Using Problem Structure for Efficient Clause Learning

Ashish Sabharwal, Paul Beame, Henry Kautz

University of Washington, Seattle

April 23, 2003

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The SAT Approach

Input p 2 D

CNF encoding f

SAT solver

f SAT f SATp : Instance

D : Domain graph problem, AI planning, model checking p bad p good

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Key Facts

• Problem instances typically have structure– Graphs, precedence relations, cause and effects– Translation to CNF flattens this structure

• Best complete SAT solvers are– DPLL based clause learners; branch and backtrack– Critical: Variable order used for branching

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Natural Questions

• Can we extract structure efficiently?– In translation to CNF formula itself– From CNF formula– From higher level description

• How can we exploit this auxiliary information?– Tweak SAT solver for each domain– Tweak SAT solver to use general “guidance”

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Our Approach

Input p 2 D

CNF encoding f

SAT solver

f SAT f SATEncode “structure”as branching sequence

p bad p good

Branching sequence

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Related Work

• Exploiting structure in CNF formula– [GMT’02] Dependent variables– [OGMS’02] LSAT (blocked/redundant clauses)– [B’01] Binary clauses– [AM’00] Partition-based reasoning

• Exploiting domain knowledge– [S’00] Model checking– [KS’96] Planning (cause vars / effect vars)

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Our Result, Informally

– Structure can be efficiently retrieved from highlevel description (pebbling graph)

– Branching sequence as auxiliary information can be easily exploited

Given a pebbling graph G, can efficiently generatea branching sequence BG that dramatically improvesthe performance of current best SAT solvers on fG.

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Preliminaries: CNF Formula

f = (x1 OR x2 OR :x9) AND (:x3 OR x9) AND (:x1 OR :x4 OR :x5 OR :x6)

Conjunctionof clauses

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Preliminaries: DPLL

DPLL(CNF formula f) {

Simplify(f);

If (conflict) return UNSAT;

If (all-vars-assigned) {return SAT assignment; exit}

Pick unassigned variable x;

Try DPLL(f |x=0), DPLL(f |x=1)

}

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Prelim: Clause Learning

• DPLL: Change “if (conflict) return UNSAT”

to “if (conflict)

{learn conflict clause; return UNSAT}”

x2 = 1, x3 = 0, x6 = 0 ) conflict

“Learn” (:x2 OR x3 OR x6)

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Prelim: Branching Sequence

• B = (x1, x4, :x3, x1, :x8, :x2, :x4, x7, :x1, x2)

• DPLL: Change “Pick unassigned var x”to “Pick next literal x from B; delete it from B;

if x already assigned, repeat”

• How “good” is B?– Depends on backtracking process, learning

scheme

Different from“branching order”

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Prelim: Pebbling Formulas

(a1 OR a2) (b1 OR b2)

(e1 OR e2)

(t1 OR t2)

(f1)

(c1 OR c2 OR c3)

Target(s)

Sources

E

A B C

F

T

Node E is pebbled if(e1 OR e2) = 1 fG = Pebbling(G)

Source axioms: A, B, C are pebbled

Pebbling axioms: A and B are pebbled ) E is pebbled …Target axioms: T is not pebbled

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Prelim: Pebbling Formulas

• Can have– Multiple targets– Unbounded fanin– Large clause labels

• Pebbling(G) is unsatisfiable

• Removing any clause from subgraph of each target makes it satisfiable

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Grid vs. Randomized Pebbling

(a1 a2) b1 (c1 c2 c3)

(d1 d2 d3)

l1

(h1 h2)

(i1 i2 i3 i4)e1

(g1 g2)

f1

(n1 n2)

m1

(a1 a2) (b1 b2) (c1 c2) (d1 d2)

(e1 e2)

(h1 h2)

(t1 t2)

(i1 i2)

(g1 g2)(f1 f2)

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Why Pebbling?

• Practically useful– precedence relations in tasks, fault propagation in

circuits, restricted planning problems

• Theoretically interesting– Used earlier for separating proof complexity classes– “Easy” to analyze

• Hard for current best SAT solvers like zChaff– Shown by our experiments

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Our Result, Again

– Efficient : (|fG|)

– zChaff : One of the current best SAT solvers

Given a pebbling graph G, can efficiently generatea branching sequence BG such that zChaff(fG, BG) is

empirically exponentially faster than zChaff(fG).

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The Algorithm

• Input:– Pebbling graph G

• Output:– Branching sequence BG, |BG| = (|fG|), that works

well for 1UIP learning scheme and fast backtracking

[fG : CNF encoding of pebbling(G)]

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The Algorithm: GenSeq(G)

1. Compute node heights

2. Foreach u 2 {unit clause labeled nodes} bottom up• Add u to G.sources• GenSubseq(u)

3. Foreach t 2 {targets} bottom up• GenSubseq(t)

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The Algorithm: GenSubseq(v)

// trivial wrapper

1. If (|v.preds| > 0)– GenSubseq(v, |v.preds|)

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The Algorithm: GenSubseq(v, i)

1. u = v.preds[i] // by increasing height

2. if i=1 // lowest pred

a. GenSubseq(u) if unvisited non-source

b. return

3. Output u.labels // higher pred

4. GenSubseq(u) if unvisitedHigh non-source

5. GenSubseq(v, i-1) // recurse on i-1

6. GenPattern(u, v, i-1) // repetitive pattern

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Results: Grid Pebbling

– Pure DPLL upto 60 variables

– DPLL + upto 60 variablesbranching seq

– Clause learning upto 4,000 variables(original zChaff)

– Clause learning upto 2,000,000 variables+ branching seq

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Results: Randomized Pebl.

– Pure DPLL upto 35 variables

– DPLL + upto 50 variablesbranching seq

– Clause learning upto 350 variables(original zChaff)

– Clause learning upto 1,000,000 variables+ branching seq

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Summary

• High level problem description is useful– Domain knowledge can help SAT solvers

• Branching sequence– One good way to encode structure

• Pebbling problems: Proof of concept– Can efficiently generate good branching sequence– Structure use improves performance dramatically

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Open Problems

• Other domains?– STRIPS planning problems (layered structure)– Bounded model checking

• Variable ordering strategies from BDDs?

• Other ways of exploiting structure?– branching “order”– something to guide learning?– Domain-based tweaking of SAT algorithms