tailoring temporal description logics for reasoning over temporal conceptual models

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

Motivations

Investigation of the Computational Complexity of reasoningover Temporal Ontologies/Conceptual Models.

Languages considered: Family of Temporally ExtendedDL-Lite languages.

ERVTThe Temporal Data Model

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.

ERVT: A Company Example

Department S InterestGroup

OrganizationalUnit

Member S

(1,∞)

mbrEmployee S

Name(String)

PaySlipNumber(Integer)

S Salary(Integer)

Manager T

TopManagerAreaManager

dex−

WorksOn T

(3,∞)

Project

ProjectCode(String)

SPropose gp

Ex-Project tex

Managesman

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].

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].

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

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!

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!

The DL-Lite Languages

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 ⊥.

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.

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.

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

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 —

The Temporal DL-LiteLanguages

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.

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.

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.

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

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

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

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 .

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

(2∗ R)I(n) =⋂k∈Z

RI(k) and (3∗ R)I(n) =⋃k∈Z

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

Temporal DL-Lite LanguagesVs.

Temporal ConceptualModelling

TFPDL-LiteNbool/T

RUDL-Lite

Nbool – Timestamping

OrganizationalUnit

Member S

(1,∞)

mbrEmployee S

Name(String)

S Salary(Integer)

Manager T

dex−

WorksOn T

(3,∞)

Project

ProjectCode(String)

SPropose gp

Ex-Project tex

Managesman

Manager v 3F3P¬Manager, Manager v 3∗ ¬ManagerEmployee v 2F2PEmployee, Employee v 2∗ Employee

Temporary Relations/Attributes: Reification

Global Relations/Attributes: Reification + Global Roles

TFPXDL-LiteNbool – Evolution and Transition Constraints

OrganizationalUnit

Member S

(1,∞)

mbrEmployee S

Name(String)

S Salary(Integer)

Manager T

dex−

WorksOn T

(3,∞)

Project

ProjectCode(String)

SPropose gp

Ex-Project tex

Managesman

Manager v 3P¬EmployeeManager v 2FManager

AreaManager v 3FTopManager

TRUDL-LiteNbool – Lifespan Cardinality Constraints

OrganizationalUnit

Member S

(1,∞)

mbrEmployee S

Name(String)

S Salary(Integer)

Manager T

dex−

WorksOn T

(3,∞)

Project

ProjectCode(String)

SPropose gp

Ex-Project tex

Managesman

A top-manager manages at most 5 different projects in her lifespanTopManager v ≤ 53∗ Manages (Lifespan Cardinalities)

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

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)

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.

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.

LT Lkrom and LT Lcore

LT Lkrom

ϕ ::= (λ1 ∨ λ2) | 2∗ (λ1 ∨ λ2) | ϕ1 ∧ ϕ2.

LT Lcore

ϕ ::= (λ1 → λ2) | (¬λ1 ∨ ¬λ2) | 2∗ (λ1 → λ2) |2∗ (¬λ1 ∨ ¬λ2) | ϕ1 ∧ ϕ2.

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]

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]

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

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.

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.

Thanks to...

Enrico Franconi

Carsten Lutz

Christine Parent

Stefano Spaccapietra

David Toman

Frank Wolter

THANK YOU!

tailoring temporal description logics for reasoning over temporal conceptual models

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