x 4 ∨ ¬ x 5 ) ∧ ( x 1 ∨ x 5 ∨ x 6 ) ∧ ( x 1 ∨ ¬ x 3 ∨ x 5 ∨ ¬ x 7 )