First Workshop on Logics for Reasoning about Preferences, Uncertainty, and Vagueness.

1. COMPUTING
K-RANK ANSWERS
WITH ONTOLOGICAL CP-NETS
* University of Oxford
** Politecnico di Bari
Tommaso Di Noia**, Thomas Lukasiewicz *,
Maria Vanina Martinez*, Gerardo I Simari *,
Oana Tifrea-Marciuska *
PRUV 2014

2. Web 3.0
Semantic DataSocial Data

3. Web 3.0 Search
Semantic DataSocial Data
Precise and rich resultsPersonalised access

4. Personalized Information Access
CONJUNCTIVE QUERY
ORDER BY user’s preferences
LIMIT k

5. Personalized Information Access
CONJUNCTIVE QUERY
ORDER BY user preferences
LIMIT k
Datalog +/- CQ
Ontological CP-Net
Top-k

6. Personalized Information Access
CONJUNCTIVE QUERY
ORDER BY user preferences
LIMIT k
Datalog +/- CQ
Ontological CP-Net
Top-k

7. Datalog +/-
¨ Database instances
¨ class(b); class(e);
¨ Datalog ∀X ∀Y Θ(X,Y) → Ψ(X)
¨ book(F,P,C) →class(C)
¨ TGD’s ∀X ∀Y Θ(X,Y) → ∃Z Ψ(X,Z)
¨ flight(F,D,A) → ∃C dPlace(C,D)
¨ Negative constraints ∀X Θ(X) → ⊥
¨ flight(F,D,D) → ⊥

8. Personalized Information Access
CONJUNCTIVE QUERY
ORDER BY user preferences
LIMIT k
Datalog +/- CQ
Ontological CP-Net
Top-k

9. Personalized Information Access
CONJUNCTIVE QUERY
ORDER BY user preferences
LIMIT k
Datalog +/- CQ
Ontological CP-Net
Top-k

10. Conditional Preferences (CP)-Nets
Compact qualitative specification of
preferences ma na
preferred
Variable

11. Conditional Preferences (CP)-Nets
Unconditional Preferences
Conditional
Preferences
Parents condition that
preference depends on

12. Conditional Preferences (CP)-Nets
Ceteris paribus
All other things being equal
E.g. if I fly on business class and on a night arrival time then I
always prefer morning departure over night departure.

13. Reasoning with CP nets
¨ Worsening flip
¤ Changing value of a variable so that it is less
preferred in some outcome (direct dominance)
md ma e ≻ md ma b
md na e ≻ md ma e

14. Reasoning with CP nets
¨ Ordering on outcomes
¤ o1 is preferred to o2 (o1 ≻ o2) iff there is a sequence of
worsening flips from o1 to o2
¨ Partial order
¤ o1 and o2 can be incomparable

15. Preference Graph
md ma b
md ma e
md na e nd ma e
md na b nd na e
nd na b
nd ma b

16. Direct Dominance
md ma e
md ma b md na e nd ma e
md na b nd na e
nd na b
nd ma b
o1
o2

17. Personalized Information Access
CONJUNCTIVE QUERY
ORDER BY user’s preferences
LIMIT k
Datalog +/- CQ
Ontological CP-Net
Top-k

18. Ontological CP-Nets
¨ Datalog +/- ontology O
¨ CP-net variables P are predicates p
¨ dom(P) = {p(t11,... , t1k),..., p(tn1,... , tnk)}

19. Consistent Ontological CP-Nets
¨ Datalog +/- ontology O
¨ CP-net variables P are predicates p
¨ dom(P) = {p(t11,... , t1k),..., p(tn1,... , tnk)} constraint by O
¨ Outcomes equivalent
¤ aircraft(X) →airplane(X)
¨ Inconsistent CP-net
¤ airplane(a1) →⊥
aircraft airplanea1 ≻ a2 a2 ≻ a1 {aicraft(a1),airplane(a2)}
{aircraft(a2), airplane(a1)}

20. Personalized Information Access
CONJUNCTIVE QUERY
ORDER BY user preferences
LIMIT k
Datalog +/- CQ
Ontological CP-Net
Top-k

21. Query Answering
Q(X,Y)→ ∃Z class(Z) ^dTime(X) ^ aTime(Y)
<md,ma,e>
the undominated outcome

22. Query Answering
md ma e
md ma b md na e nd ma e
md na b nd na e
nd na b
nd ma b
Top-k Query Answering with Ontological CP-nets
Q(X,Y)→ ∃Z class(Z) ^ dTime(X) ^ aTime(Y)
Top-1 answer to Q: <md,ma>
o1

23. Query Answering
md ma e
md ma b md na e nd ma e
md na b nd na e
nd na b
nd ma b
Top-k Query Answering with Ontological CP-nets
Q(X,Y)→ ∃Z class(Z) ^ dTime(X) ^ aTime(Y)
Top-2 answers to Q: {<md,ma>,…}
o1

24. Query Answering
md ma b md na e nd ma e
md na b nd na e
nd na b
nd ma b
Top-k Query Answering with Ontological CP-nets
Q(X,Y)→ ∃Z class(Z) ^ dTime(X) ^ aTime(Y)
Top-2 answers to Q: {<md,ma>,…}
o2 o3

25. Query Answering
md na e nd ma e
md na b nd na e
nd na b
nd ma b
Top-k Query Answering with Ontological CP-nets
Q(X,Y)→ ∃Z class(Z) ^ dTime(X) ^ aTime(Y)
Top-2 answers to Q: {<md,ma>,<md,na>}
o3
o4

26. Query Answering
nd ma e
md na b nd na e
nd na b
nd ma b
Top-k Query Answering with Ontological CP-nets
Q(X,Y)→ ∃Z class(Z) ^ dTime(X) ^ aTime(Y)
Top-3 answers to Q: {<md,ma>,<md,na>,
<nd,ma>}
o4o5

27. Query Answering
Theorems
q Computing the k-rank is PSpace (resp. 2EXPTIME) complete for linear (resp.
guarded) Datalog+/- . Computing k-rank is data complete for PSPACE.
• TGD is guarded iff it contains an atom in its body that contains all universally quantified
variables.
σ1 :P(X)∧R(X,Y)∧Q(Y)→∃Z R(Y,Z) YES.
σ2 :R(X,Y)∧R(Y,Z)→R(X,Z) NO.
• TGD is linear: rules with one body-atom
σ3 :R(X,Y) →∃Z R(X,Z)
Top-k Query Answering with Ontological CP-nets

28. Query Answering
Theorems
If the CP-net is a polytree, the query has bounded width and the
Datalog+/- ontology is guarded or linear top-k can be done in polynomial
time.
Top-k Query Answering with Ontological CP-nets

29. Conclusion
D
O
Database
Ontology
User’s preferences
K
n
o
w
le
d
g
e
b
a
s
e
Top-k CQ
l Combining Knowlege and Preference Representation
l A framework for Top-k Query Answering with preferences in Datalog +/-

32. Query Answering
md ma e
md ma b md na e nd ma e
md na b nd na e
nd na b
nd ma b
Top-2 answers to Q:
{<md,ma>,<md,na>}
Top-k Query Answering with Ontological CP-nets
Q(X,Y)→ ∃Z class(Z) ^ dTime(X) ^ aTime(Y)
Checked: {<md,ma,e>}
Outcomes: {<md,ma,e>,
<md,ma,b>, <md,na,e>}
Outcomes / Checked:
{<md,ma,b>, <md,na,e>}

33. Query Answering
md ma e
md ma b md na e nd ma e
md na b nd na e
nd na b
nd ma b
Top-3 answers to Q:
{<md,ma>,<md,na>…}
Q(X,Y)→ ∃Z class(Z) ^ dTime(X) ^ aTime(Y)
Top-k Query Answering with Ontological CP-nets

34. Query Answering
md ma e
md ma b md na e nd ma e
md na b nd na e
nd na b
nd ma b
Top-3 answers to Q:
{<md,ma>,<md,na>…}
Q(X,Y)→ ∃Z class(Z) ^ dTime(X) ^ aTime(Y)
Top-k Query Answering with Ontological CP-nets

35. Query Answering
md ma e
md ma b md na e nd ma e
nd na e
nd na b
nd ma b
Top-k Query Answering with Ontological CP-nets
md na b
Top-3 answers to Q:
{ <md,ma>,
<md,na>,
<nd,ma>}
Q(X,Y)→ ∃Z class(Z) ^ dTime(X) ^ aTime(Y)

36. Query Answering
md ma e
md ma b md na e nd ma e
nd na e
nd na b
nd ma b
Top-k Query Answering with Ontological CP-nets
md na b
Checked: {<md,ma,e>,
<md,ma,b>, <md,na,e>,
<md,na,b>, <nd,ma,b>}
Outcomes:
<md,ma,e>,
<md,ma,b>,
<md,na,e>,
<nd,ma,b>,
<md,na,b>}
Outcomes / Checked:
{<nd,ma,b>}
Top-3 answers to Q:
{ <md,ma>,
<md,na>,
<nd,ma>}
Q(X,Y)→ ∃Z class(Z) ^ dTime(X) ^ aTime(Y)