S OFT C ONSTRAINTS AND L OGICAL S TRUCTURES S OFT C ONSTRAINTS P ROCESSING C ONCLUSION Soft Constraints Processing over Divisible Residuated Lattices Simone Bova bova@dico.unimi.it Department of Computer Science University of Milan (Milan, Italy) ECSQARU 2009 1-3 July 2009, Verona (Italy)
S OFT C ONSTRAINTS AND L OGICAL S TRUCTURES S OFT C ONSTRAINTS P ROCESSING C ONCLUSION Outline Soft Constraints and Logical Structures Soft Constraint Satisfaction Problems Commutative Bounded Residuated Lattices Soft Constraints Processing Enforcing Algorithms k -Hyperarc Consistency Conclusion
S OFT C ONSTRAINTS AND L OGICAL S TRUCTURES S OFT C ONSTRAINTS P ROCESSING C ONCLUSION Outline Soft Constraints and Logical Structures Soft Constraint Satisfaction Problems Commutative Bounded Residuated Lattices Soft Constraints Processing Enforcing Algorithms k -Hyperarc Consistency Conclusion
S OFT C ONSTRAINTS AND L OGICAL S TRUCTURES S OFT C ONSTRAINTS P ROCESSING C ONCLUSION Constraint Satisfaction Problems Problem: CSP Instance: ( X , D , P ) where: ( i ) X is a finite set of variables ; ( ii ) D is a finite set of values (aka domain ); ( iii ) P = { C 1 , . . . , C q } is a finite set of constraints , that is, pairs ( x i , R i ) having x i ∈ X m as scope and R i ⊆ D m as relation . Question: Is there an assignment f : X → D satisfying all constraints, that is, such that f ( x i ) ∈ R i for all i ∈ { 1 , . . . , q } ?
S OFT C ONSTRAINTS AND L OGICAL S TRUCTURES S OFT C ONSTRAINTS P ROCESSING C ONCLUSION CSP | Example { R 1 ( x 1 , x 2 ) , R 2 ( x 1 , x 2 ) , R 3 ( x 1 , x 2 ) } with R 1 , R 2 , R 3 ⊆ { 0 , . . . , 5 } 2 : � 4,5 � � 5,5 � � 0,5 � � 1,5 � � 2,5 � � 3,5 � � 4,5 � � 5,5 � � 0,5 � � 1,5 � � 2,5 � � 3,4 � � 4,4 � � 5,4 � � 0,4 � � 1,4 � � 2,4 � � 3,4 � � 4,4 � � 5,4 � � 0,4 � � 1,4 � � 2,4 � � 3,4 � � 3,3 � � 4,3 � � 5,3 � � 0,3 � � 1,3 � � 2,3 � � 3,3 � � 4,3 � � 5,3 � � 0,3 � � 1,3 � � 2,3 � � 3,3 � � 4,3 � � 2,2 � � 3,2 � � 4,2 � � 5,2 � � 0,2 � � 1,2 � � 2,2 � � 3,2 � � 4,2 � � 0,2 � � 1,2 � � 2,2 � � 3,2 � � 4,2 � � 5,2 � � 1,1 � � 2,1 � � 3,1 � � 4,1 � � 5,1 � � 0,1 � � 0,1 � � 1,1 � � 2,1 � � 3,1 � � 4,1 � � 5,1 � � 1,0 � � 2,0 � � 3,0 � � 4,0 � � 5,0 � � 0,0 � � 1,0 � � 2,0 � � 3,0 � � 4,0 � � 5,0 � (a) R 1 . (b) R 2 . (c) R 3 .
