This document discusses methods for handling conditional preference queries using possibilistic logic. It presents three methods: 1) Weak comparative preferences which uses a minimum specificity principle to provide a well-ordered partition of preference levels. 2) Lexicographic comparison which encodes preferences as vectors and uses a leximin order. 3) A hybrid method which first uses weak comparative preferences to determine the initial levels and then applies lexicographic comparison on those levels. The document also compares these approaches and discusses using symbolic weights to capture order relations between preferences.

1. Conditional preference queries and possibilistic logic
Didier Dubois Henri Prade Fayçal Touazi
IRIT, University of Toulouse, France.
September 2013
FQAS
Didier Dubois, Henri Prade, Fayçal Touazi (IRIT) 1 September 2013 1 / 21

2. Outlines
1 Background
Possibilistic logic
Running example
2 Preference representation
Preference formats
Preference encoding in possibilistic logic
3 Handling preference queries
Weak Comparative Preferences
Lexicographic comparison
Adding constraints between symbolic weights
Hybrid method
4 Conclusion
Didier Dubois, Henri Prade, Fayçal Touazi (IRIT) 2 September 2013 2 / 21

3. Background Possibilistic logic
Possibilistic logic
Weighted extension of classical logic : (Φ, α) with Φ a proposition and
1>α>0
(Φ, α) is interpreted as N(Φ) ≥ α where :
N(Φ) = 1 − Π(¬Φ)
(Φ, α) is viewed as a constraint : it means Φ must be satisfied with
priority α
The degree of preference of a model of Φ is 1
The degree of preference of a model of ¬Φ is 1 − α
Didier Dubois, Henri Prade, Fayçal Touazi (IRIT) 3 September 2013 3 / 21

4. Background Running example
Running example
Let us consider a data base storing pieces of information about houses to
let that are described in terms of 25 attributes.
Example
We want to express the following preferences :
The number of persons accommodated should be more than 10,
imperatively.
It is preferred to have a house where animals are allowed,
It is preferred to be close to the sea by a distance between 1 and 20
km ;
If the house is far from the sea by more than 20 km, it is preferred to
have a tennis court at less than 4 km
If moreover the distance of the house to the tennis court is more than
4 km, it is desirable to have a swimming pool be at a distance less
than 6 km
Didier Dubois, Henri Prade, Fayçal Touazi (IRIT) 4 September 2013 4 / 21

5. Preference representation Preference formats
Preference formats
two types of preferences :
Unconditional preference
“q is preferred to ¬q”
Example It is preferred to have a house where animals are allowed.
Conditional preference
“in context p, q is preferred to ¬q"
Example If the house is far from the sea by more than 20 km, it is
preferred to have a tennis court at less than 4 km.
Didier Dubois, Henri Prade, Fayçal Touazi (IRIT) 5 September 2013 5 / 21

6. Preference representation Preference encoding in possibilistic logic
Representing preferences in Possibilistic logic
“in context p, q is preferred to ¬q" ⇒ Π(p ∧ q) > Π(p ∧ ¬q)
of the form Π(L(p ∧ q)) > Π(R(p ∧ q)).
Example
We want to express the following preferences :
The number of persons accommodated should be more than 10
- Π(Accomod. ≥ 10) > Π(¬(Accomod. ≥ 10))
If the house is far from the sea by more than 20 km, it is preferred to
have a tennis court at less than 4 km
- Π((Sea > 20) ∧ Tennis ≤ 4) > Π((Sea > 20) ∧ ¬(Tennis ≤ 4))
Didier Dubois, Henri Prade, Fayçal Touazi (IRIT) 6 September 2013 6 / 21

7. Handling preference queries Weak Comparative Preferences
Weak Comparative Preferences
Based on applying the minimum specificity principle.
Minimum specificity principle
consider the least informative possibility distribution ensuring all the
constraints are satisfied.
The ordering is given as a well-ordered partition (E1, ..., Em).
Algorithm WCP
The most satisfactory set Em is made of the interpretations that do
not satisfy any R(φj ).
Constraints whose left part L(φi ) is satisfied by an interpretation of
Em are deleted.
Repeat until all constraints are deleted.
Didier Dubois, Henri Prade, Fayçal Touazi (IRIT) 7 September 2013 7 / 21

