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

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


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)}


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


Outline

Introduction and motivations

The general frameworkSemirings

Soft Constraints

Soft Constraint Satisfaction Problems

A focus on (Weighted) Argumentation Frameworks

Conclusion


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


Classical instantiations

Weighted

Fuzzy

Probabilistic

Boolean

Boolean semirings can be used to represent classical crisp

problems

The Cartesian product is still a semiring


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


Examples

ca

cb

cc

cd


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


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


Argumentation in AI (Dung ‘95)

It is possible to define subsets of A with different semantics


Conflict-free extensions

No conflict in the subset: a set of coherent arguments


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


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


(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


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

Tests over small-world networks (Barabasi and Kleinberg)


Results

Finding classical not-weighted extensions (Kleinberg)

Hard problems considering a relaxation beta

Comparison with a ASPARTIX


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


Thank you for your time!

Contacts:

[email protected]

Documents

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