Computing prime implicants David D´ eharbe Pascal Fontaine Daniel Le Berre Bertrand Mazure UFRN, Brazil Inria, U. of Lorraine, France CRIL-CNRS, U. of Artois, France FMCAD 2013, October 21, Portland 1/17
Model, implicant, prime implicant Example ◮ Let φ = { a ∨ b , a ∨ c , ¬ d ∨ ¬ e ∨ ¬ f } ◮ { a , b , c , d , e , ¬ f } is a model of φ ◮ a ∧ b ∧ c ∧ d ∧ e ∧ ¬ f and a ∧ c ∧ ¬ d are implicants of φ ◮ a ∧ ¬ f , b ∧ c ∧ ¬ f are prime implicants of φ ◮ A model of a formula is an implicant of that formula ◮ From one implicant, one can derive at least one prime implicant ◮ SAT solvers compute models ◮ How to make them compute prime implicants (efficiently) ? 2/17
Clauses, cardinality constraints, pseudo-boolean constraints Various boolean constraints clauses a ∨ ¬ b ∨ c cardinality � l i {≤ , = , ≥} k a + b + c + d ≤ 1 pseudo boolean � w i × l i {≤ , = , ≥} k 4 × a + 2 × b + c + d ≥ 6 ◮ Boolean variables seen as 0/1 integer variables ◮ Normalization : ¬ a + ¬ b + ¬ c + ¬ d ≥ 3 ◮ Clauses are specific cardinality constraints with k = 1 a ∨ ¬ b ∨ c ≡ a + ¬ b + c ≥ 1 ◮ Cardinality constraints are specific PB constraints with w i = 1.
Motivation : SAT-based MAXSAT Example ◮ Let φ = {¬ a ∨ b , ¬ a ∨ c , a , ¬ b , a ∨ b , ¬ c ∨ b } ◮ MAXSAT ( φ ) = minimize � s i such that φ ′ with φ ′ = { s 1 ∨¬ a ∨ b , s 2 ∨¬ a ∨ c , s 3 ∨ a , s 4 ∨¬ b , s 5 ∨ a ∨ b , s 6 ∨¬ c ∨ b } ◮ S = { s i } called “selector variables” ◮ Use SAT solver to find models M of φ ′ ∧ ( � s i < k ) with decreasing k = | M ∩ S | until formula inconsistent. ◮ � s i < k being either a native cardinality constraints (Sat4j) or translated into CNF (QMaxSat). ◮ � w i × s i < k (pseudo-boolean constraint) for Weighted [Partial] MaxSat
Motivation : SAT-based MAXSAT Example ◮ Let φ = {¬ a ∨ b , ¬ a ∨ c , a , ¬ b , a ∨ b , ¬ c ∨ b } ◮ MAXSAT ( φ ) = minimize � s i such that φ ′ with φ ′ = { s 1 ∨¬ a ∨ b , s 2 ∨¬ a ∨ c , s 3 ∨ a , s 4 ∨¬ b , s 5 ∨ a ∨ b , s 6 ∨¬ c ∨ b } ◮ S = { s i } called “selector variables” ◮ Use SAT solver to find models M of φ ′ ∧ ( � s i < k ) with decreasing k = | M ∩ S | until formula inconsistent. ◮ � s i < k being either a native cardinality constraints (Sat4j) or translated into CNF (QMaxSat). ◮ � w i × s i < k (pseudo-boolean constraint) for Weighted [Partial] MaxSat ◮ if M = { a , b , c , s 1 , s 2 , s 3 , s 4 , ¬ s 5 , ¬ s 6 } , k = 4 The bound is not tight ! s 1 , s 2 , s 3 are satisfied while their corresponding clauses are satisfied !
Improve upper bound for SAT-based MAXSAT solvers ◮ Two possible approaches : ◮ Change the encoding : equivalence instead of implication for selector variables ¬ s i → c i becomes ¬ s i ↔ c i adds | φ | binary clauses to φ ′ ◮ Use a prime implicant instead of a model to compute the upper bound ◮ Requirements : ◮ fast : computation need to be done at each model found ◮ compatible with incremental SAT (no/small data structure overhead) ◮ should work with clauses, cardinality constraints, and pseudo-boolean constraints 5/17
Abstract computation of prime implicants 1: procedure Prime ( C , M 0 , Π 0 ) M , Π ← M 0 , Π 0 2: while ℓ ∈ M \ Π do 3: if Required ( M , ℓ, C ) then Π ← Π ∪ { ℓ } 4: else M ← M \ { ℓ } 5: return Π 6: ◮ M 0 is the model returned by the SAT solver ◮ Required () checks if a given literal ℓ is required in the implicant, i.e. ∃ c ∈ C such that satisfying ℓ is mandatory to satysfy c [Castell96]. ◮ Π 0 easy to find required literals (e.g. propagated literals). ◮ In practice, | M 0 \ Π 0 | << | Π 0 | ◮ Works for any kind of constraints ◮ Needs to be refined for efficient implementation !
Prime implicants for clauses (counter based) 1: procedure Prime ( C , M 0 , Π 0 ) M , Π ← M 0 , Π 0 2: for all ℓ ∈ M do W ( ℓ ) ← ∅ 3: for all c ∈ C do 4: N [ c ] ← 0 5: for all ℓ ∈ c do W ( ℓ ) ← W ( ℓ ) ∪ { c } 6: for all ℓ ∈ M do 7: for all c ∈ W ( ℓ ) do N [ c ] ← N [ c ] + 1 8: for all ℓ ∈ M \ Π do 9: if ∃ c ∈ W ( ℓ ) . N [ c ] = 1 then 10: Π ← Π ∪ { ℓ } 11: else 12: for all c ∈ W ( ℓ ) do N [ c ] ← N [ c ] − 1 13: M ← M \ { ℓ } 14: return Π 15:
Prime implicants for cardinality constraints (counter based) 1: procedure Prime ( C , M 0 , Π 0 ) M , Π ← M 0 , Π 0 2: for all ℓ ∈ M do W ( ℓ ) ← ∅ 3: for all c ∈ C do 4: N [ c ] ← 0 5: for all ℓ ∈ c do W ( ℓ ) ← W ( ℓ ) ∪ { c } 6: for all ℓ ∈ M do 7: for all c ∈ W ( ℓ ) do N [ c ] ← N [ c ] + 1 8: for all ℓ ∈ M \ Π do 9: if ∃ c ∈ W ( ℓ ) . N [ c ] = degree ( c ) then 10: Π ← Π ∪ { ℓ } 11: else 12: for all c ∈ W ( ℓ ) do N [ c ] ← N [ c ] − 1 13: M ← M \ { ℓ } 14: return Π 15:
About Counter-based approaches ◮ Complexity is linear in the size of C : O ( � c ∈C | c | ) ◮ Works for both clauses and cardinality constraints ◮ Easy to implement outside the solver ◮ What about early detection of required literals ? ◮ What about pseudo boolean constraints ? ◮ What about incremental SAT solving ? 9/17
Abstract propagation-based algorithm 1: procedure Prime ( C , M 0 , Π 0 ) M , Π ← M 0 , Π 0 2: Π ← Π ∪ Implied ( C , M ) ⊲ Propagates required literals 3: while ℓ ∈ M \ Π do 4: M ← M \ { ℓ } 5: Π ← Π ∪ Implied ( C , M ) ⊲ Propagates removal of ℓ 6: return Π 7: ◮ Implied () propagates truth value similarly to Unit Propagation ◮ New invariant : each literal picked up at line 4 is not required ◮ We can reuse here the data structures found in modern SAT solvers ! 10/17
Prime implicants using watched literals 1: procedure Prime ( C , M 0 , Π 0 , W ) M , Π ← M 0 , Π 0 2: for all ℓ ∈ M \ Π do ⊲ Watch satisfied literals 3: Implied W ( C , M , ¯ ℓ, Π , W ) 4: while ℓ ∈ M \ Π do 5: M ← M \ { ℓ } 6: Implied W ( C , M , ℓ, Π , W ) ⊲ Propagates removal of ℓ 7: return Π 8: 9: procedure Implied W ( C , M , ℓ, ref Π , ref W ) W ℓ ← W ( ℓ ) 10: for all c ∈ W ℓ do 11: Hdl constr ( c , M , ℓ, Π , W ) ⊲ Specific to each c 12: W ( ℓ ) = constraints “watched” for literal ℓ