8. Handling preference queries Weak Comparative Preferences
We have shown :
Proposition
Let a query Q composed of n preference constraints. The number of
elements of the well-ordered partition {E1, · · · , Em} is AT MOST
m = n + 1.
Example
Let 4 preference constraints be given as follows (L(φi ),R(φi ) are replaced
by sets of interpretations) :
φ1 = ({t1, t2, t3}, {t4, t5, t6, t7, t8}) ;
φ2 = ({t4, t5}, {t6, t7, t8}) ;
φ3 = ({t6}, {t7}) ;
φ4 = ({t7}, {t8}).
Applying Algorithm WCP gives 5 preference levels :
E1 = {t1, t2, t3}, E2 = {t4, t5}, E3 = {t6}, E4 = {t7} and E5 = {t8}.
Didier Dubois, Henri Prade, Fayçal Touazi (IRIT) 8 September 2013 8 / 21

9. Handling preference queries Lexicographic comparison
Lexicographic comparison method
Vector comparison based method
Constraints of the form Π(p ∧ q) > Π(p ∧ ¬q) can also be encoded by
constraints of the form (¬p ∨ q, αi ), the weights αi are supposed
incomparable.
Vector construction
Let Σ = {(Φ1, αi ), ..., (Φn, αn)} be a possibilistic base. For each
interpretation ω, we can build a vector ω(Σ) as follows.
For each preference constraint φi for i = 1, · · · , n :
ith
component = π((Φi , αi )) =
1, if ω |= Φi ;
1 − αi , else.
Didier Dubois, Henri Prade, Fayçal Touazi (IRIT) 9 September 2013 9 / 21

10. Handling preference queries Lexicographic comparison
Leximin order
Definition (leximin)
Let v and v be two vectors having the same number of components.
Find permuation f so as to delete the maximal number of pairs
(vi , vf (i)) such that vi = vf (i) in v and v
v leximin v iff min(r(v) ∪ r(v )) ⊆ r(v ), where r(v) contains
remaining components in v.
Example
v = (1, 1 − α2, 1, 1), v = (1, 1 − α2, 1 − α3, 1)
r = (1) and r = (1 − α3). We have v leximin v .
Proposition
If a query Q is composed of n preference constraints, then the maximal
number of levels generated by leximin is n + 1.
Didier Dubois, Henri Prade, Fayçal Touazi (IRIT) 10 September 2013 10 / 21

11. Handling preference queries Lexicographic comparison
Example (continue)
The number of persons accommodated should be more than 10 :
φ0 = (Accomod. ≥ 10, 1)
It is preferred to have a house where animals are allowed,
φ1 = (Animal, α1)
If the house is far from the sea by more than 20 km, it is preferred to
have a tennis court at less than 4 km
φ3 = (¬(Sea > 20) ∨ Tennis ≤ 4, α3)
After applying the leximin, we get 3 levels of preference ranking.
Didier Dubois, Henri Prade, Fayçal Touazi (IRIT) 11 September 2013 11 / 21

12. Handling preference queries Lexicographic comparison
Background on CP-nets
A CP-net N is a directed acyclic graph on a set of variables
V = {X1, ·, Xn} (numbered in accordance with the graph)
A CP-net preference is of the form u : xi > ¬xi (or u : ¬xi > xi ) where
u is conjunction of instances of parent variables of Xi (called a
context)
In the final preference graph, father nodes Pa(xi ) seem to be more
important than children nodes Xi
Every CP-net preference u : xi > ¬xi is encoded by (¬u ∨ xi , αi ).
Didier Dubois, Henri Prade, Fayçal Touazi (IRIT) 12 September 2013 12 / 21

13. Handling preference queries Adding constraints between symbolic weights
Constraints between weights in CP-net style
This method is inspired from CP-nets :
Idea
For each pair of formulas of the form (¬u ∨ xi , αi ) and
(¬u ∨ ¬xi ∨ xj , αj ) such that u is a context, Xi plays the role of the
father of Xj in a CP-net.
We add a constraint αi > αj
Didier Dubois, Henri Prade, Fayçal Touazi (IRIT) 13 September 2013 13 / 21

