The DL-Lite Family of Languages A FO Perspective Alessandro Artale - - PowerPoint PPT Presentation

The DL-Lite Family of Languages A FO Perspective

Alessandro Artale KRDB Research Centre – Free University of Bozen-Bolzano Joint work with D. Calvanese, R. Kontchakov, M. Zakharyaschev

TU Dresden. December 14–15, 2011

Recommended Readings

[1] A. Artale, D. Calvanese, R. Kontchakov and M. Zakharyaschev.

The DL-Lite family and relations. JAIR, 36:1–69, 2009.

[2] D. Calvanese, G. De Giacomo, D. Lembo, M. Lenzerini, and R. Rosati.

DL-Lite: Tractable description logics for ontologies. Proceedings of AAAI 2005.

[3] D. Calvanese, G. De Giacomo, D. Lembo, M. Lenzerini, R. Rosati.

Tractable reasoning and efficient query answering in DLs: The DL-Lite family. Journal of Automated Reasoning, 39:385–429, 2007.

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 1/40

Outline

• 1. Ontology based data access
• 2. The DL-Lite-family of ontology languages:

DL-Litebool, DL-Litehorn, DL-Litecore, DL-Litekrom

• 3. Translation to the one-variable fragment of First-Order Logic
• 4. Answering UCQ
• 5. Conclusions

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 2/40

Ontologies in Computer Science

• Ontologies are formal specifications of a particular domain
• Used to represent information at the conceptual level in terms of

classes/concepts/entities and relationships between them

• Typically expressed in logic:

– First Order Logic – Description Logics: a specialized formalism (typically a fragment of FOL) for expressing knowledge in terms of classes and relationships

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 3/40

Ontologies in Computer Science

• Ontologies are formal specifications of a particular domain
• Used to represent information at the conceptual level in terms of

classes/concepts/entities and relationships between them

• Typically expressed in logic:

– First Order Logic – Description Logics: a specialized formalism (typically a fragment of FOL) for expressing knowledge in terms of classes and relationships

• Share strong similarities with other representation formalisms in Computer

Science – Frame systems in Artificial Intelligence – ER diagrams in databases and information systems – UML class diagrams in software engineering – Constraints over a relational schema (inclusion and key dependencies)

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 3-a/40

Ontology based data access

Desiderata: achieve logical transparency in access to data:

• Hide to the user where and how data are stored
• Present to the user a conceptual view of the data
• Query the data sources through the conceptual model

Layer Data Layer Query over conceptual layer Conceptual Ontology

As in Data Integration, but with a rich conceptual description as the global view

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 4/40

Description Logics: The DL-Lite family

The DL-Lite DLs provide an answer to our basic question: For which ontology languages can we answer queries over an ontology efficiently (in data complexity)?

• DL-Lite is a family of DLs optimized according to the tradeoff between expressive

power and data complexity

• The DL-Lite family establishes the maximal subset of DLs constructs for which

data complexity of query answering is LOGSPACE – Query answering techniques leverage on RDBMS technology (i.e. SQL)

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 5/40

Objectives of the Lecture

• 1. To show how the basic DL-Lite in [CDLLR,AAAI05; CDLLR,KR06] can be extended

with full Booleans, cardinalities and role inclusion axioms obtaining the logic DL-Litebool and three sublanguages: DL-Litekrom, DL-Litecore and DL-Litehorn

• 2. To characterize the first-order logic nature of class of DL-Lite DLs
• 3. To provide tight combined complexity results for reasoning in the new languages

showing that:

• Cardinalities are harmless;
• Role inclusions, in most cases, destroy the nice computational behavior of

DL-Lite!

• 4. To show the LOGSPACE data complexity result of answering positive existential

queries in DL-Litehorn.

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 6/40

The simplest DL-Lite Language: DL-Litecore

DL-Litecore Ontology language:

• Concept Inclusions:

B1 ⊑ B2, B1 ⊑ ¬B2 with: B − → A | ∃R | ⊥ R − → P | P −

• ABox assertions:

A(c), ¬A(c) P (c, d), ¬P (c, d) with c, d constants

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 7/40

The most expressive DL-Lite Language: DL-LiteR,N

bool

DL-LiteR,N

bool Ontology language:

• Concept Inclusions:

C1 ⊑ C2, with: C − → B | ¬C | C1 ⊓ C2 B − → A | ≥ q R | ⊥ R − → P | P −

• Role Inclusions:

R1 ⊑ R2

• ABox assertions (A):

A(c), ¬A(c) P (c, d), ¬P (c, d), with c, d constants.

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 8/40

The most expressive DL-Lite Language: DL-LiteR,N

bool

DL-LiteR,N

bool Ontology language:

• Concept Inclusions:

C1 ⊑ C2, with: C − → B | ¬C | C1 ⊓ C2 B − → A | ≥ q R | ⊥ R − → P | P −

• Role Inclusions:

R1 ⊑ R2

• ABox assertions (A):

A(c), ¬A(c) P (c, d), ¬P (c, d), with c, d constants.

