The DL-Lite Family of Languages A FO Perspective

Transcription

1 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

2 Recommended Readings [1] A. Artale, D. Calvanese, R. Kontchakov and M. Zakharyaschev. The DL-Lite family and relations. JAIR, 36:1 69, [2] D. Calvanese, G. De Giacomo, D. Lembo, M. Lenzerini, and R. Rosati. DL-Lite: Tractable description logics for ontologies. Proceedings of AAAI [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: , Alessandro Artale The DL-Lite Family of Languages TU Dresden December /40

3 Outline 1. Ontology based data access 2. The DL-Lite-family of ontology languages: DL-Lite bool, DL-Lite horn, DL-Lite core, DL-Lite krom 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 December /40

4 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 December /40

5 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 December a/40

6 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 Query over conceptual layer Ontology Conceptual Layer Data Layer As in Data Integration, but with a rich conceptual description as the global view Alessandro Artale The DL-Lite Family of Languages TU Dresden December /40

7 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 December /40

8 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-Lite bool and three sublanguages: DL-Lite krom, DL-Lite core and DL-Lite horn 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-Lite horn. Alessandro Artale The DL-Lite Family of Languages TU Dresden December /40

9 The simplest DL-Lite Language: DL-Lite core DL-Lite core Ontology language: Concept Inclusions: B 1 B 2, B 1 B 2 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 December /40

10 DL-Lite R,N bool The most expressive DL-Lite Language: DL-Lite R,N bool Ontology language: Concept Inclusions: C 1 C 2, with: C B C C 1 C 2 B A qr R P P Role Inclusions: R 1 R 2 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 December /40

11 The most expressive DL-Lite Language: DL-Lite R,N bool DL-Lite R,N bool Ontology language: Concept Inclusions: C 1 C 2, with: C B C C 1 C 2 B A qr R P P Role Inclusions: R 1 R 2 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 December a/40

12 DL-Lite R,N core, DL-LiteR,N krom, DL-LiteR,N horn DL-Lite R,N core Ontology language: B 1 B 2, B 1 B 2 DL-Lite R,N krom Ontology language: B 1 B 2, B 1 B 2, B 1 B 2 DL-Lite R,N horn Ontology language: k B k B Alessandro Artale The DL-Lite Family of Languages TU Dresden December /40

13 DL-Lite Example Manager Employee AreaManager Manager TopManager Manager AreaManager TopManager WorksFor Employee WorksFor Project Project WorksFor 2Manages 2Manages 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 December /40

14 Semantics of DL-Lite Construct Syntax Example Semantics atomic concept A Doctor A I I atomic role P child P I I I inverse role P child {(d,e) I I (e,d) P I } empty concept conjunction C 1 C 2 Doctor Male C1 I CI 2 negation C (Doctor Male) I \ C I cardinalities nr 2 child {d I {e I (d,e) R I } n} inclusion asser. Cl Cr Father 1 child Cl I Cr I memb. asser. A(a) Father(bob) a I A I memb. asser. P(a,b) child(bob,ann) (a I,b I ) P I Alessandro Artale The DL-Lite Family of Languages TU Dresden December /40

15 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 December /40

16 FO Translation Alessandro Artale The DL-Lite Family of Languages TU Dresden December /40

17 DL-Lite N bool is NP-complete Upper Bound Class consistency for DL-Lite N bool can be reduced to formula satisfiability for the one-variable fragment QL 1 of first-order logic without equality and functions. Formula satisfiability for the one-variable fragment QL 1 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 December /40

18 DL-Lite N bool is NP-complete Upper Bound Translating C Inductive translation of Concepts, C : ( ) = (A) = A(x) ( C) = C (x) (C 1 C 2 ) = C 1 (x) C 2 (x) ( qr) =E q R(x) Alessandro Artale The DL-Lite Family of Languages TU Dresden December /40

19 DL-Lite N 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 December /40

20 DL-Lite N 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 = C 1 C 2 T x( C 1 (x) C 2 (x)) Alessandro Artale The DL-Lite Family of Languages TU Dresden December a/40

