![Page 1: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/1.jpg)
Tailoring Temporal Description Logics forReasoning over Temporal Conceptual Models
A. Artale1
R. Kontchakov2, V. Ryzhikov1, and M. Zakharyaschev2
1 KRDB Research Centre, Free University of Bozen-Bolzano2 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
![Page 2: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/2.jpg)
Motivations
Investigation of the Computational Complexity of reasoningover Temporal Ontologies/Conceptual Models.
Languages considered: Family of Temporally ExtendedDL-Lite languages.
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs
![Page 3: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/3.jpg)
ERVTThe Temporal Data Model
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs
![Page 4: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/4.jpg)
ERVT: The Proposed Temporal Conceptual Model
ERVT is the temporal extended Entity-Relationship model able tocapture Validity Time with the following temporal features:
Timestamping: to distinguish between temporal andatemporal modeling constructs.
Evolution and Transition constraints: to describe how objectscan change their class membership over time. Transitionconstraints presuppose that class migration happens in a fixedamount of time.
Lifespan cardinality constraints: temporal counterparts ofstandard cardinality constraints evaluated over the entireexistence of the object.
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs
![Page 5: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/5.jpg)
ERVT: The Proposed Temporal Conceptual Model
ERVT is the temporal extended Entity-Relationship model able tocapture Validity Time with the following temporal features:
Timestamping: to distinguish between temporal andatemporal modeling constructs.
Evolution and Transition constraints: to describe how objectscan change their class membership over time. Transitionconstraints presuppose that class migration happens in a fixedamount of time.
Lifespan cardinality constraints: temporal counterparts ofstandard cardinality constraints evaluated over the entireexistence of the object.
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs
![Page 6: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/6.jpg)
ERVT: The Proposed Temporal Conceptual Model
ERVT is the temporal extended Entity-Relationship model able tocapture Validity Time with the following temporal features:
Timestamping: to distinguish between temporal andatemporal modeling constructs.
Evolution and Transition constraints: to describe how objectscan change their class membership over time. Transitionconstraints presuppose that class migration happens in a fixedamount of time.
Lifespan cardinality constraints: temporal counterparts ofstandard cardinality constraints evaluated over the entireexistence of the object.
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs
![Page 7: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/7.jpg)
ERVT: The Proposed Temporal Conceptual Model
ERVT is the temporal extended Entity-Relationship model able tocapture Validity Time with the following temporal features:
Timestamping: to distinguish between temporal andatemporal modeling constructs.
Evolution and Transition constraints: to describe how objectscan change their class membership over time. Transitionconstraints presuppose that class migration happens in a fixedamount of time.
Lifespan cardinality constraints: temporal counterparts ofstandard cardinality constraints evaluated over the entireexistence of the object.
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs
![Page 8: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/8.jpg)
ERVT: A Company Example
Department S InterestGroup
OrganizationalUnit
d
Member S
(1,∞)
org
mbrEmployee S
Name(String)
S
PaySlipNumber(Integer)
S Salary(Integer)
T
Manager T
TopManagerAreaManager
dex−
dev
pex
WorksOn T
(3,∞)
act
emp
Project
ProjectCode(String)
SPropose gp
(0,1)
Ex-Project tex
Managesman
(1,1)
[0,5]
prj
(1,1)
1
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs
![Page 9: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/9.jpg)
Known Complexity Results for Reasoning over ERVT
Undecidability.
Theorem. Reasoning on the ERVT fragment with bothtimestamping and evolution constraints isundecidable [ :AMAI-05].
Decidability.
Theorem. Reasoning on the ERVT fragment with bothtimestamping and evolution constraints restricted to classes isExpTime [ FWZ:02].Theorem. Reasoning on the ERVT fragment withtimestamping and lifespan cardinalities is 2ExpTime [ LT:07].
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs
![Page 10: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/10.jpg)
Known Complexity Results for Reasoning over ERVT
Undecidability.
Theorem. Reasoning on the ERVT fragment with bothtimestamping and evolution constraints isundecidable [ :AMAI-05].
Decidability.
Theorem. Reasoning on the ERVT fragment with bothtimestamping and evolution constraints restricted to classes isExpTime [ FWZ:02].Theorem. Reasoning on the ERVT fragment withtimestamping and lifespan cardinalities is 2ExpTime [ LT:07].