• A TBox, T , is a set of concept and role inclusions. A TBox, T , is what we call an

Ontology, O.

• A Knowledge Base is a pair K = (T , A)

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 8-a/40

DL-LiteR,N

core , DL-LiteR,N krom, DL-LiteR,N horn

DL-LiteR,N

core Ontology language: B1 ⊑ B2,

B1 ⊑ ¬B2 DL-LiteR,N

krom Ontology language: B1 ⊑ B2,

B1 ⊑ ¬B2, ¬B1 ⊑ B2 DL-LiteR,N

horn Ontology language: k Bk ⊑ B

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 9/40

DL-Lite – Example

Manager ⊑ Employee AreaManager ⊑ Manager TopManager ⊑ Manager AreaManager ⊓ TopManager ⊑ ⊥ ∃WorksFor ⊑ Employee ∃WorksFor− ⊑ Project Project ⊑ ∃WorksFor− ≥ 2 Manages ⊑ ⊥ ≥ 2 Manages− ⊑ ⊥ . . . See [Artale et. al.,ER07] for more details on the correspondence between DL-Lite and conceptual data models

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 10/40

Semantics of DL-Lite

Construct Syntax Example Semantics atomic concept A Doctor AI ⊆ ∆I atomic role P child P I ⊆ ∆I × ∆I inverse role P − child− {(d, e) ∈ ∆I × ∆I | (e, d) ∈ P I} empty concept ⊥ ⊥ ∅ conjunction C1 ⊓ C2 Doctor ⊓ Male CI

1 ∩ CI 2

negation ¬C ¬(Doctor ⊓ Male) ∆I \ CI cardinalities ≥ nR ≥ 2 child− {d ∈ ∆I | ♯{e ∈ ∆I | (d, e) ∈ RI} ≥ n} inclusion asser. Cl ⊑ Cr Father ⊑ ≥ 1 child ClI ⊆ CrI

• memb. asser.

A(a) Father(bob) aI ∈ AI

• memb. asser.

P (a, b) child(bob, ann) (aI, bI) ∈ P I

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 11/40

Relevant reasoning tasks

We are interested in:

• 1. Checking the consistency of the ontology (Schema Consistency)
• 2. Checking the consistency of single classes in the ontology (Class Consistency)
• 3. Checking whether new constraints hold in the ontology (e.g. discovering new ISA

– Class Subsumption)

• 4. Checking the consistency of the data wrt the ontology
• 5. Answering queries expressed over the ontology by means of the underlying data

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 12/40

FO Translation

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 13/40

DL-LiteN

bool is NP-complete – Upper Bound

• Class consistency for DL-LiteN

bool can be reduced to formula satisfiability for the

• ne-variable fragment QL1 of first-order logic without equality and functions.
• Formula satisfiability for the one-variable fragment QL1 is known to be

NP-complete [BGG:97].

• First we present a lengthy yet quite ‘natural’ and ‘transparent’ reduction ·†;
• Then we shall see that this reduction can be substantially optimised to a

LOGSPACE reduction ·‡.

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 14/40

DL-LiteN

bool is NP-complete – Upper Bound – Translating C

• Inductive translation of Concepts, C∗:

(⊥)∗ = ⊥ (A)∗ = A(x) (¬C)∗ = ¬C∗(x) (C1 ⊓ C2)∗ = C∗

1(x) ∧ C∗ 2(x)

(≥q R)∗ =EqR(x)

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 15/40

DL-LiteN

bool is NP-complete – Upper Bound – Translation K†

• Translation of K=(TBox,ABox): The lengthy translation K†.

K† =

• T ∗ ∧

R∈role±(K)

• ε(R) ∧ δ(R)
• ∧
• A† ∧

R∈role±(K) R†

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 16/40

DL-LiteN

bool is NP-complete – Upper Bound – Translation K†

• Translation of K=(TBox,ABox): The lengthy translation K†.

K† =

• T ∗ ∧

R∈role±(K)

• ε(R) ∧ δ(R)
• ∧
• A† ∧

R∈role±(K) R†

T ∗ =

• C1⊑C2∈T ∀x
• C∗

1(x) → C∗ 2(x)

• Alessandro Artale

The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 16-a/40

DL-LiteN

bool is NP-complete – Upper Bound – Translation K†

• Translation of K=(TBox,ABox): The lengthy translation K†.

K† =

• T ∗ ∧

R∈role±(K)

• ε(R) ∧ δ(R)
• ∧
• A† ∧

R∈role±(K) R†

T ∗ =

• C1⊑C2∈T ∀x
• C∗

1(x) → C∗ 2(x)

• A†

=

• A(ai)∈A A(ai) ∧

P (ai,aj)∈A P aiaj

• ¬A(ai)∈A ¬A(ai) ∧

¬P (ai,aj)∈A ¬P aiaj

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 16-b/40

DL-LiteN

bool is NP-complete – Upper Bound – Translation K†

• Translation of K=(TBox,ABox): The lengthy translation K†.

