Understanding the Power of Clause Learning

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Understanding the Power ofClause Learning

• Ashish Sabharwal, Paul Beame, Henry Kautz
• University of Washington, Seattle
• IJCAI Conference Aug 14, 2003

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SAT The Satisfiability Problem

• Given a CNF formula F,
• e.g. F (a ? b) ? (?a ? c) ? ?a
• determine if F has a satisfying assignment

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SAT Solver DPLL Algorithm

• Best complete CNF (un)satisfiability algorithms
  in
• practice are extensions of the DPLL algorithm
  Davis-Putnam 60
• Davis-Logeman-Loveland 62
• Recursive backtrack search
• Search space pruning based on falsified clauses

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DPLL Algorithm
DPLL(F) // Perform unit propagation while
exists unit clause (y) ? F F Fy if F is
empty, report satisfiable and halt if F
contains the empty clause L, return else choose
a literal x DPLL(Fx) DPLL(F?x)
Remove all clauses containing y Shrink all
clauses containing ?y
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Extending DPLL Clause Learning
When backtracking in DPLL, add new clauses
corresponding to causes of failure of the
search EBL Kleer-Williams 87, Stallman-Sussman
77, Genesereth 84, Davis 84 CL
Bayardo-Schrag 97, MarquesSilva-Sakallah 96,
Zhang 97, Moskewicz et al. 01, Zhang et al.
01

• Added conflict clauses
• Capture reasons of conflicts
• Obtained via unit propagations from known ones
• Reduce future search by producing conflicts sooner

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Conflict Graphs
Known Clauses (p ? q ? a) (? a ? ? b ? ? t) (t ?
?x1) (t ? ?x2) (t ? ?x3) (x1 ? x2 ? x3 ? y) (x2 ?
?y)
Current decisions p false q false b true
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Restarts

• Clause Learning (CL) algorithms can be restarted
  at any point Baptista-Silva 00
• Unset all variables and start over
• But retain all clauses learned so far

• Avoids getting stuck in one part of the search
  space
• Evidently adds power to CL

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CL Critical to Performance

• Best current SAT algorithms rely heavily on CL
  for good behavior on real world problems
• GRASP MarquesSilva-Sakallah 96, SATO H.Zhang
  97zChaff Moskewicz et al. 01, Berkmin
  Goldberg-Novikov 02

• However,
• No good understanding of its strengths and
  weaknesses
• Not much insight on why it works when it does

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Our Contribution

• Mathematical framework
• for analyzing clause learning
• Characterization of its power
• in relation to well-studied topics in
• proof complexity theory
• Ways to improve solver performance
• based on formal analysis

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Proofs of Unsatisfiability

• When F is unsatisfiable,
• DPLL refutation of F is a proof of its
  unsatisfiability
• Size lower bound on proofs of F givestime lower
  bound on executions of DPLL(F)
• Size upper bound on proofs of F givespotential
  for quick executions of such algorithms

(with or w/o learning depending on the class of
proofs)
(with the best possible branching heuristic, the
best learning scheme, etc.)
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What about Satisfiable Formulas?

• Achlioptas-Beame-Molloy 01

Bounds on unsatisfiable formulas imply bounds on
satisfiable ones!
DPLL Tree
Satisfying assignment
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Proof System Resolution

• F (a ? b) ? (?a ? c) ? ?a ? (?b ? c) ? (a ?
  ?c)
• Unsatisfiable CNF formula

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Special Cases of Resolution

• Tree-like resolution
• Graph of inferences forms a tree ? DPLL
• Regular resolution
• Variable can be resolved on only once on any path
  from input to empty clause? Directed acyclic
  graph analog of DPLL tree
• Natural to not branch on a variable once it has
  been eliminated
• Used in original DP Davis-Putnam 60

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Proof System Hierarchy
Space of polynomial time solvable formulas
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Our Results
General RES
General RES
Regular RES
CL w/o restarts
Regular RES
DPLL Tree-like
Trivial RES Learned clauses
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Thm1. CL can beat Regular RES
Such formulas exist! GTn Ordering principlePeb
Pebbling formulas Alekhnovich et al. 02
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PT(f,?) Proof Trace Extension

• Start with
• unsatisfiable formula f with poly-size RES proof
  ?

• PT(f, ?) contains
• All clauses of f
• For each derived clause Q(a?b?c) in ?,
• Trace variable tQ
• New clauses (tQ ? ?a), (tQ ? ?b), (tQ ? ?c)

CL proof of PT(f, ?) works by branching
negatively on tQs in bottom up order of clauses
of ?
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PT(f,?) Proof Trace Extension
Formula f RES proof ?
?
Q ? (a ? b ? c)
(a ? b ? x)
(c ? ?x)




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How hard is PT(f,?)?

• Easy for CL by construction
• CL branches exactly once on each trace
  variable? branches size(?)
• Hard for Regular RES reduction argument
• Fact 1 PT(f,?)TraceVars true ? f
• Fact 2 If ? is a Regular RES proof of g,
  then ?x is a Regular RES
  proof of gx
• Fact 3 f does not have small Regular proofs!

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Implications?

• DPLL algorithms w/o clause learning arehopeless
  for some structured formulas
• CL algorithms have potential for small proofs

Can we use such analysis to harness this
potential?
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Branching Sequence

• B (x1, x4, ?x3, x1, ?x8, ?x2, ?x4, x7, ?x1)
• DPLL picks branching literals from B
• Repetitions allowed
• Different from branching order
• How good is B?
• Depends on backtracking process, learning scheme,
  etc

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From Analysis to Practice

• Maximum size of grid pebbling formulas solvable
• by zChaff (in 1 day) can be substantially
  increased!

Results extend to general randomized pebbling
formulas Sabharwal-Beame-Kautz SAT 2003
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Trivial RES
Known clauses (a ? b ? c), (?a ? x), (?b ? c),
(?c ? y) Derived clause (x ? y)

• Simple ladder structure
• Distinct variables resolved upon
• Properties
• Tree-like
• Regular
• Linear

Clauses learned by CL are those that can be
derived by trivial RES
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Our Results
CL w/o restarts
Regular RES
DPLL Tree-like
Trivial RES Learned clauses
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CL-- A Variant of CL

• Allow branching on variables whose value is
  already implied by unit propagation
• Equivalently, allow branching on variables not
  appearing at all in the residual formula

How can this possibly help?
Can learn a conflict clause that will reduce
search later!
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Error in Paper

• Incorrect Theorem 1 in the paperCL restarts
  is equivalent to RES

Correct versionCL-- restarts is equivalent
to RES
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Thm2. CL-- restarts General RES

• General RES can simulate CL restarts
• CL learns by unit propagation
• Infer learned clauses using RES derivation
• CL-- restarts can simulate General RES
• If RES proof resolves (A ? x) and (B ? ?x) to
  obtain C, branch and set all literals of A and B
  to false, learn C, restart
• Eventually learn empty clause

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Our Results
General RES CL-- restarts
CL w/o restarts
Regular RES
DPLL Tree-like
Trivial RES Learned clauses
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Summary

• Formal framework
• CL can be stronger than Regular RES
• FirstNewCut learning scheme
• CL-- restarts is equivalent to RES

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Open Problems

• Fill in complexity hierarchy gaps
• Can CL efficiently simulate Regular RES?
• Can CL restarts efficiently simulate RES?

From Analysis to Practice Can we use this
framework to improve SAT solvers for other
classes of structured formulas?E.g. Planning
as satisfiability Bounded model checking