Semiring-based Soft
Constraints
Francesco Santini
ERCIM Fellow @Projet Contraintes, INRIA – Rocquencourt, France
Dipartimento di Matematica e Informatica, Perugia, Italy
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
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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):
.,x }
6 {(x =
Sol =
1
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7), (x2 = 5)…, (x8 = 4)}
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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)
17
Multicriteria problems
E.g., to minimize the distance
in columns among queens
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Outline
 Introduction and motivations
 The general framework
Semirings
Soft Constraints
Soft Constraint Satisfaction Problems
 A focus on (Weighted) Argumentation Frameworks
 Conclusion
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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
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Classical instantiations
 Weighted
 Fuzzy
 Probabilistic
 Boolean
 Boolean semirings can be used to represent classical crisp
problems
 The Cartesian product is still a semiring
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Soft Constraints
 A constraint where each instantiation of its variables has an
associated preference
Assignment
Constraint
Semiring set!
 Sum:
 Combination:
Extensions of the
semiring operators to
assignments
 Projection:
 Entailment:
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Examples
cd
cc
cb
ca
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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
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≥ 11
We can consider an α-consistency
of the solutions to prune the search!
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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
François
4
Nicolas François
6
Nicolas
5
3
Attacks can
be
weighted
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2
9
François
6
Nicolas
Support votes
for each
attack!
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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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Mapping to CSPs and SCSPs

(α-)Conflict-free constraints
–

(α-)Admissible constraints
–

To find (α-)admissible extensions
(α-)Complete constraints
–

To find (α-)conflict free extensions
To find (α-)complete extensions
(α-)Stable constraints
–
To find (α-)stable extensions
V= {a, b, c, d, e}
 D= {0,1}

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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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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)
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Thank you for your time!
Contacts:
francesco.santini@inria.fr
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