Tutorial by NET 2 Partner - MERAKA

1. Tailoring Temporal Description Logics for
Reasoning over Temporal Conceptual Models
A. Artale1
R. Kontchakov2, V. Ryzhikov1, and M. Zakharyaschev2
1 KRDB Research Centre, Free University of Bozen-Bolzano
2 Department of Comp. Science and Inf. Sys., Birkbeck College, London
University of KwaZulu-Natal, Durban, South Africa, 30-09-11
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

2. Motivations
Investigation of the Computational Complexity of reasoning
over Temporal Ontologies/Conceptual Models.
Languages considered: Family of Temporally Extended
DL-Lite languages.
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

3. ERVT
The Temporal Data Model
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

4. ERVT: The Proposed Temporal Conceptual Model
ERVT is the temporal extended Entity-Relationship model able to
capture Validity Time with the following temporal features:
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

5. ERVT: The Proposed Temporal Conceptual Model
ERVT is the temporal extended Entity-Relationship model able to
capture Validity Time with the following temporal features:
Timestamping: to distinguish between temporal and
atemporal modeling constructs.
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

6. ERVT: The Proposed Temporal Conceptual Model
ERVT is the temporal extended Entity-Relationship model able to
capture Validity Time with the following temporal features:
Timestamping: to distinguish between temporal and
atemporal modeling constructs.
Evolution and Transition constraints: to describe how objects
can change their class membership over time. Transition
constraints presuppose that class migration happens in a fixed
amount of time.
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

7. ERVT: The Proposed Temporal Conceptual Model
ERVT is the temporal extended Entity-Relationship model able to
capture Validity Time with the following temporal features:
Timestamping: to distinguish between temporal and
atemporal modeling constructs.
Evolution and Transition constraints: to describe how objects
can change their class membership over time. Transition
constraints presuppose that class migration happens in a fixed
amount of time.
Lifespan cardinality constraints: temporal counterparts of
standard cardinality constraints evaluated over the entire
existence of the object.
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

8. ERVT: A Company Example
Department S InterestGroup
OrganizationalUnit
d
Member S
(1,∞)
org
mbr
Employee S
Name(String)
S
PaySlipNumber(Integer)
S Salary(Integer)
T
Manager T
TopManagerAreaManager
dex−
dev
pex
WorksOn T
(3,∞)
act
emp
Project
ProjectCode(String)
S
Propose gp
(0,1)
Ex-Project tex
Manages
man
(1,1)
[0,5]
prj
(1,1)
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

9. Known Complexity Results for Reasoning over ERVT
Undecidability.
Theorem. Reasoning on the ERVT fragment with both
timestamping and evolution constraints is
undecidable [ :AMAI-05].
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

10. Known Complexity Results for Reasoning over ERVT
Undecidability.
Theorem. Reasoning on the ERVT fragment with both
timestamping and evolution constraints is
undecidable [ :AMAI-05].
Decidability.
Theorem. Reasoning on the ERVT fragment with both
timestamping and evolution constraints restricted to classes is
ExpTime [ FWZ:02].
Theorem. Reasoning on the ERVT fragment with
timestamping and lifespan cardinalities is 2ExpTime [ LT:07].
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

11. Temporal Conceptual Modelling – Known Results
temporal ERVT components temporal
features ERfull (ALCQI) modalities
trans, evo ExpTime [ FWZ:02] F / P, 2F /2P
ts 2ExpTime [ LT:07] 2∗ , R
ts, evo Undec. [ :05] 2F /2P, R
ts, trans 2∗ , R, F / P
ts, lfc 2ExpTime [ LT:07] 2∗ , R
trans, lfc R, F / P
evo, lfc 2F /2P, R
ts: Timestamping lfc: Lifespan Cardinalities
evo: Evolution trans: Transition
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

12. Aims of this Work
Our Aims: Conduct an exhaustive investigation on useful
fragments of ERVT weakening either the atemporal or
temporal component.
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

13. Aims of this Work
Our Aims: Conduct an exhaustive investigation on useful
fragments of ERVT weakening either the atemporal or
temporal component.
Our Results:
We give an exhaustive picture on the complexity of reasoning
over temporal extensions of DL-Lite;
Based on these results, we show encouraging complexity
results for reasoning over temporal ontologies where practical
reasoning is feasible!
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

14. The DL-Lite Languages
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

15. DL-LiteN
bool, DL-LiteN
krom and DL-LiteN
core
DL-LiteN
bool. C1 C2, with:
R −→ P | P−
B −→ A | ≥ n R | ⊥
C −→ B | ¬C | C1 C2
DL-LiteN
krom. B1 B2, B1 B2 ⊥, ¬B1 B2.
DL-LiteN
core. B1 B2, B1 B2 ⊥.
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

16. The DL-Lite Languages - Complexity Results
Complexity Results [CDLLR:AAAI05, CKZ:JAIR09]:
Satisfiability: NP-complete/NLogSpace/NLogSpace;
Instance Checking (Data Complexity): AC0/AC0/AC0;
Query Answering (Data Complexity): coNP/coNP/AC0.
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

17. DL-Lite – Conceptual Modelling Example
Manager Employee
AreaManager Manager
TopManager Manager
AreaManager TopManager ⊥
Manager AreaManager TopManager
∃WorksFor Employee
∃WorksFor−
Project
Project ∃WorksFor−
≥ 2 Manages ⊥
≥ 2 Manages−
⊥
...
Note: We use the shortcut ∃R instead of ≥ 1 R.
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

18. The Family of EER/UML Languages [ CKRZ:ER07]
ERref
DL-LiteN
core
ERbool
DL-LiteN
krom
ERfull
DL-LiteN
bool
Construct
DL-Lite
Representation
Entities Concept Name: E
+ + + Isa E1 E2
+ + + Disjointness E1 ¬E2
– + + Covering E ≡ E1 E2
Attributes Role Name: A
+ + + Range ∃A− D
+ + + Multiplicity E ≥ nA
E ≤ mA
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

19. The Family of EER/UML Languages [ CKRZ:ER07]
ERref
DL-LiteN
core
ERbool
DL-LiteN
krom
ERfull
DL-LiteN
bool
Construct
DL-Lite
Representation
Relationships
Concept Name CR
and n Roles Ui
+ + + Typing
CR ≡ ∃Ui
≥ 2 Ui ⊥
∃U−
i Ei
+ + +
Cardinality
(Refinement)
E ≥ n U−
i
E ≤ m U−
i
– – + Isa —
– – + Disjointness —
– – + Covering —
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

20. The Temporal DL-Lite
Languages
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

21. Seminal Papers
SCHILD, K., 1993. Combining terminological logics with
tense logic. Proc. of the 6th Portuguese Conference on AI.
F. Baader and A. Laux., 1995. Terminological Logics with
Modal Operators, IJCAI-95.
Wolter, F. and Zakharyaschev, M., 1998, Temporalizing
Description Logics, FroCoS-98.
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

22. Complexity for Temporal ALC– Known Results
Temporal operators can be applied to:
Concepts, roles or axioms (they are temporalized);
Concepts, roles or axioms can have a time-invariant
interpretation (they are rigid).
The satisfiability problem has a different complexity depending
from the combination between LT L and ALC constructs:
concepts roles axioms
rigid temp rigid temp rigid temp
Undec. - yes yes - yes - [GKWZ:03]
2ExpTime∗ - yes - yes yes - [ LT:07]
2ExpTime yes - yes - yes yes [BGL:08]
ExpSpace - yes - - - yes [GKWZ:03]
ExpTime - yes - - yes - [S:93, FWZ:02
(∗) Using the S5 modalities 2∗ and R.
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

23. The Temporal Language TFPX DL-LiteN
bool
TFPX DL-LiteN
bool has the following features:
The temporal operators are:
3F /3P (sometime in the future/past),
2F /2P (always in the future/past), and
F / P (next/previous time);
Concepts can be temporalized;
Roles can be rigid or flexible;
Axioms are rigid.
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

24. The TFPX DL-LiteN
bool Temporal Languages
TFPX DL-LiteN
bool. C1 C2, with:
S ::= Pi | Gi , R ::= S | S−
,
B ::= ⊥ | Ai | ≥ q R,
C ::= B | ¬C | C1 C2 | 3F C | 3PC | 2F C | 2PC | F C | PC
Where Gi denotes rigid roles.
TFPX DL-LiteN
core. D1 D2, D1 D2 ⊥;
TFPX DL-LiteN
krom. D1 D2, D1 D2 ⊥, ¬D1 D2;
with:
D ::= B | 2F B | 2PB | F B | PB
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

25. The TFPDL-LiteN
bool Temporal Languages
TFPDL-LiteN
bool. C1 C2, with:
S ::= Pi | Gi , R ::= S | S−
,
B ::= ⊥ | Ai | ≥ q R,
C ::= B | ¬C | C1 C2 | 3F C | 3PC | 2F C | 2PC
Where Gi denotes rigid roles.
TFPDL-LiteN
core. D1 D2, D1 D2 ⊥;
TFPDL-LiteN
krom. D1 D2, D1 D2 ⊥, ¬D1 D2;
with:
D ::= B | 2F B | 2PB
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

26. Semantics of TFPX DL-LiteN
bool
A TFPX DL-LiteN
bool interpretation I is a function over Z
I(n) = ∆I
, A
I(n)
0 , . . . , P
I(n)
0 , . . . , G
I(n)
0 , . . . ,
where:
Rigid roles are time-invariant:
GI(n1)
= GI(n2)
, ∀n1, n2 ∈ Z
Temporal operators are interpreted over Z:
(3F C)I(n)
= k>n CI(k), (3PC)I(n) = k<n CI(k),
(2F C)I(n)
= k>n CI(k), (2PC)I(n) = k<n CI(k),
( F C)I(n)
= CI(n+1), ( PC)I(n) = CI(n−1).
TBox assertions are interpreted globally:
I |= C D iff CI(n)
⊆ DI(n)
, for all n ∈ Z
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

27. The Temporal Language TR
U DL-LiteN
bool
TR
U DL-LiteN
bool has the following features:
The temporal operators are:
3∗ (sometime), and
2∗ (always);
Concepts can be temporalized;
Roles can be temporalized;
Axioms are rigid;
We have the following equivalences:
2∗ C = 2F 2PC and 3∗ C = 3F 3PC.
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

28. The TR
U DL-LiteN
bool Temporal Language
TR
U DL-LiteN
bool language: Uses the universal modalities, 2∗ , 3∗ ,
on both concepts and roles.
R ::=S | S−
| 2∗ R | 3∗ R
C ::=B | ¬C | C1 C2 | 2∗ C | 3∗ C
(2∗ C)I(n)
=
k∈Z
CI(k)
and (3∗ C)I(n)
=
k∈Z
CI(k)
(2∗ R)I(n)
=
k∈Z
RI(k)
and (3∗ R)I(n)
=
k∈Z
RI(k)
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

29. The TR
X DL-LiteN
bool Temporal Language
TR
X DL-LiteN
bool language: Uses the universal modalities, 2∗ , 3∗ ,
just on roles, and the next/previous-time modalities, F , P
on concepts.
R ::=S | S−
| 2∗ R | 3∗ R
C ::=B | ¬C | C1 C2 | F C | PC
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

30. Temporal DL-Lite Languages
Vs.
Temporal Conceptual
Modelling
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

31. TFPDL-LiteN
bool/TR
U DL-LiteN
bool – Timestamping
Department S InterestGroup
OrganizationalUnit
d
Member S
(1,∞)
org
mbr
Employee S
Name(String)
S
PaySlipNumber(Integer)
S Salary(Integer)
T
Manager T
TopManagerAreaManager
dex−
dev
pex
WorksOn T
(3,∞)
act
emp
Project
ProjectCode(String)
S
Propose gp
(0,1)
Ex-Project tex
Manages
man
(1,1)
[0,5]
prj
(1,1)
Manager 3F3P¬Manager, Manager 3∗ ¬Manager
Employee 2F2PEmployee, Employee 2∗ Employee
Temporary Relations/Attributes: Reification
Global Relations/Attributes: Reification + Global Roles
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

32. TFPX DL-LiteN
bool – Evolution and Transition Constraints
Department S InterestGroup
OrganizationalUnit
d
Member S
(1,∞)
org
mbr
Employee S
Name(String)
S
PaySlipNumber(Integer)
S Salary(Integer)
T
Manager T
TopManagerAreaManager
dex−
dev
pex
WorksOn T
(3,∞)
act
emp
Project
ProjectCode(String)
S
Propose gp
(0,1)
Ex-Project tex
Manages
man
(1,1)
[0,5]
prj
(1,1)
Manager 3P¬Employee
Manager 2FManager
AreaManager 3FTopManager
Project PEx-Project
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

33. TR
U DL-LiteN
bool – Lifespan Cardinality Constraints
Department S InterestGroup
OrganizationalUnit
d
Member S
(1,∞)
org
mbr
Employee S
Name(String)
S
PaySlipNumber(Integer)
S Salary(Integer)
T
Manager T
TopManagerAreaManager
dex−
dev
pex
WorksOn T
(3,∞)
act
emp
Project
ProjectCode(String)
S
Propose gp
(0,1)
Ex-Project tex
Manages
man
(1,1)
[0,5]
prj
(1,1)
A top-manager manages at most 5 different projects in her lifespan
TopManager ≤ 5 3∗ Manages (Lifespan Cardinalities)
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