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs
![Page 11: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/11.jpg)
Temporal Conceptual Modelling – Known Results
temporal ERVT components temporalfeatures ERfull (ALCQI) modalities
trans, evo ExpTime [ FWZ:02] ©F/©P ,2F/2P
ts 2ExpTime [ LT:07] 2∗ ,Rts, evo Undec. [ :05] 2F/2P ,Rts, trans 2∗ ,R,©F/©Pts, lfc 2ExpTime [ LT:07] 2∗ ,Rtrans, lfc R,©F/©Pevo, lfc 2F/2P ,R
ts: Timestamping lfc: Lifespan Cardinalitiesevo: Evolution trans: Transition
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs
![Page 12: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/12.jpg)
Aims of this Work
Our Aims: Conduct an exhaustive investigation on usefulfragments of ERVT weakening either the atemporal ortemporal component.
Our Results:
We give an exhaustive picture on the complexity of reasoningover temporal extensions of DL-Lite;Based on these results, we show encouraging complexityresults for reasoning over temporal ontologies where practicalreasoning is feasible!
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs
![Page 13: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/13.jpg)
Aims of this Work
Our Aims: Conduct an exhaustive investigation on usefulfragments of ERVT weakening either the atemporal ortemporal component.
Our Results:
We give an exhaustive picture on the complexity of reasoningover temporal extensions of DL-Lite;Based on these results, we show encouraging complexityresults for reasoning over temporal ontologies where practicalreasoning is feasible!
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs
![Page 14: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/14.jpg)
The DL-Lite Languages
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs
![Page 15: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/15.jpg)
DL-LiteNbool, DL-LiteNkrom and DL-LiteNcore
DL-LiteNbool. C1 v C2, with:
R −→ P | P−
B −→ A | ≥ n R | ⊥C −→ B | ¬C | C1 u C2
DL-LiteNkrom. B1 v B2, B1 u B2 v ⊥, ¬B1 v B2.
DL-LiteNcore. B1 v B2, B1 u B2 v ⊥.
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs
![Page 16: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/16.jpg)
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
![Page 17: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/17.jpg)
DL-Lite – Conceptual Modelling Example
Manager v EmployeeAreaManager v ManagerTopManager v Manager
AreaManager u TopManager v ⊥Manager v AreaManager t TopManager∃WorksFor v Employee∃WorksFor− v Project
Project v ∃WorksFor−
≥ 2 Manages v ⊥≥ 2 Manages− v ⊥
...
Note: We use the shortcut ∃R instead of ≥ 1 R.
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs
![Page 18: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/18.jpg)
The Family of EER/UML Languages [ CKRZ:ER07]
ERrefDL-LiteNcore
ERboolDL-LiteNkrom
ERfullDL-LiteNbool
ConstructDL-Lite
RepresentationEntities Concept Name: E
+ + + Isa E1 v E2
+ + + Disjointness E1 v ¬E2
– + + Covering E ≡ E1 t E2
Attributes Role Name: A
+ + + Range ∃A− v D
+ + + Multiplicity E v≥ nAE v≤ mA
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs
![Page 19: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/19.jpg)
The Family of EER/UML Languages [ CKRZ:ER07]
ERrefDL-LiteNcore
ERboolDL-LiteNkrom
ERfullDL-LiteNbool
ConstructDL-Lite
Representation
RelationshipsConcept Name CR
and n Roles Ui
+ + + TypingCR ≡ ∃Ui
≥ 2 Ui v ⊥∃U−i v Ei
+ + +Cardinality
(Refinement)E v≥ n U−iE v≤ m U−i
– – + Isa —
– – + Disjointness —
– – + Covering —
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs
![Page 20: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/20.jpg)
The Temporal DL-LiteLanguages
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs
![Page 21: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/21.jpg)
Seminal Papers
SCHILD, K., 1993. Combining terminological logics withtense logic. Proc. of the 6th Portuguese Conference on AI.
F. Baader and A. Laux., 1995. Terminological Logics withModal Operators, IJCAI-95.
Wolter, F. and Zakharyaschev, M., 1998, TemporalizingDescription Logics, FroCoS-98.
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs
![Page 22: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/22.jpg)
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-invariantinterpretation (they are rigid).
The satisfiability problem has a different complexity dependingfrom the combination between LT L and ALC constructs:
concepts roles axiomsrigid 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
![Page 23: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/23.jpg)
The Temporal Language TFPXDL-LiteNbool
TFPXDL-LiteNbool 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
![Page 24: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/24.jpg)
The TFPXDL-LiteNbool Temporal Languages
TFPXDL-LiteNbool. C1 v C2, with:
S ::= Pi | Gi , R ::= S | S−,
B ::= ⊥ | Ai | ≥ q R,
C ::= B | ¬C | C1 u C2 | 3FC | 3PC | 2FC | 2PC | ©FC | ©PC
Where Gi denotes rigid roles.
