Owf 2013 rii veri t fontaine speaker4
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The solver David Déharbe, Pablo Federico Dobal, Pascal Fontaine,... VeriDis 4 octobre 2013 1/7
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Transcript of Owf 2013 rii veri t fontaine speaker4
The solver
David Déharbe, Pablo Federico Dobal,Pascal Fontaine, . . .
VeriDis
4 octobre 2013 1 / 7
SAT and SMT solvers : what can they do for you
Automated reasoning, constraint solvingReason on (reasonably large) sets of factsDepending on the facts, more or less expressivity is requiredE.g. Sudokus can be expressed in SAT logic
Besides solving puzzles, SAT/SMT solvers are used for :planningconfiguration checkingplacementmodel checkingformal verification. . .
4 octobre 2013 2 / 7
Using a SAT solver : a toy example
1
3
2
2
“or” : ∨“and” : ∧negation : ¬line i, column j, there is x (pi,j,x)
Line 1, column 1, there is “1” :
p1,1,1
Line 2, column 4, there is “3” :
p2,4,3
At position (1, 2), there is only one number
¬p1,2,1 ∨ ¬p1,2,2¬p1,2,1 ∨ ¬p1,2,3¬p1,2,1 ∨ ¬p1,2,4¬p1,2,2 ∨ ¬p1,2,3¬p1,2,2 ∨ ¬p1,2,4¬p1,2,3 ∨ ¬p1,2,4
4 octobre 2013 3 / 7
Using a SAT solver : a toy example
1
3
2
2
“or” : ∨“and” : ∧negation : ¬line i, column j, there is x (pi,j,x)
Line 1, column 1, there is “1” :
p1,1,1
Line 2, column 4, there is “3” :
p2,4,3
At position (1, 2), there is only one number
¬p1,2,1 ∨ ¬p1,2,2¬p1,2,1 ∨ ¬p1,2,3¬p1,2,1 ∨ ¬p1,2,4¬p1,2,2 ∨ ¬p1,2,3¬p1,2,2 ∨ ¬p1,2,4¬p1,2,3 ∨ ¬p1,2,4
4 octobre 2013 3 / 7
Using a SAT solver : a toy example
1
3
2
2
“or” : ∨“and” : ∧negation : ¬line i, column j, there is x (pi,j,x)
Line 1, column 1, there is “1” : p1,1,1
Line 2, column 4, there is “3” :
p2,4,3
At position (1, 2), there is only one number
¬p1,2,1 ∨ ¬p1,2,2¬p1,2,1 ∨ ¬p1,2,3¬p1,2,1 ∨ ¬p1,2,4¬p1,2,2 ∨ ¬p1,2,3¬p1,2,2 ∨ ¬p1,2,4¬p1,2,3 ∨ ¬p1,2,4
4 octobre 2013 3 / 7
Using a SAT solver : a toy example
1
3
2
2
“or” : ∨“and” : ∧negation : ¬line i, column j, there is x (pi,j,x)
Line 1, column 1, there is “1” : p1,1,1
Line 2, column 4, there is “3” :
p2,4,3
At position (1, 2), there is only one number
¬p1,2,1 ∨ ¬p1,2,2¬p1,2,1 ∨ ¬p1,2,3¬p1,2,1 ∨ ¬p1,2,4¬p1,2,2 ∨ ¬p1,2,3¬p1,2,2 ∨ ¬p1,2,4¬p1,2,3 ∨ ¬p1,2,4
4 octobre 2013 3 / 7
Using a SAT solver : a toy example
1
3
2
2
“or” : ∨“and” : ∧negation : ¬line i, column j, there is x (pi,j,x)
Line 1, column 1, there is “1” : p1,1,1
Line 2, column 4, there is “3” : p2,4,3
At position (1, 2), there is only one number
¬p1,2,1 ∨ ¬p1,2,2¬p1,2,1 ∨ ¬p1,2,3¬p1,2,1 ∨ ¬p1,2,4¬p1,2,2 ∨ ¬p1,2,3¬p1,2,2 ∨ ¬p1,2,4¬p1,2,3 ∨ ¬p1,2,4
4 octobre 2013 3 / 7
Using a SAT solver : a toy example
1
3
2
2
“or” : ∨“and” : ∧negation : ¬line i, column j, there is x (pi,j,x)
Line 1, column 1, there is “1” : p1,1,1
Line 2, column 4, there is “3” : p2,4,3
At position (1, 2), there is only one number
¬p1,2,1 ∨ ¬p1,2,2¬p1,2,1 ∨ ¬p1,2,3¬p1,2,1 ∨ ¬p1,2,4¬p1,2,2 ∨ ¬p1,2,3¬p1,2,2 ∨ ¬p1,2,4¬p1,2,3 ∨ ¬p1,2,4
4 octobre 2013 3 / 7
Using a SAT solver : a toy example
1
3
2
2
“or” : ∨“and” : ∧negation : ¬line i, column j, there is x (pi,j,x)
Line 1, column 1, there is “1” : p1,1,1
Line 2, column 4, there is “3” : p2,4,3
At position (1, 2), there is only one number¬p1,2,1 ∨ ¬p1,2,2¬p1,2,1 ∨ ¬p1,2,3¬p1,2,1 ∨ ¬p1,2,4¬p1,2,2 ∨ ¬p1,2,3¬p1,2,2 ∨ ¬p1,2,4¬p1,2,3 ∨ ¬p1,2,4
4 octobre 2013 3 / 7
Using a SAT solver : a toy example
1
3
2
2
“or” : ∨“and” : ∧negation : ¬line i, column j, there is x (pi,j,x)
At position (1, 2), there is either 1, 2, 3, or 4
p1,2,1 ∨ p1,2,2 ∨ p1,2,3 ∨ p1,2,4
Number “1” should be somewhere on line 1
p1,1,1 ∨ p1,2,1 ∨ p1,3,1 ∨ p1,4,1
Number “1” should be at most once on line 1
¬p1,1,1 ∨ ¬p1,2,1¬p1,1,1 ∨ ¬p1,3,1¬p1,1,1 ∨ ¬p1,4,1¬p1,2,1 ∨ ¬p1,3,1¬p1,2,1 ∨ ¬p1,4,1¬p1,3,1 ∨ ¬p1,4,1
4 octobre 2013 4 / 7
Using a SAT solver : a toy example
1
3
2
2
“or” : ∨“and” : ∧negation : ¬line i, column j, there is x (pi,j,x)
At position (1, 2), there is either 1, 2, 3, or 4
p1,2,1 ∨ p1,2,2 ∨ p1,2,3 ∨ p1,2,4Number “1” should be somewhere on line 1
p1,1,1 ∨ p1,2,1 ∨ p1,3,1 ∨ p1,4,1
Number “1” should be at most once on line 1
¬p1,1,1 ∨ ¬p1,2,1¬p1,1,1 ∨ ¬p1,3,1¬p1,1,1 ∨ ¬p1,4,1¬p1,2,1 ∨ ¬p1,3,1¬p1,2,1 ∨ ¬p1,4,1¬p1,3,1 ∨ ¬p1,4,1
4 octobre 2013 4 / 7
Using a SAT solver : a toy example
1
3
2
2
“or” : ∨“and” : ∧negation : ¬line i, column j, there is x (pi,j,x)
At position (1, 2), there is either 1, 2, 3, or 4p1,2,1 ∨ p1,2,2 ∨ p1,2,3 ∨ p1,2,4
Number “1” should be somewhere on line 1
p1,1,1 ∨ p1,2,1 ∨ p1,3,1 ∨ p1,4,1
Number “1” should be at most once on line 1
¬p1,1,1 ∨ ¬p1,2,1¬p1,1,1 ∨ ¬p1,3,1¬p1,1,1 ∨ ¬p1,4,1¬p1,2,1 ∨ ¬p1,3,1¬p1,2,1 ∨ ¬p1,4,1¬p1,3,1 ∨ ¬p1,4,1
4 octobre 2013 4 / 7
Using a SAT solver : a toy example
1
3
2
2
“or” : ∨“and” : ∧negation : ¬line i, column j, there is x (pi,j,x)
At position (1, 2), there is either 1, 2, 3, or 4p1,2,1 ∨ p1,2,2 ∨ p1,2,3 ∨ p1,2,4Number “1” should be somewhere on line 1
p1,1,1 ∨ p1,2,1 ∨ p1,3,1 ∨ p1,4,1Number “1” should be at most once on line 1
¬p1,1,1 ∨ ¬p1,2,1¬p1,1,1 ∨ ¬p1,3,1¬p1,1,1 ∨ ¬p1,4,1¬p1,2,1 ∨ ¬p1,3,1¬p1,2,1 ∨ ¬p1,4,1¬p1,3,1 ∨ ¬p1,4,1
4 octobre 2013 4 / 7
Using a SAT solver : a toy example
1
3
2
2
“or” : ∨“and” : ∧negation : ¬line i, column j, there is x (pi,j,x)
At position (1, 2), there is either 1, 2, 3, or 4p1,2,1 ∨ p1,2,2 ∨ p1,2,3 ∨ p1,2,4Number “1” should be somewhere on line 1p1,1,1 ∨ p1,2,1 ∨ p1,3,1 ∨ p1,4,1
Number “1” should be at most once on line 1
