Successfully reported this slideshow.
Upcoming SlideShare
×

# Conditional preference queries and possibilistic logic

211 views

Published on

Presentation of the paper "Conditional preference queries and possibilistic logic" in the conference Flexible Query Answering Systems 2013

Published in: Science
• Full Name
Comment goes here.

Are you sure you want to Yes No
• Be the first to comment

• Be the first to like this

### Conditional preference queries and possibilistic logic

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