From satisfaction to optimization, and beyond SAT-based guided - PowerPoint PPT Presentation

From satisfaction to optimization, and beyond SAT-based guided problem solving Daniel Le Berre joint work with Mutsunori Banbara, Tiago de Lima, Jean-Marie Lagniez, Valentin Montmirail, Stéphanie Roussel, Naoyuki Tamura, Takehide Soh CNRS, Université d’Artois, FRANCE {leberre}@cril.univ-artois.fr SAT+SMT school, IIT Bombay, India, 10 December 2019

2 Purpose of this talk ◮ Using SAT solvers are black boxes ◮ Importance of the interaction with the solver ◮ When encodings are too large

3 The SAT problem: textbook definition Definition Input : A set of clauses C built from a propositional language with n variables. Output : Is there an assignment of the n variables that satisfies all those clauses?

3 The SAT problem: textbook definition Definition Input : A set of clauses C built from a propositional language with n variables. Output : Is there an assignment of the n variables that satisfies all those clauses? Example C 1 = {¬ a ∨ b , ¬ b ∨ c } = ( ¬ a ∨ b ) ∧ ( ¬ b ∨ c ) = ( a ′ + b ) . ( b ′ + c ) C 2 = C 1 ∪ { a , ¬ c } = C 1 ∧ a ∧ ¬ c For C 1 , the answer is yes, for C 2 the answer is no C 1 | = ¬ ( a ∧ ¬ c ) = ¬ a ∨ c

4 The SAT problem solver: practical point of view 1/3 Definition Input : A set of clauses C built from a propositional language with n variables. Output : If there is an assignment of the n variables that satisfies all those clauses, provide such assignment, else answer UNSAT.

4 The SAT problem solver: practical point of view 1/3 Definition Input : A set of clauses C built from a propositional language with n variables. Output : If there is an assignment of the n variables that satisfies all those clauses, provide such assignment, else answer UNSAT. Example C 1 = {¬ a ∨ b , ¬ b ∨ c } = ( ¬ a ∨ b ) ∧ ( ¬ b ∨ c ) = ( a ′ + b ) . ( b ′ + c ) C 2 = C 1 ∪ { a , ¬ c } = C 1 ∧ a ∧ ¬ c For C 1 , one answer is { a , b , c } , for C 2 the answer is UNSAT.

4 The SAT problem solver: practical point of view 1/3 Definition Input : A set of clauses C built from a propositional language with n variables. Output : If there is an assignment of the n variables that satisfies all those clauses, provide such assignment, else answer UNSAT. Example C 1 = {¬ a ∨ b , ¬ b ∨ c } = ( ¬ a ∨ b ) ∧ ( ¬ b ∨ c ) = ( a ′ + b ) . ( b ′ + c ) C 2 = C 1 ∪ { a , ¬ c } = C 1 ∧ a ∧ ¬ c For C 1 , one answer is { a , b , c } , for C 2 the answer is UNSAT. SAT answers can be checked: trusted model oracle

5 The SAT problem solver: practical point of view 2/3 Definition Input : A set of clauses C built from a propositional language with n variables. Output : If there is an assignment of the n variables that satisfies all those clauses, provide such assignment, else provide a subset of C which cannot be satisfied.

5 The SAT problem solver: practical point of view 2/3 Definition Input : A set of clauses C built from a propositional language with n variables. Output : If there is an assignment of the n variables that satisfies all those clauses, provide such assignment, else provide a subset of C which cannot be satisfied. Example C 1 = {¬ a ∨ b , ¬ b ∨ c } = ( ¬ a ∨ b ) ∧ ( ¬ b ∨ c ) = ( a ′ + b ) . ( b ′ + c ) C 2 = C 1 ∪ { a , ¬ c } = C 1 ∧ a ∧ ¬ c For C 1 , one answer is { a , b , c } , for C 2 the answer is C 2

5 The SAT problem solver: practical point of view 2/3 Definition Input : A set of clauses C built from a propositional language with n variables. Output : If there is an assignment of the n variables that satisfies all those clauses, provide such assignment, else provide a subset of C which cannot be satisfied. Example C 1 = {¬ a ∨ b , ¬ b ∨ c } = ( ¬ a ∨ b ) ∧ ( ¬ b ∨ c ) = ( a ′ + b ) . ( b ′ + c ) C 2 = C 1 ∪ { a , ¬ c } = C 1 ∧ a ∧ ¬ c For C 1 , one answer is { a , b , c } , for C 2 the answer is C 2 UNSAT core may explain inconsistency if much smaller than C : informative UNSAT oracle

6 The SAT problem solver: practical point of view 3/3 Definition Allow the solver to decide the satisfiability of a formula with: ◮ increasing number of constraints ◮ provided some “assumptions" are satisfied

6 The SAT problem solver: practical point of view 3/3 Definition Allow the solver to decide the satisfiability of a formula with: ◮ increasing number of constraints ◮ provided some “assumptions" are satisfied Example C = { s 1 ∨ ¬ a ∨ b , s 1 ∨ ¬ b ∨ c , s 2 ∨ a , s 2 ∨ ¬ c } C 1 ≡ C ∧ ¬ s 1 ∧ s 2 C 2 ≡ C ∧ ¬ s 1 ∧ ¬ s 2

6 The SAT problem solver: practical point of view 3/3 Definition Allow the solver to decide the satisfiability of a formula with: ◮ increasing number of constraints ◮ provided some “assumptions" are satisfied Example C = { s 1 ∨ ¬ a ∨ b , s 1 ∨ ¬ b ∨ c , s 2 ∨ a , s 2 ∨ ¬ c } C 1 ≡ C ∧ ¬ s 1 ∧ s 2 C 2 ≡ C ∧ ¬ s 1 ∧ ¬ s 2 The solver is considered as a stateful system: as long as the constraints are satisfiable, learn clauses can be kept: incremental SAT oracle

7 How to solve MaxSat MinUnsat with SAT? ◮ Associate to each clause a weight (penalty) w i taken into account if the clause is violated: Soft clauses S . ◮ Special weight ( ∞ ) for clauses that cannot be violated: hard clauses H Definition (Partial Weighted MaxSat) Find a model M of H that minimizes weight ( M , S ) such that: ◮ weight ( M , ( c i , w i )) = 0 if M satisfies c i , else w i . ◮ weight ( M , S ) = � wc ∈ S weight ( M , wc ) Simply called MaxSAT if k = 1 and H = ∅

7 How to solve MaxSat MinUnsat with SAT? ◮ Associate to each clause a weight (penalty) w i taken into account if the clause is violated: Soft clauses S . ( ¬ a ∨ b , 6 ) ∧ ( ¬ b ∨ c , 8 ) ◮ Special weight ( ∞ ) for clauses that cannot be violated: hard clauses H Definition (Partial Weighted MaxSat) Find a model M of H that minimizes weight ( M , S ) such that: ◮ weight ( M , ( c i , w i )) = 0 if M satisfies c i , else w i . ◮ weight ( M , S ) = � wc ∈ S weight ( M , wc ) Simply called MaxSAT if k = 1 and H = ∅

7 How to solve MaxSat MinUnsat with SAT? ◮ Associate to each clause a weight (penalty) w i taken into account if the clause is violated: Soft clauses S . ( ¬ a ∨ b , 6 ) ∧ ( ¬ b ∨ c , 8 ) ◮ Special weight ( ∞ ) for clauses that cannot be violated: hard clauses H ( a , ∞ ) ∧ ( ¬ c , ∞ ) Definition (Partial Weighted MaxSat) Find a model M of H that minimizes weight ( M , S ) such that: ◮ weight ( M , ( c i , w i )) = 0 if M satisfies c i , else w i . ◮ weight ( M , S ) = � wc ∈ S weight ( M , wc ) Simply called MaxSAT if k = 1 and H = ∅

7 How to solve MaxSat MinUnsat with SAT? ◮ Associate to each clause a weight (penalty) w i taken into account if the clause is violated: Soft clauses S . ( ¬ a ∨ b , 6 ) ∧ ( ¬ b ∨ c , 8 ) ◮ Special weight ( ∞ ) for clauses that cannot be violated: hard ( a , ∞ ) ∧ ( ¬ c , ∞ ) clauses H Definition (Partial Weighted MaxSat) Find a model M of H that minimizes weight ( M , S ) such that: ◮ weight ( M , ( c i , w i )) = 0 if M satisfies c i , else w i . ◮ weight ( M , S ) = � wc ∈ S weight ( M , wc ) weight of { a , ¬ b , ¬ c } is 6 Simply called MaxSAT if k = 1 and H = ∅

8 Linear Search for solving MaxSAT x 6 , x 2 ¬ x 6 , x 2 ¬ x 2 , x 1 ¬ x 1 ¬ x 6 , x 8 x 6 , ¬ x 8 ¬ x 4 , x 5 x 2 , x 4 x 7 , x 5 ¬ x 7 , x 5 ¬ x 5 , x 3 ¬ x 3 Example CNF formula ( k = 1 for each clause, not displayed)

8 Linear Search for solving MaxSAT x 6 , x 2 , b 7 ¬ x 6 , x 2 , b 8 ¬ x 2 , x 1 , b 1 ¬ x 1 , b 2 ¬ x 6 , x 8 , b 9 x 6 , ¬ x 8 , b 10 x 2 , x 4 , b 3 ¬ x 4 , x 5 , b 4 x 7 , x 5 , b 11 ¬ x 7 , x 5 , b 12 ¬ x 5 , x 3 , b 5 ¬ x 3 , b 6 Add selector or blocking variables b i

8 Linear Search for solving MaxSAT x 6 , x 2 , b 7 ¬ x 6 , x 2 , b 8 ¬ x 2 , x 1 , b 1 ¬ x 1 , b 2 ¬ x 6 , x 8 , b 9 x 6 , ¬ x 8 , b 10 x 2 , x 4 , b 3 ¬ x 4 , x 5 , b 4 x 7 , x 5 , b 11 ¬ x 7 , x 5 , b 12 ¬ x 5 , x 3 , b 5 ¬ x 3 , b 6 Formula is SAT; eg model M contains b 1 , ¬ b 2 , b 3 , ¬ b 4 , b 5 , ¬ b 7 , ¬ b 8 , ¬ b 9 , b 10 , ¬ b 11 , b 12

8 Linear Search for solving MaxSAT x 6 , x 2 , b 7 ¬ x 6 , x 2 , b 8 ¬ x 2 , x 1 , b 1 ¬ x 1 , b 2 ¬ x 6 , x 8 , b 9 x 6 , ¬ x 8 , b 10 x 2 , x 4 , b 3 ¬ x 4 , x 5 , b 4 x 7 , x 5 , b 11 ¬ x 7 , x 5 , b 12 ¬ x 5 , x 3 , b 5 ¬ x 3 , b 6 � 12 i = 1 b i < 5 Bound the number of constraints to be relaxed: | M ∩ B | = 5

8 Linear Search for solving MaxSAT x 6 , x 2 , b 7 ¬ x 6 , x 2 , b 8 ¬ x 2 , x 1 , b 1 ¬ x 1 , b 2 ¬ x 6 , x 8 , b 9 x 6 , ¬ x 8 , b 10 x 2 , x 4 , b 3 ¬ x 4 , x 5 , b 4 x 7 , x 5 , b 11 ¬ x 7 , x 5 , b 12 ¬ x 5 , x 3 , b 5 ¬ x 3 , b 6 � 12 i = 1 b i < 5 Formula is (again) SAT; eg model contains b 1 , ¬ b 2 , ¬ b 3 , ¬ b 4 , ¬ b 5 , ¬ b 7 , ¬ b 8 , ¬ b 9 , ¬ b 10 , ¬ b 11 , b 12

8 Linear Search for solving MaxSAT x 6 , x 2 , b 7 ¬ x 6 , x 2 , b 8 ¬ x 2 , x 1 , b 1 ¬ x 1 , b 2 ¬ x 6 , x 8 , b 9 x 6 , ¬ x 8 , b 10 x 2 , x 4 , b 3 ¬ x 4 , x 5 , b 4 x 7 , x 5 , b 11 ¬ x 7 , x 5 , b 12 ¬ x 5 , x 3 , b 5 ¬ x 3 , b 6 � 12 i = 1 b i < 2 Bound the number of constraints to be relaxed | M ∩ B | = 2

8 Linear Search for solving MaxSAT x 6 , x 2 , b 7 ¬ x 6 , x 2 , b 8 ¬ x 2 , x 1 , b 1 ¬ x 1 , b 2 ¬ x 6 , x 8 , b 9 x 6 , ¬ x 8 , b 10 x 2 , x 4 , b 3 ¬ x 4 , x 5 , b 4 x 7 , x 5 , b 11 ¬ x 7 , x 5 , b 12 ¬ x 5 , x 3 , b 5 ¬ x 3 , b 6 � 12 i = 1 b i < 2 Instance is now UNSAT