1. Faster Extraction of High-Level Minimal Unsatisfiable Cores
Ryvchin Vadim and Ofer Strichman
Technion, Israel
SAT’11 Conference Ann Arbor, USA
June 21, 2011

2. Agenda Introduction and motivation Optimizations Experimental results
A. Partial Resolution
B. Selective clause minimization
C . Postponed IC-propagation
E. Selective learning of IC-clauses
G. Removal Strategy
Experimental results
Resolution vs. Selector variables

3. High-Level UC
Given:
A set of interesting constraints (IC)  = { IC1, IC2 , …, ICm }, and
The remainder 
The set    is a high-level UC (HLUC) if    is unsatisfiable
HLUC is minimal (HLMUC) if removal of any IC makes    satisfiable

4. Examples Abstraction-refinement in model checking [MA’03]
Latches in core define the next abstraction.
Compositional Formal Equivalence Checking (FEC) [CGLNR’10]
Decompose the compared circuits to blocks.
Assume inputs to blocks are the same
Assumptions in the core need to be proved.
A very popular model checking technique is based on abstraction/refinement. A BDD-based model checker checks an abstract model, where the abstraction is defined by the set of latches that we replace with free inputs. B. If the property fails, a SAT solver is ran on the concrete model to the depth of the ce. C. If there is no BUG - the latches that participate in the proof define the abstraction in the next iteration. D: clearly, then we want to have as few latches in the proof as possible

5. Traditional UC Extraction: Stage 1: Translate to Clauses
An interesting constraint
The remainder (the rest of the formula)
Each small square is a propositional clause, e.g. (a + b’)

6. Traditional UC Extraction: Stage 2: Extract a Clause-Level UC
An interesting constraint
The remainder (the rest of the formula)
Colored squares belong to the clause-level UC

7. Traditional UC Extraction: Stage 3: Map UC back to ICs
An interesting constraint
The remainder (the rest of the formula)
The UC contains three interesting constraints!

8. A Mismatch between Mainstream Research and the Needs of Real-World Applications
Real-world applications: reduce # interesting constraints in the core
Latches/gates for abstraction refinement
Assumptions for compositional FEC
Vast majority of existing algorithms: reduce # of clauses in the core
19/21 papers on UC extraction only consider clause-level UC extraction

9. Small/Minimal Clause-Level UC  Small/Minimal High-Level UC
A small clause-level UC, but the high-level UC is the largest possible:
A large clause-level UC, but the high-level UC is empty:

10. Resolution Refutation
C22
C23=()
C17
C18
C19
C20
C21
C10
C11
C12
C13
C14
C15
C16 C1 C2 C3 C4 C5 C6 C7 C8 C9
Legend:
Input clauses
Derived clauses

11. Resolution Refutation
C22
C23=()
C17
C18
C19
C20
C21
C10
C11
C12
C13
C14
C15
C16 C1 C2 C3 C4 C5 C6 C7 C8 C9
Legend:
Input clauses
Derived clauses
Empty clause cone: { C4 C5 C6 C7 C13 C14 C19 C20 C23 }
Unsat Core: { C4 C5 C6 C7 }

12. Resolution Refutation
C22
C23=()
C17
C18
C19
C20
C21
C10
C11
C12
C13
C14
C15
C16 C1 C2 C3 C4 C5 C6 C7 C8 C9
Legend:
Empty Clause Cone
Unsat Core
Empty clause cone: { C4 C5 C6 C7 C13 C14 C19 C20 C23 }
Unsat Core: { C4 C5 C6 C7 }

13. Resolution with ICs C22 C23=() C17 C18 C19 C20 C21 C10 C11 C12 C13 C14
Legend:
Input Clauses
IC1
IC2
Remainder
Derived Clauses
Derived clauses

14. Resolution with ICs C22 C23=() C17 C18 C19 C20 C21 C10 C11 C12 C13 C14
Legend:
Input Clauses
IC1
IC2
Remainder
Derived Clauses
IC1
IC2
Remainder

15. HLUC C22 C23=() C17 C18 C19 C20 C21 C10 C11 C12 C13 C14 C15 C16 C1 C2
Legend:
IC1
IC2
Remainder
HLUC: { IC2 }

16. HLMUC Algorithm [N’10]
Solve  Æ 
’ = HLUC
= ’
 =  \ ICi
UNSAT
SAT
Remove one ICi ϵ 
that wasn’t already removed 
No unchecked ICs
 - remainder,  - ICs Initialization: ’ =  =  Assumption:  Æ  is UNSAT

17. Contribution of this Work
Seven optimizations for single HLMUC.
improved run time and smaller HLMUC
Comparison between resolution and selector variables solvers.

18. A. Partial Resolution Observations: Suggestion: Result:
IC-clauses usually between 5-15% of the problem clauses
We do not need the whole resolution table
Suggestion:
Keep only clauses relevant to IC resolutions
Result:
The size of the resolution graph reduced
Very effective on large CNFs

19. A. Partial Resolution C22 C23=() C17 C18 C19 C20 C21 C10 C11 C12 C13
Legend:
IC1
IC2
Remainder

20. A. Partial Resolution C22 C23=() C17 C18 C19 C20 C21 C10 C11 C12 C13
Legend:
IC1
IC2
Not Needed

21. A. Partial Resolution C22 C23=() C17 C20 C21 C10 C15 C16 C1 C7 C8
Legend:
IC1
IC2

22. A. Partial Resolution - Summary
C22
C23=()
C17
C20
C21
C10
C15
C16 C1 C7 C8
Legend:
IC1
IC2
Keep only the needed resolutions

23. B. Selective clause minimization
Technique for shrinking conflict clauses
The algorithm is based on traversing the resolution DAG backward from each literal in the learned clause
The problem:
May turn a non-IC-clause into a shorter IC-clause

24. B. Selective clause minimization
c1= (¬v1 Ç v2) c2= (¬v2 Ç v3) c3= (¬v4 Ç v5) c4= (¬v5 Ç v6) c5= (¬v1 Ç ¬v3 Ç ¬v4 Ç ¬v6)

25. B. Selective clause minimization
c1= (¬v1 Ç v2) c2= (¬v2 Ç v3) c3= (¬v4 Ç v5) c4= (¬v5 Ç v6) c5= (¬v1 Ç ¬v3 Ç ¬v4 Ç ¬v6) c1 c2 v1 v2 v3

26. B. Selective clause minimization
c1= (¬v1 Ç v2) c2= (¬v2 Ç v3) c3= (¬v4 Ç v5) c4= (¬v5 Ç v6) c5= (¬v1 Ç ¬v3 Ç ¬v4 Ç ¬v6) c1 c2 v1 v2 v3 c5 c5
¬v6 c5 v4 c3 c4 v5 v6

27. B. Selective clause minimization
c1= (¬v1 Ç v2) c2= (¬v2 Ç v3) c3= (¬v4 Ç v5) c4= (¬v5 Ç v6) c5= (¬v1 Ç ¬v3 Ç ¬v4 Ç ¬v6) c1 c2 v1 v2 v3 c5 c5
¬v6 c5 v4 c3 c4 v5 v6

28. B. Selective clause minimization
c1= (¬v1 Ç v2) c2= (¬v2 Ç v3) c3= (¬v4 Ç v5) c4= (¬v5 Ç v6) c5= (¬v1 Ç ¬v3 Ç ¬v4 Ç ¬v6) 1-UIP based conflict analysis: c6= (¬v1 Ç ¬v3 Ç ¬v4) c1 c2 v1 v2 v3 c5 c5
¬v6 c5 v4 c3 c4 v5 v6

29. B. Selective clause minimization
c1= (¬v1 Ç v2) c2= (¬v2 Ç v3)
v1  v3
¬v3  ¬v1

30. B. Selective clause minimization
c1= (¬v1 Ç v2) c2= (¬v2 Ç v3) c6= (¬v1 Ç ¬v3 Ç ¬v4) c6= (¬v1 Ç ¬v4)
v1  v3
¬v3  ¬v1

31. B. Selective clause minimization
c1= (¬v1 Ç v2) c2= (¬v2 Ç v3) c3= (¬v4 Ç v5) c4= (¬v5 Ç v6) c5= (¬v1 Ç ¬v3 Ç ¬v4 Ç ¬v6) c6= (¬v1 Ç ¬v4) c1 c2 v1 v2 v3 c5 c5
¬v6 c5 v4 c3 c4 v5 v6

32. B. Selective clause minimization
c1= (¬v1 Ç v2) c2= (¬v2 Ç v3) (IC) c3= (¬v4 Ç v5) c4= (¬v5 Ç v6) c5= (¬v1 Ç ¬v3 Ç ¬v4 Ç ¬v6) c1 c2 v1 v2 v3 c5 c5
¬v6 c5 v4 c3 c4 v5 v6

33. B. Selective clause minimization
c1= (¬v1 Ç v2) c2= (¬v2 Ç v3) (IC) With minimization using c2: c6= (¬v1 Ç ¬v4) (IC) Without minimization: c6= (¬v1 Ç ¬v3 Ç ¬v4) (remainder)

34. B. Selective clause minimization
Suggested solution:
Disable minimization if it adds dependency on IC-clause.
c6= (¬v1 Ç ¬v3 Ç ¬v4) instead of c6= (¬v1 Ç ¬v4)
Disabling minimization 
reduces #derived IC-clauses 
reduces #IC-clauses in UC and
finds HLMUC faster

35. C. Postpone IC-propagations
Change BCP order
Run BCP
Conflict
Analyze Conflict
no implications
Next Operations

36. C. Postpone IC-propagations
Change BCP order
Run BCP over non IC-clause
Conflict
Analyze Conflict
found implication
no implications
Propagate a single IC-clause
Conflict
no implications
Next Operations

37. C. Postpone IC-propagations
Increase chances to get conflicts in remainder
Decreases number of derived IC-clauses
Decreases number of IC-clauses in UC.

38. E. Selective Learning implication IC-clause implication X @2 @2 @5 @5@3 @3

39. E. Selective Learning implication IC-clause implication X @2 @2 @5 @5@3 @3

40. E. Selective Learning implication IC-clause implication X@2 @2 @5 @5 @5 @5 @5 @5 @5 X @5 @5 @5 @3 @3
Learnt clause should be marked “IC-clause”

41. E. Selective Learning We refrain from learning IC-clauses Instead,
do not learn it
learn a (non-asserting) remainder clause
make a decision
How ?

42. E. Selective Learning How ?
Treat the last IC-clause implication as decision
Perform new 1-UIP conflict analysis
The learnt clause is ‘remainder’

43. E. Selective Learning implication IC-clause implication X @2 @2 @5 @5@3 @3

44. E. Selective Learning implication IC-clause implication X @2 @2 @5 @5@6 @5 @5 @6 X @5 @5 @6 @3 @3

45. E. Selective Learning implication IC-clause implication X @2 @2 @5 @5@6 @5 @5 @6 X @5 @5 @6 @3 @3

46. E. Selective Learning implication IC-clause implication X @2 @2 @5 @5@6 @5 @5 @6 X @5 @5 @6 @3 @3

47. G. Removal Strategy Recall:
Solve  Æ 
’ = HLUC
= ’
 =  \ ICi
UNSAT
SAT
Remove one ICi ϵ 
that wasn’t already removed 
No unchecked ICs
Recall:
In each iteration one IC is chosen to be removed.

48. G. Removal Strategy What is the effect of the removal order?
Which IC should we remove first?

49. G. Removal Strategy Criterion: #clauses in UC Choose the one
that contains least clauses in UC
If UNSAT (not necessary), will converge faster
If UNSAT (not necessary), will likely allow further removals
that contains most clauses in UC
If SAT (necessary), clauses are added as ‘remainder’ fast

50. Experimental Results Benchmark Set: Machines:
Industrial set of problems from Intel
Average #clauses = 2,572,270
Average #ICs = 3804
Average #IC-clauses = (6% of #clauses)
Machines:
Intel® Xeon® 4Ghz 32Gb of memory

51. Experimental Results

52. Experimental Results (zoom)

53. Experimental Results Improves N’10:
Time: sec vs sec (x2.19 speedup)
HLMUC size: vs (73.2% reduction)

54. Resolution vs. Selector variables
Main cause for performance gap: clause minimization.
When clause minimization is off: similar results (time + size of HLMUC).
Selector variables: minimization does not turn non-IC- clause into IC-clause
Selector variable prevents the minimization

55. Resolution vs. Selector variables
c1= (¬v1 Ç v2) c2= (¬v2 Ç v3) (IC) c3= (¬v4 Ç v5) c4= (¬v5 Ç v6) c5= (¬v1 Ç ¬v3 Ç ¬v4 Ç ¬v6) c1 c2 v1 v2 v3 c5 c5
¬v6 c5 v4 c3 c4 v5 v6

56. Resolution vs. Selector variables
c1= (¬v1 Ç v2) c2= (¬v2 Ç v3 Ç ¬s) (IC) c3= (¬v4 Ç v5) c4= (¬v5 Ç v6) c5= (¬v1 Ç ¬v3 Ç ¬v4 Ç ¬v6) Selector variables are pure  cannot be removed  every IC-clause have at least one c1 c1 c2 c2 v1 v2 v3 c5 c5
¬v6 c5 v4 c3 c4 v5 v6

57. Resolution vs. Selector variables
Resolution-based: c1= (¬v1 Ç v2) c2= (¬v2 Ç v3) (IC) Selector-variables-based: c2= (¬v2 Ç v3 Ç ¬s) (IC)
v1  v3
¬v3  ¬v1
Cannot be resolved away
v1  v3

58. Resolution vs. Selector variables

59. Resolution vs. Selector variables

60. Resolution vs. Selector variables
Comparison to selector-variables:
Time: sec vs sec (x1.37 speedup)
HLMUC size: vs (57.8% reduction)

61. Thank You!