Soft Constraints Processing over Divisible Residuated Lattices - - PowerPoint PPT Presentation

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Soft Constraints Processing over Divisible Residuated Lattices - - PowerPoint PPT Presentation

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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,

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SOFT CONSTRAINTS AND LOGICAL STRUCTURES SOFT CONSTRAINTS PROCESSING CONCLUSION

Soft Constraints Processing

  • ver 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)

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SOFT CONSTRAINTS AND LOGICAL STRUCTURES SOFT CONSTRAINTS PROCESSING CONCLUSION

Outline

Soft Constraints and Logical Structures Soft Constraint Satisfaction Problems Commutative Bounded Residuated Lattices Soft Constraints Processing Enforcing Algorithms k-Hyperarc Consistency Conclusion

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SOFT CONSTRAINTS AND LOGICAL STRUCTURES SOFT CONSTRAINTS PROCESSING CONCLUSION

Outline

Soft Constraints and Logical Structures Soft Constraint Satisfaction Problems Commutative Bounded Residuated Lattices Soft Constraints Processing Enforcing Algorithms k-Hyperarc Consistency Conclusion

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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 = {C1, . . . , Cq} is a finite set of constraints, that is, pairs (xi, Ri) having xi ∈ Xm as scope and Ri ⊆ Dm as relation. Question: Is there an assignment f : X → D satisfying all constraints, that is, such that f(xi) ∈ Ri for all i ∈ {1, . . . , q}?

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CSP | Example

{R1(x1, x2), R2(x1, x2), R3(x1, x2)} with R1, R2, R3 ⊆ {0, . . . , 5}2:

5,0 5,1 5,2 5,3 5,4 5,5 4,0 4,1 4,2 4,3 4,4 4,5 3,0 3,1 3,2 3,3 3,4 2,0 2,1 2,2 1,0 1,1

(a) R1.

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

(b) R2.

0,0 0,1 0,2 0,3 0,4 0,5 1,0 1,1 1,2 1,3 1,4 1,5 2,0 2,1 2,2 2,3 2,4 2,5 3,0 3,1 3,2 3,3 3,4 4,0 4,1 4,2 4,3 5,0 5,1 5,2

(c) R3.

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CSP | Example

{R1(x1, x2), R2(x1, x2), R3(x1, x2)} with R1, R2, R3 ⊆ {0, . . . , 5}2:

5,0 5,1 5,2 5,3 5,4 5,5 4,0 4,1 4,2 4,3 4,4 4,5 3,0 3,1 3,2 3,3 3,4 2,0 2,1 2,2 1,0 1,1

(a) R1.

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

(b) R2.

0,0 0,1 0,2 0,3 0,4 0,5 1,0 1,1 1,2 1,3 1,4 1,5 2,0 2,1 2,2 2,3 2,4 2,5 3,0 3,1 3,2 3,3 3,4 4,0 4,1 4,2 4,3 5,0 5,1 5,2

(c) R3.

Is there f : {x1, x2} → {0, . . . , 5} satisfying all constraints?

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CSP | Example

There are several such f’s. . .

2,2 3,3 1,1 3,2 4,2 4,3 3,4

(a) R1 ∩ R2 ∩ R3.

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CSP | Example

There are several such f’s. . . what if they pay f(x1) + f(x2) euro?

2,2 3,3 1,1 3,2 4,2 4,3 3,4

(a) R1 ∩ R2 ∩ R3.

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CSP | Example

There are several such f’s. . . what if they pay f(x1) + f(x2) euro?

2,2 3,3 1,1 3,2 4,2 4,3 3,4

(a) R1 ∩ R2 ∩ R3.

2 4 5 6 7 6 7

(b) f’ venue.

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CSP | Example

There are several such f’s. . . what if they pay f(x1) + f(x2) euro?

2,2 3,3 1,1 3,2 4,2 4,3 3,4

(a) R1 ∩ R2 ∩ R3.

2 4 5 6 7 6 7

(b) f’ venue.

7 7

(c) Optimal f’s.

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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.

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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).

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Soft Constraints | Example (Cont’d)

Step 2: Soften crisp constraints (map assignments to the structure).

5,0 5,1 5,2 5,3 5,4 5,5 4,0 4,1 4,2 4,3 4,4 4,5 3,0 3,1 3,2 3,3 3,4 2,0 2,1 2,2 1,0 1,1

(a) Crisp R1.

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Soft Constraints | Example (Cont’d)

Step 2: Soften crisp constraints (map assignments to the structure).

5,0 5,1 5,2 5,3 5,4 5,5 4,0 4,1 4,2 4,3 4,4 4,5 3,0 3,1 3,2 3,3 3,4 2,0 2,1 2,2 1,0 1,1

(a) Crisp R1.

5 6 7 8 9 10 4 5 6 7 8 9 3 4 5 6 7 2 3 4 1 2

(b) Soft R1.

Figure: R1 : {0, . . . , 5}2 → {0, . . . , 10}.

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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 R2.

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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 R2.

5 6 7 8 9 10 4 5 6 7 8 9 3 4 5 6 7 8 2 3 4 5 6 1

(b) Soft R2 image.

Figure: R2 : {0, . . . , 5}2 → {0, . . . , 10}.

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Soft Constraints | Example (Cont’d)

Step 2: Soften crisp constraints (map assignments to the structure).

0,0 0,1 0,2 0,3 0,4 0,5 1,0 1,1 1,2 1,3 1,4 1,5 2,0 2,1 2,2 2,3 2,4 2,5 3,0 3,1 3,2 3,3 3,4 4,0 4,1 4,2 4,3 5,0 5,1 5,2

(a) Crisp R3.

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Soft Constraints | Example (Cont’d)

Step 2: Soften crisp constraints (map assignments to the structure).

0,0 0,1 0,2 0,3 0,4 0,5 1,0 1,1 1,2 1,3 1,4 1,5 2,0 2,1 2,2 2,3 2,4 2,5 3,0 3,1 3,2 3,3 3,4 4,0 4,1 4,2 4,3 5,0 5,1 5,2

(a) Crisp R3.

5 6 7 4 5 6 7 3 4 5 6 7 2 3 4 5 6 7 6 1 2 3 4 5 6 1 1 2 3 4 5

(b) Soft R3 image.

Figure: R3 : {0, . . . , 5}2 → {0, . . . , 10}.

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Combination and Maximization | Example (Cont’d)

Step 3: Maximize constraints combination. For instance,

(2, 4) ⇒ min{R1(2, 4), R2(2, 4), R3(2, 4)} = min{0, 6, 6} = 0, (3, 2) ⇒ min{R1(3, 2), R2(3, 2), R3(3, 2)} = min{5, 5, 5} = 5, . . .

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Combination and Maximization | Example (Cont’d)

Step 3: Maximize constraints combination. For instance,

(2, 4) ⇒ min{R1(2, 4), R2(2, 4), R3(2, 4)} = min{0, 6, 6} = 0, (3, 2) ⇒ min{R1(3, 2), R2(3, 2), R3(3, 2)} = min{5, 5, 5} = 5, . . .

2,2 3,3 1,1 3,2 4,2 4,3 3,4

(a) Crisp solutions.

2 4 5 6 7 6 7

(b) Soft solutions.

7 7

(c) Optimal solutions.

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SOFT CONSTRAINTS AND LOGICAL STRUCTURES SOFT CONSTRAINTS PROCESSING CONCLUSION

Definition

Definition (Soft CSP) A soft CSP is a tuple P = (X, D, P, A) with: (i) variables X = {1, . . . , n} = [n]; (ii) finite domains D = (Di)i∈[n] where i ranges over Di; (iii) valuation structure A = (A, ≤, ⊙, ⊤, ⊥) st (A, ≤, ⊤, ⊥) is a bounded poset, (A, ⊙, ⊤) is a commutative monoid, ⊙ is monotone over ≤ (that is, x ≤ y implies z ⊙ x ≤ z ⊙ y); (iv) P finite multiset of constraints of the form CY :

  • i∈Y

Di → A, where Y ⊆ X is the scope of CY.

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Definition

Notation (Y ⊆ X): l(Y) =

i∈Y Di; t|Y projects t ∈ l(X) onto Y.

Definition (Solution, Inconsistence, Equivalence) Any t ∈ l(X) such that

CY∈P CY(t|Y) is maximal wrt ≤ in

S(P) = {

  • CY∈P

CY(t|Y) | t ∈ l(X)} ⊆ A is a solution to P, and P is inconsistent if S(P) = {⊥}. P = (X, D, P, A) is equivalent to P′ = (X, D, P′, A) iff for every t ∈ l(X),

  • CY∈P

CY(t|Y) =

  • CY∈P′

CY(t|Y).

