Efficient MUS Extraction with Resolution

1
Efficient MUS Extraction with Resolution
Minimal Unsatisfiable Set

• Alexander Nadel Intel, Haifa
• Vadim Ryvchin Intel Technion, Haifa
• Ofer Strichman Technion, Haifa

2
Minimal Unsatisfiable Set (MUS)

• Given an unsatisfiable CNF ?, find a minimal
  (irreducible) set of clauses à µ ? such that à is
  unsatisfiable.
• Two main schools of finding cores
• Assumptions-based (Een Sorensson, 2003)
• Resolution-based (Zhang et al. 2003)

3
Deletion-based minimization
Initially Roots are unmarked
? Roots
Return Roots
All marked
Choose unmarked clause c 2 Roots
Remove(?, c)
SAT(?) ?
no
yes
mark c
Roots core
Works for both assumptions-based and
resolution-based
4
Derived clauses are also marked
Optimization 1
?

• Maintain partial resolution proofs
• Only part of proof emanating from unmarked clauses

5
Derived clauses are also marked
Optimization 2
?
m
m
m
m
m
m

• Postpone propagation of unmarked clauses
• These pull unmarked clauses into the proof

6
Derived clauses are also marked
Optimizations 3,4
?
m
m
m
m
m
m
and two other optimizations we used for group
MUS RS12
7
Model Rotation

• Belov, A., Marques-Silva, J. Accelerating MUS
  extraction with recursive model rotation. In
  FMCAD11. (2011)

? Roots
Return Roots
All marked
Choose unmarked clause c 2 Roots
Remove(?, c)
Rotation mark more clauses
SAT(?) ?
no
yes
mark c
Roots core
8
Model Rotation

• ? is unsat, but ² ?/c
• l à l for some l 2 c
• if (UnsatSet(?, ) c Æ c is unmarked) then
• Mark c
• Apply recursively with (?, c, )

Searching for other clauses that can be marked
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Model Rotation, eager
Optimization 5

• ? is unsat, but ² ?/c
• l à l for some l 2 c
• if (UnsatSet(?, ) c Æ c is unmarked) then
• Mark c
• Apply recursively with (?, c, )

in the current call
New assignment ) new starting point for the search
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The impact of E-rotation
Optimization 5
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Redundancy removal BS11
Optimization 6

• S is unsat ) (SAT(S / c) , SAT((S / c) Æ c)
• Hence, add c literals as assumptions.
• This is implemented already in MUSer-2 BS11.
• Our improvement (path falsification)
• Add c, c1, cn literals as assumptions.

12
Path falsification
Optimization 6

• Vertex cut separates ? from the roots
• Implies whats on its left
• ) must be unsat

?
c
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Path falsification
SAT(C/c)
Optimization 6

• Consider , ² Roots / c
• cannot satisfy a cut )
• ² (c Æ c1 Æ c2) Ç (c Æ c1 Æ c3)
• ) ² c Æ c1

?
c2
c1
c
c3
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Results
295 benchmarks of the 2011 MUS competition
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HaifaMUC vs. Minisatbb
Minisatbb
HaifaMUC
Lagniez, Biere, SAT13
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Heuristics, Heuristics,

• Other heuristics
• Selective Clause minimization
• Refrain from marking derived clauses
• Reclassify derived unmarked clauses
• Based on similar techniques for the Group MUS
  problem in RS - SAT11.

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Model Rotation BS11

• Precondition ? is unsat, ² ?/c
• Rotate(formula S, clause c, assignment )
• for all x 2 Var(c) do
• x à x // swap assignment of x
• if (UnsatSet(S, ) c Æ c is unmarked)
  then
• Mark c
• Rotate (S, c, )

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Two main schools of finding cores

• Assumptions-based
• Assumptions clause selectors

• Resolution-based
• Recreate the resolution proof
• Core roots