SAT Genealogy
Alexander Nadel,
Intel, Haifa, Israel
The Technion, Haifa, Israel
July, 3 2012
1
Agenda



Introduction
Early Days of SAT Solving
Core SAT Solving






Conflict Analysis and Learning
Boolean Constraint Propagation
Decision Heuristics
Restart Strategies
Inprocessing
Extensions to SAT




Incremental SAT Solving under Assumptions
Simultaneous Satisfiability (SSAT)
Diverse Solutions Generation
High-level (group-oriented) MUC Extraction
2
Agenda



Introduction
Early Days of SAT Solving
Core SAT Solving






Conflict Analysis and Learning
Boolean Constraint Propagation
Decision Heuristics
Restart Strategies
Inprocessing
Extensions
SAT
We won’tto
use
implication graphs for explanation, but:




Incremental
SAT search
Solvingand
under
Assumptions
Duality between
resolution
Simultaneous Satisfiability (SSAT)
Diverse Solutions Generation
High-level (group-oriented) MUC Extraction
3
What is SAT?


Find a variable assignment (AKA solution or
model) that satisfies a propositional formula
or prove that there are no solutions
SAT solvers operate on CNF formulas:

Any formula can be reduced to a CNF
CNF Formula:
F = ( a + c ) ( b + c ) (a’ + b’ + c’ )
clause
negative
literal
positive literal
4
SAT: Theory and Practice

Theory:

SAT is the first known NP-complete problem



One can check a solution in polynomial time
Can one find a solution in polynomial time?


Stephen Cook, 1971
The P=NP question…
Practice:


Amazingly, nowadays SAT solvers can solve
industrial problems having millions of clauses and
variables
SAT has numerous applications in formal verification,
planning, bioinformatics, combinatorics, …
5
Approaches to SAT Solving

Backtrack search: DFS search for a solution


Look-ahead: BFS search for a solution



The baseline approach for industrial-strength
solvers. In focus today.
Helpful for certain classes of formulas
Recently, there were attempts of combining it with
backtrack search
Local search

Helpful mostly for randomly generated formulas
6
Early Days of SAT Solving
Agenda


Resolution
Backtrack Search
7
Resolution: a Way to Derive New
Valid Clauses

Resolution over a pair of clauses with exactly one pivot variable: a
variable appearing in different polarities:
•Known to be invented by Davis&Putnam, 1960
•Had been invented independently by Lowenheim in early 1900’s (as well as the DP
algorithm, presented next)
•According to Chvatal&Szemeredy, 1988 (JACM)
a + b + c’ + f
g + h’ + c + f
a + b + g + h’ + f
- The resolvent clause is a logical consequence of the two source clauses
DP Algorithm: Davis&Putnam,
1960
Remove the variables one-by-one by resolution
over all the clauses containing that variable
DP is sound and complete
(a + b + c)(b + c’ + f’)(b’ + e)
(a + c + e)(c’ + e + f)
(a + e + f)
SAT
(a + b) (a + b’) (a’ + c)(a’ + c’)
(a) (a’ + c)(a’ + c’)
(c)(c’)
()
UNSAT
9
Backtrack Search or DLL: DavisLogemann-Loveland, 1962
a+b
b’ + c
b’ + c’
a’ + b
Backtrack Search or DLL: DavisLogemann-Loveland, 1962
a+b
b’ + c
b’ + c’
a’ + b
a’
Backtrack Search or DLL: DavisLogemann-Loveland, 1962
a+b
a’
Decision level 1
b’ + c
b’ + c’
a’ + b
a is the decision variable;
a’ is the decision literal
Backtrack Search or DLL: DavisLogemann-Loveland, 1962
a+b
a’
b’ + c
b’ + c’
a’ + b
b’
Decision level 2
Backtrack Search or DLL: DavisLogemann-Loveland, 1962
a+b
a’
b’ + c
b’ + c’
a’ + b
b’
a+b
A conflict. A blocking clause – a clause, falsified by the
current assignment – is encountered.
Backtrack Search or DLL: DavisLogemann-Loveland, 1962
a+b
a’
b’ + c
b’ + c’
a’ + b
b’
a+b
b
Backtrack
and flip
Backtrack Search or DLL: DavisLogemann-Loveland, 1962
a+b
a’
b’ + c
b’ + c’
a’ + b
Decision level 1
b’
a+b
b
c’
b’ + c
Decision level 2
Backtrack Search or DLL: DavisLogemann-Loveland, 1962
a+b
a’
b’ + c
b’ + c’
a’ + b
b’
b
Decision level 1
a+b
c’
b’ + c
c
b’ + c’
Backtrack Search or DLL: DavisLogemann-Loveland, 1962
a+b
a’
a
b’ + c
b’ + c’
a’ + b
b’
a+b
b
c’
b’ + c
c
b’ + c’
Backtrack Search or DLL: DavisLogemann-Loveland, 1962
a+b
a’
a
b’ + c
b’ + c’
a’ + b
b’
a+b
b
c’
b’ + c
b
c
b’ + c’
Backtrack Search or DLL: DavisLogemann-Loveland, 1962
a+b
a’
a
b’ + c
b’ + c’
a’ + b
b’
a+b
b
c’
b’ + c
b
c
b’ + c’
c’
b’ + c
Backtrack Search or DLL: DavisLogemann-Loveland, 1962
a+b
a’
a
b’ + c
b’ + c’
a’ + b
b’
a+b
b
c’
b’ + c
b
c
b’ + c’
c’
b’ + c
c
b’ + c’
Backtrack Search or DLL: DavisLogemann-Loveland, 1962
a+b
a’
a
b’ + c
b’ + c’
a’ + b
b’
a+b
b
c’
b’ + c
b’
b
c
a’ + b
b’ + c’
c’
b’ + c
c
b’ + c’
Backtrack Search or DLL: DavisLogemann-Loveland, 1962
UNSAT!
a+b
a’
a
b’ + c
b’ + c’
a’ + b
b’
a+b
b
b’
b
c’
b’ + c
c
a’ + b
b’ + c’
c’
b’ + c
c
b’ + c’
Core SAT Solving: the
Principles