14. Handling preference queries Adding constraints between symbolic weights
Example (continue)
Let us consider the previous running example, the preference constraints
are encoded as follows :
φ0 = (Accomod. ≥ 10, 1)
φ1 = (Animal, α1)
φ2 = (1 ≤ Sea ≤ 20, α2)
φ3 = (¬(Sea > 20) ∨ Tennis ≤ 4, α3)
φ4 = (¬(Sea > 20) ∨ ¬(Tennis ≤ 4) ∨ Pool ≤ 5, α4)
We add the following constraints : α2 > α3 and α3 > α4
After applying the leximin, we get 4 levels of preference ranking.
Didier Dubois, Henri Prade, Fayçal Touazi (IRIT) 14 September 2013 14 / 21

15. Handling preference queries Adding constraints between symbolic weights
Constraints between weights in CP-theories style
This method is inspired from CP-theories a generalization of CP-nets :
idea
Preference constraint is of the form : u : xi > ¬xi [W ] (irrespective of
values of w ∈ W ).
The same encoding of CP-net is used.
If we have :(¬u ∨ xi , αi ), we add αi > αj for any αj , such that
(¬u ∨ w, αj ) is a possibilistic preference statement, w ∈ W .
Proposition
Let Q be a query made of n preference constraints, then the maximal
number of levels generated by leximin with additional constraints over
weights is 2n.
Didier Dubois, Henri Prade, Fayçal Touazi (IRIT) 15 September 2013 15 / 21

16. Handling preference queries Adding constraints between symbolic weights
Example (continue)
In addition to the previous preference constraints, let us consider the
preference for animals allowance holds irrespectively of the preference
concerning the distance to the sea :
Animals > ¬Animals[Sea]
In addition of the previous constraints added inspiring from CP-nets :
α2 > α3 and α3 > α4. We add the following constraint : α1 > α2
After applying the leximin, we get 5 levels of preference ranking.
Didier Dubois, Henri Prade, Fayçal Touazi (IRIT) 16 September 2013 16 / 21

17. Handling preference queries Hybrid method
Hybridizing weak comparative preferences and lexicographic
methods
Complexity results
Based on theoretical complexity and a small experimental results we have
noticed :
Weak preference comparison : Polynomial ;
Lexicographic method ΠP
2 − complete.
We have shown that the first level is the same for the both methods.
Didier Dubois, Henri Prade, Fayçal Touazi (IRIT) 17 September 2013 17 / 21

18. Handling preference queries Hybrid method
Hybridizing Algorithm
Hybride method
First, we use Weak preference comparison
Use the levels except the first one as a data base and apply the
Lexicographic method on them.
The use of these methods
WPC is perfectly adapted to capture the Skyline order
The hybrid method is adapted to Top − K order
Didier Dubois, Henri Prade, Fayçal Touazi (IRIT) 18 September 2013 18 / 21

19. Handling preference queries Hybrid method
Comparison between different existing approaches
Table : Comparative Table of different approaches dealing with preference queries
Formulation Context Ranking
Quali. Quanti. Uncond. Cond. Skyli. Top-k
Lacroix Lavency
Chomicki 2002
Kießling 2002
Fagin et al 2001
Fuzzy logic
Symbolic PL
Didier Dubois, Henri Prade, Fayçal Touazi (IRIT) 19 September 2013 19 / 21

20. Conclusion
Conclusion
One merit of the possibilistic approach lies in its logical nature
The use of symbolic weights gives more freedom to introduce order
relation between preference contraints.
The three proposed methods are characterized by an increasing
refinement power with manageable complexity.
Future works
The use of symbolic weights is really advantageous but we still miss
some properties of numerical weights. One may think of combining
these two formats to be as much expressive as possible.
this approach should be able to deal with null values, which create
specific difficulties in preference queries.
Didier Dubois, Henri Prade, Fayçal Touazi (IRIT) 20 September 2013 20 / 21

21. Conclusion
Didier Dubois, Henri Prade, Fayçal Touazi (IRIT) 21 September 2013 21 / 21