21 DL-Lite N 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 = C 1 C 2 T x( C 1 (x) C 2 (x)) A = A(a i ) A A(a i) P(a i,a j ) A Pa ia j A(a i ) A A(a i) P(a i,a j ) A Pa ia j Alessandro Artale The DL-Lite Family of Languages TU Dresden December b/40

22 DL-Lite N 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 = C 1 C 2 T x( C 1 (x) C 2 (x)) A = A(a i ) A A(a i) P(a i,a j ) A Pa ia j R = q T q=1 A(a i ) A A(a i) P(a i,a j ) A Pa ia j a i,a j ob(a) a,a j1,...,a jq ob(a) ( q j i j i for i i ( ) Rai a j inv(r)a j a i (q T is the maximum cardinality number in T ) ) i=1 Raa j i E q R(a) Alessandro Artale The DL-Lite Family of Languages TU Dresden December c/40

23 DL-Lite N 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 December /40

24 DL-Lite N 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 ( E 1 R(x) inv(e 1 R(dr)) ) Alessandro Artale The DL-Lite Family of Languages TU Dresden December d/40

25 DL-Lite N 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 ( E 1 R(x) inv(e 1 R(dr)) ) δ(r) = q T 1 q=1 x( E q+1 R(x) E q R(x) ) Alessandro Artale The DL-Lite Family of Languages TU Dresden December e/40

26 DL-Lite N bool is NP-complete FO Model T = {A P, A 2P, 1P, P A}, A = {A(a), P(a,a )} T = ( A(x) E 1 P (x) ) ( A(x) E 2 P(x) ) E 2 P (x) ( E 1 P(x) A(x) ) A M = {a, } E 1 P M = { } E 1 P M = { Alessandro Artale The DL-Lite Family of Languages TU Dresden December /40

27 DL-Lite N bool is NP-complete FO Model T = {A P, A 2P, 1P, P A}, A = {A(a), P(a,a )} T = ( A(x) E 1 P (x) ) ( A(x) E 2 P(x) ) E 2 P (x) ( E 1 P(x) A(x) ) R : Paa P a a, E 1 P(a) E 1 P (a ) A M = {a, } E 1 P M = { } E 1 P M = { Alessandro Artale The DL-Lite Family of Languages TU Dresden December a/40

28 DL-Lite N bool is NP-complete FO Model T = {A P, A 2P, 1P, P A}, A = {A(a), P(a,a )} T = ( A(x) E 1 P (x) ) ( A(x) E 2 P(x) ) E 2 P (x) ( E 1 P(x) A(x) ) R : Paa P a a, E 1 P(a) E 1 P (a ) ε(r) : E 1 P(a), E 1 P (a ) E 1 P (dp ), E 1 P(dp) A M = {a, } E 1 P M = {a, } E 1 P M = {a, Alessandro Artale The DL-Lite Family of Languages TU Dresden December b/40

29 DL-Lite N bool is NP-complete FO Model T = {A P, A 2P, 1P, P A}, A = {A(a), P(a,a )} T = ( A(x) E 1 P (x) ) ( A(x) E 2 P(x) ) E 2 P (x) ( E 1 P(x) A(x) ) R : Paa P a a, E 1 P(a) E 1 P (a ) ε(r) : E 1 P(a), E 1 P (a ) E 1 P (dp ), E 1 P(dp) T : E 1 P(x) A(x) A(dp) A M = {a, } E 1 P M = {a,dp, } E 1 P M = {a,dp, Alessandro Artale The DL-Lite Family of Languages TU Dresden December c/40

30 DL-Lite N bool is NP-complete FO Model T = {A P, A 2P, 1P, P A}, A = {A(a), P(a,a )} T = ( A(x) E 1 P (x) ) ( A(x) E 2 P(x) ) E 2 P (x) ( E 1 P(x) A(x) ) R : Paa P a a, E 1 P(a) E 1 P (a ) ε(r) : E 1 P(a), E 1 P (a ) E 1 P (dp ), E 1 P(dp) T : E 1 P(x) A(x) A(dp) T : A(x) E 1 P (x) E 1 P (a), E 1 P (dp), A(a ), A(dp ) A M = {a,dp, } E 1 P M = {a,dp, } E 1 P M = {a,dp, Alessandro Artale The DL-Lite Family of Languages TU Dresden December d/40