Hdl constr for clause or cardinality constraints 1: procedure Hdl constr ( c , M , ℓ, ref Π , ref W ) if ∃ ℓ ′ ∈ c ∩ M . ℓ ′ / ∈ W − 1 ( c ) then 2: W ← ( W ∪ { ℓ ′ �→ c } ) \ { ℓ �→ c } 3: else Π ← Π ∪ ( W − 1 ( c ) \ { ℓ } ) 4: ◮ W − 1 ( c ) = literals “watched” in constraint c ◮ Just like lazy data structure management during unit propagation (in clauses or cardinality constraints) ◮ Watches satisfied literals : there is at least one such literal per clause. ◮ One important difference : constraints are traversed only once. that condition must hold to achieve linear time !
Prime Implicant specific propagation ◮ Triggered when a literal is removed from M ◮ Procedure looks for a satisfied literal ◮ Some literals may be deleted ◮ Propagates a required literal . . . p q a c e r s b d w1 w2 13/17
Prime Implicant specific propagation ◮ Triggered when a literal is removed from M ◮ Procedure looks for a satisfied literal ◮ Some literals may be deleted ◮ Propagates a required literal . . . p q a c e r s b d w1 w2 ? 13/17
Prime Implicant specific propagation ◮ Triggered when a literal is removed from M ◮ Procedure looks for a satisfied literal ◮ Some literals may be deleted ◮ Propagates a required literal . . . p q a c c e r s b d w1 w2 13/17
Prime Implicant specific propagation ◮ Triggered when a literal is removed from M ◮ Procedure looks for a satisfied literal ◮ Some literals may be deleted ◮ Propagates a required literal . . . p q a c c e r s b d w1 w2 ? 13/17
Prime Implicant specific propagation ◮ Triggered when a literal is removed from M ◮ Procedure looks for a satisfied literal ◮ Some literals may be deleted ◮ Propagates a required literal . . . p q a c c e r s b d d w1 w2 13/17
Prime Implicant specific propagation ◮ Triggered when a literal is removed from M ◮ Procedure looks for a satisfied literal ◮ Some literals may be deleted ◮ Propagates a required literal . . . p q a c c e r s b d d d b w1 w2 13/17
Prime Implicant specific propagation ◮ Triggered when a literal is removed from M ◮ Procedure looks for a satisfied literal ◮ Some literals may be deleted ◮ Propagates a required literal . . . p q a c e r s b d d b w1 w2 13/17
Prime Implicant specific propagation ◮ Triggered when a literal is removed from M ◮ Procedure looks for a satisfied literal ◮ Some literals may be deleted ◮ Propagates a required literal . . . p q a c e r s b d d b w1 w2 ? 13/17
Prime Implicant specific propagation ◮ Triggered when a literal is removed from M ◮ Procedure looks for a satisfied literal ◮ Some literals may be deleted ◮ Propagates a required literal . . . p q a c e e r s b d d b w1 w2 13/17
Prime Implicant specific propagation ◮ Triggered when a literal is removed from M ◮ Procedure looks for a satisfied literal ◮ Some literals may be deleted ◮ Propagates a required literal . . . p q a c e e r s b d d b w1 w2 ? 13/17
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