semiring-based soft constraints francesco santini ercim fellow @projet contraintes, inria –...
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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
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Outline
Introduction and motivations
The general frameworkSemirings
Soft Constraints
Soft Constraint Satisfaction Problems
A focus on (Weighted) Argumentation Frameworks
Conclusion
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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
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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
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Argumentation in AI (Dung ‘95)
It is possible to define subsets of A with different semantics
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Conflict-free extensions
No conflict in the subset: a set of coherent arguments
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Admissible extensions A set that can defend itself against all the attacks
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Stable extensions Having one more argument in the subset leads to a conflict
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(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
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ConArg (Arg. with constraints)The tool imports JaCoP, Java Constraint Solver
Tests over small-world networks (Barabasi and Kleinberg)
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Results
Finding classical not-weighted extensions (Kleinberg)
Hard problems considering a relaxation beta
Comparison with a ASPARTIX
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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