31 DL-Lite N bool is NP-complete FO Model T = {A P, A 2P, 1P, P A}, A = {A(a), P(a,a )} T = ( A(x) E 1 P (x) ) ( A(x) E 2 P(x) ) E 2 P (x) ( E 1 P(x) A(x) ) R : Paa P a a, E 1 P(a) E 1 P (a ) ε(r) : E 1 P(a), E 1 P (a ) E 1 P (dp ), E 1 P(dp) T : E 1 P(x) A(x) A(dp) T : A(x) E 1 P (x) E 1 P (a), E 1 P (dp), A(a ), A(dp ) T : A(x) E 2 P(x) E 2 P M = A M A M = {a,dp,a,dp } E 1 P M = {a,dp, E 1 P M = {a,dp,a,dp} Alessandro Artale The DL-Lite Family of Languages TU Dresden December e/40

32 DL-Lite N bool is NP-complete FO Model T = {A P, A 2P, 1P, P A}, A = {A(a), P(a,a )} T = ( A(x) E 1 P (x) ) ( A(x) E 2 P(x) ) E 2 P (x) ( E 1 P(x) A(x) ) R : Paa P a a, E 1 P(a) E 1 P (a ) ε(r) : E 1 P(a), E 1 P (a ) E 1 P (dp ), E 1 P(dp) T : E 1 P(x) A(x) A(dp) T : A(x) E 1 P (x) E 1 P (a), E 1 P (dp), A(a ), A(dp ) T : A(x) E 2 P(x) E 2 P M = A M δ(r) : E 2 P(x) E 1 P(x) E 1 P M = E 2 P M A M = {a,dp,a,dp }= E 2 P M E 1 P M = {a,dp, E 1 P M = {a,dp,a,dp} Alessandro Artale The DL-Lite Family of Languages TU Dresden December f/40

33 DL-Lite N bool is NP-complete FO Model T = {A P, A 2P, 1P, P A}, A = {A(a), P(a,a )} T = ( A(x) E 1 P (x) ) ( A(x) E 2 P(x) ) E 2 P (x) ( E 1 P(x) A(x) ) R : Paa P a a, E 1 P(a) E 1 P (a ) ε(r) : E 1 P(a), E 1 P (a ) E 1 P (dp ), E 1 P(dp) T : E 1 P(x) A(x) A(dp) T : A(x) E 1 P (x) E 1 P (a), E 1 P (dp), A(a ), A(dp ) T : A(x) E 2 P(x) E 2 P M = A M δ(r) : E 2 P(x) E 1 P(x) E 1 P M = E 2 P M A M = {a,dp,a,dp }= E 2 P M = D, E 1 P M = {a,dp,a,dp}= D, E 1 P M = {a,dp,a,dp } = D. Alessandro Artale The DL-Lite Family of Languages TU Dresden December g/40

34 DL-Lite N bool is NP-complete Upper Bound Lemma 1 Lemma 1. A DL-Lite N bool KB K is satisfiable iff the QL 1 -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-Lite N bool based on some domain D inductively defined as: = m=0 W m, where W 0 = D = constants in K Alessandro Artale The DL-Lite Family of Languages TU Dresden December /40

35 DL-Lite N bool is NP-complete Upper Bound Lemma 1 Lemma 1. A DL-Lite N bool KB K is satisfiable iff the QL 1 -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-Lite N bool based on some domain D inductively defined as: = m=0 W m, where W 0 = D = constants in K Each set W m+1, for m 0, is constructed by adding to W m some new elements, w, that are fresh copies of certain elements, d, from W 0 = D, i.e., cp(w ) = d. A I = { w M = A[cp(w)] } P I k = m=0 Pm k, where Pm k W m W m For the basis of induction we set, for each role P k : P 0 = { (a M i,a M j ) W 0 W 0 M = Pa i a j }. Alessandro Artale The DL-Lite Family of Languages TU Dresden December a/40

36 DL-Lite N bool is NP-complete Upper Bound Lemma 1 (cont.) V 0 a V 1 V 2 a. dp dp. Figure 1: Unravelling model M (first three steps). Alessandro Artale The DL-Lite Family of Languages TU Dresden December /40

37 DL-Lite N 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 December /40

38 DL-Lite N 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 Q R T, q >q ( Eq R(x) E q R(x) ) q >q >q for no q Q R T Alessandro Artale The DL-Lite Family of Languages TU Dresden December a/40

39 DL-Lite N 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 Q R T, q >q ( Eq R(x) E q R(x) ) q >q >q for no q Q R T A = A(a) A A(a) A(a) A A(a) a ob(a) R role ± (K) a ob(a) R(a,a ) A E qr,a R(a) Alessandro Artale The DL-Lite Family of Languages TU Dresden December b/40

40 DL-Lite N 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 Q R T, q >q ( Eq R(x) E q R(x) ) q >q >q for no q Q R T A = A(a) A A(a) A(a) A A(a) a ob(a) R role ± (K) a ob(a) R(a,a ) A E qr,a R(a) A = if P(a i,a j ) A and P(a i,a j ) A Alessandro Artale The DL-Lite Family of Languages TU Dresden December c/40

41 DL-Lite N 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 Q R T, q >q ( Eq R(x) E q R(x) ) q >q >q for no q Q R T A = A(a) A A(a) A(a) A A(a) a ob(a) R role ± (K) a ob(a) R(a,a ) A E qr,a R(a) A = if P(a i,a j ) A and P(a i,a j ) A K can be computed in LOGSPACE. Alessandro Artale The DL-Lite Family of Languages TU Dresden December d/40

42 Satifiability Checking Combined Complexity Results Theorem 1. The satisfiability problem for DL-Lite N bool, DL-LiteF bool and DL-Lite bool knowledge bases is NP-complete. Alessandro Artale The DL-Lite Family of Languages TU Dresden December /40

43 Satifiability Checking Combined Complexity Results Theorem 1. The satisfiability problem for DL-Lite N bool, DL-LiteF bool and DL-Lite bool knowledge bases is NP-complete. 2. The satisfiability problem for DL-Lite N krom, DL-LiteF krom, DL-Lite krom, as well as DL-Lite N core, DL-LiteF core and DL-Lite core knowledge bases, is NLOGSPACE-complete. Alessandro Artale The DL-Lite Family of Languages TU Dresden December a/40

44 Satifiability Checking Combined Complexity Results Theorem 1. The satisfiability problem for DL-Lite N bool, DL-LiteF bool and DL-Lite bool knowledge bases is NP-complete. 2. The satisfiability problem for DL-Lite N krom, DL-LiteF krom, DL-Lite krom, as well as DL-Lite N core, DL-LiteF core and DL-Lite core knowledge bases, is NLOGSPACE-complete. 3. The satisfiability problem for DL-Lite N horn, DL-LiteF horn and DL-Lite horn knowledge bases is P-complete. Alessandro Artale The DL-Lite Family of Languages TU Dresden December b/40

45 Sub-Roles Vs Cardinality Constraints Alessandro Artale The DL-Lite Family of Languages TU Dresden December /40

46 DL-Lite R,F horn DL-Lite R,F horn Ontology language: Role Inclusions: R 1 R 2, is EXPTIME-complete Functionality Axioms: Concept Inclusions: 2 R k B k 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 December /40

47 DL-Lite R,F horn DL-Lite R,F horn Ontology language: Role Inclusions: R 1 R 2, is EXPTIME-complete Functionality Axioms: Concept Inclusions: 2 R k B k 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-Lite R,F horn EXPTIME-complete. is a sub-language of SHIQ which is Lower Bound: DL-Lite R,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 December a/40

48 DL-Lite R,F core is EXPTIME-complete The only difference between DL-Lite R,F core and DL-LiteR,F horn conjunction on the left of axioms in DL-Lite R,F horn. is the possibility to express Elimination of axioms of the form A 1 A 2 C. Define a new KB K by replacing this axiom in K with the following set of new axioms, where R 1,R 2,R 3,R 12,R 23 are fresh role names: A 1 R 1 A 2 R 2, (1) R 1 R 12, R 2 R 12, (2) 2R 12, R 1 R 3, (4) R 3 C, R 3 R 23, R 2 R 23, (6) 2R 23. (3) (5) (7) Alessandro Artale The DL-Lite Family of Languages TU Dresden December /40

49 Cardinality + Sub-Roles: Regaining Tractability TBox assertions: C 1 C 2, R 1 R 2 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 ( qr) T } ) rank(r,t ) = max ( lbound(r,t ), R dsub T (R) rank(r,t ) ) rank(r,a) = max ( {0} {n R i (a,a j ) A,R i T R, for distinct a 1,...,a n } ) (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 December /40

50 Query Language Alessandro Artale The DL-Lite Family of Languages TU Dresden December /40

51 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 Query over conceptual layer Ontology Conceptual Layer Data Layer As in Data Integration, but with a rich conceptual description as the global view Alessandro Artale The DL-Lite Family of Languages TU Dresden December /40

52 Query Language We consider positive existential queries extending UCQs with unrestricted interaction of conjunction and disjunction over the terms of the ontology: t ::= y i a i q ::= A k (t) P k (t 1,t 2 ) q 1 q 2 q 1 q 2 y i 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 December /40

53 Query Answering Since we work under the Open World Semantics then Query answering over an ontology O wrt an ABox A amounts to computing certain answers: cert(q,o,a) = { t t q I 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 December /40

54 Query Answering Since we work under the Open World Semantics then Query answering over an ontology O wrt an ABox A amounts to computing certain answers: cert(q,o,a) = { t t q I 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 December a/40

55 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 December /40

56 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 December a/40

57 Data Complexity of Query Answering in DL-Lite N hornis in LOGSPACE Given a positive existential query, q( x), and a DL-Lite N horn KB, T, then: 1. First, we construct a single, but possibly infinite, model I 0 which provides all answers to all positive existential queries with respect to the DL-Lite N horn KB, K: The canonical model I 0 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 I 0 the size of which does not depend on the given ABox. Assume that, in the query q( x) = yϕ( x, y), we have y = y 1,...,y k, then, to check whether I 0 = q( a) it suffices to consider only the points of depth m 0 in I 0, with m 0 = k + role ± (T ). Alessandro Artale The DL-Lite Family of Languages TU Dresden December /40

58 Data Complexity of Query Answering in DL-Lite N hornis in LOGSPACE The LOGSPACE query answering algorithm will consider then all assignments in this finite part of I 0 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 ϕ T,q ( x): ϕ T,q ( x) depends on T and q but not on A, A = ϕ T,q ( a) iff I 0 = q( a). Alessandro Artale The DL-Lite Family of Languages TU Dresden December /40

59 Data Complexity of Query Answering in DL-Lite N hornis in LOGSPACE Formulas ψ B (x), for B Bcon(K), describe the types of the elements of ob(a) in the model I 0 : A = ψ B [a i ] iff a I 0 i B I 0, for a i ob(a) ψ 0 B (x) = A(x), if B = A, E q R T (x), if B = qr ψ i B (x) = ψ0 B (x) ( ψ i 1 B 1 (x) ψ i 1 B k (x) ), for i 1 B 1 B k B ext(t ) ( E q R T (x) = y 1... y q (y i y j ) 1 i<j q 1 i q R(x,y i ) ) Alessandro Artale The DL-Lite Family of Languages TU Dresden December /40

