Joao Marques-Silva
School of Electronics and Computer Science
University of Southampton

• Many (practical) applications:
– Formal methods:
◮
Hardware model checking ; Software model checking; Termination analysis of term-rewrite systems; Test pattern generation (testing of software & hardware); etc.
– Artificial intelligence:
◮
Planning ; Knowledge representation; Games (n-queens, sudoku, social golpher’s, etc.)
– Bioinformatics:
◮
Haplotype inference ; Pedigree checking; Maximum quartet consistency; etc.
– Design automation:
◮
Equivalence checking ; Delay computation; Fault diagnosis; Noise analysis; etc.
– Security:
◮
Cryptanalysis; Inversion attacks on hash functions; etc.
– Computationally hard problems:
◮
Graph coloring; Traveling salesperson; etc.
– Mathematical problems:
◮ van der Waerden numbers; etc.

• Core engine for (many) other problem domains:
– Extensions of Satisfiability :
◮
◮
◮
◮
◮
0-1 Integer Linear Programming
Quantified Boolean Formulas
Model counting
Maximum Satisfiability
Satisfiability Modulo Theories
– Integration with theorem provers
◮
HOL; Isabelle; ...
• In this talk:
– Boolean Satisfiability and extensions
– Modeling with Satisfiability and extensions
– Representative applications of Satisfiability

What is Boolean Satisfiability?
Extensions of SAT
Modeling with SAT
Practical Applications
Combinational Equivalence Checking (CEC)
Automatic Test Pattern Generation (ATPG)

• Boolean formula ϕ is defined over a set of propositional variables x
1 , . . . , x n
, using the standard propositional connectives
→ , ↔ , and parenthesis
¬ , ∧ , ∨ ,
– The domain of propositional variables is { 0
,
1 }
– Example: ϕ
( x
1 , . . . , x
3
) = (( ¬ x
1
∧ x
2
) ∨ x
3
) ∧ ( ¬ x
2
∨ x
3
)
• Conjunctive normal form (CNF) formula ϕ
: conjunction of disjunctions ( clauses ) of literals , where a literal is a variable or its complement
– Example: ϕ
( x
1 , . . . , x
3
) = ( ¬ x
1
∨ x
2
) ∧ ( ¬ x
2
∨ x
3
)
• Boolean satisfiability (SAT):
– Find an assignment to the variables x
1 , . . . , x n such that ϕ
( x
1 , . . . , x n
) = 1, or prove that no such assignment exists
• SAT is an NP-complete decision problem
[Cook’71]

• Is it possible to assign one of K colors to each of the vertices of graph G = ( V
,
E ) such that adjacent vertices are assigned different colors
Valid coloring Invalid coloring
• SAT model:
– Given N = | V | vertices and K colors, create N × K variables: x ij
= 1 iff vertex i is assigned color j ; 0 otherwise
– For each edge ( u
, v ), require different assigned colors to u and v :
1 ≤ j ≤ K
,
( ¬ x uj
∨ ¬ x vj
)
– Each vertex is assigned
1 ≤ exactly i ≤ N
, one color:
K j =1 x ij
= 1

What is Boolean Satisfiability?
Extensions of SAT
Modeling with SAT
Practical Applications
Combinational Equivalence Checking (CEC)
Automatic Test Pattern Generation (ATPG)

• Consider extended constraints
– Pseudo-Boolean formulas (PBS/PBO) :
Linear inequalities over Boolean variables w/o or w/ cost function
– Also known as 0-1 Integer Linear Programming
• Maximize number of satisfied clauses
– ( Weighted ) ( Partial ) Maximum Satisfiability
• Quantify the variables
– Quantified Boolean Formulas (QBF) :
Boolean formulas w/ variables existentially or universally quantified
• Decidable fragments of FOL
– Satisfiability Modulo Theories
– Decision procedures for a number of theories exist
◮
Linear Integer Arithmetic; Difference Arithmetic; Uninterpreted
Functions; ...
• Some extensions promising; still far from the impact of SAT solvers

• The problem:
– Graph G = ( V
,
E )
– Vertex cover U ⊆ V , such that for all edges ( v i v j
∈ U .
, v j
), either v
– Minimum vertex cover: vertex cover U of minimum size i
∈ U or
• Partial MaxSAT formulation:
– Associate x i otherwise v i with each
∈ V − U .
v i
∈ V , such that x i
= 1 iff v i
∈ U ,
– Must satisfy (i.e.
hard ) clauses: ( x i
∨ x j
) for each edge ( v
– Preferable to satisfy (i.e.
soft ) clauses: ( ¬ x i i , v j
)
) for each vertex v i
∈ E
◮
I.e. give preference not to include vertices in U v
2 v
3 v
1 ϕ H
= { ( x
1
∨ x
2
)
,
( x
1
∨ x
3
)
,
( x
1
∨ x
4
) } ϕ
S
= { ( ¬ x
1
)
,
( ¬ x
2
)
,
( ¬ x
3
)
,
( ¬ x
4
) } v
4

What is Boolean Satisfiability?
Extensions of SAT
Modeling with SAT
Practical Applications
Combinational Equivalence Checking (CEC)
Automatic Test Pattern Generation (ATPG)

•
•
Satisfiability problems can be defined on Boolean circuits/formulas
Can represent circuits/formulas as CNF formulas
– For each (simple) gate, CNF formula encodes the consistent assignments to the gate’s inputs and output
[Tseitin’68]
◮
Given z = OP( x
, y ) , represent in CNF z ↔ OP( x
, y )
– CNF formula for the circuit is the conjunction of CNF formula for each gate ϕ ϕ c t
= (
= ( a
¬ r
∨
∨ c ) t )
∧
∧
( b
(
∨
¬ s c
∨
) t
∧
)
(
∧
¬
( a r
∨ ¬
∨ s b ∨ ¬
∨ ¬ t ) c ) a b r s t c

a b c a b c ϕ c
(a,b,c)
0 0 0
0 0 1
0
1
0 1 0
0 1 1
1 0 0
0
1
0
1 0 1
1 1 0
1 1 1
1
1
0 ϕ c
= ( a ∨ c ) ∧ ( b ∨ c ) ∧ ( ¬ a ∨ ¬ b ∨ ¬ c )

• CNF formula for the circuit is the conjunction of the CNF formula for each gate
– Can specify objectives with additional clauses a b x c y d z = 1?
ϕ
= ( a ∨ x ) ∧ ( b ∨ x ) ∧ ( ¬ a ∨ ¬ b ∨ ¬ x ) ∧
( x ∨ ¬ y ) ∧ ( c ∨ ¬ y ) ∧ ( ¬ x ∨ ¬ c ∨ y ) ∧
( ¬ y ∨ z ) ∧ ( ¬ d ∨ z ) ∧ ( y ∨ d ∨ ¬ z ) ∧
( z )
• Note: z = d ∨ ( c ∧ ( ¬ ( a ∧ b )))
– No distinction between Boolean circuits and formulas

•
•
•
How to represent in CNF the constraint
P
N j =1 x j
≥ 1 ?
– Standard solution: ( x
1
∨
. . .
∨ x
N
)
How to represent in CNF the constraint
P
N j =1 x ij
≤ 1 ?
– Naive solution: ∀ j
1
=1
..
N
∀ j
2
= j
1
+1
..
N
◮
( ¬ x ij
1
∨ ¬ x ij
2
)
Number of clauses grows quadratically with N
– More compact (e.g. linear) solutions possible
How to represent in CNF the constraint
P
N j =1 x ij
= 1 ?
– Standard solution: one AtMost 1 and one AtLeast 1 constraints
• Additional constraints:
P
N j =1 x j
≥ k ;
P
N j =1 a j x j
≥ k ;
P
N j =1 x j
≤ k
P
N j =1 a j x j
≤ k

• Encode
P n j =1 x j
≤ 1 with sequential counter :
V x
1
∨
1
< i
< n s
1
((
)
¬
∧ x i
( ¬
∨ x s n i
∨ ¬ s n
−
1
)
) ∧ ( ¬ s i
−
1
∧
∨ s i
) ∧ ( ¬ x i
∨ ¬ s i
−
1
))
– If x j
◮
= 1, all s i variables assigned
All other x variables take value 0
– If all x j
= 0, can find consistent assignment to s i variables
– O ( n ) clauses ; O ( n ) auxiliary variables

What is Boolean Satisfiability?
Extensions of SAT
Modeling with SAT
Practical Applications
Combinational Equivalence Checking (CEC)
Automatic Test Pattern Generation (ATPG)

• Combinational equivalence checking (CEC)
• Automatic test pattern generation (ATPG)
• Hardware model checking (BMC)
• AI Planning
• Haplotyping
(see paper)
(see paper)
(see paper)

•
•
Combinational circuit C
A
, with n inputs and m outputs
Combinational circuit C
B
, with n inputs and m outputs
• Are the two circuits equivalent?
– A complicated problem formulation:
◮
Are the outputs equivalent for all input values?
– A simpler alternative:
◮
Are there (at least one) input values that distinguish outputs of the two circuits?

Miter x
1
Circuit A y
1 x n
Circuit B y m w
1 o = 1?
w m
• Can easily be represented in CNF, and solved with SAT solver
– Note: CEC is a difficult problem, and more sophisticated techniques are often used

• Digital integrated circuits can exhibit defects
•
•
Physical defects are modeled as logical faults
Most often used fault model: single stuck-at fault model
– Circuit lines stuck-at a fixed logic value
◮
◮ sa-0 : Fixed at 0 sa-1 : Fixed at 1
• Example: a b v c x x sa−0 d x’ y w
– Fault x stuck-at 0 causes line x to be fixed at value 0 z

Miter x
1 y
1
Good circuit x n y m
Faulty circuit w
1 o = 1?
w m
• Can easily be represented in CNF, and solved with SAT solver
– Note 1: Simplistic model; actual SAT formulation significantly more complex
– Note 2: Dedicated algorithms can be competitive with SAT

Miter
a b v c v c d y x w d x = 0 y w z
Good z
Faulty o = 1 ?

• Overview of practical uses of SAT
– Techniques for modeling computational problems with SAT
– Brief survey of SAT extensions
– Representative SAT applications
◮
Paper describes several additional applications
• The future of SAT and SAT extensions
– Steady improvements to core algorithms (SAT, PB, MaxSAT, SMT, etc.)
◮
See results of recent competitions, SAT, PB, etc.
– Increasing number of practical applications