Proofs from SAT Solvers
Yeting Ge
ACSys NYU
Nov 20
2007
SAT solvers and proofs
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SAT problem and solvers
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Given a propositional logic formula, a SAT solver
outputs sat or unsat
Proofs from SAT solvers are needed
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A certificate to show the solver is correct
Required by applications
 Planning
 Calculation of interpolants
 ……
2
A bunch of SAT solvers
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Complete / incomplete
Internal representation
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Search method
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BDD, circuit, CNF,……
Depth first, breadth first
Most modern SAT solvers
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Complete, CNF, breadth first
DPLL based
3
Naïve SAT algorithm
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To solve:
( x1  x 2)  (x1  x 2)  (x1  x 2)  ( x1  x 2)
x1  Τ
UNSAT
x1  F
x2  x2
UNSAT
UNSAT
x2  Τ
UNSAT
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x2  F
UNSAT
x2  Τ
UNSAT
x2  x2
x2  F
UNSAT
4
DPLL algorithm
dpll(Clauses C){
C = simplify(C);
if ( C contains contradictions ) return UNSAT ;
if ( no more free variables in C) return SAT ;
choose a free variable v in C ;
C1 = substitute(C, v, T ) ;
if (SAT == dpll(Clauses C1))
return SAT ;
else {
C2 = substitute(C, v, F ) ;
return dpll(Clauses C2) ;
}
}
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DPLL algorithm
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Two rules to simplify the CNF clauses
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Unit propagation rule
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and { x3 }, then x3  F
One literal rule
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If there is a clause contains only one literal, the
literal is forced to be true. Propagate this new
assignment immediately.
Given { x1  x3 , x1 }, deduce x1  F
Not used in most modern SAT solvers
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Modern SAT solvers
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Efficient unit propagation
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Efficient back-tracking
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Iterative algorithm
Almost constant cost back-tracking
Better heuristics on what to do next
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BCP(Boolean Constraints Propagation)
Conflict analysis
Look ahead heuristics
Fine tuning
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Restart, preprocessing,…
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Resolution and SAT problem
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Resolution
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Given two clauses l1  l 2  ...  lk and
j1  j 2  ...  lk , derive l1  l 2  ...  j1  j 2  ...
From x and x, derive empty clause
Theorem
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A set of CNF clauses is unsatisfiable if and only if
there is a resolution derivation of empty clause
Proof: Based on induction.
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From DPLL search tree to
resolution proof
( x1  x 2)  (x1  x 2)  (x1  x 2)  ( x1  x 2)
UNSAT
x1  x 2
x2  x2
1 
x
x 21

x1x
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x1  Τ
x1 , x1
x1  F
UNSAT
UNSAT
x1  x1
Observation:
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A contraction  a resolution
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Naïve proof generation
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Record the entire proof tree and reconstruct
the resolution proof
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Dump search trace
Modern SAT solvers employ unit propagation
Modern SAT solvers employ some learning
techniques
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New clauses are learned and added into the CNF
clause set.
New clause could be used later
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Unit propagation (BCP) and
resolution
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Given a unit clause x , only if there is a
clause x  y1  y  ... , we can generate a
new clause y1  y  ...
2
2
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A special case of resolution
New unit clauses will be linked to their
source clauses (implication graph)
Modern SAT solvers spent most of the
time on BCP
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Learning and resolution:
Implication graph
x5  T
......
x2  T
1) x1  x 2
2) x1  x3
3) x 2  x3  x 4
4) x 4  x5
5) x 4  x5  x 6
......
1)
5)
3)
x1  F
5)
x5  F
x4  T
2)
3)
4)
x5  F
x3  T
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Implication graph
x6  F
......
x2  T
1) x1  x 2
2) x1  x3
3) x 2  x3  x 4
4) x 4  x5
5) x 4  x5  x 6
......
1)
5)
3)
x1  F
5)
x5  F
x4  T
2)
3)
4)
x5  F
x3  T
The contraction is due to: ( x 4  x 6) or x4  x6
We can learn a clause x4  x6, which is the result of resolution of
clause 4) and 5)
contraction  resolution
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Implication graph:
more learned clause
x6  F
......
x2  T
1) x1  x 2
2) x1  x3
3) x 2  x3  x 4
4) x 4  x5
5) x 4  x5  x 6
......
1)
5)
3)
x1  F
5)
x5  F
x4  T
2)
3)
4)
x5  F
x3  T
Another clause
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( x 2  x3  x 6)
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Implication graph:
more learned clause
x6  F
......
x2  T
1) x1  x 2
2) x1  x3
3) x 2  x3  x 4
4) x 4  x5
1)
5)
3)
x1  F
5) x 4  x5  x 6
5)
x5  F
x4  T
2)
......
3)
4)
x5  F
x3  T
Yet another clause
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x1  x6
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Implication graph:
more learned clause
x6  F
......
x2  T
1) x1  x 2
2) x1  x3
3) x 2  x3  x 4
4) x 4  x5
1)
3)
x1  F
......
6) x 4  x 6
4) 5)
7) x 2  x 3  x 6
6) 3)
8) x1  x 2  x 6
7) 2)
9) x1  x 6
8) 1)
5)
x5  F
x4  T
2)
5) x 4  x5  x 6
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5)
3)
4)
x5  F
x3  T
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Resolution and learned clauses
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Conflict clause
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The learned clause that are fed back into SAT
solver
Conflict clauses can be seen as the result of
some resolution
Conflict clauses are redundant
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Could be deleted later
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Proof generation
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Whenever a learned clause is generated,
record the clause and dump the reason for
that clause
Recode all variables assigned at root level
and the reasons
Re-construct the resolution proof from the
last contradiction by searching the dumped
trace and recorded information
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Learned clauses are constructed if necessary
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Application of proofs:
Small unsat core
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Some applications require small unsat core
Given a set of unsatisfiable CNF clauses C, if
S  C and t.t  S  t is satisfiabl e , then S is a
minimal unsat core of C
If s is the smallest among all minimal unsat
core, then s is the minimum unsat core
It is difficult to obtain a minimum unsat core
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Small unsat core from proofs
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By Zhang et al.
Given a unsatisfiable set of clauses C
Collect all clauses appears in the unsat proof,
say C1, which must be unsatifiable
Run SAT solver on C1 and collect all clauses
appears in the unsat proof of C1, say C2
Repeat until reach a fixpoint
There is no guarantee that the fixpoint is
minimal
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Another kind of proof
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Given a set of unsatisfiable CNF clauses F and all
conflict clauses C found by the SAT solver, a conflict
pair {x, x} can be derived by performing BCP on
F C only
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If a set of CNF clauses is unsatisfiable, a conflict pair can be
derive by resolution
All the result of non-BCP resolution are in C
The proof checking is time consuming
The checking procedure could give a small unsat core
as a by-product
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Some future problems
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A standard proof format
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Dump of the resolution proof
Dump of conflict clauses
Better proof generation and checking
Big proofs
Proofs from SMT solvers
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More difficult
There is no general standard format
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Summary
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Proofs from SAT solvers are useful
Modern CNF based SAT solvers can
generate proofs with little overhead
Construct the resolution proof might be
a problem for large cases
Small unsat core could be obtained
from proofs
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