On the Relative Efficiency of DPLL and OBDDs with Axiom and Join - - PowerPoint PPT Presentation

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 {Fn}n of unsatisfiable CNF formulas s.t. for any n:

◮ there is a polynomial S′-proof of Fn 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. 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 {Fn}n of unsatisfiable CNF formulas s.t. for any n:

◮ there is a polynomial S′-proof of Fn 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. 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: OBDDaj

“OBDD apply” with Axiom and Join does not simulate (unrestricted) resolution [TveretinaSZ JSAT’10]

Here we concentrate on the weaker OBDDaj

• 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: OBDDaj

“OBDD apply” with Axiom and Join does not simulate (unrestricted) resolution [TveretinaSZ JSAT’10]

Here we concentrate on the weaker OBDDaj

• M. J¨

arvisalo (U. Helsinki) DPLL and OBDDs September 16, 2011 @ CP 3 / 15

Goals

Main Question

Pinpoint the power of OBDDaj 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 OBDDaj?

• 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 OBDDaj (using a suitable variable ordering) OBDDaj 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 OBDDaj (using a suitable variable ordering) OBDDaj 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(Fx) DPLL(F¬x) Fx: Unit propagated F; remove all clauses containing x and all

• ccurrences 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 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 y x 1 (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. J¨

arvisalo (U. Helsinki) DPLL and OBDDs September 16, 2011 @ CP 7 / 15

OBDDaj Proofs of CNFs

Given an unsat CNF F and a variable order ≺ over vars(F): An OBDDaj derivation of the OBDD for 0 A sequence ρ = (B1(φ1, ≺), . . . , Bm(φm, ≺)) of OBDDs, where

◮ Bm(φm, ≺) is the single-node OBDD representing 0 ◮ for each i = 1..m, either

Axiom φi is a clause in F, or Join φi = φj ∧ φk for some Bj(φj, ≺) and Bk(φk, ≺), 1 ≤ j < k < i, in ρ.

Size of OBDDaj proof ρ: Σm

i=1size(Bi(φi, ≺)).

• M. J¨

arvisalo (U. Helsinki) DPLL and OBDDs September 16, 2011 @ CP 8 / 15

Example variable ordering x ≺ y ≺ z

z 1 z y x y z 1 1 y x 1 y x 1 y x 1 (x ∨ y ∨ z) (¬z) (¬x ∨ y) (¬z)

• M. J¨

arvisalo (U. Helsinki) DPLL and OBDDs September 16, 2011 @ CP 9 / 15

DPLL does not Polynomially Simulate OBDDaj

Pebbling contradictions [Ben-SassonW’01] as witnessing formulas Peb(G) for a given DAG G:

◮ (xi,0 ∨ xi,1) for each source node (in-degree 0) i of G; ◮ (¬xi,0) and (¬xi,1) for each sink node (out-degree 0) i of G; ◮ (¬xi1,a1 ∨ · · · ∨ ¬xik,ak ∨ xj,0 ∨ xj,1) for each non-source node j, where

i1, . . . , ik are the predecessors of j, and for each (a1, . . . , ak) ∈ {0, 1}k.

Minimum-size tree-like resolution proofs of Peb(Gn) are 2Ω(n/ log n) [Ben-SassonW JACM’01] for a specific infinite family {Gn} 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 OBDDaj Proofs for log-Bounded In-degree: Idea

Similar strategy as in short ordered resolution proofs for Peb(Gn) [Buresh-OppenheimP’07] Let G be a DAG on n nodes, and j a node in G with parents i1, . . . , ik where k = O(log n).

1 Label each source j of G with axiom B((xj,0 ∨ xj,1), ≺). 2 Following an topological ordering ≺ of Gn: ◮ Poly-size OBDDaj derivation of B((xj,0 ∨ xj,1), ≺) for non-source j ⋆ OBDD of any n-variable formula is of size O(2n/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((xt,0 ∨ xt,1), ≺) for the sink t of G 3 Join B((xt,0 ∨ xt,1), ≺) with axioms B((¬xt,0), ≺) and B((¬xt,1), ≺).