DLL could solve problems with <2000 clauses
How can modern SAT solvers solve problems
with millions of clauses and variables?
The major principles:

Learning and pruning


Locality and dynamicity


Block already explored paths
Focus the search on the relevant data
Well-engineered data structures

Extremely fast propagation
24
Agenda



Introduction
Early Days of SAT Solving
Core SAT Solving






Conflict Analysis and Learning
Boolean Constraint Propagation
Decision Heuristics
Restart Strategies
Inprocessing
Extensions to SAT




Incremental SAT Solving under Assumptions
Simultaneous Satisfiability (SSAT)
Diverse Solutions Generation
High-level (group-oriented) MUC Extraction
25
Duality between Basic Backtrack
Search and Resolution
One can associate a resolution derivation with
every invocation of DLL over an unsatisfiable
formula
Duality between Basic Backtrack
Search and Resolution
a+b
b’ + c
b’ + c’
a’ + b
Duality between Basic Backtrack
Search and Resolution
a+b
b’ + c
b’ + c’
a’ + b
a’
Duality between Basic Backtrack
Search and Resolution
a+b
a’
b’ + c
b’ + c’
a’ + b
b’
a+b
Duality between Basic Backtrack
Search and Resolution
a+b
a’
b’ + c
b’ + c’
a’ + b
b’
b
a+b
•A parent clause P(x) is associated with every flip
operation for variable x. It contains:
•The flipped literal
•A subset of previously assigned falsified literals
•The parent clause justifies the flip: its existence
proves that the explored subspace has no solutions
Duality between Basic Backtrack
Search and Resolution
a+b
a’
b’ + c
b’ + c’
a’ + b
b’
a+b
b
c’
b’ + c
Duality between Basic Backtrack
Search and Resolution
a+b
a’
b’ + c
b’ + c’
a’ + b
b’
a+b
b
c’
b’ + c
c
Duality between Basic Backtrack
Search and Resolution
a+b
a’
b’ + c
b’ + c’
a’ + b
b’
a+b
b
c’
b’ + c
c
b’ + c’
Duality between Basic Backtrack
Search and Resolution
a+b
a’
b’ + c
b’ + c’
a’ + b
b’
b
Pnew
b’
a+b
c’
P(c)
b’ + c
c
Pold
b’ + c’
• Backtracking over a flipped
variable x can be associated with
a resolution operation:
• P = P(x)  P
• P is to become the parent
clause for the upcoming flip
• P is initialized with the last
blocking clause
Duality between Basic Backtrack
Search and Resolution
a+b
a’
b’ + c
b’ + c’
a’ + b
b’
a
P(b)
a+b
Pnew
b
b’
c’
b’ + c
Pold
c
b’ + c’
• Backtracking over a flipped
variable x can be associated with
a resolution operation:
• P = P(x)  P
• P is to become the parent
clause for the upcoming flip
• P is initialized with the last
blocking clause
Duality between Basic Backtrack
Search and Resolution
a+b
b’ + c
b’ + c’
a’ + b
a’
(a)
b’
a+b
a
a
b
b’
c’
b’ + c
c
b’ + c’
•The parent clause P(a) is derived by resolution.
•The resolution proof (a) of the parent clause is called parent resolution
Duality between Basic Backtrack
Search and Resolution
a+b
a’
a
b’ + c
b’ + c’
a’ + b
b’
a+b
a
b
b’
c’
b’ + c
b
c
b’ + c’
Duality between Basic Backtrack
Search and Resolution
a+b
a’
a
b’ + c
b’ + c’
a’ + b
b’
a+b
a
b
b’
c’
b’ + c
b
c
b’ + c’
c’
b’ + c
Duality between Basic Backtrack
Search and Resolution
a+b
a’
a
b’ + c
b’ + c’
a’ + b
b’
a+b
a
b
b’
c’
b’ + c
b
c
b’ + c’
c’
P(c)
b’ + c
c
Duality between Basic Backtrack
Search and Resolution
a+b
a’
a
b’ + c
b’ + c’
a’ + b
b’
a+b
a
b
b’
c’
b’ + c
b
c
b’ + c’
c’
b’ + c
c
b’ + c’
Duality between Basic Backtrack
Search and Resolution
a+b
a’
a
b’ + c
b’ + c’
a’ + b
b’
a+b
a
b
b’
c’
b’ + c
b
c
b’
b’ + c’
c’
P(c)
b’ + c
Pnew
c
Pold
b’ + c’
Duality between Basic Backtrack
Search and Resolution
a+b
a’
a
b’ + c
b’ + c’
a’ + b
b’
a+b
a
b
b’
c’
b’ + c
b’
b
c
b’ + c’
(b) b’
c’
b’ + c
c
b’ + c’
Duality between Basic Backtrack
Search and Resolution
a+b
a’
a
b’ + c
b’ + c’
a’ + b
b’
a+b
a
b
b’
c’
b’ + c
b’
b
c
b’
b’ + c’
c’
b’ + c
a’ + b
c
b’ + c’
Duality between Basic Backtrack
Search and Resolution
a+b
a’
a
b’ + c
b’ + c’
a’ + b
b’
a+b
a
a’
b
b’
c’
b’ + c
Pnew
c
b’
b
P(b)
Pold
b’
b’ + c’
c’
b’ + c
a’ + b
c
b’ + c’
Duality between Basic Backtrack
Search and Resolution
a+b
a’
a’ + b
a
P(a)
b’ + c
b’ + c’

Pnew
b’
a+b
Pold
a
a’
b
b’
c’
b’ + c
b’
b
c
b’
b’ + c’
c’
b’ + c
a’ + b
c
b’ + c’
Duality between Basic Backtrack
Search and Resolution
a+b
a’

a
b’ + c
b’ + c’
a’ + b
b’
a+b
a
a’
b
b’
c’
b’ + c
b’
b
c
b’
b’ + c’
c’
b’ + c
a’ + b
c
b’ + c’
Duality between Basic Backtrack
Search and Resolution
The final trace of DLL is both a decision tree
(top-down view) and a resolution refutation
(bottom-up view)
Variables associated with the edges are both
decision variables in the tree and pivot
variables for the resolution
A forest of parent resolutions is maintained
The forest converges to one resolution
refutation in the end (for an UNSAT formula)
a’
b’
a+b
a

a
a’
b
b’
b’
b
c’
c
b’ + c
b’ + c’
b’
c’
b’ + c
a’ + b
c
b’ + c’
Conflict Clause Recording

The idea: update the instance with
conflict clauses, that is some of the
clauses generated by resolution

Introduced in SAT by Bayardo&Schrag, 1997
(rel_sat)
a’
b’
a+b
a

a
a’
b
b’
b’
b
c’
c
b’ + c
b’ + c’
b’
c’
b’ + c
a’ + b
c
b’ + c’
Conflict Clause Recording

Assume the brown clause below was
recorded
a’
b’
a+b
a

a
a’
b
b’
b’
b
c’
c
b’ + c
b’ + c’
b’
c’
b’ + c
a’ + b
c
b’ + c’
Conflict Clause Recording


Assume the brown clause below was
recorded
The violet part would not have been
explored

It is redundant
a’
b’
a+b
a

a
a’
b
b’
b’
b
c’
c
b’ + c
b’ + c’
b’
c’
b’ + c
a’ + b
c
b’ + c’
Conflict Clause Recording


Assume the brown clause below was
recorded
The violet part would not have been
explored

It is redundant
a’
b’
a+b
a

a
a’
b
b
b’
c’
c
b’ + c
b’ + c’
b’
a’ + b
Conflict Clause Recording

Most of the modern solvers record every
non-trivial parent clause (since Chaff)
: recorded
: not recorded


a’
b’
c’
c
a
e’
b
d’
d
f’
f
e
g’
g
Enhancing CCR: Local Conflict
Clause Recording

The parent-based scheme is asymmetric
w.r.t polarity selection
a’
b’
c’
c
a
e’
b
d’
d
f’
f
e
g’
g
Enhancing CCR: Local Conflict
Clause Recording


The parent-based scheme is asymmetric w.r.t polarity selection
Solution: record an additional local conflict clause: a would-be
conflict clause if the last polarity selection was flipped


Dershowitz&Hanna&Nadel, 2007 (Eureka)
: local conflict clause
a’
b’
c’
c
a
e’
b
d’
d
f’
f
e
g’
g
Managing Conflict Clauses


Keeping too many clauses slows down the
solver
Deleting irrelevant clauses is very important.
Some of the strategies:

Size-based: remove too long clauses


Age-based: remove clauses that weren’t used for BCP


Marques-Silva&Sakallah, 1996 (GRASP)
Goldberg&Novikov, 2002 (Berkmin)
Locality-based (glue): remove clauses, whose literals
are assigned far away in the search tree

Audemard&Simon, 2009 (Glucose)
55
Modern Conflict Analysis

Next, we present the following two
techniques, commonly used in modern SAT
solvers:

Non-chronological backtracking (NCB)


1UIP scheme


GRASP
GRASP&Chaff
Both techniques prune the search tree and
the associated forest of parent resolutions
Non-Chronological Backtracking
(NCB)
NCB is an additional pruning operation before flipping: eliminate all the
decision levels adjacent to the decision level of the flipped literal, so that
the parent clause is still falsified
e’
e
d’
(e)
a+b
a’
b’ + c
b’ + c’
a’ + b
…
b’
a+b
a
•Assume we are about to flip a
b
b’
c’
b’ + c
c
b’ + c’
Non-Chronological Backtracking
(NCB)
NCB is an additional pruning operation before flipping: eliminate all the
decision levels adjacent to the decision level of the flipped literal, so that
the parent clause is still falsified
e’
e
d’
(e)
a+b
a’
b’ + c
b’ + c’
a’ + b
…
b’
a+b
a
•Assume we are about to flip a
•Eliminate irrelevant decision levels
b
b’
c’
b’ + c
c
b’ + c’
Non-Chronological Backtracking
(NCB)
NCB is an additional pruning operation before flipping: eliminate all the
decision levels adjacent to the decision level of the flipped literal, so that
the parent clause is still falsified
a+b
a’
b’ + c
b’ + c’
a’ + b
…
b’
a+b
a
a
b
b’
c’
b’ + c
c
b’ + c’
•Assume we are about to flip a
•Eliminate irrelevant decision levels
•Flip
1UIP Scheme
1UIP Scheme
1UIP scheme consists of:
 A stopping condition for backtracking: stop whenever P contains one
variable of the last decision level, called the 1UIP variable
1UIP Scheme
1UIP scheme consists of:
 A stopping condition for backtracking: stop whenever P contains one
variable of the last decision level, called the 1UIP variable
a+b
a’
b’ + c
b’ + c’
a’ + b
b’
a+b
b
P
b’
c’
b’ + c
c
b’ + c’
1UIP Scheme
1UIP scheme consists of:
 A stopping condition for backtracking: stop whenever P contains one
variable of the last decision level, called the 1UIP variable
 A rewriting operation: consider the 1UIP variable as a decision
variable and P as its parent clause
a+b
a’
b’ + c
b’ + c’
a’ + b
b’
a+b
b
P
b’
c’
b’ + c
c
b’ + c’
1UIP Scheme
1UIP scheme consists of:
 A stopping condition for backtracking: stop whenever P contains one
variable of the last decision level, called the 1UIP variable
 A rewriting operation: consider the 1UIP variable as a decision
variable and P as its parent clause
a+b
a’
b’ + c
b’ + c’
a’ + b
b’
a+b
b
P
b’
c’
b’ + c
c
b’ + c’
1UIP Scheme
1UIP scheme consists of:
 A stopping condition for backtracking: stop whenever P contains one
variable of the last decision level, called the 1UIP variable
 A rewriting operation: consider the 1UIP variable as a decision
variable and P as its parent clause
a+b
a’
b’ + c
b’ + c’
a’ + b
b’
a+b
b
b’
c’
b’ + c
c
b’ + c’
1UIP Scheme
1UIP scheme consists of:
 A stopping condition for backtracking: stop whenever P contains one
variable of the last decision level, called the 1UIP variable
 A rewriting operation: consider the 1UIP variable as a decision
variable and P as its parent clause
 A pruning technique: eliminate all the disconnected variables of the
last decision level (along with their parent resolutions)
a+b
a’
b’ + c
b’ + c’
a’ + b
b’
a+b
b
b’
c’
b’ + c
c
b’ + c’
1UIP Scheme
1UIP scheme consists of:
 A stopping condition for backtracking: stop whenever P contains one
variable of the last decision level, called the 1UIP variable
 A rewriting operation: consider the 1UIP variable as a decision
variable and P as its parent clause
 A pruning technique: eliminate all the disconnected variables of the
last decision level (along with their parent resolutions)
a+b
b’ + c
b’ + c’
b
a’ + b
b’
b’
c’
b’ + c
c
b’ + c’
Agenda



