On the Relative Efficiency of DPLL and OBDDs with Axiom and Join Matti J¨ arvisalo University of Helsinki, Finland September 16, 2011 @ CP M. J¨ arvisalo (U. Helsinki) DPLL and OBDDs September 16, 2011 @ CP 1 / 15
Background Two main approaches to industrial Boolean satisfiability solving ◮ Complete search-based methods: here DPLL and CDCL ◮ Compilation-based approaches: here OBDDs Understanding the relative efficiency of these approaches Study the power of the proof systems underlying solvers ◮ CDCL (with restarts) → Resolution [PipatsrisawatD AIJ’10] ◮ DPLL → tree-like resolution Separating CNF Proof Systems Proof system S does not polynomially simulate system S ′ : there is an infinite family { F n } n of unsatisfiable CNF formulas s.t. for any n : ◮ there is a polynomial S ′ -proof of F n w.r.t. n ◮ minimum-size S proofs of F n are of exponential w.r.t. n For example: DPLL does not polynomially simulate CDCL [BeameKS JAIR’04; PipatsrisawatD AIJ’10] M. J¨ arvisalo (U. Helsinki) DPLL and OBDDs September 16, 2011 @ CP 2 / 15
Background Two main approaches to industrial Boolean satisfiability solving ◮ Complete search-based methods: here DPLL and CDCL ◮ Compilation-based approaches: here OBDDs Understanding the relative efficiency of these approaches Study the power of the proof systems underlying solvers ◮ CDCL (with restarts) → Resolution [PipatsrisawatD AIJ’10] ◮ DPLL → tree-like resolution Separating CNF Proof Systems Proof system S does not polynomially simulate system S ′ : there is an infinite family { F n } n of unsatisfiable CNF formulas s.t. for any n : ◮ there is a polynomial S ′ -proof of F n w.r.t. n ◮ minimum-size S proofs of F n are of exponential w.r.t. n For example: DPLL does not polynomially simulate CDCL [BeameKS JAIR’04; PipatsrisawatD AIJ’10] M. J¨ arvisalo (U. Helsinki) DPLL and OBDDs September 16, 2011 @ CP 2 / 15
Previous Results Interest in the relative efficiency of SAT solving methods based on resolution and OBDDs [GrooteZ’03; AtseriasKV’04; SinzB’06; Segerlind’08; Peltier’08; TveretinaSZ’10; ...] Power of OBDDs depends on the set of construction rules ◮ With quantifier elimination (+weakening): (unrestricted) resolution does not polynomially simulate OBDDs [AtseriasKV CP’04] ◮ Without quantifier elimination: OBDD aj “OBDD apply” with Axiom and Join does not simulate (unrestricted) resolution [TveretinaSZ JSAT’10] Here we concentrate on the weaker OBDD aj M. J¨ arvisalo (U. Helsinki) DPLL and OBDDs September 16, 2011 @ CP 3 / 15
Previous Results Interest in the relative efficiency of SAT solving methods based on resolution and OBDDs [GrooteZ’03; AtseriasKV’04; SinzB’06; Segerlind’08; Peltier’08; TveretinaSZ’10; ...] Power of OBDDs depends on the set of construction rules ◮ With quantifier elimination (+weakening): (unrestricted) resolution does not polynomially simulate OBDDs [AtseriasKV CP’04] ◮ Without quantifier elimination: OBDD aj “OBDD apply” with Axiom and Join does not simulate (unrestricted) resolution [TveretinaSZ JSAT’10] Here we concentrate on the weaker OBDD aj M. J¨ arvisalo (U. Helsinki) DPLL and OBDDs September 16, 2011 @ CP 3 / 15
Goals Main Question Pinpoint the power of OBDD aj more exactly: Does it even polynomially simulate the Davis-Putnam-Logemann-Loveland procedure (DPLL) that is known to be exponentially weaker than clause learning / resolution? Does DPLL polynomially simulate OBDD aj ? M. J¨ arvisalo (U. Helsinki) DPLL and OBDDs September 16, 2011 @ CP 4 / 15
Contributions of the Paper Main Theorem OBDDs constructed using the Axiom and Join rules and DPLL (equivalently, tree-like resolution) are polynomially incomparable. DPLL (with an optimal branching heuristic) does not polynomially simulate OBDD aj (using a suitable variable ordering) OBDD aj proof system (under any variable ordering) does not polynomially simulate DPLL Results from combining and extending previous results M. J¨ arvisalo (U. Helsinki) DPLL and OBDDs September 16, 2011 @ CP 5 / 15
Contributions of the Paper Main Theorem OBDDs constructed using the Axiom and Join rules and DPLL (equivalently, tree-like resolution) are polynomially incomparable. DPLL (with an optimal branching heuristic) does not polynomially simulate OBDD aj (using a suitable variable ordering) OBDD aj proof system (under any variable ordering) does not polynomially simulate DPLL Results from combining and extending previous results M. J¨ arvisalo (U. Helsinki) DPLL and OBDDs September 16, 2011 @ CP 5 / 15
DPLL [DavisPutnam’60; Davis-Logemann-Loveland’62] DPLL( F ) If F is empty report satisfiable and halt If F contains the empty clause return Else choose a variable x ∈ vars( F ) DPLL( F x ) DPLL( F ¬ x ) F x : Unit propagated F ; remove all clauses containing x and all occurrences of ¬ x from F ; repeating until fixpoint for all unit clauses. Practical implementations deterministic: implement a branching heuristic for choosing a variable ◮ here we do not restrict this non-deterministic choice. DPLL proof of unsat CNF F : a search tree of DPLL( F ) Size of a DPLL proof: the number of nodes in the tree DPLL and tree-like resolution are polynomially equivalent M. J¨ arvisalo (U. Helsinki) DPLL and OBDDs September 16, 2011 @ CP 6 / 15
OBDDs Binary decision diagram (BDD) over a ( x ∨ y ∨ z ) set of Boolean variables V ◮ Rooted DAG with x ⋆ decision nodes labelled with y distinct variables from V ⋆ two terminal nodes 0 and 1 z ◮ Each decision node v has two children, low( v ) and high( v ). 0 1 ◮ Edge v → low( v ) (high( v ), resp.) represents assigning v = 0 (1, resp.). Ordered (O)BDD: ◮ a total variable order ≺ enforced on on all paths from root to terminals Reduced OBDD: ◮ isomorphic subgraphs merged ◮ nodes with isomorphic children eliminated Unique (R)OBDD B( φ, ≺ ) for any CNF φ size(B( φ, ≺ )): the number of nodes. M. J¨ arvisalo (U. Helsinki) DPLL and OBDDs September 16, 2011 @ CP 7 / 15
OBDD aj Proofs of CNFs Given an unsat CNF F and a variable order ≺ over vars( F ): An OBDD aj derivation of the OBDD for 0 A sequence ρ = (B 1 ( φ 1 , ≺ ) , . . . , B m ( φ m , ≺ )) of OBDDs, where ◮ B m ( φ m , ≺ ) is the single-node OBDD representing 0 ◮ for each i = 1 .. m , either Axiom φ i is a clause in F , or Join φ i = φ j ∧ φ k for some B j ( φ j , ≺ ) and B k ( φ k , ≺ ), 1 ≤ j < k < i , in ρ . Size of OBDD aj proof ρ : Σ m i =1 size(B i ( φ i , ≺ )). M. J¨ arvisalo (U. Helsinki) DPLL and OBDDs September 16, 2011 @ CP 8 / 15