S OFT C ONSTRAINTS AND L OGICAL S TRUCTURES S OFT C ONSTRAINTS P ROCESSING C ONCLUSION CSP | Example { R 1 ( x 1 , x 2 ) , R 2 ( x 1 , x 2 ) , R 3 ( x 1 , x 2 ) } with R 1 , R 2 , R 3 ⊆ { 0 , . . . , 5 } 2 : � 4,5 � � 5,5 � � 0,5 � � 1,5 � � 2,5 � � 3,5 � � 4,5 � � 5,5 � � 0,5 � � 1,5 � � 2,5 � � 3,4 � � 4,4 � � 5,4 � � 0,4 � � 1,4 � � 2,4 � � 3,4 � � 4,4 � � 5,4 � � 0,4 � � 1,4 � � 2,4 � � 3,4 � � 3,3 � � 4,3 � � 5,3 � � 0,3 � � 1,3 � � 2,3 � � 3,3 � � 4,3 � � 5,3 � � 0,3 � � 1,3 � � 2,3 � � 3,3 � � 4,3 � � 2,2 � � 3,2 � � 4,2 � � 5,2 � � 0,2 � � 1,2 � � 2,2 � � 3,2 � � 4,2 � � 0,2 � � 1,2 � � 2,2 � � 3,2 � � 4,2 � � 5,2 � � 1,1 � � 2,1 � � 3,1 � � 4,1 � � 5,1 � � 0,1 � � 0,1 � � 1,1 � � 2,1 � � 3,1 � � 4,1 � � 5,1 � � 1,0 � � 2,0 � � 3,0 � � 4,0 � � 5,0 � � 0,0 � � 1,0 � � 2,0 � � 3,0 � � 4,0 � � 5,0 � (a) R 1 . (b) R 2 . (c) R 3 . Is there f : { x 1 , x 2 } → { 0 , . . . , 5 } satisfying all constraints?
S OFT C ONSTRAINTS AND L OGICAL S TRUCTURES S OFT C ONSTRAINTS P ROCESSING C ONCLUSION CSP | Example There are several such f ’s . . . � 3,4 � � 3,3 � � 4,3 � � 2,2 � � 3,2 � � 4,2 � � 1,1 � (a) R 1 ∩ R 2 ∩ R 3 .
S OFT C ONSTRAINTS AND L OGICAL S TRUCTURES S OFT C ONSTRAINTS P ROCESSING C ONCLUSION CSP | Example There are several such f ’s . . . what if they pay f ( x 1 ) + f ( x 2 ) euro? � 3,4 � � 3,3 � � 4,3 � � 2,2 � � 3,2 � � 4,2 � � 1,1 � (a) R 1 ∩ R 2 ∩ R 3 .
S OFT C ONSTRAINTS AND L OGICAL S TRUCTURES S OFT C ONSTRAINTS P ROCESSING C ONCLUSION CSP | Example There are several such f ’s . . . what if they pay f ( x 1 ) + f ( x 2 ) euro? 0 0 0 0 0 0 � 3,4 � 0 0 0 7 0 0 � 3,3 � � 4,3 � 0 0 0 6 7 0 � 2,2 � � 3,2 � � 4,2 � 0 0 4 5 6 0 � 1,1 � 0 2 0 0 0 0 0 0 0 0 0 0 (a) R 1 ∩ R 2 ∩ R 3 . (b) f ’ venue.
S OFT C ONSTRAINTS AND L OGICAL S TRUCTURES S OFT C ONSTRAINTS P ROCESSING C ONCLUSION CSP | Example There are several such f ’s . . . what if they pay f ( x 1 ) + f ( x 2 ) euro? 0 0 0 0 0 0 � 3,4 � 0 0 0 7 0 0 7 � 3,3 � � 4,3 � 0 0 0 6 7 0 7 � 2,2 � � 3,2 � � 4,2 � 0 0 4 5 6 0 � 1,1 � 0 2 0 0 0 0 0 0 0 0 0 0 (a) R 1 ∩ R 2 ∩ R 3 . (b) f ’ venue. (c) Optimal f ’s.
S OFT C ONSTRAINTS AND L OGICAL S TRUCTURES S OFT C ONSTRAINTS P ROCESSING C ONCLUSION Feasibility vs. Optimization The crisp CSP is a feasibility problem (any satisfying assignment is equally good). The soft CSP is an optimization problem: each constraint maps assignments to a valuation structure , that is, a bounded poset equipped with a suitable combination operator; the goal is to find an assignment such that the combination of its images under all the constraints is maximal in the structure.
S OFT C ONSTRAINTS AND L OGICAL S TRUCTURES S OFT C ONSTRAINTS P ROCESSING C ONCLUSION Valuation Structure | Example (Cont’d) Step 1: Design valuation structure. A = ( { 0 , . . . , 10 } , ⊥ = 0 < · · · < 10 = ⊤ , min ) . min : ( i ) associative, commutative (no precedence, no order); ( ii ) monotone over ≤ (more constraints, worst solutions); ( iii ) min { x , ⊥} = ⊥ (unsatisfiability marker); ( iv ) min { x , ⊤} = x (triviality marker).