Introduction
Early Days of SAT Solving
Core SAT Solving






Conflict Analysis and Learning
Boolean Constraint Propagation
Decision Heuristics
Restart Strategies
Inprocessing
Extensions to SAT




Incremental SAT Solving under Assumptions
Simultaneous Satisfiability (SSAT)
Diverse Solutions Generation
High-level (group-oriented) MUC Extraction
68
Boolean Constraint
Propagation

The unit clause rule

A clause is unit if all of its literals but one are assigned to 0.
The remaining literal is unassigned, e.g.:
a = 0, b = 1, c is unassigned
a + b’ + c

Boolean Constraint Propagation (BCP)



Pick unassigned variables of unit clauses as decisions
whenever possible
80-90% of running time of modern SAT solvers is spent in
BCP
Introduced already in the original DLL
69
Data Structures for Efficient
BCP



Naïve: for each clause hold pointers to all its literals
How to minimize the number of clause visits?
When can a clause become unit?




All literals in a clause but one are assigned to 0
For an N-literal clause, this can only occur after N-1 of the
literals have been assigned to 0
So, theoretically, one could completely ignore the first N-2
assignments to this clause.
The solution: one picks two literals in each clause to watch
and thus can ignore any assignments to the other literals in
the clause.

Introduced by Zhang, 1997 (SATO solver); enhanced by
Moskewicz& Madigan&Zhao&Zhang&Malik, 2001 (Chaff)
70
Watched Lists : Example
a
W
b
c
d
e
f
g
h
W
71
Watched Lists : Example
a
W
b
c
d
e
f
g
h
a’
W
72
Watched Lists : Example
a
b
c
W
d
e
f
g
h
a’
W
•The clause is visited
•The corresponding watch moves to any unassigned literal
•No pointers to the previously visited literals are saved
73
Watched Lists : Example
a
b
W
c
d
e
f
g
h
W
a’
c’
74
Watched Lists : Example
a
b
c
d
W
e
f
g
h
W
a’
c’
•The clause is not visited!
75
Watched Lists : Example
a
b
W
c
d
e
f
g
h
a’
W
c’
e’
g’
76
Watched Lists : Example
a
b
c
d
W
e
f
g
h
a’
W
c’
e’
g’
•The clause is not visited!
77
Watched Lists : Example
a
b
c
d
e
f
g
W
h
a’
W
c’
e’
g’
h’
78
Watched Lists : Example
a
b
c
W
d
e
f
g
h
a’
W
c’
e’
g’
h’
•The clause is visited
•The corresponding watch moves to any unassigned literal
•No pointers to the previously visited literals are saved
79
Watched Lists : Example
a
b
c
d
e
f
W
g
h
a’
W
c’
e’
g’
h’
f’
80
Watched Lists : Example
a
b
W
c
d
e
f
g
h
a’
W
c’
e’
g’
h’
f’
81
Watched Lists : Example
a
b
W
c
d
e
f
g
h
a’
W
c’
e’
g’
h’
f’
b’
82
Watched Lists : Example
a
b
W
c
d
e
f
g
h
a’
W
c’
e’
g’
h’
f’
b’
•The watched literal b is visited. It is identified that the clause
became unit!
83
Watched Lists : Example
a
b
W
c
d
e
f
g
h
a’
W
c’
e’
g’
h’
f’
b’
 Backtrack
• b is unassigned : the watches do not move
• No need to visit the clause during backtracking!
84
Watched Lists : Example
a
b
W
c
d
e
f
g
h
a’
W
c’
e’
g’
h’
f’
 Backtrack
b’
• f is unassigned : the watches do not move
85
Watched Lists : Example
a
b
d
e
f
g
h
a’
W
c’
e’
 Backtrack
W
c
g’
h’
f’
b’
• When all the literals are unassigned, the watches pointers do not
get back to their initial positions
86
Watched Lists : Caching
Chu&Harwood&Stuckey, 2008

Divide the clauses into various cache levels
to improve cache performance


Most of the modern solvers put one literal of each
clause in the WL
Special data structures for clauses of length 2
and 3
87
Agenda



Introduction
Early Days of SAT Solving
Core SAT Solving






Conflict Analysis and Learning
Boolean Constraint Propagation
Decision Heuristics
Restart Strategies
Inprocessing
Extensions to SAT




Incremental SAT Solving under Assumptions
Simultaneous Satisfiability (SSAT)
Diverse Solutions Generation
High-level (group-oriented) MUC Extraction
88
Decision Heuristics


Which literal should be chosen at each
decision point?
Critical for performance!
Old-Days’ Static Decision
Heuristics



Go over all clauses that are not satisfied
Compute some function f(A) for each literal—
based on frequency
Choose literal with maximal f(A)
Variable-based Dynamic
Heuristics: VSIDS