Example variable ordering x ≺ y ≺ z ( x ∨ y ∨ z ) ( ¬ z ) ( ¬ z ) ( ¬ x ∨ y ) z y x x y y 1 0 1 0 z 0 1 0 1 x x y y z 0 1 1 0 0 M. J¨ arvisalo (U. Helsinki) DPLL and OBDDs September 16, 2011 @ CP 9 / 15
DPLL does not Polynomially Simulate OBDD aj Pebbling contradictions [Ben-SassonW’01] as witnessing formulas Peb( G ) for a given DAG G : ◮ ( x i , 0 ∨ x i , 1 ) for each source node (in-degree 0) i of G ; ◮ ( ¬ x i , 0 ) and ( ¬ x i , 1 ) for each sink node (out-degree 0) i of G ; ◮ ( ¬ x i 1 , a 1 ∨ · · · ∨ ¬ x i k , a k ∨ x j , 0 ∨ x j , 1 ) for each non-source node j , where i 1 , . . . , i k are the predecessors of j , and for each ( a 1 , . . . , a k ) ∈ { 0 , 1 } k . Minimum-size tree-like resolution proofs of Peb ( G n ) are 2 Ω( n / log n ) [Ben-SassonW JACM’01] for a specific infinite family { G n } of DAGs with constant node in-degree [PaulTC’77] Equivalently for DPLL M. J¨ arvisalo (U. Helsinki) DPLL and OBDDs September 16, 2011 @ CP 10 / 15
Short OBDD aj Proofs for log-Bounded In-degree: Idea Similar strategy as in short ordered resolution proofs for Peb( G n ) [Buresh-OppenheimP’07] Let G be a DAG on n nodes, and j a node in G with parents i 1 , . . . , i k where k = O (log n ). 1 Label each source j of G with axiom B(( x j , 0 ∨ x j , 1 ) , ≺ ). 2 Following an topological ordering ≺ of G n : ◮ Poly-size OBDD aj derivation of B(( x j , 0 ∨ x j , 1 ) , ≺ ) for non-source j ⋆ OBDD of any n -variable formula is of size O (2 n / n ) [LiawL’92] ⋆ G has log-bounded node in-degree ⇒ each derivation contains O (log n ) variables ⇒ each derivation polynomial-size wrt n ◮ ⇒ poly-size OBDD aj derivation of B(( x t , 0 ∨ x t , 1 ) , ≺ ) for the sink t of G 3 Join B(( x t , 0 ∨ x t , 1 ) , ≺ ) with axioms B(( ¬ x t , 0 ) , ≺ ) and B(( ¬ x t , 1 ) , ≺ ). Result: polynomial-size OBDD aj -proof of Peb( G n ) M. J¨ arvisalo (U. Helsinki) DPLL and OBDDs September 16, 2011 @ CP 11 / 15
Short OBDD aj Proofs for log-Bounded In-degree: Idea Similar strategy as in short ordered resolution proofs for Peb( G n ) [Buresh-OppenheimP’07] Let G be a DAG on n nodes, and j a node in G with parents i 1 , . . . , i k where k = O (log n ). 1 Label each source j of G with axiom B(( x j , 0 ∨ x j , 1 ) , ≺ ). 2 Following an topological ordering ≺ of G n : ◮ Poly-size OBDD aj derivation of B(( x j , 0 ∨ x j , 1 ) , ≺ ) for non-source j ⋆ OBDD of any n -variable formula is of size O (2 n / n ) [LiawL’92] ⋆ G has log-bounded node in-degree ⇒ each derivation contains O (log n ) variables ⇒ each derivation polynomial-size wrt n ◮ ⇒ poly-size OBDD aj derivation of B(( x t , 0 ∨ x t , 1 ) , ≺ ) for the sink t of G 3 Join B(( x t , 0 ∨ x t , 1 ) , ≺ ) with axioms B(( ¬ x t , 0 ) , ≺ ) and B(( ¬ x t , 1 ) , ≺ ). Result: polynomial-size OBDD aj -proof of Peb( G n ) M. J¨ arvisalo (U. Helsinki) DPLL and OBDDs September 16, 2011 @ CP 11 / 15
Recommend
More recommend