60 Data Complexity of Query Answering in DL-Lite N hornis in LOGSPACE Formulas θ B,dr, for B Bcon(K) and dr dr(t ), describe the types of elements dr(t ) in the model I 0 : A = θ B,dr iff w B I 0, for w I 0 with cp(w) = dr. For each B Bcon(K) and each dr dr(k), we inductively define a sequence θ 0 B,dr,θ1 B,dr,... by taking: θ 0 B,dr =, if B = R, and θ0 B,dr ( θ i 1 B 1,dr θ i B,dr = θi 1 B,dr B 1 B k B ext(t ) =, otherwise θi 1 B k,dr), for i 1 Alessandro Artale The DL-Lite Family of Languages TU Dresden December /40

61 Data Complexity of Query Answering in DL-Lite N hornis in LOGSPACE path = {(R i,r j ) T = inv(r i ) qr j } Σ T,m0 = { ε } role ± (K) { (R 1,...,R n ) 2 n m 0,(R j,r j+1 ) path } Every existential variable, y i, is evaluated using σ Σ T,m0 ; If σ = ǫ, then y i is assigned to an ABox element; σ ǫ, then y i is evaluated as w using the pair (a,σ) s.t. a I 0 is the root of the tree T a containing w, and σ is the sequence of roles on the path from a I 0 to w. For σ ǫ s.t. σ = (R i,...), this formula ensures that path σ exists in I 0 : η σ (a) = ( ψ qri (a) ψ 0 qr i (a) ) q Q R i T Alessandro Artale The DL-Lite Family of Languages TU Dresden December /40

62 Data Complexity of Query Answering in DL-Lite N hornis in LOGSPACE Let q( x) = y 1,...,y k.ϕ( x,y 1,...,y k ), then for every σ Σ k T,m 0, concept name A and role name R, we define: ψ A σ A (t), if t σ = ε, (t) = θ A,inv(ds), if t σ = σ.[s], for some σ Σ T,m0, R T (t 1,t 2 ), if t σ 1 = t σ 2 = ε, R σ (t 1,t 2 ) = (t 1 = t 2 ), if t σ R 1 t σ 2 and either t σ 1 ε or t σ 2 ε,, otherwise. The first-order rewriting of q( x) is then: ϕ T,q ( x) = y (ϕ σ ( x, y) η σ ( y)) σ Σ k T,m 0 Alessandro Artale The DL-Lite Family of Languages TU Dresden December /40

63 complexity language combined data satisfiability inst. checking query answering DL-Lite core NLOGSPACE [Log] in LOGSPACE in LOGSPACE DL-Lite F core NLOGSPACE in LOGSPACE in LOGSPACE DL-Lite N core NLOGSPACE in LOGSPACE in LOGSPACE DL-Lite R core NLOGSPACE in LOGSPACE in LOGSPACE DL-Lite R,F core EXPTIME P P DL-Lite R,N core EXPTIME conp conp DL-Lite krom NLOGSPACE in LOGSPACE conp [B] DL-Lite F krom NLOGSPACE in LOGSPACE conp DL-Lite N krom NLOGSPACE in LOGSPACE conp DL-Lite R krom NLOGSPACE in LOGSPACE conp DL-Lite R,F krom EXPTIME conp conp DL-Lite R,N krom EXPTIME conp conp Alessandro Artale The DL-Lite Family of Languages TU Dresden December /40

64 complexity language combined data satisfiability inst. checking query answering DL-Lite horn P [Log] in LOGSPACE in LOGSPACE DL-Lite F horn P in LOGSPACE in LOGSPACE DL-Lite N horn P in LOGSPACE in LOGSPACE DL-Lite R horn P in LOGSPACE in LOGSPACE [C] DL-Lite R,F horn EXPTIME P P [D] DL-Lite R,N horn EXPTIME conp conp DL-Lite bool NP [Log] in LOGSPACE conp DL-Lite F bool NP in LOGSPACE conp DL-Lite N bool NP in LOGSPACE conp DL-Lite R bool NP in LOGSPACE conp DL-Lite R,F bool EXPTIME conp conp DL-Lite R,N bool EXPTIME conp conp [E] Alessandro Artale The DL-Lite Family of Languages TU Dresden December /40

65 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 December /40

66 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 December a/40