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

Junior Seminar 13 th December 2012  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 Introduction: Constraints

Junior Seminar 13 th December 2012 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 {x 1,…,x 8 } -Dom(x i ) = {1,…,8} -Assigning a value j to a variable x i means placing a queen in row j, column i -Between each pair of variables x i x j, a constraint c(x i, x j ):., x 6 } Sol = {(x 1 = 7), (x 2 = 5)…, (x 8 = 4)}

Junior Seminar 13 th December 2012  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 Motivations on semiring-based Soft Constraints (≠ crisp ones) E.g., to minimize the distance in columns among queens 23 17

Junior Seminar 13 th December 2012 Outline  Introduction and motivations  The general framework Semirings Soft Constraints Soft Constraint Satisfaction Problems  A focus on (Weighted) Argumentation Frameworks  Conclusion

Junior Seminar 13 th December 2012 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

Junior Seminar 13 th December 2012 Classical instantiations  Weighted  Fuzzy  Probabilistic  Boolean  Boolean semirings can be used to represent classical crisp problems  The Cartesian product is still a semiring

Junior Seminar 13 th December 2012 Soft Constraints  A constraint 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

Junior Seminar 13 th December 2012 Examples caca cbcb c cdcd

Junior Seminar 13 th December 2012 A Soft CSP (graphic) We can consider an α-consistency of the solutions to prune the search! P = C 1 and C 3 : unary constraints C 2 : binary constraint ≥ 11

Junior Seminar 13 th December 2012 Argumentation Your country does not want to cooperate Your country does not want either Your country is a rogue state Rogue state is a controversial term Support votes for each attack! NicolasFrançois Nicolas François Nicolas Attacks can be weighted

Junior Seminar 13 th December 2012 Argumentation in AI (Dung ‘95)  It is possible to define subsets of A with different semantics

Junior Seminar 13 th December 2012 Conflict-free extensions  No conflict in the subset: a set of coherent arguments

Junior Seminar 13 th December 2012 Admissible extensions  A set that can defend itself against all the attacks

Junior Seminar 13 th December 2012 Stable extensions  Having one more argument in the subset leads to a conflict

Junior Seminar 13 th December 2012  (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} Mapping to CSPs and SCSPs a= 1, c= 1, b,d,e=0 is conflict-free a=1, b=1 c,d,e =0 is 7-conflict free

Junior Seminar 13 th December 2012 ConArg (Arg. with constraints)  The tool imports JaCoP, Java Constraint Solver  Tests over small-world networks (Barabasi and Kleinberg)

Junior Seminar 13 th December 2012 Results  Finding classical not-weighted extensions (Kleinberg)  Hard problems considering a relaxation beta  Comparison with a ASPARTIX

Junior Seminar 13 th December 2012  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) Conclusion

Junior Seminar 13 th December 2012 Thank you for your time! Contacts: