1. Efficient MUS Extraction with Resolution
Minimal Unsatisfiable Set
Efficient MUS Extraction with Resolution
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])
Assumptions-based: all clauses are guarded by clause selectors which are given as assumptions. Removing a clause = assuming the selector is true.
Resolution-based: analyzing the proof

3. Deletion-based minimization
Initially Roots are unmarked
 = Roots
Return Roots
All marked
Choose unmarked clause c 2 Roots
Remove(, c);
SAT() ?
yes no
Incremental setting.
Remove(\phi, c) means we remove it incrementally.
We will focus from hereon on resolution-based.
We want the core to use as little as possible ‘unmarked clauses’
mark c
Roots := core
Works for both ‘assumptions-based’ and ‘resolution-based’

4. Derived clauses are also marked
Optimization 1 ? m
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 [RS’12]

7. Choose unmarked clause c 2 Roots
Model Rotation*
* Belov, A., Marques-Silva, J.: Accelerating MUS extraction with recursive model rotation. In: FMCAD’11. (2011)
 = Roots
Return Roots
All marked
Choose unmarked clause c 2 Roots
Remove(, c);
Rotation: mark
more clauses
SAT() ?
yes no
mark c
Roots := core

8. Searching for other clauses that can be marked
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’, ®’); }
Note that \varphi \ c’ is SAT
The difference is: we are starting with a different assignment; hence we w
\begin{tabular}{|lll|}\hline
& E-rotation & Rotation \\
%Run-time & & \\
MUS size & & \\
Added clauses & & \\
Initial calls & & \\
Iterations & & \\
Iterations/calls & 1.19 & 1.09 \\
Clauses/iterations & 7.27 & 3.63 \\
Clauses/calls & 8.72 & 3.98 \\ \hline
\end{tabular}
\caption{Statistics comparing E-rotation and Rotation. Since the corresponding functions \alg{ERMR} and \alg{RMR} are recursive, we make the distinction between Initial calls (calls from line~\ref{step-call} in \alg{MUS}) and Iterations, which is the total number of calls to these functions.}\label{fig:erot}ant to keep developing that clause.
Searching for other clauses that can be marked

9. New assignment ) new starting point for the search
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
Note that \varphi \ c’ is SAT
The difference is: we are starting with a different assignment; hence we w
\begin{tabular}{|lll|}\hline
& E-rotation & Rotation \\
%Run-time & & \\
MUS size & & \\
Added clauses & & \\
Initial calls & & \\
Iterations & & \\
Iterations/calls & 1.19 & 1.09 \\
Clauses/iterations & 7.27 & 3.63 \\
Clauses/calls & 8.72 & 3.98 \\ \hline
\end{tabular}
\caption{Statistics comparing E-rotation and Rotation. Since the corresponding functions \alg{ERMR} and \alg{RMR} are recursive, we make the distinction between Initial calls (calls from line~\ref{step-call} in \alg{MUS}) and Iterations, which is the total number of calls to these functions.}\label{fig:erot}ant to keep developing that clause.
New assignment ) new starting point for the search

10. The impact of E-rotation
Optimization 5

11. Redundancy removal [BS’11]
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 [BS’11].
Our improvement (“path falsification”):
Add :c, :c1…, :cn literals as assumptions.
c = (1 2 3)
c1 = (-1 4)
c2 = (-4)

12. Path falsification Vertex cut: separates ? from the roots
Optimization 6
Vertex cut: separates ? from the roots
Implies what’s on its left
) must be unsat ? c

13. Path falsification Consider ®, ® ² Roots / c ® cannot satisfy a cut )
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

14. Results 295 benchmarks of the 2011 MUS competition
Base – HaifaMUC with deletion-based + clause-set refinement.
MUSer 2 – Belov & Marques-Silva, which includes rotation
Minisatbb – Lagniez & Biere – “Factoring-out assumptions to speed-up MUS extraction”
HaifaMUC (erot_AB…) – this article
295 benchmarks of the 2011 MUS competition

15. HaifaMUC vs. Minisatbb*
* Lagniez, Biere, SAT’13