S OFT C ONSTRAINTS AND L OGICAL S TRUCTURES S OFT C ONSTRAINTS P ROCESSING C ONCLUSION Soft Constraints | Example (Cont’d) Step 2: Soften crisp constraints (map assignments to the structure). � 4,5 � � 5,5 � � 3,4 � � 4,4 � � 5,4 � � 3,3 � � 4,3 � � 5,3 � � 2,2 � � 3,2 � � 4,2 � � 5,2 � � 1,1 � � 2,1 � � 3,1 � � 4,1 � � 5,1 � � 1,0 � � 2,0 � � 3,0 � � 4,0 � � 5,0 � (a) Crisp R 1 .
S OFT C ONSTRAINTS AND L OGICAL S TRUCTURES S OFT C ONSTRAINTS P ROCESSING C ONCLUSION Soft Constraints | Example (Cont’d) Step 2: Soften crisp constraints (map assignments to the structure). � 4,5 � � 5,5 � 0 0 0 0 9 10 � 3,4 � � 4,4 � � 5,4 � 0 0 0 7 8 9 � 3,3 � � 4,3 � � 5,3 � 0 0 0 6 7 8 � 2,2 � � 3,2 � � 4,2 � � 5,2 � 0 0 4 5 6 7 � 1,1 � � 2,1 � � 3,1 � � 4,1 � � 5,1 � 0 2 3 4 5 6 � 1,0 � � 2,0 � � 3,0 � � 4,0 � � 5,0 � 0 1 2 3 4 5 (a) Crisp R 1 . (b) Soft R 1 . Figure: R 1 : { 0 , . . . , 5 } 2 → { 0 , . . . , 10 } .
S OFT C ONSTRAINTS AND L OGICAL S TRUCTURES S OFT C ONSTRAINTS P ROCESSING C ONCLUSION Soft Constraints | Example (Cont’d) Step 2: Soften crisp constraints (map assignments to the structure). � 0,5 � � 1,5 � � 2,5 � � 3,5 � � 4,5 � � 5,5 � � 0,4 � � 1,4 � � 2,4 � � 3,4 � � 4,4 � � 5,4 � � 0,3 � � 1,3 � � 2,3 � � 3,3 � � 4,3 � � 5,3 � � 0,2 � � 1,2 � � 2,2 � � 3,2 � � 4,2 � � 0,1 � (a) Crisp R 2 .
S OFT C ONSTRAINTS AND L OGICAL S TRUCTURES S OFT C ONSTRAINTS P ROCESSING C ONCLUSION Soft Constraints | Example (Cont’d) Step 2: Soften crisp constraints (map assignments to the structure). � 0,5 � � 1,5 � � 2,5 � � 3,5 � � 4,5 � � 5,5 � 5 6 7 8 9 10 � 0,4 � � 1,4 � � 2,4 � � 3,4 � � 4,4 � � 5,4 � 4 5 6 7 8 9 � 0,3 � � 1,3 � � 2,3 � � 3,3 � � 4,3 � � 5,3 � 3 4 5 6 7 8 � 0,2 � � 1,2 � � 2,2 � � 3,2 � � 4,2 � 2 3 4 5 6 0 � 0,1 � 1 0 0 0 0 0 0 0 0 0 0 0 (a) Crisp R 2 . (b) Soft R 2 image. Figure: R 2 : { 0 , . . . , 5 } 2 → { 0 , . . . , 10 } .
S OFT C ONSTRAINTS AND L OGICAL S TRUCTURES S OFT C ONSTRAINTS P ROCESSING C ONCLUSION Soft Constraints | Example (Cont’d) Step 2: Soften crisp constraints (map assignments to the structure). � 0,5 � � 1,5 � � 2,5 � � 0,4 � � 1,4 � � 2,4 � � 3,4 � � 0,3 � � 1,3 � � 2,3 � � 3,3 � � 4,3 � � 0,2 � � 1,2 � � 2,2 � � 3,2 � � 4,2 � � 5,2 � � 0,1 � � 1,1 � � 2,1 � � 3,1 � � 4,1 � � 5,1 � � 0,0 � � 1,0 � � 2,0 � � 3,0 � � 4,0 � � 5,0 � (a) Crisp R 3 .
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