VSIDS was the first dynamic heuristic (Chaff)

Each literal is associated with a counter





Initialized to number of occurrences in input
Counter is increased when the literal participates
in a conflict clause
Occasionally, counters are halved
Literal with the maximal counter is chosen
Breakthrough compared to static heuristics:


Dynamic: focuses search on recently used variables
and clauses
Extremely low overhead
Enhancements to VSIDS


Adjusting the scope: increase the scores for
every literal in the newly generated parent
resolution (Berkmin)
Additional dynamicity: multiply scores by 95%
after each conflict, rather than occasionally
halve the scores

Eén&Sörensson, 2003 (Minisat)
92
The Clause-Based Heuristic
(CBH)

The idea: use relevant clauses for guiding the
decision heuristic

The Clause-Based Heuristic or CBH (Eureka)



All the clauses (both initial and conflict clauses) are
organized in a list
The next variable is chosen from the top-most unsatisfied
clause
After a conflict:



All the clauses that participate in the newly derived parent
resolution are moved to the top, then
The conflict clause is placed at the top
Partial clause-based heuristics: Berkmin, HaifaSAT
CBH: More

CBH is even more dynamic than VSIDS:
prefers variables from very recent conflicts

CBH tends to pick interrelated variables:

Variables whose joint assignment increases the
chances of:
Satisfying clauses in satisfiable branches
 Quickly reaching conflicts in unsatisfiable branches
Variables appearing in the same clause are interrelated:
 Picking variables from the same clause, results in either
that:




the clause becomes satisfied, or
there’s a contradiction
94
Polarity Selection

Phase Saving:


Strichman, 2000; Pipatsrisawat&Darwiche, 2007
(RSAT)
Assign a new decision variable the last polarity it
was assigned: dynamicity rules again
95
Decision Heuristics: the
Current Status




Everybody uses phase saving
Most of the SAT solvers use VSIDS
Intel’s Eureka uses CBH for most of the
instances and VSIDS for tiny instances only
We plan to compare VSIDS and CBH
thoroughly in our new solver Fiver
96
Core SAT Solving: the Major
Enhancements to DLL





Boolean Constraint Propagation
Conflict Analysis and Learning
Decision Heuristics
Restart Strategies
Pre- and Inter- Processing
The slides on restarts are based on
Vadim Ryvchin’s SAT’08 presentation
97
Agenda



Introduction
Early Days of SAT Solving
Core SAT Solving






Conflict Analysis and Learning
Boolean Constraint Propagation
Decision Heuristics
Restart Strategies
Inprocessing
Extensions to SAT




Incremental SAT Solving under Assumptions
Simultaneous Satisfiability (SSAT)
Diverse Solutions Generation
High-level (group-oriented) MUC Extraction
98
Restarts

Restarts: the solver backtracks to decision
level 0, when certain criteria are met


crucial impact on performance
Motivation:

Dynamicity: refocus the search on relevant data


Variables identified as important will be pick first by
the decision heuristic after the restart
Avoid spending too much time in ‘bad’ branches
99
Restart Criteria

Restart after a certain number of conflicts has
been encountered either:

Since the previous restart: global


Higher than a certain decision level: local


Gomes&Selman&Kautz, 1998
Ryvchin&Strichman, 2008
Next: methods to calculate the threshold on
the number of conflicts

Holds for both global and local schemes
100
Restarts Strategies
Arithmetic (or fixed) series.



Parameters: x, y.
Init(t) = x
Next(t)=t+y
Arithm(2000, 0) , Arithm(1000, 10)
Threshold
1.
3500
3000
2500
2000
1500
1000
500
0
1
21
41 61
81 101 121 141 161 181 201
Restart Number
101
Restarts Strategies (cont.)
Luby et al. series.



Parameter: x.
Init(t) = x
Next(t) = ti*x
Ruan&Horvitz&Kautz,
2003
Luby(512)
20000
Threshold
2.
15000
10000
5000
0
1
7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97
Restart Number
ti =1 1 2 1 1 2 4 1 1 2 1 1 2 4 8 1 1 2 1 1 2 4 1 1 2 1 1 2
4 8 16 1 1 2 1 1 2 4 1 1 2 1 1 2 4 8 …
102
Restarts Strategies (cont.)
Inner-Outer Geometric series.



Parameters: x, y, z.
Init(t) = x
if (t*y < z)


else



Next(t) = t*y
Inner-Outer (100, 1.1, 100)
2000
Threshold
3.
1500
1000
500
0
1 17 33 49 65 81 97 113 129 145 161 177 193
Next(t) = x
Next(z) = z*y
Restart Number
Armin Biere, 2007 (Picosat)
103
Agenda



Introduction
Early Days of SAT Solving
Core SAT Solving






Conflict Analysis and Learning
Boolean Constraint Propagation
Decision Heuristics
Restart Strategies
Inprocessing
Extensions to SAT




Incremental SAT Solving under Assumptions
Simultaneous Satisfiability (SSAT)
Diverse Solutions Generation
High-level (group-oriented) MUC Extraction
104
Preprocessing and
Inprocessing

The idea:


Simplify the formula prior (pre-) and during (in-) the search
History:




Freeman, 1995 (POSIT): first mentioning of preprocessing
in the context of SAT
Eén&Biere, 2005 (SatELite): a commonly used efficient
preprocessing procedure
Heule&Järvisalo&Biere (2010-2012): a series of papers on
inprocessing
 Used in the current state-of-the-art solvers Lingeling and
CryptoMinisat
Nadel&Ryvchin&Strichman (2012): apply SatELite in
incremental SAT solving
105
Inprocessing Techniques

SatELite:



Subsumption: remove clause (C+D) if (C) exists
Self-subsuming resolution: replace (D+l’) by (D), if (C+l)
exists, such that C  D
Variable elimination: apply DP for variables, whose
elimination does not increase the number of clauses


Example: (a+b)(a+b’)(a’+c)(a’+c’)  (a)(a’+c)(a’+c’)
Example of other techniques:

Failed literal elimination with BCP:

Repeat for a certain subset of literals on decision level 0:


Propagate a literal l with BCP.
If a conflict emerges, l must be 0  the formula can be simplified
106
Agenda



Introduction
Early Days of SAT Solving
Core SAT Solving






Conflict Analysis and Learning
Boolean Constraint Propagation
Decision Heuristics
Restart Strategies
Inprocessing
Extensions to SAT




Incremental SAT Solving under Assumptions
Simultaneous Satisfiability (SSAT)
Diverse Solutions Generation
High-level (group-oriented) MUC Extraction
107
Extensions to SAT


Nowadays, SAT solving is much more than finding one
solution to a given problem
Extensions to SAT:










Incremental SAT under assumptions
Simultaneous SAT (SSAT): SAT over multiple properties at once
Diverse solution generation
Minimal Unsatisfiable Core (MUC) extraction
Push/pop support
Model minimization
ALL-SAT
XOR clauses support
ISSAT: assumptions are implications
…
108
Agenda



Introduction
Early Days of SAT Solving
Core SAT Solving






Conflict Analysis and Learning
Boolean Constraint Propagation
Decision Heuristics
Restart Strategies
Inprocessing
Extensions to SAT




Incremental SAT Solving under Assumptions
Simultaneous Satisfiability (SSAT)
Diverse Solutions Generation
High-level (group-oriented) MUC Extraction
109
Incremental SAT Solving under
Assumptions



The challenge: speed-up solving of related
SAT instances by enabling re-use of relevant
data
Incremental SAT solving has numerous
applications
Next, we review a prominent application in
Formal Verification of Hardware
110
Reasoning about Circuit Properties with
SAT-based Bounded Model Checking
(BMC)

BMC: given a circuit and a property, does the
property holds for the first n cycles?


Unroll: generate a combinational instantiation of
the circuit for each cycle
Run a SAT solver for each cycle over:



The translation of unrolled circuit to CNF
The negation of the property at that cycle
The property holds for n cycles iff all the SAT
solver invocations return UNSAT
111
BMC Example
a
b
g
h
c
The property: b’h’
BMC Example: Cycle 0
g
a
b
h
c
The property: b’h’
a
b
ci
g
h
A user-given initial value
BMC Example: Cycle 0
g
a
b
h
c
The property: b’h’
g + a’ + b’
g’ + a
g’ + b
g
a
b
ci
The negation of the property b’h’:
h
h + g’ + ci’
h’ + g
h’ + ci
b’
h
UNSAT!
BMC Example: Cycle 1
g
a
b
h
c
The property: b’h’
a
b
ci
g
h
cx
ax
bx
gx
hx
BMC Example: Cycle 1
g
a
b
h
c
The property: b’h’
The negation of the property bx’hx’:
g + a’ + b’
g’ + a
g’ + b
g
a
b
ci
h
h + g’ + ci’
h’ + g
h’ + ci
cx
gx + ax’ + bx’
gx’ + ax
gx’ + bx
ax
gx
bx
cx + h’
cx’ + h
bx’
hx
hx
hx + gx’ + cx’
hx’ + gx
hx’ + cx
UNSAT!
Re-Using Relevant Information from
Previous Cycles


C0 and C1 hold globally
S0 and S1 hold solely
for a particular cycle
a
b
ci
g
h
C0: cycle 0
C1: cycle 1
g + a’ + b’
g’ + a
g’ + b
gx + ax’ + bx’
gx’ + ax
gx’ + bx
h + g’ + ci’
h’ + g
h’ + ci
hx + gx’ + cx’
hx’ + gx
hx’ + cx
b’
h
bx’
hx
cx
S0: cycle
0-specific
bx
hx
cx + h’
cx’ + h
S1: cycle 1-specific
The property: b’h’
117
Pervasive Clause Learning; MarquesSilva&Sakallah, 1997 (GRASP); Strichman, 2001
C0: cycle 0

g + a’ + b’
g’ + a
g’ + b
gx + ax’ + bx’
gx’ + ax
gx’ + bx
h + g’ + ci’
h’ + g
h’ + ci
hx + gx’ + cx’
hx’ + gx
hx’ + cx
b’
h
bx’
hx
S0: cycle
0-specific
cx + h’
cx’ + h
S1: cycle 1-specific
Cycle 0: create a CNF instance C0  S0 and solve it


C1: cycle 1
Let C0* be the set of pervasive conflict clauses, that is conflict clauses
that depend only on C0
Cycle 1: create a CNF instance C0  C1  S1  C0* and solve it
118
Pervasive Clause Learning; MarquesSilva&Sakallah, 1997 (GRASP); Strichman, 2001
C0: cycle 0
g + a’ + b’
g’ + a
g’ + b
C1: cycle 1
C0*
a + h’
h + g’ + ci’
h’ + g
h’ + ci
b’
h

hx + gx’ + cx’
hx’ + gx
hx’ + cx
bx’
hx
cx + h’
cx’ + h
S1: cycle 1-specific
Cycle 0: create a CNF instance C0  S0 and solve it


g
S0: cycle
0-specific
gx + ax’ + bx’
gx’ + ax
gx’ + bx
Let C0* be the set of pervasive conflict clauses, that is conflict clauses
that depend only on C0
Cycle 1: create a CNF instance C0  C1  S1  C0* and solve it
119
Incremental SAT Solving under
Assumptions; Eén&Sörensson, 2003
(Minisat)
C0: cycle 0

g + a’ + b’
g’ + a
g’ + b
gx + ax’ + bx’
gx’ + ax
gx’ + bx
h + g’ + ci’
h’ + g
h’ + ci
hx + gx’ + cx’
hx’ + gx
hx’ + cx
b’
h
bx’
hx
S0: cycle
0-specific
cx + h’
cx’ + h
S1: cycle 1-specific
Cycle 0: create a CNF instance C0 and solve it under the assumptions S0




