semiring-based soft constraints francesco santini ercim fellow @projet contraintes, inria –...

Semiring-based Soft Constraints

Francesco Santini

ERCIM Fellow @Projet Contraintes, INRIA – Rocquencourt, France

Dipartimento di Matematica e Informatica, Perugia, Italy

Junior Seminar 13th 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 13th 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 {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 xi xj, a constraint c(xi, xj):

. , x6

}Sol = {(x1= 7), (x2 = 5)…, (x8 = 4)}

Junior Seminar 13th December 2012

A formal framework: constraints are associated with valuesOver-constrained problems

Preference-driven problems (Constraint Optimization Problems)

Mixed with crisp constraints

Benefits from semiring-like structuresFormal 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 13th December 2012

Outline

Introduction and motivations

The general frameworkSemirings

Soft Constraints

Soft Constraint Satisfaction Problems

A focus on (Weighted) Argumentation Frameworks

Conclusion

Junior Seminar 13th December 2012

C-semirings

A c-semiring is a tupleA 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 13th 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 13th 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 13th December 2012

Examples

ca

cb

cc

cd

Junior Seminar 13th December 2012

A Soft CSP (graphic)

<x = a, y= a> 11<x = a, y= b> 7<x = b, y = a> 16<x = b, y = b> 16

We can consider an α-consistency of the solutions to prune the search!

P = <V, D, C> C1 and C3: unary constraintsC2: binary constraint

≥ 11

Junior Seminar 13th 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

9

4

5

6

6

23

Support votes for each attack!

Nicolas François

Nicolas

François

François

Nicolas

Attacks can be

weighted

Junior Seminar 13th December 2012

Argumentation in AI (Dung ‘95)

It is possible to define subsets of A with different semantics

Junior Seminar 13th December 2012

Conflict-free extensions

No conflict in the subset: a set of coherent arguments

Junior Seminar 13th December 2012

Admissible extensions A set that can defend itself against all the attacks

Junior Seminar 13th December 2012

Stable extensions Having one more argument in the subset leads to a conflict

Junior Seminar 13th 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 13th December 2012

ConArg (Arg. with constraints)The tool imports JaCoP, Java Constraint Solver

Tests over small-world networks (Barabasi and Kleinberg)

Junior Seminar 13th December 2012

Results

Finding classical not-weighted extensions (Kleinberg)

Hard problems considering a relaxation beta

Comparison with a ASPARTIX

Junior Seminar 13th 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 13th December 2012

Thank you for your time!

Contacts:

francesco.santini@inria.fr