Mark Worth
The Satisfiability Problem and the MiniSat
Solver
1
– What is the Satisfiability problem?
– The MiniSat solver
– Algorithms used by solvers
2
The Satisfiability Problem (SAT)
– A boolean function
(X1 + X2 + ¬X3)^(¬X1 + X2 + ¬X3)
– Are there assignments to X1, X2, and X3 that
make the function TRUE?
– If there are, the function is Satisfiable.
3
(
¬X1 + X2) ^ (X1 + X3)
4
(X1 + X2 + ¬X3)^(¬X1 + X2 + ¬X3)
– One assignment is X1=T, X2=F, X3=F
– This function is Satisfiable
5
SAT problems can come in
– Conjunctive Normal Form(CNF) or
– Disjunctive Normal Form(DNF)
6
(X1 + X2 + ¬X3)^(¬X1 + X2 + ¬X3)
This is in Conjunctive Normal Form (CNF)
– A conjunction of disjunctions
– Each clause is the OR of variables
– Clauses are ANDed together
7
(X5^X6^X7) + (¬X5^X8^¬X9)
This is in Disjunctive Normal Form (DNF)
– A disjunction of conjunctions
– Each clause is the AND of variables
– Clauses are ORed together
8
SAT in DNF can be easier to solve
(X5^X6^X7) + (¬X5^X8^¬X9)
Why not use DNF?
9
– Some problems present themselves in CNF
– – Circuit analysis
– Converting to DNF is an NP problem.
10
Complexity Classes
– P Can be solved in polynomial time
– NP Verifiable in polynomial time
– NP-C As “hard” as any NP problem
11
12
SAT solvers like MiniSat usually accept input in
CNF form.
13
There are different types of SAT problems
– 2SAT
2 literals per clause
– 3SAT
3 literals per clause
– kSAT
k literals per clause
– Horn SAT
14
(X1 + X2 + ¬X3)^(¬X1 + X2 + ¬X3)
Example of 3SAT problem
– Exactly 3 variables per clause
– Can have any number of clauses
15
– 2SAT can be solved in polynomial time
– 3SAT and over are NP-Complete
16
Horn SAT
– Special case of SAT
– Each clause has at most 1 positive variable
(X1+¬X2+¬X3)^(X2+¬X3+¬X4)
17
– SAT is the first problem known to be NPComplete
– Stephen Cook proved this in 1971
– This was the origin of the concept of an NPComplete problem
18
Uses of SAT
19
– Since the 1990's, great improvements have
been made to SAT solvers
– This has caused SAT to be applied to a wide
range of uses
20
SAT is used in:
– Model Checking of Finite State Systems
– Test Pattern Generation in Digital Systems
– Combinational Equivalence Checking
– Planning in AI
– – Constraint satisfaction problem
– Automated Theorem Proving
– Software Verification
– Haplotype inference in bioinformatics
21
MiniSat Solver
22
MiniSat Solver
– Accepts problems in CNF
– Any number of variables in each clause
23
$ ./minisat --help
WARNING: for repeatability, setting FPU to use double precision
USAGE: ./minisat [options] <input-file> <result-output-file>
where input may be either in plain or gzipped DIMACS.
CORE OPTIONS:
-rnd-init, -no-rnd-init
-luby, -no-luby
(default: off)
(default: on)
24
-rnd-freq = <double> [ 0 .. 1] (default: 0)
-rnd-seed = <double> ( 0 .. inf) (default: 9.16483e+07)
-var-decay = <double> ( 0 .. 1) (default: 0.95)
-cla-decay = <double> ( 0 .. 1) (default: 0.999)
-rinc
= <double> ( 1 .. inf) (default: 2)
-gc-frac
= <double> ( 0 .. inf) (default: 0.2)
-- garbage collection trigger (wasted memory)
-rfirst
= <int32> [ 1 .. imax] (default: 100)
-ccmin-mode = <int32> [ 0 .. 2] (default: 2)
-phase-saving = <int32> [ 0 .. 2] (default: 2)
25
MAIN OPTIONS:
-verb
= <int32> [ 0 .. 2] (default: 1)
-cpu-lim
= <int32> [ 0 .. imax] (default: 2147483647) secs
-mem-lim
= <int32> [ 0 .. imax] (default: 2147483647) MB
HELP OPTIONS:
--help
Print help message.
--help-verb Print verbose help message.
26
c All comments begin with a c
p cnf 5 3
1 -5 4 0
-1 5 3 4 0
-3 -4 0
– p cnf 5 3 → problem is in CNF, 5 variables, 3
clauses
– clauses end with 0
(X1+¬X5+X4)^(¬X1+X5+X3+X4)^(¬X3+¬X4)
27
(X1+¬X5+X4)^(¬X1+X5+X3)^(¬X3+¬X4)
Output:
SAT
-1 -2 -3 -4 -5 0
X1=F, X2=F, X3=F, X4=F, X5=F
28
– Only one solution generated
– Purpose is show the problem is satisfiable, not
to find all solutions
– Can force a new solution by adding the
negation of the output as a new clause to the
input
29
c All comments begin with a c
p cnf 5 4
1 -5 4 0
-1 5 3 4 0
-3 -4 0
123450
30
Solver Algorithms and Strategies
31
• Davis-Putnam (DP)
• Davis-Logemann-Loveland (DLL/DPLL)
• Stalmarck’s algorithm
• Stochastic search
32
Davis-Putnam (DP)
– Based on resolution
33
(A  B)  (B  ¬C)  (A  ¬B)
A=0
A=1
B  (B  ¬C)  ¬B
B=0
()
B  ¬C
B=1
B=0
()
B=1

¬C
C=0
C=1

()
34
– Good at reducing the function
– Still exponential complexity for worst case
35
• Davis-Logemann-Loveland (DLL/DPLL)
– Used by MiniSat
– Search-based
– Basis for current most successful solvers
36
• Davis-Logemann-Loveland (DLL/DPLL) (cont.)
Step 1
– Loop through the variables
– – set each variable (true or false)
– – minisat initially sets unknowns to false
– – if all variables are set → Satisfiable
37
Step 2
– Calculate the value of other variables as you
go, looking for conflicts
(X1 + X2) ^ (¬X2 + X3)
If X1 is false, X2 has to be true
X3 has to be true
Or
(X1 + X2) ^ (¬X2 + X3) ^ (¬X3)
– until you find a conflict → backtrack and
choose a different value
38
39
– SAT
The Satisfiability Problem
-- Theoretical applications
-- Real world applications
– NP-Complete
– Solvers exist
40
Resources
Joao Marques-Silva, “Practical Applications of Boolean Satisfiability”
www.cs.ucf.edu/courses/cot4810.spr2003/dhand1.ppt
Thomas Cormen, Charles Leiserson, “Introduction to Algorithms”
41