Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models
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Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models

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Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models Tailoring Temporal Description Logics for Reasoning over Temporal Conceptual Models Presentation Transcript

  • 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
  • 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
  • ERVT The Temporal Data Model Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • The DL-Lite Languages Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs
  • 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
  • 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
  • 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
  • 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
  • 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
  • The Temporal DL-Lite Languages Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • Temporal DL-Lite Languages Vs. Temporal Conceptual Modelling Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • THANK YOU! Alessandro Artale – KRDB Tailoring Temporal DLs for Reasoning over Temporal CMs