C1: cycle 1
S0 clauses are not part of the instance, instead:
The literals of S0 are used as the first decision, or assumptions
The solver stops, whenever one of the assumptions must be flipped
Cycle 1: add the clauses C1 to the same instance and solve under the
assumptions S1
120
Incremental SAT Solving: More

Minisat’s method is the state-of-the-art

Advantages:



GRASP’s method advantage



Re-uses a single solver instance: heuristics are incremental
All the clauses are re-used
Assumptions are unit clauses: preprocessing can use them
to simplify the formula
Incremental SAT solving was not compatible with
preprocessing
Nadel&Ryvchin&Strichman 2012:


Make incremental SAT solving compatible with SatELite
Show a way to treat assumptions efficiently
121
Agenda



Introduction
Early Days of SAT Solving
Core SAT Solving






Conflict Analysis and Learning
Boolean Constraint Propagation
Decision Heuristics
Restart Strategies
Inprocessing
Extensions to SAT




Incremental SAT Solving under Assumptions
Simultaneous Satisfiability (SSAT)
Diverse Solutions Generation
High-level (group-oriented) MUC Extraction
122
Simultaneous SAT (SSAT)

A SAT-based algorithm to efficiently solve
chunks of related properties in one SAT
solver invocation

For example, one can solve multiple properties
during BMC
Khasidashvili&Nadel&Palti&Hanna, 2005
Khasidashvili&Nadel, 2011
123
Example: Solve Both p1 and p2
p1
C1
p2
C2
Incremental SAT-based Approach
p1
C1




p2
C2
Translate C1 to CNF formula F
Solve F under the assumption p1’
Update F with clause projection of C2\C1
Solve F under the assumption p2’
SSAT Approach
p1
C1


p2
C2
Translate both C1 and C2 to CNF formula F
Find the status of both p1 and p2 in the same
invocation of the SAT solver
Advantages of SSAT approach to
Incremental SAT-based Approach

Looks at all the properties at once


One solution can falsify more than one property
May find conflict clauses (lemmas) relevant for
solving many POs
SSAT: the Algorithm Interface

Input



A combinational formula F (in CNF)
A list of proof objectives (POs) p1,p2,…,pn
Output

Each pi is either

falsifiable


A model to F, such that pi = 0, exists (F  pi’ is SAT)
valid

pi always holds, given F (F  pi’ is UNSAT)
128
SSAT Algorithm Interface
Example
F = (a + b)  c’  a’
POs: a, b, c, a’, b’, c’
a is falsifiable: a = 0; b = 1; c = 0 is the model
b is valid: there is no model to F, where b = 0
In another words, (a + b)  c’  a’  b’ is UNSAT
c is falsifiable: a = 0; b = 1; c = 0 is the model
a’ is valid: no model to F where a = 1
b’ is falsifiable with a = 0; b = 1; c = 0
c’ is valid: no model to F where c = 1
• Both l and l’ may be falsifiable
•Example: F = a + b; PO: a
129
Basic SSAT Algorithm
SSAT(F; P={p1,p2,…,pn})

While (P is non-empty)



Pick any s  P
Solve F under the assumption s’
If satisfiable by a satisfying assignment 




Initialized with clause
projection of the union of
cones of all the properties
T:={s  other POs in P falsified by }
Return to the user that the POs T are falsifiable
P := P \ T
If unsatisfiable


Return that s is valid
P := P \ {p}
SSAT: More

How to boost SSAT

Take further advantage of reasoning about all the POs at
once


Pick all the POs as decision variables and assign them 0
Fairness: rotate unsolved POs


Set an internal time threshold for an attempt to solve one PO
When the threshold expires:



Move the unresolved PO to the end of unsolved POs list
Switch to another PO
SSAT is widely used at Intel

Applied as the core reasoning engine for simultaneous
model checking algorithms we developed
Agenda



Introduction
Early Days of SAT Solving
Core SAT Solving






Conflict Analysis and Learning
Boolean Constraint Propagation
Decision Heuristics
Restart Strategies
Inprocessing
Extensions to SAT




Incremental SAT Solving under Assumptions
Simultaneous Satisfiability (SSAT)
Diverse Solutions Generation
High-level (group-oriented) MUC Extraction
132
DiversekSet: Generating
Diverse Solutions

DiversekSet in SAT: generate a user-given
number of diverse solutions, given a CNF
formula


Nadel, 2011
The problem has multiple applications at Intel
133
Application: Semi-formal FPV
deep bugs
New Initial
states
New Initial
states
Max
FV bound
initial
states
New Initial
states
Multi-Threaded Search to
Enhance Coverage


Choosing a single path through waypoints
may miss the bug
Must search along multiple diverse paths
calculated:
Diversification Quality as the
Average Hamming Distance

Quality: the average Hamming distance
between the solutions, normalized to [0…1]
Hamming distances matrix
abc
1 0 0 0
2 1 1 0
3 0 1 1
4 1 0 0
1
1
2
3
4
2
2
3
4
Diversification Quality as the
Average Hamming Distance

Quality: the average Hamming distance
between the solutions, normalized to [0…1]
Hamming distances matrix
abc
1 0 0 0
2 1 1 0
3 0 1 1
4 1 0 0
1
1
2
3
4
2
3
2
2
4
Diversification Quality as the
Average Hamming Distance

Quality: the average Hamming distance
between the solutions, normalized to [0…1]
Hamming distances matrix
abc
1 0 0 0
2 1 1 0
3 0 1 1
4 1 0 0
1
1
2
3
4
2
3
4
2
2
1
Diversification Quality as the
Average Hamming Distance

Quality: the average Hamming distance
between the solutions, normalized to [0…1]
Hamming distances matrix
abc
1 0 0 0
2 1 1 0
3 0 1 1
4 1 0 0
1
1
2
3
4
2
3
4
2
2
2
1
Diversification Quality as the
Average Hamming Distance

Quality: the average Hamming distance
between the solutions, normalized to [0…1]
Hamming distances matrix
abc
1 0 0 0
2 1 1 0
3 0 1 1
4 1 0 0
1
1
2
3
4
2
3
4
2
2
2
1
1
Diversification Quality as the
Average Hamming Distance

Quality: the average Hamming distance
between the solutions, normalized to [0…1]
Hamming distances matrix
abc
1 0 0 0
2 1 1 0
3 0 1 1
4 1 0 0
1
1
2
3
4
2
3
4
2
2
2
1
1
3
Diversification Quality as the
Average Hamming Distance

Quality: the average Hamming distance
between the solutions, normalized to [0…1]
Hamming distances matrix
1
abc
1 0 0 0
2 1 1 0
3 0 1 1
4 1 0 0
m
Q
1
2
3
3
4
2
2
2
1
1
3
4
m
  D(  , 
i 1 j i 1
Variables
2
 m
q
2

 
i
j
)
2  2  1  2  1  3 11
Hamming Distance

18
 4


3
Solutions
 2
 

Diversification Quality as the
Average Hamming Distance

Quality: the average Hamming distance
between the solutions, normalized to [0…1]
Hamming distances matrix
1
abc
1 0 0 0
2 1 1 0
3 0 1 1
4 1 0 0
m
Q
1
2
3
2
3
4
2
2
2
1
1
3
4
m
  D(  , 
i 1 j i 1
 m
q
2

 
i
j
)

2  2  1  2  1  3 11

18
 4

3
 2
 
Algorithms for DiversekSet in
SAT in a Glance

The idea:



Adapt a modern CDCL SAT solver for
DiversekSet
Make minimal changes to remain efficient
Compact algorithms:



Invoke the SAT solver once to generate all the
solutions
Restart after a solution is generated
Modify the polarity and variable selection
heuristics for generating diverse solutions
Algorithms for DiversekSet in
SAT in a Glance Cont.

Polarity-based algorithms:



Change solely the polarity selection heuristic
pRand: pick the polarity randomly
pGuide: pick the polarity so as to improve the
diversification quality

pGuide outperforms pRand in terms of both
diversification quality and performance
Quality can be improved further by taking BCP into
account and adapting the variable ordering


Balance the number of 0’s and 1’s assigned to a variable by
picking  {0,1} when variable was assigned ’ more times
Agenda



Introduction
Early Days of SAT Solving
Core SAT Solving






Conflict Analysis and Learning
Boolean Constraint Propagation
Decision Heuristics
Restart Strategies
Inprocessing
Extensions to SAT




Incremental SAT Solving under Assumptions
Simultaneous Satisfiability (SSAT)
Diverse Solutions Generation
High-level (group-oriented) MUC Extraction
146
Unsatisfiable Core Extraction

An unsatisfiable core is an unsatisfiable
subset of an unsatisfiable set of constraints

An unsatisfiable core is minimal if removal of
any constraint makes it satisfiable (local
minima)

Has numerous applications
Example Application: Proof-based Abstraction
Refinement for Model Checking;
McMillan et al.,’03; Gupta et al.,’03
Turn latches/
gates into free
inputs
Inputs: model M, property P
Output: does P hold under M?
Abstract model A  { }
A  A  latches/gates in the
UNSAT core of BMC(M,P,k)
Model Check A
Valid
No Bug
Cex C at depth k
BMC(M,P,k)


Yes
Spurious?
No
The UNSAT core is used for refinement
The UNSAT core is required in terms of latches/gates
Bug
Example Application 2: Assumption Minimization for
Compositional Formal Equivalence Checking (FEC);
Cohen et al.,’10



Assumption
Assertion
Outputs

FEC verifies the equivalence between the design (RTL) and its
implementation (schematics).
The whole design is too large to be verified at once.
FEC is done on small sub-blocks, restricted with assumptions.
Assumptions required for the proof of equivalence of subblocks must be proved relative to the driving logic.
MUC extraction in terms of assumptions is vital for feasibility.
Inputs

Traditionally, a Clause-Level
UC Extractor is the Workhorse

Clause-level UC extraction: given a CNF
formula, extract an unsatisfiable subset of
its clauses
F = ( a + b ) ( b’ + c ) (c’ ) (a’ + c ) ( b + c ) ( a + b + c’ )
U1 = ( a + b ) (b’ + c ) ( c’ ) ( a’ + c ) ( b + c ) ( a + b + c’ )
U2 = ( a + b ) ( b’ + c ) ( c’ ) ( a’ + c ) ( b + c ) ( a + b + c’ )
U3 = ( a + b ) ( b’ + c ) ( c’ ) ( a’ + c ) ( b + c ) ( a + b + c’ )

Dozens of papers on clause-level UC
extraction since 2002
Traditional UC Extraction for
Practical Needs: the Input
The user is interested in a MUC
in terms of these constraints
An interesting constraint
The remainder (the rest of the formula)
Traditional UC Extraction:
Example Input 1
Proof-based abstraction refinement
An unrolled latch
The rest of the unrolled circuit
Traditional UC Extraction:
Example Input 1
Assumption minimization for FEV
An assumption
Equivalence between sub-block RTL
and implementation
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’)
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
Traditional UC Extraction:
Stage 3: Map the Clause-Level UC Back to the
Interesting Constraints
An interesting constraint
The remainder (the rest of the formula)
The UC contains three interesting constraints
High-Level Unsatisfiable Core
Extraction

Real-world applications require reducing the
number of interesting constraints in the core
rather than clauses




Latches for abstraction refinement
Assumptions for compositional FEV
Most of the algorithms for UC extraction are
clause-level
High-level UC: extracting a UC in terms of
interesting constraints only

Liffiton&Sakallah, 2008; Nadel, 2010;
Ryvchin&Strichman, 2011
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:
High-Level Unsatisfiable Core
Extraction: Main Results

Minimal UC extraction: high-level algorithms
solve Intel families that are out of reach for
clause-level algorithms

Non-minimal UC extraction: high-level
algorithms are preferable

2-3x boost on difficult benchmarks
Thanks!
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