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### Using Problem Structure for Efficient Clause Learning

Ashish Sabharwal, Paul Beame, Henry Kautz

University of Washington, Seattle

April 23, 2003

The SAT Approach

CNF encoding f

SAT solver

Input p2D

p : Instance

D : Domaingraph problem,

AI planning, model checking

f SAT

f SAT

p bad

p good

University of Washington

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

University of Washington

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”

University of Washington

Our Approach

CNF encoding f

Branching sequence

SAT solver

Input p2D

f SAT

f SAT

Encode “structure”

as branching sequence

p bad

p good

University of Washington

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)

University of Washington

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 graphG, can efficiently generate

a branching sequenceBG that dramatically improves

the performance of current best SAT solvers on fG.

University of Washington

Preliminaries: CNF Formula

Conjunction

of clauses

f = (x1ORx2OR:x9) AND (:x3ORx9)AND (:x1OR:x4OR:x5OR:x6)

University of Washington

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)

}

University of Washington

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” (:x2ORx3ORx6)

University of Washington

Prelim: Branching Sequence

- B = (x1, x4, :x3, x1, :x8, :x2,:x4, x7, :x1, x2)
- DPLL: Change “Pick unassigned var x”to “Pick next literal xfrom B; delete it from B; if x already assigned, repeat”
- How “good” is B?
- Depends on backtracking process, learning scheme

Different from

“branching order”

University of Washington

Prelim: Pebbling Formulas

Node E is pebbled if(e1ORe2) = 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

Target(s)

(t1ORt2)

T

(e1ORe2)

E

F

(f1)

A

B

C

(c1ORc2ORc3)

(a1ORa2)

(b1ORb2)

Sources

University of Washington

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

University of Washington

Grid vs. Randomized Pebbling

(n1 n2)

m1

(t1 t2)

l1

(h1 h2)

(h1 h2)

(i1 i2)

e1

(i1 i2 i3 i4)

f1

(e1 e2)

(f1 f2)

(g1 g2)

(d1 d2 d3)

(g1 g2)

(a1 a2)

(b1 b2)

(c1 c2)

(d1 d2)

(a1 a2)

(c1 c2 c3)

b1

University of Washington

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

University of Washington

Our Result, Again

- Efficient : Q(|fG|)
- zChaff : One of the current best SAT solvers

Given a pebbling graphG, can efficiently generate

a branching sequenceBG such that zChaff(fG, BG) is

empirically exponentially faster than zChaff(fG).

University of Washington

The Algorithm

- Input:
- Pebbling graphG
- Output:
- Branching sequenceBG, |BG| = Q(|fG|), that works well for 1UIP learning scheme and fast backtracking[fG : CNF encoding of pebbling(G)]

University of Washington

The Algorithm: GenSeq(G)

- Compute node heights
- Foreach u2 {unit clause labeled nodes} bottom up
- Add u to G.sources
- GenSubseq(u)
- Foreach t2 {targets} bottom up
- GenSubseq(t)

University of Washington

The Algorithm: GenSubseq(v)

// trivial wrapper

- If (|v.preds| > 0)
- GenSubseq(v, |v.preds|)

University of Washington

The Algorithm: GenSubseq(v, i)

- u = v.preds[i] // by increasing height
- if i=1 // lowest pred
- GenSubseq(u) if unvisited non-source
- return
- Output u.labels // higher pred
- GenSubseq(u) if unvisitedHigh non-source
- GenSubseq(v, i-1) // recurse on i-1
- GenPattern(u, v, i-1) // repetitive pattern

University of Washington

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

University of Washington

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

University of Washington

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

University of Washington

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

University of Washington

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