Faster Extraction of High-Level
Minimal Unsatisfiable Cores
1
Agenda

Introduction and motivation

Optimizations
◦ 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
2
High-Level UC

Given:
1.
2.
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
3
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.
4
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’)
7
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
8
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!
9
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
10
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:
11
Resolution Refutation
C22
C17
Legend:
C2
C19
C18
C11
C10
C1
C23=()
C3
Input clauses
C4
C21
C5
C16
C15
C14
C13
C12
C20
C6
C7
C8
C9
Derived clauses
12
Resolution Refutation
C22
C17
Legend:
C2
C19
C18
C11
C10
C1
C23=()
C3
Input clauses
C4
C21
C5
C16
C15
C14
C13
C12
C20
C6
C7
C8
C9
Derived clauses
Empty clause cone: { C4 C5 C6 C7 C13 C14 C19 C20 C23 }
Unsat Core: { C4 C5 C6 C7 }
13
Resolution Refutation
C22
C17
Legend:
C19
C18
C11
C10
C1
C23=()
C2
C3
C4
Empty Clause Cone
C21
C5
C16
C15
C14
C13
C12
C20
C6
C7
C8
C9
Unsat Core
Empty clause cone: { C4 C5 C6 C7 C13 C14 C19 C20 C23 }
Unsat Core: { C4 C5 C6 C7 }
14
Resolution with ICs
C22
C17
C23=()
C11
C10
C19
C18
C1
C2
C3
Legend:
Input Clauses
Derived Clauses
C4
IC1
C21
C5
IC2
C16
C15
C14
C13
C12
C20
C6
C7
C8
C9
Remainder
Derived clauses
15
Resolution with ICs
C22
C17
C23=()
C11
C10
C19
C18
C3
C4
C5
C16
C15
C14
C13
C12
C21
C20
C1
C2
Legend:
Input Clauses
IC1
IC2
Remainder
Derived Clauses
IC1
IC2
Remainder
C6
C7
C8
C9
16
HLUC
C22
C17
C2
Legend:
C19
C18
C11
C10
C1
C23=()
C3
IC1
IC2
C4
C21
C5
C16
C15
C14
C13
C12
C20
C6
C7
C8
C9
Remainder
HLUC: { IC2 }
17
HLMUC Algorithm [N’10]
 - remainder,
 - ICs
Solve  Æ 
UNSAT
Initialization:
’ =  = 
’ = HLUC
SAT
Assumption:
 Æ  is UNSAT
 = ’
 =  \ ICi

No unchecked ICs
Remove one ICi ϵ 
that wasn’t already removed
18
Contribution of this Work

Seven optimizations for single HLMUC.
◦ improved run time and smaller HLMUC

Comparison between resolution and selector variables
solvers.
19
A. Partial Resolution

Observations:
◦ 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
20
A. Partial Resolution
C22
C17
Legend:
C19
C18
C11
C10
C1
C23=()
C2
C3
IC1
IC2
C4
C21
C5
C16
C15
C14
C13
C12
C20
C6
C7
C8
C9
Remainder
21
A. Partial Resolution
C22
C17
Legend:
C19
C18
C11
C10
C1
C23=()
C2
C3
IC1
IC2
C4
C21
C5
C16
C15
C14
C13
C12
C20
C6
C7
C8
C9
Not Needed
22
A. Partial Resolution
C22
C23=()
C17
C20
C21
C1
Legend:
C16
C15
C10
C7
IC1
C8
IC2
23
A. Partial Resolution - Summary
C22
C23=()
C17
C20
C21
C1
Legend:
C16
C15
C10
C7
IC1
C8
IC2
Keep only the needed resolutions
24
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
25
B. Selective clause minimization
v
c = (¬v
c = (¬v
c = (¬v
c = (¬v
c1= (¬
1
2
2
3
4
4
5
5
1
v)
Çv )
Çv )
Çv )
Ç ¬v Ç ¬v
Ç
2
3
5
6
3
4Ç
v
¬ 6)
26
B. Selective clause minimization
v
c = (¬v
c = (¬v
c = (¬v
c = (¬v
c1= (¬
1
2
2
3
4
4
5
5
1
v)
Çv )
Çv )
Çv )
Ç ¬v Ç ¬v
Ç
2
v1
c1
v2
c2
v3
3
5
6
3
4Ç
v
¬ 6)
27
B. Selective clause minimization
v
c = (¬v
c = (¬v
c = (¬v
c = (¬v
c1= (¬
1
2
2
3
4
4
5
5
1
v)
Çv )
Çv )
Çv )
Ç ¬v Ç ¬v
Ç
2
c1
v1
v2
3
5
v4
6
3
4Ç
v
c2
v3
c5
c5
c5
¬v6
c3
v5
c4
v6
¬ 6)
28
B. Selective clause minimization
v
c = (¬v
c = (¬v
c = (¬v
c = (¬v
c1= (¬
1
2
2
3
4
4
5
5
1
v)
Çv )
Çv )
Çv )
Ç ¬v Ç ¬v
Ç
2
c1
v1
v2
3
5
v4
6
3
4Ç
v
c2
v3
c5
c5
c5
¬v6
c3
v5
c4
v6
¬ 6)
29
B. Selective clause minimization
v
c = (¬v
c = (¬v
c = (¬v
c = (¬v
c1= (¬
1
2
2
3
4
4
5
5
1
v)
Çv )
Çv )
Çv )
Ç ¬v Ç ¬v
Ç
2
c1
v1
v2
3
5
v4
6
3
4Ç
v
c2
v3
c5
c5
c5
¬v6
c3
v5
c4
v6
¬ 6)
1-UIP based conflict analysis:
c6= (¬
v
1
Ǭ
v Ç ¬v )
3
4
30
B. Selective clause minimization
v Çv )
c = (¬v Ç v )
c1= (¬
2
1
2
2
v v
1
3
¬v
3

¬v
1
3
31
B. Selective clause minimization
v Çv )
c = (¬v Ç v )
c1= (¬
1
2
c6= (¬
2
2
v
1
v v
1
¬v
3
3

¬v
1
3
Ǭ
v Ç ¬v )
3
4
c6= (¬
v
1
v
Ç ¬ 4)
32
B. Selective clause minimization
v
c = (¬v
c = (¬v
c = (¬v
c = (¬v
c = (¬v
c1= (¬
1
2
2
3
4
4
5
5
1
6
1
v)
Çv )
Çv )
Çv )
Ç ¬v Ç ¬v
Ç ¬v )
Ç
2
c1
v1
v2
3
5
v4
6
3
4Ç
v
c2
v3
c5
c5
c5
¬v6
c3
v5
c4
v6
¬ 6)
4
33
B. Selective clause minimization
v
c = (¬v
c = (¬v
c = (¬v
c = (¬v
c1= (¬
1
2
2
3
4
4
5
5
1
v)
Ç v ) (IC)
Çv )
Çv )
Ç ¬v Ç ¬v
Ç
2
c1
v1
v2
3
5
v4
6
3
4Ç
v
c2
v3
c5
c5
c5
¬v6
c3
v5
c4
v6
¬ 6)
34
B. Selective clause minimization
v Çv )
c = (¬v Ç v ) (IC)
c1= (¬
2
1
2
2
3
With minimization using c2:
c6= (¬
v
1
v
Ç ¬ 4) (IC)
Without minimization:
c6= (¬
v
1
Ǭ
v Ç ¬v ) (remainder)
3
4
35
B. Selective clause minimization

Suggested solution:
◦ Disable minimization if it adds dependency on IC-clause.
v
v Ç ¬v ) instead of c = (¬v

c6= (¬

Disabling minimization 
reduces #derived IC-clauses 
reduces #IC-clauses in UC and
finds HLMUC faster
1
Ǭ
3
4
6
1
v
Ç ¬ 4)
36
C. Postpone IC-propagations

Change BCP order
Run BCP
Conflict
Analyze Conflict
no
implications
Next Operations
37
C. Postpone IC-propagations

Change BCP order
Run BCP over non ICclause
found
implication
Conflict
no
implications
Propagate a single ICclause
Analyze Conflict
Conflict
no
implications
Next Operations
38
C. Postpone IC-propagations
Increase chances to get conflicts in remainder
 Decreases number of derived IC-clauses
 Decreases number of IC-clauses in UC.

39
E. Selective Learning
implication
IC-clause implication
@2
@2
@5
@5
@5
@5
@5
@5
@5
@5
@5
@3
@5
X
@3
40
E. Selective Learning
implication
IC-clause implication
@2
@2
@5
@5
@5
@5
@5
@5
@5
@5
@5
@3
@5
X
@3
41
E. Selective Learning
implication
IC-clause implication
@2
@2
@5
@5
@5
@5
@5
@5
@5
@5
@5
@3
@5
X
@3
Learnt clause should be marked “IC-clause”
42
E. Selective Learning
We refrain from learning IC-clauses
 Instead,

◦ do not learn it
◦ learn a (non-asserting) remainder clause
◦ make a decision

How ?
43
E. Selective Learning

How ?
◦ Treat the last IC-clause implication as decision
◦ Perform new 1-UIP conflict analysis
◦ The learnt clause is ‘remainder’
44
E. Selective Learning
implication
IC-clause implication
@2
@2
@5
@5
@5
@5
@5
@5
@5
@5
@5
@3
@5
X
@3
45
E. Selective Learning
implication
IC-clause implication
@2
@2
@5
@5
@5
@6
@5
@5
@5
@6
@5
@3
@6
X
@3
46
E. Selective Learning
implication
IC-clause implication
@2
@2
@5
@5
@5
@6
@5
@5
@5
@6
@5
@3
@6
X
@3
47
E. Selective Learning
implication
IC-clause implication
@2
@2
@5
@5
@5
@6
@5
@5
@5
@6
@5
@3
@6
X
@3
48
G. Removal Strategy
Recall:
In each iteration one IC is
chosen to be removed.
Solve  Æ 
UNSAT
’ = HLUC
SAT
 = ’
 =  \ ICi

No unchecked ICs
Remove one ICi ϵ 
that wasn’t already removed
G. Removal Strategy
What is the effect of the removal order?
 Which IC should we remove first?

50
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
51
Experimental Results

Benchmark Set:
◦
◦
◦
◦

Industrial set of problems from Intel
Average #clauses = 2,572,270
Average #ICs = 3804
Average #IC-clauses = 96568 (6% of #clauses)
Machines:
◦ Intel® Xeon® 4Ghz 32Gb of memory
52
Experimental Results
3600
3000
Picomus
2400
N'10
A
AB
1800
ABC
ABCE
ABCDE
1200
ABCDEF
ABCDEFG
600
0
1
11
21
31
41
51
61
71
81
91
53
Experimental Results (zoom)
3600
3000
2400
N'10
A
AB
ABC
1800
ABCE
ABCDE
ABCDEF
1200
ABCDEFG
600
0
60
65
70
75
80
85
90
95
100
54
Experimental Results
N'10 vs. Optimized
3600
3000
A-F
2400
1800
1200
600
0
0
600
1200
1800
2400
3000
3600
Nadel'10

Improves N’10:
◦ Time: 44913 sec vs. 20453 sec (x2.19 speedup)
◦ HLMUC size: 18995 vs. 5090 (73.2% reduction)
55
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-ICclause into IC-clause
 Selector variable prevents the minimization
56
Resolution vs. Selector variables
v
c = (¬v
c = (¬v
c = (¬v
c = (¬v
c1= (¬
1
2
2
3
4
4
5
5
1
v)
Ç v ) (IC)
Çv )
Çv )
Ç ¬v Ç ¬v
Ç
2
c1
v1
v2
3
5
v4
6
3
4Ç
v
c2
v3
c5
c5
c5
¬v6
c3
v5
c4
v6
¬ 6)
57
Resolution vs. Selector variables
s
v
c = (¬v
c = (¬v
c = (¬v
c = (¬v
c1= (¬
1
2
2
3
4
4
5
5
1
v)
Ç v Ç ¬s) (IC)
Çv )
Çv )
Ç ¬v Ç ¬v Ç ¬v )
Ç
2
c1
v1
v2
3
5
6
3
4
v4
c2
v3
c5
c5
c5
¬v6
c3
v5
c4
v6
6
Selector variables are pure 
cannot be removed 
every IC-clause have at least one
58
Resolution vs. Selector variables
Resolution-based:
v Çv )
c = (¬v Ç v ) (IC)
c1= (¬
2
1
2
2
v v
1
v Çv )
c = (¬v Ç v Ç ¬s) (IC)
2
¬v
3

¬v
1
3
Cannot be resolved
away
Selector-variables-based:
c1= (¬
3
1
2
2
3
v v
1
3
59
Resolution vs. Selector variables
3600
3000
2400
Selector-based
N'10
1800
A
AB
ABCDEF
1200
600
0
1
11
21
31
41
51
61
71
81
91
60
Resolution vs. Selector variables
3600
3000
2400
Selector-based
N'10
1800
A
AB
ABCDEF
1200
600
0
60
70
80
90
61
Resolution vs. Selector variables
Selector-variables vs. Resolution A-F
3600
3000
A-F
2400
1800
1200
600
0
0
600
1200
1800
2400
3000
3600
Selector-variables

Comparison to selector-variables:
◦ Time: 28032 sec vs. 20453 sec (x1.37 speedup)
◦ HLMUC size: 12090 vs. 5090 (57.8% reduction)
62
Thank You!
63