K† =

• T ∗ ∧

R∈role±(K)

• ε(R) ∧ δ(R)
• ∧
• A† ∧

R∈role±(K) R†

T ∗ =

• C1⊑C2∈T ∀x
• C∗

1(x) → C∗ 2(x)

• A†

=

• A(ai)∈A A(ai) ∧

P (ai,aj)∈A P aiaj

• ¬A(ai)∈A ¬A(ai) ∧

¬P (ai,aj)∈A ¬P aiaj

R† = qT

q=1

• a, aj1, . . . , ajq ∈ ob(A)

ji = ji′ for i = i′ q

i=1 Raaji → EqR(a)

• ∧
• ai,aj∈ob(A)
• Raiaj → inv(R)ajai
• (qT is the maximum cardinality number in T )

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 16-c/40

DL-LiteN

bool is NP-complete – Upper Bound – Translation K† (cont.)

• Translation of K=(TBox,ABox): The lengthy translation K†.

K† =

• T ∗ ∧

R∈role±(K)

• ε(R) ∧ δ(R)
• ∧
• A† ∧

R∈role±(K) R†

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 17/40

DL-LiteN

bool is NP-complete – Upper Bound – Translation K† (cont.)

• Translation of K=(TBox,ABox): The lengthy translation K†.

K† =

• T ∗ ∧

R∈role±(K)

• ε(R) ∧ δ(R)
• ∧
• A† ∧

R∈role±(K) R†

ε(R) = ∀x

• E1R(x) → inv(E1R(dr))
• Alessandro Artale

The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 17-d/40

DL-LiteN

bool is NP-complete – Upper Bound – Translation K† (cont.)

• Translation of K=(TBox,ABox): The lengthy translation K†.

K† =

• T ∗ ∧

R∈role±(K)

• ε(R) ∧ δ(R)
• ∧
• A† ∧

R∈role±(K) R†

ε(R) = ∀x

• E1R(x) → inv(E1R(dr))
• δ(R)

= qT −1

q=1

∀x

• Eq+1R(x) → EqR(x)
• Alessandro Artale

The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 17-e/40

DL-LiteN

bool is NP-complete – FO Model

T = {A ≡ ∃P −, A ⊑ ≥ 2 P, ⊤ ⊑ ≤ 1 P −, ∃P ⊑ A}, A = {A(a), P (a, a′)} T ∗ =

• A(x)↔E1P −(x)
• ∧
• A(x)→E2P (x)
• ∧¬E2P −(x)∧
• E1P (x)→A(x)
• AM = {a,

} E1P −M = { } E1P M = {

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 18/40

DL-LiteN

bool is NP-complete – FO Model

T = {A ≡ ∃P −, A ⊑ ≥ 2 P, ⊤ ⊑ ≤ 1 P −, ∃P ⊑ A}, A = {A(a), P (a, a′)} T ∗ =

• A(x)↔E1P −(x)
• ∧
• A(x)→E2P (x)
• ∧¬E2P −(x)∧
• E1P (x)→A(x)
• R† : P aa′

⇒ P −a′a, E1P (a) ⇒ E1P −(a′) AM = {a, } E1P −M = { } E1P M = {

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 18-a/40

DL-LiteN

bool is NP-complete – FO Model

T = {A ≡ ∃P −, A ⊑ ≥ 2 P, ⊤ ⊑ ≤ 1 P −, ∃P ⊑ A}, A = {A(a), P (a, a′)} T ∗ =

• A(x)↔E1P −(x)
• ∧
• A(x)→E2P (x)
• ∧¬E2P −(x)∧
• E1P (x)→A(x)
• R† : P aa′

⇒ P −a′a, E1P (a) ⇒ E1P −(a′) ε(R) : E1P (a), E1P −(a′) ⇒ E1P −(dp−), E1P (dp) AM = {a, } E1P −M = {a′, } E1P M = {a,

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 18-b/40

DL-LiteN

bool is NP-complete – FO Model

T = {A ≡ ∃P −, A ⊑ ≥ 2 P, ⊤ ⊑ ≤ 1 P −, ∃P ⊑ A}, A = {A(a), P (a, a′)} T ∗ =

• A(x)↔E1P −(x)
• ∧
• A(x)→E2P (x)
• ∧¬E2P −(x)∧
• E1P (x)→A(x)
• R† : P aa′

⇒ P −a′a, E1P (a) ⇒ E1P −(a′) ε(R) : E1P (a), E1P −(a′) ⇒ E1P −(dp−), E1P (dp) T ∗ : E1P (x)→A(x) ⇒ A(dp) AM = {a, } E1P −M = {a′,dp−, } E1P M = {a,dp,

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 18-c/40

DL-LiteN

bool is NP-complete – FO Model

T = {A ≡ ∃P −, A ⊑ ≥ 2 P, ⊤ ⊑ ≤ 1 P −, ∃P ⊑ A}, A = {A(a), P (a, a′)} T ∗ =

• A(x)↔E1P −(x)
• ∧
• A(x)→E2P (x)
• ∧¬E2P −(x)∧
• E1P (x)→A(x)
• R† : P aa′

⇒ P −a′a, E1P (a) ⇒ E1P −(a′) ε(R) : E1P (a), E1P −(a′) ⇒ E1P −(dp−), E1P (dp) T ∗ : E1P (x)→A(x) ⇒ A(dp) T ∗ : A(x)↔E1P −(x) ⇒ E1P −(a), E1P −(dp), A(a′), A(dp−) AM = {a, dp, } E1P −M = {a′,dp−, } E1P M = {a,dp,

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 18-d/40

DL-LiteN

bool is NP-complete – FO Model

T = {A ≡ ∃P −, A ⊑ ≥ 2 P, ⊤ ⊑ ≤ 1 P −, ∃P ⊑ A}, A = {A(a), P (a, a′)} T ∗ =

• A(x)↔E1P −(x)
• ∧
• A(x)→E2P (x)
• ∧¬E2P −(x)∧
• E1P (x)→A(x)
• R† : P aa′

⇒ P −a′a, E1P (a) ⇒ E1P −(a′) ε(R) : E1P (a), E1P −(a′) ⇒ E1P −(dp−), E1P (dp) T ∗ : E1P (x)→A(x) ⇒ A(dp) T ∗ : A(x)↔E1P −(x) ⇒ E1P −(a), E1P −(dp), A(a′), A(dp−) T ∗ : A(x)→E2P (x) ⇒ E2P M = AM AM = {a, dp,a′, dp−} E1P −M = {a′,dp−,a, dp} E1P M = {a,dp,

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 18-e/40

DL-LiteN

bool is NP-complete – FO Model

T = {A ≡ ∃P −, A ⊑ ≥ 2 P, ⊤ ⊑ ≤ 1 P −, ∃P ⊑ A}, A = {A(a), P (a, a′)} T ∗ =

• A(x)↔E1P −(x)
• ∧
• A(x)→E2P (x)
• ∧¬E2P −(x)∧
• E1P (x)→A(x)
• R† : P aa′

⇒ P −a′a, E1P (a) ⇒ E1P −(a′) ε(R) : E1P (a), E1P −(a′) ⇒ E1P −(dp−), E1P (dp) T ∗ : E1P (x)→A(x) ⇒ A(dp) T ∗ : A(x)↔E1P −(x) ⇒ E1P −(a), E1P −(dp), A(a′), A(dp−) T ∗ : A(x)→E2P (x) ⇒ E2P M = AM δ(R) : E2P (x) → E1P (x) ⇒ E1P M = E2P M AM = {a, dp,a′, dp−}= E2P M E1P −M = {a′,dp−,a, dp} E1P M = {a,dp,

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 18-f/40

DL-LiteN

bool is NP-complete – FO Model

T = {A ≡ ∃P −, A ⊑ ≥ 2 P, ⊤ ⊑ ≤ 1 P −, ∃P ⊑ A}, A = {A(a), P (a, a′)} T ∗ =

• A(x)↔E1P −(x)
• ∧
• A(x)→E2P (x)
• ∧¬E2P −(x)∧
• E1P (x)→A(x)
• R† : P aa′

⇒ P −a′a, E1P (a) ⇒ E1P −(a′) ε(R) : E1P (a), E1P −(a′) ⇒ E1P −(dp−), E1P (dp) T ∗ : E1P (x)→A(x) ⇒ A(dp) T ∗ : A(x)↔E1P −(x) ⇒ E1P −(a), E1P −(dp), A(a′), A(dp−) T ∗ : A(x)→E2P (x) ⇒ E2P M = AM δ(R) : E2P (x) → E1P (x) ⇒ E1P M = E2P M AM = {a, dp,a′, dp−}= E2P M= D, E1P −M = {a′,dp−,a, dp}= D, E1P M = {a,dp,a′, dp−} = D.

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 18-g/40

DL-LiteN

bool is NP-complete – Upper Bound – Lemma 1

Lemma 1. A DL-LiteN

bool KB K is satisfiable iff the QL1-sentence K† is satisfiable.

(⇐) Starting from a model M = (D, ·M), with D = ob(A) ∪ dr(K), of K†, we construct an interpretation I for DL-LiteN

bool based on some domain ∆ ⊇ D

inductively defined as: ∆ =

∞

• m=0

Wm, where W0 = D = constants in K†

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 19/40

DL-LiteN

bool is NP-complete – Upper Bound – Lemma 1

Lemma 1. A DL-LiteN

bool KB K is satisfiable iff the QL1-sentence K† is satisfiable.

