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Semiring -based Soft Constraints

Semiring -based Soft Constraints. Francesco Santini. ERCIM Fellow @Projet Contraintes, INRIA – Rocquencourt, France Dipartimento di Matematica e Informatica, Perugia, Italy. Introduction: Constraints.

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Semiring -based Soft Constraints

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  1. Semiring-based Soft Constraints Francesco Santini ERCIM Fellow @Projet Contraintes, INRIA – Rocquencourt, France Dipartimento di Matematica e Informatica, Perugia, Italy

  2. Introduction: Constraints • Constraint programming is a programming paradigm wherein relations between variables are stated in the form of constraints (yes/no) • A form of declarative programming in form of: • Constraint Satisfaction Problems: P = list of variables/constraints • Constraint Logic Programming: A(X,Y) :- X+Y>0, B(X), C(Y) • Mixed with other paradigms, e.g. Imperative Languages • To solve hard problems (i.e., NP-complete), related to AI • Applied to scheduling and planning, vehicle routing, component configuration, networks and bioinformatics

  3. A Classic Example of CSP • The n-queens problem (proposed in 1848), with n ≥ 4 • N=8, 4,426,165,368 arrangements, but 92 solutions! • Manageable for n = 8, intractable for problems of n ≥ 20 • A possible model: • A variable for each board column {x1,…,x8} • Dom(xi) = {1,…,8} • Assigning a value j to a variable xi means placing a queen in row j, column i • Between each pair of variables xixj, a constraint c(xi, xj): • . , x6 } Sol = {(x1= 7), (x2 = 5)…, (x8 = 4)}

  4. Motivations on semiring-based Soft Constraints (≠ crisp ones) • A formal framework: constraints are associated with values • Over-constrained problems • Preference-driven problems (Constraint Optimization Problems) • Mixed with crisp constraints • Benefits from semiring-like structures • Formal properties • Parametrical with the chosen semirings (general, replaceable metrics, elegant) • Multicriteria problems 17 E.g., to minimize the distance in columns among queens 23

  5. Outline • Introduction and motivations • The general framework • Semirings • Soft Constraints • Soft Constraint Satisfaction Problems • A focus on (Weighted) Argumentation Frameworks • Conclusion

  6. C-semirings • A c-semiring is a tuple • A is the (possibly infinite) set of preference values • 0 and 1 represent the bottom and top preference values • + defines a partial order ( ≥S ) over A such that a ≥S b iff a+b = a • + is commutative, associative, and idempotent, it is closed, 0 is its unit element and 1 is its absorbing element • closed, associative, commutative, and distributes over +, 1 is its unit element and 0 is its absorbing element • is a complete lattice to compose the preferences and + to find the best one

  7. Classical instantiations • Weighted • Fuzzy • Probabilistic • Boolean • Boolean semirings can be used to represent classical crisp problems • The Cartesian product is still a semiring

  8. Soft Constraints • Aconstraint where each instantiation of its variables has an associated preference • Assignment • Constraint • Sum: • Combination: • Projection: • Entailment: Semiring set! Extensions of the semiring operators to assignments

  9. Examples cd cc cb ca

  10. A Soft CSP (graphic) C1 and C3: unary constraints C2: binary constraint P = <V, D, C> <x = a, y= a> 11 <x = a, y= b> 7 <x = b, y = a> 16 <x = b, y = b> 16 ≥ 11 We can consider an α-consistency of the solutions to prune the search!

  11. Argumentation Your country doesnotwant to cooperate Your country doesnotwanteither Your country is a rogue state Rogue state is a controversialterm François Nicolas François 4 6 Nicolas 5 3 2 9 Supportvotes for eachattack! Attacks can be weighted François 6 Nicolas

  12. Argumentation in AI (Dung ‘95) • Itispossible to definesubsets of A with differentsemantics

  13. Conflict-free extensions • No conflict in the subset: a set of coherentarguments

  14. Admissible extensions • A set that can defenditselfagainstall the attacks

  15. Stable extensions • Havingone more argument in the subset leads to a conflict

  16. Mapping to CSPs and SCSPs • (α-)Conflict-free constraints • To find (α-)conflict free extensions • (α-)Admissible constraints • To find (α-)admissible extensions • (α-)Complete constraints • To find (α-)complete extensions • (α-)Stable constraints • To find (α-)stable extensions • V= {a, b, c, d, e} • D= {0,1} a= 1, c= 1, b,d,e=0 isconflict-free a=1, b=1 c,d,e =0 is 7-conflict free

  17. ConArg (Arg. with constraints) • The tool imports JaCoP, Java Constraint Solver • Tests over small-world networks (Barabasi and Kleinberg)

  18. Results • Finding classical not-weighted extensions (Kleinberg) • Hard problems considering a relaxation beta • Comparison with a ASPARTIX

  19. Conclusion • Soft constraints are able to model several hard problems considering preference values (of users). • The semiring-based framework may be used to have a formal and parametrical mean to solve these problems • Links with Operational Research and (Combinatorial) Optimization Problems (Soft CSP)

  20. Thank you for your time! Contacts: francesco.santini@inria.fr

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