Result: polynomial-size OBDDaj-proof of Peb(Gn)

• M. J¨

arvisalo (U. Helsinki) DPLL and OBDDs September 16, 2011 @ CP 11 / 15

Short OBDDaj Proofs for log-Bounded In-degree: Idea

Similar strategy as in short ordered resolution proofs for Peb(Gn) [Buresh-OppenheimP’07] Let G be a DAG on n nodes, and j a node in G with parents i1, . . . , ik where k = O(log n).

1 Label each source j of G with axiom B((xj,0 ∨ xj,1), ≺). 2 Following an topological ordering ≺ of Gn: ◮ Poly-size OBDDaj derivation of B((xj,0 ∨ xj,1), ≺) for non-source j ⋆ OBDD of any n-variable formula is of size O(2n/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((xt,0 ∨ xt,1), ≺) for the sink t of G 3 Join B((xt,0 ∨ xt,1), ≺) with axioms B((¬xt,0), ≺) and B((¬xt,1), ≺).

Result: polynomial-size OBDDaj-proof of Peb(Gn)

• M. J¨

arvisalo (U. Helsinki) DPLL and OBDDs September 16, 2011 @ CP 11 / 15

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]

OBDDaj 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 OBDDaj proof of size s ⇒ F has a OBDDaj proof of size s. Generalization of lemma in [TveretinaSZ JSAT’10]

• M. J¨

arvisalo (U. Helsinki) DPLL and OBDDs September 16, 2011 @ CP 12 / 15

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 πF = (C1, . . . , Cm = ∅) of F. Define extension of F: [Krajicek]

◮ Take new variables ei ↔ Ci for i = 1, . . . , m −1 using the extension rule

For any unsat CNF F: There is a DPLL proof of F ∧ E(πF) of size O(|πF|).

◮ Branch according to e1 ≺ · · · ≺ em−1 ⋆ ei = 0 ⇒ immediate conflict with unit propagation

(due to soundness of resolution)

⋆ All e′ i s 1 ⇒ conflicting unit clauses in πF

• M. J¨

arvisalo (U. Helsinki) DPLL and OBDDs September 16, 2011 @ CP 13 / 15

Needed: An explicit witness {Fn}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

PHPn+1

n

:=

n+1

• i=1
• n
• j=1

pi,j

• ∧

n

• j=1

n

• i=1

n+1

• i′=i+1
• ¬pi,j ∨ ¬pi′,j
• ,

pi,j = 1 ↔ “ith pigeon sits in the jth hole” There is no polynomial-size OBDDaj proof [TveretinaSZ JSAT’10] There is no polynomial-size RES proof of PHPn+1

n

[Haken’86] There is an extension E such that PHPn+1

n

∧ E has polynomial-size resolution proof πF∧E [Cook’79]

◮ Explicit short proofs: [J¨

arvisaloJ Constraint’09; TveretinaSZ JSAT’10]

Final witness:

• PHPn+1

n

∧ E

• ∧ E(πF∧E)
• M. J¨

arvisalo (U. Helsinki) DPLL and OBDDs September 16, 2011 @ CP 14 / 15

Needed: An explicit witness {Fn}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

PHPn+1

n

:=

n+1

• i=1
• n
• j=1

pi,j

• ∧

n

• j=1

n

• i=1

n+1

• i′=i+1
• ¬pi,j ∨ ¬pi′,j
• ,

pi,j = 1 ↔ “ith pigeon sits in the jth hole” There is no polynomial-size OBDDaj proof [TveretinaSZ JSAT’10] There is no polynomial-size RES proof of PHPn+1

n

[Haken’86] There is an extension E such that PHPn+1

n

∧ E has polynomial-size resolution proof πF∧E [Cook’79]

◮ Explicit short proofs: [J¨

arvisaloJ Constraint’09; TveretinaSZ JSAT’10]

Final witness:

• PHPn+1

n

∧ E

• ∧ E(πF∧E)
• M. J¨

arvisalo (U. Helsinki) DPLL and OBDDs September 16, 2011 @ CP 14 / 15

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 OBDDaj (using a suitable variable ordering) OBDDaj proof system (under any variable ordering) does not polynomially simulate DPLL

• M. J¨

arvisalo (U. Helsinki) DPLL and OBDDs September 16, 2011 @ CP 15 / 15