TFPXDL-LiteNcore. D1 v D2, D1 u D2 v ⊥;
TFPXDL-LiteNkrom. D1 v D2, D1 u D2 v ⊥, ¬D1 v D2;
with:D ::= B | 2FB | 2PB | ©FB | ©PB
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs
![Page 25: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/25.jpg)
The TFPDL-LiteNbool Temporal Languages
TFPDL-LiteNbool. C1 v C2, with:
S ::= Pi | Gi , R ::= S | S−,
B ::= ⊥ | Ai | ≥ q R,
C ::= B | ¬C | C1 u C2 | 3FC | 3PC | 2FC | 2PC
Where Gi denotes rigid roles.
TFPDL-LiteNcore. D1 v D2, D1 u D2 v ⊥;
TFPDL-LiteNkrom. D1 v D2, D1 u D2 v ⊥, ¬D1 v D2;
with:D ::= B | 2FB | 2PB
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs
![Page 26: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/26.jpg)
Semantics of TFPXDL-LiteNbool
A TFPXDL-LiteNbool interpretation I is a function over Z
I(n) =(∆I ,AI(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:
(3FC )I(n) =⋃
k>n CI(k), (3PC )I(n) =⋃
k<n CI(k),
(2FC )I(n) =⋂
k>n CI(k), (2PC )I(n) =⋂
k<n CI(k),
(©FC )I(n) = CI(n+1), (©PC )I(n) = CI(n−1).
TBox assertions are interpreted globally:
I |= C v D iff CI(n) ⊆ DI(n), for all n ∈ Z
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs
![Page 27: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/27.jpg)
The Temporal Language TRUDL-LiteNbool
TRU DL-LiteNbool has the following features:
The temporal operators are:
3∗ (sometime), and2∗ (always);
Concepts can be temporalized;
Roles can be temporalized;
Axioms are rigid;
We have the following equivalences:
2∗ C = 2F2PC and 3∗ C = 3F3PC .
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs
![Page 28: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/28.jpg)
The TRUDL-LiteNbool Temporal Language
TRU DL-LiteNbool language: Uses the universal modalities, 2∗ , 3∗ ,on both concepts and roles.
R ::=S | S− | 2∗ R | 3∗ R
C ::=B | ¬C | C1 u 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
![Page 29: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/29.jpg)
The TRXDL-LiteNbool Temporal Language
TRX DL-LiteNbool language: Uses the universal modalities, 2∗ , 3∗ ,just on roles, and the next/previous-time modalities, ©F , ©Pon concepts.
R ::=S | S− | 2∗ R | 3∗ R
C ::=B | ¬C | C1 u C2 | ©FC | ©PC
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs
![Page 30: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/30.jpg)
Temporal DL-Lite LanguagesVs.
Temporal ConceptualModelling
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs
![Page 31: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/31.jpg)
TFPDL-LiteNbool/T
RUDL-Lite
Nbool – Timestamping
Department S InterestGroup
OrganizationalUnit
d
Member S
(1,∞)
org
mbrEmployee S
Name(String)
S
PaySlipNumber(Integer)
S Salary(Integer)
T
Manager T
TopManagerAreaManager
dex−
dev
pex
WorksOn T
(3,∞)
act
emp
Project
ProjectCode(String)
SPropose gp
(0,1)
Ex-Project tex
Managesman
(1,1)
[0,5]
prj
(1,1)
1
Manager v 3F3P¬Manager, Manager v 3∗ ¬ManagerEmployee v 2F2PEmployee, Employee v 2∗ Employee
Temporary Relations/Attributes: Reification
Global Relations/Attributes: Reification + Global Roles
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs
![Page 32: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/32.jpg)
TFPXDL-LiteNbool – Evolution and Transition Constraints
Department S InterestGroup
OrganizationalUnit
d
Member S
(1,∞)
org
mbrEmployee S
Name(String)
S
PaySlipNumber(Integer)
S Salary(Integer)
T
Manager T
TopManagerAreaManager
dex−
dev
pex
WorksOn T
(3,∞)
act
emp
Project
ProjectCode(String)
SPropose gp
(0,1)
Ex-Project tex
Managesman
(1,1)
[0,5]
prj
(1,1)
1
Manager v 3P¬EmployeeManager v 2FManager
AreaManager v 3FTopManager
Project v ©PEx-Project
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs
![Page 33: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/33.jpg)
TRUDL-LiteNbool – Lifespan Cardinality Constraints
Department S InterestGroup
OrganizationalUnit
d
Member S
(1,∞)
org
mbrEmployee S
Name(String)
S
PaySlipNumber(Integer)
S Salary(Integer)
T
Manager T
TopManagerAreaManager
dex−
dev
pex
WorksOn T
(3,∞)
act
emp
Project
ProjectCode(String)
SPropose gp
(0,1)
Ex-Project tex
Managesman
(1,1)
[0,5]
prj
(1,1)
1
A top-manager manages at most 5 different projects in her lifespanTopManager v ≤ 53∗ Manages (Lifespan Cardinalities)
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs
![Page 34: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/34.jpg)
Temporal DL-Lite – Obtained Complexity Results
The following original complexity results have been used to showupper bounds for reasoning over Temporal Conceptual Models.
concepttemporaloperators
flexible & rigid roles onlytemporalized
roles (R)DL-LiteNbool DL-LiteNkrom/core DL-LiteNbool
2F/P ,©F/P(FPX)
PSpace NPin PTime
Undec.
2F/P(FP)
NP NPin PTime
?
2∗ ,©F/P(UX)
PSpace NPin PTime
Undec.(R X)
2∗(U)
NP NLogSpace NP(R U)
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs
![Page 35: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/35.jpg)
Complexity: TFPXDL-LiteNbool and Fragments
1 We reduce satisfiability in TFPXDL-LiteNbool KBs tosatisfiability in QT L1, i.e., the one-variable fragment offirst-order temporal logic over (Z, <).
2 We then show how to remove existential quantifiers from suchQT L1 formulas, thus reducing to LT L formulas.
3 Complexity results for temporal extensions of DL-LiteNboolfollow from the corresponding LT L results.
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs
![Page 36: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/36.jpg)
Complexity: TFPXDL-LiteNkrom/core and Fragments
1 We reduce satisfiability in TFPXDL-LiteNkrom/core KBs tosatisfiability in QT L1, i.e., the one-variable fragment offirst-order temporal logic over (Z, <).
2 We then show how to remove existential quantifiers from suchQT L1 formulas, thus reducing to two fragments of LT L, i.e.,propositional temporal logic of binary clauses, i.e., LT Lkromand 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
![Page 37: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/37.jpg)
LT Lkrom and LT Lcore
LT Lkrom
λ ::= p | ¬λ | ©Fλ | ©Pλ | 2Fλ | 2Pλ | 2∗ λ,
ϕ ::= (λ1 ∨ λ2) | 2∗ (λ1 ∨ λ2) | ϕ1 ∧ ϕ2.
LT Lcore
λ ::= p | ©Fλ | ©Pλ | 2Fλ | 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
![Page 38: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/38.jpg)
Complexity: DL-Lite with Temporalized Roles
TRX DL-LiteNbool is Undecidable is proved by encoding the tilingproblem.
TRU DL-LiteNbool is NP-complete and the upper bound isshowed 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
![Page 39: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/39.jpg)
Complexity: DL-Lite with Temporalized Roles
TRX DL-LiteNbool is Undecidable is proved by encoding the tilingproblem.
TRU DL-LiteNbool is NP-complete and the upper bound isshowed 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
![Page 40: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/40.jpg)
Temporal Ontologies – Obtained Complexity Results
temporal EER componentfeatures 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 Cardinalitiesevo: Evolution trans: Transition
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs
![Page 41: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/41.jpg)
Conclusions
We showed that:
By dropping ISA between relations (ERbool/DL-LiteNbool) andcovering (ERref/DL-LiteNcore) we obtained bettercomputational behavior for reasoning over temporalschemas/DL-Lite ontologies.
Both ERbool and ERref have been extended with timestamping,evolution and transition constraints, lifespan cardinalities.
DL-LiteNbool, DL-LiteNkrom and DL-LiteNcore have been extendedwith past and future temporal operators, and with theuniversal modality (both over concepts and roles).
We presented a nearly complete picture for reasoning overtemporal CMs/Ontologies/DL-Lite KBs.
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs
![Page 42: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/42.jpg)
Future Work
Few cases involving lifespan cardinalities/universal modalityare still open.
Investigating the problem of temporal queries over temporalontologies/conceptual schemas.
Investigating the possibility to use standard and implementedtemporal reasoners for practical reasoning over temporalschemas.
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs
![Page 43: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/43.jpg)
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
![Page 44: Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models](https://reader034.vdocument.in/reader034/viewer/2022051610/5495f6a1b47959514d8b4ea7/html5/thumbnails/44.jpg)
THANK YOU!
Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs