On the Relative Efficiency of DPLL and OBDDs with Axiom and Join

Transcription

1 On the Relative Efficiency of DPLL and OBDDs with Axiom and Join Matti Järvisalo University of Helsinki, Finland September 16, CP M. Järvisalo (U. Helsinki) DPLL and OBDDs September 16, CP 1 / 15

2 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 Fn are of exponential w.r.t. n For example: DPLL does not polynomially simulate CDCL [BeameKS JAIR 04; PipatsrisawatD AIJ 10] M. Järvisalo (U. Helsinki) DPLL and OBDDs September 16, CP 2 / 15

3 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 Fn are of exponential w.r.t. n For example: DPLL does not polynomially simulate CDCL [BeameKS JAIR 04; PipatsrisawatD AIJ 10] M. Järvisalo (U. Helsinki) DPLL and OBDDs September 16, CP 2 / 15

4 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 Without quantifier elimination: OBDD aj OBDD apply with Axiom and Join does not simulate (unrestricted) resolution Here we concentrate on the weaker OBDD aj [AtseriasKV CP 04] [TveretinaSZ JSAT 10] M. Järvisalo (U. Helsinki) DPLL and OBDDs September 16, CP 3 / 15

5 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 Without quantifier elimination: OBDD aj OBDD apply with Axiom and Join does not simulate (unrestricted) resolution Here we concentrate on the weaker OBDD aj [AtseriasKV CP 04] [TveretinaSZ JSAT 10] M. Järvisalo (U. Helsinki) DPLL and OBDDs September 16, CP 3 / 15

6 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. Järvisalo (U. Helsinki) DPLL and OBDDs September 16, CP 4 / 15

7 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. Järvisalo (U. Helsinki) DPLL and OBDDs September 16, CP 5 / 15

8 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. Järvisalo (U. Helsinki) DPLL and OBDDs September 16, CP 5 / 15

9 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. Järvisalo (U. Helsinki) DPLL and OBDDs September 16, CP 6 / 15

10 OBDDs Binary decision diagram (BDD) over a set of Boolean variables V Rooted DAG with decision nodes labelled with distinct variables from V two terminal nodes 0 and 1 Each decision node v has two children, low(v) and high(v). Edge v low(v) (high(v), resp.) represents assigning v = 0 (1, resp.). z (x y z) 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. Järvisalo (U. Helsinki) DPLL and OBDDs September 16, CP 7 / 15 0 y x 1

11 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. Järvisalo (U. Helsinki) DPLL and OBDDs September 16, CP 8 / 15

12 Example variable ordering x y z (x y z) ( z) ( z) ( x y) x z y x y y z x x y z y M. Järvisalo (U. Helsinki) DPLL and OBDDs September 16, CP 9 / 15

13 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; ( xi1,a 1 x ik,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 2Ω(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. Järvisalo (U. Helsinki) DPLL and OBDDs September 16, CP 10 / 15

14 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 OBDDaj 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 OBDDaj 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. Järvisalo (U. Helsinki) DPLL and OBDDs September 16, CP 11 / 15

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 OBDDaj 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 OBDDaj 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. Järvisalo (U. Helsinki) DPLL and OBDDs September 16, CP 11 / 15

16 OBDDaj does not Polynomially Simulate DPLL (1/3) Main Lemma OBDDaj cannot exploit a specific form of redundancy in CNFs Any iteratively added redundant clauses of the form x a b Introduce new variable x to stand for a b, where a and b are variables in the current formula The extension rule of Extended Resolution [Tseitin 68] OBDD aj cannot exploit any extensions Let F be an unsatisfiable CNF formula and E an extension to F. For any variable order over vars(f ) vars(e): F E has a OBDD aj proof of size s F has a OBDD aj proof of size s. Generalization of lemma in [TveretinaSZ JSAT 10] M. Järvisalo (U. Helsinki) DPLL and OBDDs September 16, CP 12 / 15

17 OBDDaj does not Polynomially Simulate DPLL (2/3) Tree-like Resolution/DPLL simulates Resolution under extensions Known Arbitrary unsat CNF formula F Take any resolution proof π F = (C 1,..., C m = ) of F. Define extension of F : [Krajicek] Take new variables ei C i for i = 1,..., m 1 using the extension rule For any unsat CNF F : There is a DPLL proof of F E(π F ) of size O( π F ). Branch according to e1 e m 1 e i = 0 immediate conflict with unit propagation (due to soundness of resolution) All e i s 1 conflicting unit clauses in π F M. Järvisalo (U. Helsinki) DPLL and OBDDs September 16, CP 13 / 15

18 Needed: An explicit witness {F n } n for which OBDDaj proofs are exponential wrt n Fn has polynomial DPLL proofs under suitable extensions Pigeon-Holes n + 1 pigeons cannot get individual n holes PHP n+1 n := n+1 i=1 ( n ) n n p i,j j=1 n+1 j=1 i=1 i =i+1 p i,j = 1 i th pigeon sits in the j th hole ( ) pi,j p i,j, There is no polynomial-size OBDD aj proof [TveretinaSZ JSAT 10] There is no polynomial-size RES proof of PHP n+1 n [Haken 86] There is an extension E such that PHP n+1 n E has polynomial-size resolution proof π F E [Cook 79] Explicit short proofs: [JärvisaloJ Constraint 09; TveretinaSZ JSAT 10] Final witness: ( PHP n+1 n E ) E(π F E ) M. Järvisalo (U. Helsinki) DPLL and OBDDs September 16, CP 14 / 15

19 Needed: An explicit witness {F n } n for which OBDDaj proofs are exponential wrt n Fn has polynomial DPLL proofs under suitable extensions Pigeon-Holes n + 1 pigeons cannot get individual n holes PHP n+1 n := n+1 i=1 ( n ) n n p i,j j=1 n+1 j=1 i=1 i =i+1 p i,j = 1 i th pigeon sits in the j th hole ( ) pi,j p i,j, There is no polynomial-size OBDD aj proof [TveretinaSZ JSAT 10] There is no polynomial-size RES proof of PHP n+1 n [Haken 86] There is an extension E such that PHP n+1 n E has polynomial-size resolution proof π F E [Cook 79] Explicit short proofs: [JärvisaloJ Constraint 09; TveretinaSZ JSAT 10] Final witness: ( PHP n+1 n E ) E(π F E ) M. Järvisalo (U. Helsinki) DPLL and OBDDs September 16, CP 14 / 15

20 Conclusions 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 M. Järvisalo (U. Helsinki) DPLL and OBDDs September 16, CP 15 / 15