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Exploring Semiring-based Soft Constraints in Constraint Programming

This presentation introduces semiring-based soft constraints, a framework blending constraint programming with preference-driven optimization. It illustrates how constraints can express nuanced relationships between variables, useful in applications like scheduling, vehicle routing, and bioinformatics. The concept is grounded in classical problems such as the n-queens problem, emphasizing how soft constraints address over-constrained and multicriteria scenarios. By examining semirings, preference values, and argumentation frameworks, we demonstrate a comprehensive approach to solving complex problems in AI and operational research.

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Exploring Semiring-based Soft Constraints in Constraint Programming

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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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