Query Answering in DL-Lite with Datatypes: A Non-Uniform Approach

Transcription

1 1 / 20 Department of Computer Science Query Answering in DL-Lite with Datatypes: A Non-Uniform Approach February 2, 2017 André Hernich, Julio Lemos, Frank Wolter

2 OBDM In Ontology-Based Data Management, data is stored in a database A (an ABox, in the DL setting), intensional knowledge is stored in an ontology T (a TBox ), queries against A are answered taking T into account. query: q(x) A(x), r(x, y), U(y, z) TBox A s T r U ABox A One of the main aims in OBDM: tractable query answering. 2 / 20

3 3 / 20 EXAMPLE Suppose we have a company ontology/database: T = {Employee hasemployer, Manager Employee} A = {Employee(Anna), Manager(Barbara), hasemployer(cid, IBM)} Since ontologies use open world semantics, we look for answers that hold in all models. Suppose we query A w.r.t. T : q(x) Employee(x) We will get as answers Anna, Barbara and Cid.

4 4 / 20 THE QUERY EVALUATION PROBLEM OVER DL-LITE Given a DL-Lite TBox T and a conjunctive query q: Query evaluation of q w.r.t. T Input: an ABox A, a tuple c Question: T, A = q( c)? Complexity of query answering with T, q fixed is called data complexity (here the default). Otherwise combined complexity. This problem is in PTIME (actually FO-rewritable).

5 5 / 20 OBDM WITH DATATYPES We add datatypes D = (D, R 1,..., R n ) to ontologies and queries. E.g. fix D = (Z, ((a, b) a, b Z {, })), set: T = T {Employee salary, Manager salary.(40000, )} Query: q(x) Employee(x), salary(x, v), v (40000, ), and then Barbara is the answer! Query answering with such queries is NP-hard for DL-Lite: [Savkovic-Calvanese 2012], [Artale-Ryzhikov-Kontchakov 2012].

6 6 / 20 OBDM WITH DATATYPES We go further and allow datatypes with binary predicates. E.g. for T and A as above, add the predicate < to D and consider the query q(x, y) Employee(x), Employee(y), salary(x, v), salary(y, w), v < w

7 7 / 20 OBDM WITH DATATYPES We go further and allow datatypes with binary predicates. E.g. for T and A as above, add the predicate < to D and consider the query q(x, y) Employee(x), Employee(y), salary(x, v), salary(y, w), v < w It turns out...

8 8 / 20 OBDM WITH DATATYPES We go further and allow datatypes with binary predicates. E.g. for T and A as above, add the predicate < to D and consider the query q(x, y) Employee(x), Employee(y), salary(x, v), salary(y, w), v < w It turns out... Theorem Query answering in DL-Lite is undecidable in combined complexity for the datatypes (Z, ), (Z, <), (Z, ), (Q, ), (Q, <).

9 Tractable/intractable based on how datatype atoms occur in T and q. 9 / 20 OBDM WITH DATATYPES We go further and allow datatypes with binary predicates. E.g. for T and A as above, add the predicate < to D and consider the query q(x, y) Employee(x), Employee(y), salary(x, v), salary(y, w), v < w It turns out... Theorem Query answering in DL-Lite is undecidable in combined complexity for the datatypes (Z, ), (Z, <), (Z, ), (Q, ), (Q, <). In face of these negative results, our goal is a classification: PTIME NP-hard

10 10 / 20 DL-LITE AND OUR EXTENSION DL-Lite: Fix a datatype D. In DL-Lite we allow concepts B := A r U where r can be a role or an inverse role and U an attribute over D = (D, R 1,..., R n ). The language supports axioms of the form B 1 ( )B 2.

11 11 / 20 DL-LITE AND OUR EXTENSION DL-Lite: Fix a datatype D. In DL-Lite we allow concepts B := A r U where r can be a role or an inverse role and U an attribute over D = (D, R 1,..., R n ). The language supports axioms of the form B 1 ( )B 2. We then extend DL-Lite with (non-negated) concepts on the right-hand side restricting the range of attributes U: B U.ϕ(x) B U.ϕ(x) with ϕ(x) R 1 ( z 1 ),..., R m ( z m ).

12 DL-LITE AND OUR EXTENSION DL-Lite: Fix a datatype D. In DL-Lite we allow concepts B := A r U where r can be a role or an inverse role and U an attribute over D = (D, R 1,..., R n ). The language supports axioms of the form B 1 ( )B 2. We then extend DL-Lite with (non-negated) concepts on the right-hand side restricting the range of attributes U: B U.ϕ(x) B U.ϕ(x) with ϕ(x) R 1 ( z 1 ),..., R m ( z m ). Query language: q( x) A(x), r(x, y), U(x, z),..., R 1 ( z 1 ),..., R m ( z m ), where each R( z i ) is a datatype atom over D; the red part is called the datatype pattern of q. 12 / 20

13 13 / 20 OUR RESULTS Our main results are: 1. A framework for transferring classification results from constraint satisfaction problems (CSPs); 2. An instantiation of the datatype (Q, ), that with the use of a recent result gives us a classification of PTIME / NP-hard cases.

14 14 / 20 RESULTS I: QUERY EVALUATION CSPS Framework (intuition behind theorem): given a query evaluation problem in DL-Lite over D, we can translate it into a CSP over a similar structure D (and vice-versa). T, q over datatype D CSP over D reduction Therefore if we have a classification for CSPs, we can transfer it to query evaluation.

15 15 / 20 CLASSIFICATION FOR (Q, ) Bodirsky and Kára 1 recently obtained a deep classification result for temporal CSPs, that is, CSPs over (Q, R 1, R 2,... ) where each relation R i is FO-definable on (Q, <). Example of temporal CSP is the one over D = (Q, Betw) where Betw = {(a, b, c) Q 3 a < b < c c < b < a}. The classification is in terms of a certain algebraic property of the relations: If all R i satisfy this property, then the CSP is in PTIME; Otherwise it is NP-complete. 1 The complexity of temporal CSPs, Journal of the ACM 2010.

16 16 / 20 RESULTS II: INSTANTIATION OF (Q, ) query evaluation of q w.r.t. T CSP over (Q, R T, R q) PTIME T = {Manager salary. 40k,...}? q(x) Employee(x), Employee(y), salary(x, v), salary(y, w), v w reduction? NP-hard

17 17 / 20 CLASSIFICATION OF DATATYPE PATTERNS FOR (Q, ) Using our framework and Bodirsky and Kára s result, using the same algebraic approach we showed a classification based on the datatype pattern q 0 = x 1 y 1... x n y n. For instance, given the datatype pattern q 0 = (x y) (x z): for all T, q where q q 0 PTIME? (x z) (x z)? co-np-hard there exists T and q where q q 0

18 18 / 20 SYNTHESIS OF OUR CLASSIFICATION RESULT We denote: min-pattern: x 0 x 1 x 0 x 2... x 0 x n max-pattern: x 1 x 0 x 1 x 0... x n x 0 Let q 0 be a datatype pattern over (Q, ). Then: 1. If q 0 is a max-pattern or a min-pattern, then evaluating (T, q), where q q 0, is in PTIME in data complexity. 2. Otherwise there is an OMQ (T, q) with q q 0 such that evaluating (T, q) is co-np-complete in data complexity.

19 19 / 20 for all T, q where q q 0 PTIME q 0 is a min- or a max-pattern otherwise NP-hard there exists T and q where q q 0

20 20 / 20 CONCLUSION AND FUTURE WORK We created a bridge to CSPs and showed a classification result for the evaluation of OMQs for DL-Lite over (Q, ), providing a syntactical criterion for that purpose. Natural next steps are: Using our general framework and tools: transfer results for different datatypes. Developing practical query answering algorithms, in particular using constraint solvers as part of the query engines.