1. Deriving small unsatisfiable cores with dominators
Ofer Strichman
Joint work with
Maya Koifman and Roman Gershman

2. Unsatisfiable cores
Problem: given an unsatisfiable CNF, find a ‘small’ subset of clauses that is still unsatisfiable.
Hard problems:
Minimum Unsatisfiable core (2-complete) [G05]
Minimal Unsatisfiable core (Dp-complete) [LS04, OMASM04, H05, NDH06]

3. Goal
Typically part of a larger system (e.g., proof-based abstraction/refinement)
Our Goal: a cost-effective algorithm for reducing the core size
Reduction size
`Velocity’ = clauses / sec.

4. A fixpoint approach [ZM03]
run-till-fix(φ) {
Repeat forever:
U := unsatcore(φ)
if U = φ return U;
φ = U }

5. SAT solvers are complete resolution engines
Specifically, if the formula is unsatisfiable:
… they can output a resolution proof ending with the empty clause.
Our approach is based on analyzing this graph.

6. Resolution graph L : Empty clause Inferred clauses Original clauses
learning
Original clauses
L :
Unsatisfiable core

7. Dominators
A vertex d dominates another vertex m ≠ d, if every path from m to the sink-node contains d.
m is called a minion of d
( ) d 12
Finding dominance relation:
Lengauer – Tarjan: O(|E|log|V|) m m

8. Transforming the resolution graph
Observation: If d is derivable without its minions M, then M can be removed from the core. d 12
L :
L’(d)

9. Transforming the resolution graph
Is there a proof of node d from L’(d)? d 12
L :
L’(d)

10. An alternative proof.
Suppose we have an alternative proof of d from L’(d): d 12
L :
L’(d)

11. Then we can... Embed the new proof, and re-iterate.
But: how do we find a resolution proof of d? 12
L :
L’(d)

12. Transforming the resolution graph
Q: How do we find a resolution proof L’(d) ` d?
Recall: SAT can only generate a proof of unsatisfiability (the empty clause).
A: Rely on the equivalence L’(d) ` d , L’(d) ^ !d ` ()
Problem: proof transformation

13. Proof transformation !d d = (z1 z2) L’(d) ^ !(z1 z2) ` ()

14. Rewriting the proof For every assumption (:z): For every addition of z
rewrite
rewrite

15. Proof transformation !d d = (z1 z2) L’(d) ^ !(z1 z2) ` ()

16. Proof transformation
L’(d) ^ !(z1 z2) ` () L’(d) ` (z1 z2)
L’ (d) !d

17. Proof transformation
L’(d) ^ !(z1 z2) ` () L’(d) ` (z1 z2)
L’ (d) !d

18. Proof transformation
L’(d) ^ !(z1 z2) ` () L’(d) ` (z1 z2)
L’ (d) !d

19. Proof transformation
L’(d) ^ !(z1 z2) ` () L’(d) ` (z1 z2)
L’ (d) !d

20. Proof transformation !d L’(d) ^ !(z1 z2) ` () L’(d) ` (z1 z2)
Result is a ‘Clause Implication Graph’
L’ (d) !d

21.  L’ (d) ^ !d ` () L’ (d) ` d Create resolution graph R
Create priority queue of R’s dominators
None
Select next
dominator d
Embed new proof in R
Output: current
leaves of R
SAT (L’(d) ^ !d)
yes 
Remove old proof from R No
L’ (d) ^ !d ` ()
Transform
proof into
L’ (d) ` d

22. Using Incremental SAT
Reuse all clauses not on a path from the minions to the sink node. 14 d

23. Results – Accumulated # removed clauses
71 industrial benchmarks (timeout – 1 hour)

24. Trim-till-fix
run-till-fix

25. 71 industrial benchmarks (timeout – 1 hour)
Core reduction:
Trim-till-fix
Run-till-fix

26. Conclusions Trim-till-fix Room for further research More continues
Removes more clauses
Room for further research
Various combinations with Run-till-fix
Different orderings of the dominators
Removing variables, not clauses …

27. Acceleration