(⇐) Starting from a model M = (D, ·M), with D = ob(A) ∪ dr(K), of K†, we construct an interpretation I for DL-LiteN

bool based on some domain ∆ ⊇ D

inductively defined as: ∆ =

∞

• m=0

Wm, where W0 = D = constants in K† Each set Wm+1, for m ≥ 0, is constructed by adding to Wm some new elements, w′, that are fresh copies of certain elements, d, from W0 = D, i.e., cp(w′) = d. AI =

• w ∈ ∆ | M |

= A[cp(w)]

• P I

k

= ∞

m=0 P m k ,

where P m

k

⊆ Wm × Wm For the basis of induction we set, for each role Pk: P 0 =

• (aM

i , aM j ) ∈ W0 × W0 | M |

= P aiaj

• .

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 19-a/40

DL-LiteN

bool is NP-complete – Upper Bound – Lemma 1 (cont.)

. . a a′ dp dp− V0 V1 V2 Figure 1: Unravelling model M (first three steps).

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 20/40

DL-LiteN

bool is NP-complete – Upper Bound – Translation K‡

• Translation of K=(TBox,ABox): The lengthy vs. short translation K‡.

K† =

• T ∗ ∧

R∈role±(K)

• ε(R) ∧ δ(R)
• ∧
• A† ∧

R∈role±(K) R†

K‡ =

• T ∗ ∧

R∈role±(K)

• ε(R) ∧ δ‡(R)
• ∧ A‡ ∧ A⊥

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 21/40

DL-LiteN

bool is NP-complete – Upper Bound – Translation K‡

• Translation of K=(TBox,ABox): The lengthy vs. short translation K‡.

K† =

• T ∗ ∧

R∈role±(K)

• ε(R) ∧ δ(R)
• ∧
• A† ∧

R∈role±(K) R†

K‡ =

• T ∗ ∧

R∈role±(K)

• ε(R) ∧ δ‡(R)
• ∧ A‡ ∧ A⊥

δ‡

R(x)

=

• q,q′∈QR

T ,

q′>q q′>q′′>q for no q′′∈QR

T

• Eq′R(x) → EqR(x)
• Alessandro Artale

The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 21-a/40

DL-LiteN

bool is NP-complete – Upper Bound – Translation K‡

• Translation of K=(TBox,ABox): The lengthy vs. short translation K‡.

K† =

• T ∗ ∧

R∈role±(K)

• ε(R) ∧ δ(R)
• ∧
• A† ∧

R∈role±(K) R†

K‡ =

• T ∗ ∧

R∈role±(K)

• ε(R) ∧ δ‡(R)
• ∧ A‡ ∧ A⊥

δ‡

R(x)

=

• q,q′∈QR

T ,

q′>q q′>q′′>q for no q′′∈QR

T

• Eq′R(x) → EqR(x)
• A‡

=

• A(a)∈A A(a)

∧

• ¬A(a)∈A ¬A(a)

∧

• a∈ob(A)
• R∈role±(K)

∃a′∈ob(A) R(a,a′)∈A

EqR,aR(a)

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 21-b/40

DL-LiteN

bool is NP-complete – Upper Bound – Translation K‡

• Translation of K=(TBox,ABox): The lengthy vs. short translation K‡.

K† =

• T ∗ ∧

R∈role±(K)

• ε(R) ∧ δ(R)
• ∧
• A† ∧

R∈role±(K) R†

K‡ =

• T ∗ ∧

R∈role±(K)

• ε(R) ∧ δ‡(R)
• ∧ A‡ ∧ A⊥

δ‡

R(x)

=

• q,q′∈QR

T ,

q′>q q′>q′′>q for no q′′∈QR

T

• Eq′R(x) → EqR(x)
• A‡

=

• A(a)∈A A(a)

∧

• ¬A(a)∈A ¬A(a)

∧

• a∈ob(A)
• R∈role±(K)

∃a′∈ob(A) R(a,a′)∈A

EqR,aR(a) A⊥ = ⊥ if P (ai, aj) ∈ A and ¬P (ai, aj) ∈ A

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 21-c/40

DL-LiteN

bool is NP-complete – Upper Bound – Translation K‡

• Translation of K=(TBox,ABox): The lengthy vs. short translation K‡.

K† =

• T ∗ ∧

R∈role±(K)

• ε(R) ∧ δ(R)
• ∧
• A† ∧

R∈role±(K) R†

K‡ =

• T ∗ ∧

R∈role±(K)

• ε(R) ∧ δ‡(R)
• ∧ A‡ ∧ A⊥

δ‡

R(x)

=

• q,q′∈QR

T ,

q′>q q′>q′′>q for no q′′∈QR

T

• Eq′R(x) → EqR(x)
• A‡

=

• A(a)∈A A(a)

∧

• ¬A(a)∈A ¬A(a)

∧

• a∈ob(A)
• R∈role±(K)

∃a′∈ob(A) R(a,a′)∈A

EqR,aR(a) A⊥ = ⊥ if P (ai, aj) ∈ A and ¬P (ai, aj) ∈ A K‡ can be computed in LOGSPACE.

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 21-d/40

Satifiability Checking – Combined Complexity Results

Theorem

• 1. The satisfiability problem for DL-LiteN

bool, DL-LiteF bool and DL-Litebool knowledge

bases is NP-complete.

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 22/40

Satifiability Checking – Combined Complexity Results

Theorem

• 1. The satisfiability problem for DL-LiteN

bool, DL-LiteF bool and DL-Litebool knowledge

bases is NP-complete.

• 2. The satisfiability problem for DL-LiteN

krom, DL-LiteF krom, DL-Litekrom, as well as

DL-LiteN

core, DL-LiteF core and DL-Litecore knowledge bases, is

NLOGSPACE-complete.

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 22-a/40

Satifiability Checking – Combined Complexity Results

Theorem

• 1. The satisfiability problem for DL-LiteN

bool, DL-LiteF bool and DL-Litebool knowledge

bases is NP-complete.

• 2. The satisfiability problem for DL-LiteN

krom, DL-LiteF krom, DL-Litekrom, as well as

DL-LiteN

core, DL-LiteF core and DL-Litecore knowledge bases, is

NLOGSPACE-complete.

• 3. The satisfiability problem for DL-LiteN

horn, DL-LiteF horn and DL-Litehorn knowledge

bases is P-complete.

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 22-b/40

Sub-Roles Vs Cardinality Constraints

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 23/40

DL-LiteR,F

horn is EXPTIME-complete

DL-LiteR,F

horn Ontology language:

• Role Inclusions:

R1 ⊑ R2,

• Functionality Axioms:

≥ 2 R ⊑ ⊥

• Concept Inclusions:
• k Bk ⊑ B, with:

B − → A | ∃R | ⊥ R − → P | P −

• ABox assertions:

A(c), ¬A(c) P (c, d), ¬P (c, d) with c, d constants

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 24/40

DL-LiteR,F

horn is EXPTIME-complete

DL-LiteR,F

horn Ontology language:

• Role Inclusions:

R1 ⊑ R2,

• Functionality Axioms:

≥ 2 R ⊑ ⊥

• Concept Inclusions:
• k Bk ⊑ B, with:

B − → A | ∃R | ⊥ R − → P | P −

• ABox assertions:

A(c), ¬A(c) P (c, d), ¬P (c, d) with c, d constants

• Upper Bound: DL-LiteR,F

horn is a sub-language of SHIQ which is

EXPTIME-complete.

• Lower Bound: DL-LiteR,F

horn KBs can encode the behaviour of

polynomial-space-bounded alternating Turing machines (ATMs).

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 24-a/40

DL-LiteR,F

core is EXPTIME-complete

The only difference between DL-LiteR,F

core and DL-LiteR,F horn is the possibility to express

conjunction on the left of axioms in DL-LiteR,F

horn .

Elimination of axioms of the form A1 ⊓ A2 ⊑ C. Define a new KB K′ by replacing this axiom in K with the following set of new axioms, where R1, R2, R3, R12, R23 are fresh role names: A1 ⊑ ∃R1 A2 ⊑ ∃R2, (1) R1 ⊑ R12, R2 ⊑ R12, (2) ≥ 2 R12 ⊑ ⊥, (3) ∃R−

1

⊑ ∃R−

3 ,

(4) ∃R3 ⊑ C, (5) R3 ⊑ R23, R2 ⊑ R23, (6) ≥ 2 R−

23 ⊑ ⊥.

(7)

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 25/40

Cardinality + Sub-Roles: Regaining Tractability

• TBox assertions: C1 ⊑ C2, R1 ⊑ R2

Definition 1 (Relaxing cardinality constraints) Given a TBox T and a role R ∈ role±(T ), we define the following parameters: ubound(R, T ) = min

• {∞} ∪ {q | C ⊑ (≤ q R) ∈ T }
• lbound(R, T ) = max
• {0} ∪ {q | C ⊑ (≥ q R) ∈ T }
• rank(R, T ) = max
• lbound(R, T ),

R′∈dsubT (R) rank(R′, T )

• rank(R, A) = max
• {0}∪{n | Ri(a, aj) ∈ A, Ri ⊑∗

T R,

for distinct a1, . . . , an}

• (inter1) If R has a proper sub-role in T then the TBox contains no at-most cardinality

restrictions on R. (inter2) If R has a proper sub-role in T then ubound(R, T ) ≥ rank(R, T ) + max{1, rank(R, A)}

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 26/40

Query Language

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 27/40

Ontology based data access

Desiderata: achieve logical transparency in access to data:

• Hide to the user where and how data are stored
• Present to the user a conceptual view of the data
• Query the data sources through the conceptual model

Layer Data Layer Query over conceptual layer Conceptual Ontology

As in Data Integration, but with a rich conceptual description as the global view

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 28/40

Query Language

We consider positive existential queries—extending UCQs with unrestricted interaction of conjunction and disjunction—over the terms of the ontology: t ::= yi | ai q ::= Ak(t) | Pk(t1, t2) | q1 ∧ q2 | q1 ∨ q2 | ∃yi q Example: q(x) = { x | ∃y, p. Employee(x) ∧ WorksFor(x, p) ∧ Project(p) ∧ Boss(y, x) ∧ Employee(y) ∧ WorksFor(y, p) }

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 29/40

Query Answering

Since we work under the Open World Semantics then Query answering over an

• ntology O wrt an ABox A amounts to computing certain answers:

cert(q, O, A) = { t | t ∈ qI for every I ∈ mod(O, A) } i.e., the tuples that are answers to q in all models of the ABox A w.r.t. the ontology O.

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 30/40

Query Answering

Since we work under the Open World Semantics then Query answering over an

• ntology O wrt an ABox A amounts to computing certain answers:

cert(q, O, A) = { t | t ∈ qI for every I ∈ mod(O, A) } i.e., the tuples that are answers to q in all models of the ABox A w.r.t. the ontology O. Computing certain answers is a form of logical implication:

• t ∈ cert(q, O, A)

iff (O, A) | = q( t)

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 30-a/40

Data complexity Vs. Combined complexity

When considering a setting where the size of the data largely dominates the size of the conceptual layer ❀ We look at data complexity When both the size of the ontology and the size of the underlying data are comparable ❀ We look at combined complexity

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 31/40

Data complexity Vs. Combined complexity

When considering a setting where the size of the data largely dominates the size of the conceptual layer ❀ We look at data complexity When both the size of the ontology and the size of the underlying data are comparable ❀ We look at combined complexity Basic questions:

• How complex becomes reasoning over both an ontology and a data source?

(both data and combined complexity) – In particular, for which ontology language can we answer queries to DB sources through an ontology efficiently? (data complexity)

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 31-a/40

Data Complexity of Query Answering in DL-LiteN

hornis in LOGSPACE

Given a positive existential query, q( x), and a DL-LiteN

horn KB, T , then:

• 1. First, we construct a single, but possibly infinite, model I0 which provides all

answers to all positive existential queries with respect to the DL-LiteN

horn KB, K:

• The canonical model I0 is obtained starting from the minimal Herbrand model

for K‡.

• 2. Second, to find all answers to a given query it is enough to consider some finite

part of I0 the size of which does not depend on the given ABox.

• Assume that, in the query q(

x) = ∃ y ϕ( x, y), we have y = y1, . . . , yk, then, to check whether I0 | = q( a) it suffices to consider only the points of depth ≤ m0 in ∆I0, with m0 = k + |role±(T )|.

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 32/40

Data Complexity of Query Answering in DL-LiteN

hornis in LOGSPACE

• The LOGSPACE query answering algorithm will consider then all assignments in

this finite part of I0 to the variables x, y, compute the corresponding types (the concepts that contain these elements), and, finally, encode the problem ‘K | = q( a)?’ as a model checking problem for the first-order formula ϕT ,q( x): – ϕT ,q( x) depends on T and q but not on A, – A | = ϕT ,q( a) iff I0 | = q( a).

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 33/40

Data Complexity of Query Answering in DL-LiteN

hornis in LOGSPACE

Formulas ψB(x), for B ∈ Bcon(K), describe the types of the elements of ob(A) in the model I0: A | = ψB[ai] iff aI0

i

∈ BI0, for ai ∈ ob(A) ψ0

B(x) =

   A(x), if B = A, EqRT (x), if B = ≥ q R ψi

B(x) = ψ0 B(x) ∨

• B1⊓···⊓Bk⊑B∈ext(T )
• ψi−1

B1 (x) ∧ · · · ∧ ψi−1 Bk (x)

• ,

for i ≥ 1 EqRT (x) = ∃y1 . . . ∃yq

• 1≤i<j≤q

(yi = yj) ∧

• 1≤i≤q

R(x, yi)

• Alessandro Artale

The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 34/40

Data Complexity of Query Answering in DL-LiteN

hornis in LOGSPACE

Formulas θB,dr, for B ∈ Bcon(K) and dr ∈ dr(T ), describe the types of elements dr(T ) in the model I0: A | = θB,dr iff w ∈ BI0, for w ∈ ∆I0 with cp(w) = dr. For each B ∈ Bcon(K) and each dr ∈ dr(K), we inductively define a sequence θ0

B,dr, θ1 B,dr, . . . by taking:

θ0

B,dr = ⊤, if B = ∃R, and θ0 B,dr = ⊥, otherwise

θi

B,dr = θi−1 B,dr ∨

• B1⊓···⊓Bk⊑B∈ext(T )
• θi−1

B1,dr ∧ · · · ∧ θi−1 Bk,dr

• , for i ≥ 1

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 35/40

Data Complexity of Query Answering in DL-LiteN

hornis in LOGSPACE

path = {(Ri, Rj) | T | = ∃inv(Ri) ⊑ ≥ q Rj} ΣT ,m0 =

• ε
• ∪ role±(K) ∪
• (R1, . . . , Rn) | 2 ≤ n ≤ m0, (Rj, Rj+1) ∈ path
• Every existential variable, yi, is evaluated using σ ∈ ΣT ,m0;
• If σ = ǫ, then yi is assigned to an ABox element;
• σ = ǫ, then yi is evaluated as w using the pair (a, σ) s.t. aI0 is the root of the

tree Ta containing w, and σ is the sequence of roles on the path from aI0 to w. For σ = ǫ s.t. σ = (Ri, . . . ), this formula ensures that path σ exists in I0: ησ(a) =

• q∈Q

Ri T

• ψ≥q Ri(a) ∧ ¬ψ0

≥q Ri(a)

• Alessandro Artale

The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 36/40

Data Complexity of Query Answering in DL-LiteN

hornis in LOGSPACE

Let q( x) = ∃y1, . . . , yk.ϕ( x, y1, . . . , yk), then for every σ ∈ Σk

T ,m0, concept name

A and role name R, we define: A

σ(t) =

   ψA(t), if t

σ = ε,

θA,inv(ds), if t

σ = σ′.[S], for some σ′ ∈ ΣT ,m0,

R

σ(t1, t2) =

       RT (t1, t2), if t

σ 1 = t σ 2 = ε,

(t1 = t2), if t

σ 1 R

→ t

σ 2 and either t σ 1 = ε or t σ 2 = ε,

⊥,

• therwise.

The first-order rewriting of q( x) is then: ϕT ,q( x) = ∃ y

• σ∈Σk

T ,m0

(ϕ

σ(

x, y) ∧ η

σ(

y))

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 37/40

complexity language combined data satisfiability

• inst. checking

query answering DL-Litecore NLOGSPACE ≥ [Log] in LOGSPACE in LOGSPACE DL-LiteF

core

NLOGSPACE in LOGSPACE in LOGSPACE DL-LiteN

core

NLOGSPACE in LOGSPACE in LOGSPACE DL-LiteR

core

NLOGSPACE in LOGSPACE in LOGSPACE DL-LiteR,F

core

EXPTIME ≥ P ≥ P DL-LiteR,N

core

EXPTIME coNP ≥ coNP DL-Litekrom NLOGSPACE in LOGSPACE coNP ≥ [B] DL-LiteF

krom

NLOGSPACE in LOGSPACE coNP DL-LiteN

krom

NLOGSPACE ≤ in LOGSPACE coNP DL-LiteR

krom

NLOGSPACE ≤ in LOGSPACE coNP DL-LiteR,F

krom

EXPTIME coNP ≥ coNP DL-LiteR,N

krom

EXPTIME coNP coNP

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 38/40

complexity language combined data satisfiability

• inst. checking

query answering DL-Litehorn P ≥ [Log] in LOGSPACE in LOGSPACE DL-LiteF

horn

P in LOGSPACE in LOGSPACE DL-LiteN

horn

P ≤ in LOGSPACE in LOGSPACE ≤ DL-LiteR

horn

P ≤ in LOGSPACE in LOGSPACE ≤ [C] DL-LiteR,F

horn

EXPTIME P P ≤ [D] DL-LiteR,N

horn

EXPTIME coNP coNP DL-Litebool NP ≥ [Log] in LOGSPACE coNP DL-LiteF

bool

NP in LOGSPACE coNP DL-LiteN

bool

NP ≤ in LOGSPACE ≤ coNP DL-LiteR

bool

NP ≤ in LOGSPACE ≤ coNP DL-LiteR,F

bool

EXPTIME coNP coNP DL-LiteR,N

bool

EXPTIME coNP coNP ≤ [E]

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 39/40

Conclusions & Ongoing Work

• Ontologies based data access is an important problem we have to consider
• Expressive power of ontology languages heavily influences (data) complexity of

query answering

• Reasonable expressiveness in the ontology and efficiency of query answering

can be reconciled ❀ DL-Lite-family

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 40/40

Conclusions & Ongoing Work

• Ontologies based data access is an important problem we have to consider
• Expressive power of ontology languages heavily influences (data) complexity of

query answering

• Reasonable expressiveness in the ontology and efficiency of query answering

can be reconciled ❀ DL-Lite-family

• The DL-Lite DLs do not enjoy the finite model property: What if we want to

restrict the attention to finite models only?

• Developing efficient algorithms for answering positive existential queries for the

LOGSPACE languages that rely on relational database techniques (i.e., SQL).

Alessandro Artale The DL-Lite Family of Languages—TU Dresden 14-15 December 2011 40-a/40