¬p1,1,1 ∨ ¬p1,2,1¬p1,1,1 ∨ ¬p1,3,1¬p1,1,1 ∨ ¬p1,4,1¬p1,2,1 ∨ ¬p1,3,1¬p1,2,1 ∨ ¬p1,4,1¬p1,3,1 ∨ ¬p1,4,1
4 octobre 2013 4 / 7
Using a SAT solver : a toy example
1
3
2
2
“or” : ∨“and” : ∧negation : ¬line i, column j, there is x (pi,j,x)
At position (1, 2), there is either 1, 2, 3, or 4p1,2,1 ∨ p1,2,2 ∨ p1,2,3 ∨ p1,2,4Number “1” should be somewhere on line 1p1,1,1 ∨ p1,2,1 ∨ p1,3,1 ∨ p1,4,1Number “1” should be at most once on line 1
¬p1,1,1 ∨ ¬p1,2,1¬p1,1,1 ∨ ¬p1,3,1¬p1,1,1 ∨ ¬p1,4,1¬p1,2,1 ∨ ¬p1,3,1¬p1,2,1 ∨ ¬p1,4,1¬p1,3,1 ∨ ¬p1,4,1
4 octobre 2013 4 / 7
Using a SAT solver : a toy example
1
3
2
2
“or” : ∨“and” : ∧negation : ¬line i, column j, there is x (pi,j,x)
At position (1, 2), there is either 1, 2, 3, or 4p1,2,1 ∨ p1,2,2 ∨ p1,2,3 ∨ p1,2,4Number “1” should be somewhere on line 1p1,1,1 ∨ p1,2,1 ∨ p1,3,1 ∨ p1,4,1Number “1” should be at most once on line 1¬p1,1,1 ∨ ¬p1,2,1¬p1,1,1 ∨ ¬p1,3,1¬p1,1,1 ∨ ¬p1,4,1¬p1,2,1 ∨ ¬p1,3,1¬p1,2,1 ∨ ¬p1,4,1¬p1,3,1 ∨ ¬p1,4,1
4 octobre 2013 4 / 7
SMT = SAT + expressiveness
SAT solvers
¬[(p⇒ q)⇒
[(¬p⇒ q)⇒ q
]]Congruence closure (uninterpreted symbols + equality)
a = b ∧[f (a) 6= f (b) ∨ (p(a) ∧ ¬p(b))
]adding arithmetic
a ≤ b ∧ b ≤ a + x ∧ x = 0 ∧[f (a) 6= f (b) ∨ (p(a) ∧ ¬p(b + x))
]. . .
Some examples : Barcelogic, CVC4 (SVC, CVC, CVC-lite, CVC3),MathSAT, OpenSMT, Yices, Z3 . . .
The solver
4 octobre 2013 5 / 7
SMT internals
SMT formula
SMT solver
4 octobre 2013 6 / 7
SMT internals
SMT formula
SMT solver
SAT solver
4 octobre 2013 6 / 7
SMT internals
SMT formula
SMT solver
Boolean Model
SAT solver
4 octobre 2013 6 / 7
SMT internals
SMT formula
SMT solver
Boolean Model
Theoryreasoner SAT solver
4 octobre 2013 6 / 7
SMT internals
SMT formula
SMT solver
Boolean Model
Conflict clause
Theoryreasoner SAT solver
4 octobre 2013 6 / 7
SMT internals
SMT formula
SMT solverQuantifier-free SMT solver
Boolean Model
Conflict clause
Theoryreasoner SAT solver
4 octobre 2013 6 / 7
SMT internals
SMT formula
SMT solverQuantifier-free SMT solver
Boolean Model
Conflict clause
Theoryreasoner SAT solver
Model
4 octobre 2013 6 / 7
SMT internals
SMT formula
SMT solverQuantifier-free SMT solver
Boolean Model
Conflict clause
Theoryreasoner SAT solver
Instantiationmodule
Instance
Model
4 octobre 2013 6 / 7
SMT internals
SMT formula
SMT solverQuantifier-free SMT solver
Boolean Model
Conflict clause
Theoryreasoner SAT solver
Instantiationmodule
Instance
Model
Model UNSAT (proof/core)
4 octobre 2013 6 / 7
Conclusion
Reasoning on large formulas with arithmetic, functions, equalities,. . .
Applicationsplanningplacementbounded model-checkingverification. . .
The solver
http://www.verit-solver.org
BSD license
4 octobre 2013 7 / 7