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Logical Structures

Fact A CSP is a soft CSP (X, D, P, A) where: (i) D = (Di)i∈X with |{Di | i ∈ X}| = 1; (ii) A = ({0, 1}, 0 < 1, min, 1, 0). In the crisp CSP, A is a reduct of the Boolean algebra 2, the algebraic counterpart of classical logic.

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Logical Structures

Fact A CSP is a soft CSP (X, D, P, A) where: (i) D = (Di)i∈X with |{Di | i ∈ X}| = 1; (ii) A = ({0, 1}, 0 < 1, min, 1, 0). In the crisp CSP, A is a reduct of the Boolean algebra 2, the algebraic counterpart of classical logic. Proposal: Adopt algebraic counterparts of nonclassical logics as valuation structures for the soft CSP.

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Residuated Lattices

In Boolean logic the relation between conjunction, ∧, and implication, →, is given by the residuation equivalences, x ∧ y ≤ z iff x ≤ y → z iff y ≤ x → z, which imply many of the properties of ∧ and → (commutativity

  • f ∧, distributivity of ∧ over ∨, left-distributivity of → over ∨,

and right-distributivity of → over ∧). The prominent approach in generalizing Boolean logic relies upon generalizing Boolean conjunction, by means of a binary

  • peration, ⊙, called fusion, and imposing the residuation

equivalences with ∧ replaced by ⊙.

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Residuated Lattices

Definition (Commutative Bounded Residuated Lattice, CBRL) A (commutative bounded) residuated lattice is an algebra (A, ∨, ∧, ⊙, →, ⊤, ⊥) of type (2, 2, 2, 2, 0, 0) st: (i) (A, ⊙, ⊤) is a commutative monoid; (ii) (A, ∨, ∧, ⊤, ⊥) is a bounded lattice; (iii) residuation holds, that is x ⊙ y ≤ z if and only if y ≤ x → z. The monotonicity of fusion over the order follows.

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Lattice Orders and Nonidempotent Combinations

Y ⊆ X, t, t′ ∈ l(Y), A CBRL.

  • CY(t) ≤ CY(t′) says that t′ is preferred to t (the distance

between CY(t) and CY(t′) gives the degree of such preference, ranging over A’s depth).

  • CY(t) CY(t′) says that t′ and t are incomparable (A’s width

gives the number of simultaneous rankings supported by A).

  • ∧’s and ∨’s required by algorithmics (tentative).
  • CY(t) ⊙ CY(t) < CY(t) says that repetitions matter.
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Outline

Soft Constraints and Logical Structures Soft Constraint Satisfaction Problems Commutative Bounded Residuated Lattices Soft Constraints Processing Enforcing Algorithms k-Hyperarc Consistency Conclusion

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Soft CSP

Problem: SOFT-CSP Instance: (X, D, P, A) Goal: Find t ∈ l(X) maximizing

CY∈P CY(t|Y) in A.

The SOFT-CSP is NP-hard: (i) characterize tractable cases (theoretical side); (ii) leverage exhaustive search (enforcing algorithms, applicative side).

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Enforcing Algorithms

Given a soft CSP, an enforcing algorithm enforces over it a local consistency property, in polynomial time. Either the input problem is found locally (hence, globally) inconsistent, or it is transformed into an equivalent problem, possibly inconsistent but easier (with a smaller solution space). Despite their incompleteness as inconsistency test, enforcing algorithms are useful as subprocedures in exhaustive search methods (branch and bound).

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Divisible Residuated Lattices

What is the additional structure required to implement enforcing algorithms over CBRL?

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Divisible Residuated Lattices

What is the additional structure required to implement enforcing algorithms over CBRL? Divisibility is necessary. . . Definition (GBL-algebra) A GBL-algebra is a CBRL where divisibility holds, that is, x ∧ y = x ⊙ (x → y).

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Divisible Residuated Lattices

What is the additional structure required to implement enforcing algorithms over CBRL? Divisibility is necessary. . . Definition (GBL-algebra) A GBL-algebra is a CBRL where divisibility holds, that is, x ∧ y = x ⊙ (x → y). GBL-algebras have a natural logical interpretation, the intersection of Basic (fuzzy) logic and intuitionistic logic. Adopting valuation structures with a logical interpretation, enforcing algorithms reduce to logical deductions (refutations).

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k-Hyperarc Consistency

A soft CSP is k-hyperarc consistent if it is possible to extend any consistent assignment of a variable i to an assignment of any other ≤ k − 1 variables, constrained by i, avoiding additional costs [BG06, CS04, LS04]. Notation (Y ⊆ X, i ∈ Y, a ∈ Di, t ∈ l(Y \ {i})): (t · a) = t′ ∈ l(Y) st t′|{i} = a and t′|Y\{i} = t. Definition (k-Hyperarc Consistency) P = (X, D, P, A) soft CSP, Y ⊆ X st 2 ≤ |Y| ≤ k and CY ∈ P. Y is k-hyperarc consistent if for each i ∈ Y and each a ∈ Di such that C{i}(a) > ⊥, there exists t ∈ l(Y \ {i}) such that, CY(t · a) = ⊤. P is k-hyperarc consistent if every Y ⊆ X st 2 ≤ |Y| ≤ k and CY ∈ P is k-hyperarc consistent.

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Specification

Algorithm: k-HYPERARCCONSISTENCY Input: A soft CSP P = (X, D, P, A), where A is GBL-algebra. Output: ⊥, or a k-hyperarc consistent soft CSP, equivalent to P.

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Pseudocode | 1

k-HYPERARCCONSISTENCY((X, D, P, A)) 1 Q ← {1, . . . , n} 2 while Q = ∅ do 3 i ← POP(Q) 4 foreach Y ⊆ X such that 2 ≤ |Y| ≤ k, i ∈ Y and CY ∈ P do 5 domainShrink ← PROJECT(Y, i) 6 if C{i}(a) = ⊥ for each a ∈ Di then 7 return ⊥ 8 else if domainShrink then 9 PUSH(Q, i) 10 endif 11 endforeach 12 endwhile 13 return (X, D, P′, A)

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Pseudocode | 2

PROJECT(Y, i) 14 domainShrink ← false 15 foreach a ∈ Di such that C{i}(a) > ⊥ do 16 x ← a maximal element in {CY(t · a) | t ∈ l(Y \ {i})} 17 C{i}(a) ← C{i}(a) ⊙ x 18 if C{i}(a) = ⊥ then 19 domainShrink ← true 20 endif 21 foreach t ∈ l(Y \ {i}) do 22 CY(t · a) ← (x → CY(t · a)) 23 ⊲ by divisibility, z ≤ x implies (y ⊙ x) ⊙ (x → z) = y ⊙ z 24 endforeach 25 endforeach 26 return domainShrink

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Correctness and Complexity

Lemma (Complexity) Let P = (X, D, P, A) be soft CSP with X = [n], d = maxi∈[n] |Di| and e = |P|. Then, k-HYPERARCCONSISTENCY(P) runs in O(e2 · dk+1) time. Lemma (Soundness) Let P = (X, D, P, A) be a soft CSP. Consider the output of k-HYPERARCCONSISTENCY(P): (i) if it is ⊥, then P is inconsistent; (ii) ow it is a k-hyperarc consistent soft CSP equivalent to P.

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Outline

Soft Constraints and Logical Structures Soft Constraint Satisfaction Problems Commutative Bounded Residuated Lattices Soft Constraints Processing Enforcing Algorithms k-Hyperarc Consistency Conclusion

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Summary

We presented certain subvarieties of commutative bounded residuated lattices as natural valuation structures for soft CSP’s. These structures constitute the algebraic counterparts of a large family of nonclassical logics, and provide a uniform logical interpretation of enforcing procedures. Divisibility supports a sound implementation of standard enforcing procedures.

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References

  • S. Bistarelli and F. Gadducci.

Enhancing Constraints Manipulation in Semiring-Based Formalisms. ECAI 2006, 63–67, 2006.

  • M. C. Cooper and T. Schiex.

Arc Consistency for Soft Constraints. Artificial Intellingence, 154(1-2):199–227, 2004.

  • N. Galatos, P. Jipsen, T. Kowalski, and H. Ono.

Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Elsevier, 2007.

  • J. Larrosa and T. Schiex.

Solving Weighted CSP by Maintaining Arc Consistency. Artificial Intellingence, 159(1-2):1–26, 2004.

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Thanks!