34. Temporal DL-Lite – Obtained Complexity Results
The following original complexity results have been used to show
upper bounds for reasoning over Temporal Conceptual Models.
concept
temporal
operators
flexible & rigid roles only
temporalized
roles (R)
DL-LiteN
bool DL-LiteN
krom/core DL-LiteN
bool
2F/P, F/P
(FPX)
PSpace NP
in PTime
Undec.
2F/P
(FP)
NP NP
in PTime
?
2∗ , F/P
(UX)
PSpace NP
in PTime
Undec.
(R X)
2∗
(U)
NP NLogSpace NP
(R U)
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

35. Complexity: TFPX DL-LiteN
bool and Fragments
1 We reduce satisfiability in TFPX DL-LiteN
bool KBs to
satisfiability in QT L1
, i.e., the one-variable fragment of
first-order temporal logic over (Z, <).
2 We then show how to remove existential quantifiers from such
QT L1
formulas, thus reducing to LT L formulas.
3 Complexity results for temporal extensions of DL-LiteN
bool
follow from the corresponding LT L results.
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

36. Complexity: TFPX DL-LiteN
krom/core and Fragments
1 We reduce satisfiability in TFPX DL-LiteN
krom/core KBs to
satisfiability in QT L1
, i.e., the one-variable fragment of
first-order temporal logic over (Z, <).
2 We then show how to remove existential quantifiers from such
QT L1
formulas, thus reducing to two fragments of LT L, i.e.,
propositional temporal logic of binary clauses, i.e., LT Lkrom
and LT Lcore.
3 We show that:
LT Lkrom is NP-complete;
LT Lcore is in PTime.
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

37. LT Lkrom and LT Lcore
LT Lkrom
λ ::= p | ¬λ | F λ | Pλ | 2F λ | 2Pλ | 2∗ λ,
ϕ ::= (λ1 ∨ λ2) | 2∗ (λ1 ∨ λ2) | ϕ1 ∧ ϕ2.
LT Lcore
λ ::= p | F λ | Pλ | 2F λ | 2Pλ | 2∗ λ,
ϕ ::= (λ1 → λ2) | (¬λ1 ∨ ¬λ2) | 2∗ (λ1 → λ2) |
2∗ (¬λ1 ∨ ¬λ2) | ϕ1 ∧ ϕ2.
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

38. Complexity: DL-Lite with Temporalized Roles
TR
X DL-LiteN
bool is Undecidable is proved by encoding the tiling
problem.
TR
U DL-LiteN
bool is NP-complete and the upper bound is
showed by construction of Quasimodels/Mosaics.
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

39. Complexity: DL-Lite with Temporalized Roles
TR
X DL-LiteN
bool is Undecidable is proved by encoding the tiling
problem.
TR
U DL-LiteN
bool is NP-complete and the upper bound is
showed by construction of Quasimodels/Mosaics.
All the complexity results are shown in [ KRZ:xx]
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

40. Temporal Ontologies – Obtained Complexity Results
temporal EER component
features ERfull ERbool ERref
ts 2ExpTime [ LT:IJCAI07] NP NLogSpace
trans ExpTime [ FWZ:JELIA02] PSpace in PTime
ts, trans Undec. PSpace in PTime
evo ExpTime [ FWZ:JELIA02] NP NP
ts, evo Undec. [ :AMAI05] NP NP
trans, evo ExpTime [ FWZ:JELIA02] PSpace NP
ts, trans, evo Undec. [ :AMAI05] PSpace NP
ts, lfc 2ExpTime [ LT:IJCAI07] NP† in NP†
trans, lfc Undec. Undec. ?
evo, lfc Undec. ? ?
(†) This result is proved only for binary relationships.
ts: Timestamping lfc: Lifespan Cardinalities
evo: Evolution trans: Transition
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

41. Conclusions
We showed that:
By dropping ISA between relations (ERbool/DL-LiteN
bool) and
covering (ERref/DL-LiteN
core) we obtained better
computational behavior for reasoning over temporal
schemas/DL-Lite ontologies.
Both ERbool and ERref have been extended with timestamping,
evolution and transition constraints, lifespan cardinalities.
DL-LiteN
bool, DL-LiteN
krom and DL-LiteN
core have been extended
with past and future temporal operators, and with the
universal modality (both over concepts and roles).
We presented a nearly complete picture for reasoning over
temporal CMs/Ontologies/DL-Lite KBs.
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

42. Future Work
Few cases involving lifespan cardinalities/universal modality
are still open.
Investigating the problem of temporal queries over temporal
ontologies/conceptual schemas.
Investigating the possibility to use standard and implemented
temporal reasoners for practical reasoning over temporal
schemas.
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

43. Thanks to...
Enrico Franconi
Carsten Lutz
Christine Parent
Stefano Spaccapietra
David Toman
Frank Wolter
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs

44. THANK YOU!
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs