Abstract
We give solutions to two fundamental computational problems in ontology-based data access with the W3C standard ontology language OWL 2 QL: the succinctness problem for first-order rewritings of ontology-mediated queries (OMQs) and the complexity problem for OMQ answering. We classify OMQs according to the shape of their conjunctive queries (treewidth, the number of leaves) and the existential depth of their ontologies. For each of these classes, we determine the combined complexity of OMQ answering and whether all OMQs in the class have polynomial-size first-order, positive existential, and nonrecursive datalog rewritings. We obtain the succinctness results using hypergraph programs, a new computational model for Boolean functions, which makes it possible to connect the size of OMQ rewritings and circuit complexity.
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- S. Abiteboul, R. Hull, and V. Vianu. 1995. Foundations of Databases. Addison-Wesley. Google ScholarDigital Library
- M. Agrawal, E. Allender, R. Impagliazzo, T. Pitassi, and S. Rudich. 2001. Reducing the complexity of reductions. Comput. Complex. 10, 2 (2001), 117--138. Google ScholarDigital Library
- M. Agrawal, E. Allender, and S. Rudich. 1998. Reductions in circuit complexity: An isomorphism theorem and a gap theorem. J. Comput. System Sci. 57, 2 (1998), 127--143. Google ScholarDigital Library
- N. Alon and R. Boppana. 1987. The monotone circuit complexity of boolean functions. Combinatorica 7, 1 (1987), 1--22. Google ScholarDigital Library
- S. Arora and B. Barak. 2009. Computational Complexity: A Modern Approach. Cambridge University Press, New York. Google ScholarDigital Library
- A. Artale, D. Calvanese, R. Kontchakov, and M. Zakharyaschev. 2009. The DL-Lite family and relations. J. Artif. Intell. Res. (JAIR) 36 (2009), 1--69. Google ScholarDigital Library
- B. Aspvall, M. Plass, and R. Tarjan. 1979. A linear-time algorithm for testing the truth of certain quantified Boolean formulas. Inform. Process. Lett. 8, 3 (1979), 121--123.Google ScholarCross Ref
- J. Avigad. 2003. Eliminating definitions and Skolem functions in first-order logic. ACM Trans. Comput. Logic 4, 3 (2003), 402--415. Google ScholarDigital Library
- J.-F. Baget, M. Leclère, M.-L. Mugnier, and E. Salvat. 2011. On rules with existential variables: Walking the decidability line. Artif. Intell. 175, 9--10 (2011), 1620--1654. Google ScholarDigital Library
- M. Bienvenu, S. Kikot, R. Kontchakov, V. V. Podolskii, V. Ryzhikov, and M. Zakharyaschev. 2017. The complexity of ontology-based data access with OWL 2 QL and bounded treewidth queries. In Proc. of the 36th ACM SIGMOD-SIGACT-SIGAI Symp. on Principles of Database Systems (PODS’17). ACM, 201--216. Google ScholarDigital Library
- M. Bienvenu, S. Kikot, and V. V. Podolskii. 2015. Tree-like queries in OWL 2 QL: Succinctness and complexity results. In Proc. of the 30th Annual ACM/IEEE Symp. on Logic in Computer Science (LICS’15). IEEE Computer Society, 317--328. Google ScholarDigital Library
- M. Bienvenu, C. Lutz, and F. Wolter. 2013. First-order rewritability of atomic queries in horn description logics. In Proc. of the 23nd Int. Joint Conf. on Artificial Intelligence (IJCAI’13). IJCAI/AAAI, 754--760. Google ScholarDigital Library
- M. Bienvenu, M. Ortiz, and M. Simkus. 2015. Regular path queries in lightweight description logics: Complexity and algorithms. J. Artif. Intell. Res. (JAIR) 53 (2015), 315--374. Google ScholarDigital Library
- M. Bienvenu, M. Ortiz, M. Simkus, and G. Xiao. 2013. Tractable queries for lightweight description logics. In Proc. of the 23nd Int. Joint Conf. on Artificial Intelligence (IJCAI’13). IJCAI/AAAI, 768--774. Google ScholarDigital Library
- M. Bienvenu and R. Rosati. 2015. Query-based comparison of OBDA specifications. In Proc. of the 28th Int. Workshop on Description Logics, DL 2015 (CEUR), Vol. 1350. CEUR-WS, 55--66.Google Scholar
- M. Bienvenu, B. ten Cate, C. Lutz, and F. Wolter. 2014. Ontology-based data access: A study through disjunctive Datalog, CSP, and MMSNP. ACM Trans. Database Syst. 39, 4 (2014), 33:1--33:44. Google ScholarDigital Library
- E. Botoeva, D. Calvanese, V. Santarelli, D. F. Savo, A. Solimando, and G. Xiao. 2016. Beyond OWL 2 QL in OBDA: Rewritings and approximations. In Proc. of the AAAI Conf. on Artificial Intelligence (AAAI’16). Google ScholarDigital Library
- A. Brandstädt, V. B. Le, and J. P. Spinrad. 1999. Graph Classes: A Survey. SIAM, Philadelphia. Google ScholarDigital Library
- A. Bretto. 2013. Hypergraph Theory: An Introduction. Springer. Google ScholarCross Ref
- A. Calì, G. Gottlob, and T. Lukasiewicz. 2012. A general datalog-based framework for tractable query answering over ontologies. J. Web Semantics 14 (2012), 57--83. Google ScholarDigital Library
- A. Calì, G. Gottlob, and A. Pieris. 2012. Towards more expressive ontology languages: The query answering problem. Artif. Intell. 193 (2012), 87--128. Google ScholarDigital Library
- D. Calvanese, G. De Giacomo, D. Lembo, M. Lenzerini, A. Poggi, M. Rodriguez-Muro, R. Rosati, M. Ruzzi, and D. F. Savo. 2011. The MASTRO system for ontology-based data access. Semantic Web 2, 1 (2011), 43--53. Google ScholarCross Ref
- D. Calvanese, G. De Giacomo, D. Lembo, M. Lenzerini, and R. Rosati. 2007. Tractable reasoning and efficient query answering in description logics: The DL-Lite family. J. Autom. Reason. 39, 3 (2007), 385--429. Google ScholarDigital Library
- A. Chandra and P. Merlin. 1977. Optimal implementation of conjunctive queries in relational data bases. In Conf. Record of the 9th Annual ACM Symp. on Theory of Computing (STOC’77). ACM, 77--90. Google ScholarDigital Library
- C. Chekuri and A. Rajaraman. 2000. Conjunctive query containment revisited. Theor. Comput. Sci. 239, 2 (2000), 211--229. Google ScholarDigital Library
- A. Chortaras, D. Trivela, and G. Stamou. 2011. Optimized query rewriting for OWL 2 QL. In Proc. of the 23rd Int. Conf. on Automated Deduction (CADE-23) (LNCS), Vol. 6803. Springer, 192--206. Google ScholarDigital Library
- C. Civili and R. Rosati. 2012. A broad class of first-order rewritable tuple-generating dependencies. In Proc. of the 2nd Int. Datalog 2.0 Workshop (LNCS), Vol. 7494. Springer, 68--80. Google ScholarDigital Library
- M. Console, J. Mora, R. Rosati, V. Santarelli, and D. F. Savo. 2014. Effective computation of maximal sound approximations of description logic ontologies. In Proc. of the 13th Int. Semantic Web Conf. (ISWC’14), Part II (LNCS), Vol. 8797. Springer, 164--179. Google ScholarDigital Library
- S. A. Cook. 1971. Characterizations of pushdown machines in terms of time-bounded computers. J. ACM 18, 1 (1971), 4--18. Google ScholarDigital Library
- T. Eiter, M. Ortiz, M. Šimkus, T.-K. Tran, and G. Xiao. 2012. Query rewriting for Horn-SHIQ plus rules. In Proc. of the 26th AAAI Conf. on Artificial Intelligence (AAAI’12). AAAI, 726--733. Google ScholarDigital Library
- C. Flament. 1978. Hypergraphes arborés. Discrete Math. 21, 3 (1978), 223--227. Google ScholarDigital Library
- J. Flum and M. Grohe. 2006. Parameterized Complexity Theory. Springer. Google ScholarDigital Library
- M. Giese, A. Soylu, G. Vega-Gorgojo, A. Waaler, P. Haase, E. Jiménez-Ruiz, D. Lanti, M. Rezk, G. Xiao, Ö. Özçep, and R. Rosati. 2015. Optique: Zooming in on big data. IEEE Comput. 48, 3 (2015), 60--67.Google ScholarDigital Library
- G. Gottlob, S. Kikot, R. Kontchakov, V. V. Podolskii, T. Schwentick, and M. Zakharyaschev. 2014. The price of query rewriting in ontology-based data access. Artif. Intell. 213 (2014), 42--59. Google ScholarDigital Library
- G. Gottlob, N. Leone, and F. Scarcello. 1999. Computing LOGCFL certificates. In Proc. of the 26th Int. Coll. on Automata, Languages 8 Programming (ICALP’99) (LNCS), Vol. 1644. Springer, 361--371. Google ScholarDigital Library
- G. Gottlob, N. Leone, and F. Scarcello. 2001. The complexity of acyclic conjunctive queries. J. ACM 48, 3 (2001), 431--498. Google ScholarDigital Library
- G. Gottlob, M. Manna, and A. Pieris. 2015. Polynomial rewritings for linear existential rules. In Proc. of the 24th Int. Joint Conf. on Artificial Intelligence (IJCAI’15). AAAI, 2992--2998. Google ScholarDigital Library
- G. Gottlob, G. Orsi, and A. Pieris. 2011. Ontological queries: Rewriting and optimization. In Proc. of the 27th Int. Conf. on Data Engineering (ICDE’11). IEEE Computer Society, 2--13. Google ScholarDigital Library
- G. Gottlob and T. Schwentick. 2012. Rewriting ontological queries into small nonrecursive Datalog programs. In Proc. of the 13th Int. Conf. on Principles of Knowledge Representation 8 Reasoning (KR’12). AAAI, 254--263. Google ScholarDigital Library
- S. A. Greibach. 1973. The hardest context-free language. SIAM J. Comput. 2, 4 (1973), 304--310.Google ScholarDigital Library
- M. Grigni and M. Sipser. 1992. Monotone complexity. In Proc. of the London Mathematical Society Symp. on Boolean Function Complexity. Cambridge University Press, 57--75. Google ScholarDigital Library
- M. Grohe, T. Schwentick, and L. Segoufin. 2001. When is the evaluation of conjunctive queries tractable? In Proc. of the 33rd Annual ACM Symp. on Theory of Computing (STOC’01). ACM, 657--666. Google ScholarDigital Library
- V. Gutiérrez-Basulto, Y. Ibáñez-García, R. Kontchakov, and E. V. Kostylev. 2015. Queries with negation and inequalities over lightweight ontologies. J. Web Semantics 35 (2015), 184--202. Google ScholarDigital Library
- P. Hansen, C. Lutz, I. Seylan, and F. Wolter. 2015. Efficient query rewriting in the description logic EL and beyond. In Proc. of the 24th Int. Joint Conf. on Artificial Intelligence (IJCAI’15). AAAI, 3034--3040. Google ScholarDigital Library
- D. A. Huffman. 1952. A method for the construction of minimum-redundancy codes. Proc. Inst. Radio Eng. 40, 9 (1952), 1098--1101.Google ScholarCross Ref
- N. Immerman. 1988. Nondeterministic space is closed under complementation. SIAM J. Comput. 17, 5 (1988), 935--938. Google ScholarDigital Library
- D. S. Johnson. 1990. A catalog of complexity classes. In Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity (A). 67--161. Google ScholarDigital Library
- D. S. Johnson and A. C. Klug. 1982. Testing containment of conjunctive queries under functional and inclusion dependencies. In Proc. of the ACM Symp. on Principles of Database Systems (PODS’82). ACM, 164--169. Google ScholarDigital Library
- S. Jukna. 2012. Boolean Function Complexity — Advances and Frontiers. Algorithms and Combinatorics, Vol. 27. Springer. Google ScholarDigital Library
- M. Kaminski, Y. Nenov, and B. Cuenca Grau. 2014. Datalog rewritability of disjunctive datalog programs and its applications to ontology reasoning. In Proc. of the 28th AAAI Conference on Artificial Intelligence (AAAI’14). AAAI, 1077--1083. Google ScholarDigital Library
- M. Karchmer and A. Wigderson. 1988. Monotone circuits for connectivity require super-logarithmic depth. In Proc. of the 20th Annual ACM Symp. on Theory of Computing (STOC’88). ACM, 539--550. Google ScholarDigital Library
- E. Kharlamov, D. Bilidas, D. Hovland, E. Jiménez-Ruiz, D. Lanti, H. Lie, M. Rezk, M. Skjæveland, A. Soylu, G. Xiao, D. Zheleznyakov, M. Giese, Y. Ioannidis, Y. Kotidis, M. Koubarakis, and A. Waaler. 2017. Ontology based data access in Statoil. J. Web Semantics 44 (2017), 3--36. Google ScholarDigital Library
- E. Kharlamov, T. Mailis, G. Mehdi, C. Neuenstadt, Ö. Özçep, M. Roshchin, N. Solomakhina, A. Soylu, C. Svingos, S. Brandt, M. Giese, Y. Ioannidis, S. Lamparter, R. Möller, Y. Kotidis, and A. Waaler. 2017. Semantic access to streaming and static data at Siemens. J. Web Semantics 44 (2017), 54--74. Google ScholarDigital Library
- S. Kikot, R. Kontchakov, V. V. Podolskii, and M. Zakharyaschev. 2012. Exponential lower bounds and separation for query rewriting. In Proc. of the 39th Int. Coll. on Automata, Languages 8 Programming (ICALP’12) (LNCS), Vol. 7392. Springer, 263--274. Google ScholarDigital Library
- S. Kikot, R. Kontchakov, V. V. Podolskii, and M. Zakharyaschev. 2014. On the succinctness of query rewriting over shallow ontologies. In Proc. of the Joint Meeting of the 23rd EACSL Annual Conf. on Computer Science Logic (CSL) and the 29th Annual ACM/IEEE Symp. on Logic in Computer Science (LICS) (CSL-LICS’14). ACM, 57:1--57:10. Google ScholarDigital Library
- S. Kikot, R. Kontchakov, and M. Zakharyaschev. 2011. On (in)tractability of OBDA with OWL 2 QL. In Proc. of the 24th Int. Workshop on Description Logics (DL’11), Vol. 745. CEUR-WS, 224--234.Google Scholar
- S. Kikot, R. Kontchakov, and M. Zakharyaschev. 2012. Conjunctive query answering with OWL 2 QL. In Proc. of the 13th Int. Conf. on Principles of Knowledge Representation 8 Reasoning (KR’12). AAAI, 275--285. Google ScholarDigital Library
- M. König, M. Leclère, and M.-L. Mugnier. 2015. Query rewriting for existential rules with compiled preorder. In Proc. of the 24th Int. Joint Conf. on Artificial Intelligence (IJCAI’15). AAAI, 3106--3112. Google ScholarDigital Library
- M. König, M. Leclère, M.-L. Mugnier, and M. Thomazo. 2015. Sound, complete and minimal UCQ-rewriting for existential rules. Semantic Web 6, 5 (2015), 451--475.Google ScholarCross Ref
- R. Kontchakov, C. Lutz, D. Toman, F. Wolter, and M. Zakharyaschev. 2010. The combined approach to query answering in DL-Lite. In Proc. of the 12th Int. Conf. on Principles of Knowledge Representation 8 Reasoning (KR’10). AAAI, 247--257. Google ScholarDigital Library
- R. Kontchakov, M. Rezk, M. Rodriguez-Muro, G. Xiao, and M. Zakharyaschev. 2014. Answering SPARQL queries over databases under OWL 2 QL entailment regime. In Proc. of the 13th Int. Semantic Web Conf. (ISWC’14), Part I (LNCS), Vol. 8796. Springer, 552--567. Google ScholarDigital Library
- E. V. Kostylev, J. L. Reutter, and D. Vrgo. 2015. XPath for DL ontologies. In Proc. of the 29th AAAI Conf. on Artificial Intelligence (AAAI’15). AAAI, 1525--1531. Google ScholarDigital Library
- D. Lembo, J. Mora, R. Rosati, D. F. Savo, and E. Thorstensen. 2015. Mapping analysis in ontology-based data access: Algorithms and complexity. In Proc. of the 14th Int. Semantic Web Conf. (ISWC’15) (LNCS), Vol. 9366. Springer, 217--234. Google ScholarDigital Library
- L. Libkin. 2004. Elements of Finite Model Theory. Springer. Google ScholarDigital Library
- C. Lutz. 2008. The complexity of conjunctive query answering in expressive description logics. In Proc. of the 4th Int. Joint Conf. on Automated Reasoning (IJCAR’08) (LNAI). Springer, 179--193. Google ScholarDigital Library
- C. Lutz, R. Piro, and F. Wolter. 2011. Description logic TBoxes: Model-theoretic characterizations and rewritability. In Proc. of the 22nd Int. Joint Conf. on Artificial Intelligence (IJCAI’11). IJCAI/AAAI, 983--988. Google ScholarDigital Library
- C. Lutz, I. Seylan, D. Toman, and F. Wolter. 2013. The combined approach to OBDA: Taming role hierarchies using filters. In Proc. of the 12th Int. Semantic Web Conf. (ISWC’13), Part I (LNCS), Vol. 8218. Springer, 314--330. Google ScholarDigital Library
- C. Lutz, D. Toman, and F. Wolter. 2009. Conjunctive query answering in the description logic EL using a relational database system. In Proc. of the 21st Int. Joint Conf. on Artificial Intelligence (IJCAI’09). 2070--2075. Google ScholarDigital Library
- J. Mora, R. Rosati, and Ó. Corcho. 2014. Kyrie2: Query rewriting under extensional constraints in ELHIO. In Proc. of the 13th Int. Semantic Web Conf. (ISWC’14) (LNCS), Vol. 8796. Springer, 568--583. Google ScholarDigital Library
- H. Pérez-Urbina, B. Motik, and I. Horrocks. 2009. A comparison of query rewriting techniques for DL-lite. In Proc. of the 22nd Int. Workshop on Description Logics, DL 2009 (CEUR), Vol. 477. CEUR-WS.Google Scholar
- H. Pérez-Urbina, E. Rodríguez-Díaz, M. Grove, G. Konstantinidis, and E. Sirin. 2012. Evaluation of query rewriting approaches for OWL 2. In Proc. of the Joint Workshop on Scalable and High-Performance Semantic Web Systems (SSWS+HPCSW 2012) (CEUR), Vol. 943. CEUR-WS.Google Scholar
- A. Poggi, D. Lembo, D. Calvanese, G. De Giacomo, M. Lenzerini, and R. Rosati. 2008. Linking data to ontologies. J. Data Semantics 10 (2008), 133--173. Google ScholarDigital Library
- R. Raz and A. Wigderson. 1992. Monotone circuits for matching require linear depth. J. ACM 39, 3 (1992), 736--744. Google ScholarDigital Library
- A. Razborov. 1985. Lower bounds for the monotone complexity of some boolean functions. Dokl. Akad. Nauk SSSR 281, 4 (1985), 798--801.Google Scholar
- A. A. Razborov. 1991. Lower bounds for deterministic and nondeterministic branching programs. In Proc. of the 8th Int. Symp. on Fundamentals of Computation Theory (FCT’91) (LNCS), Vol. 529. Springer, 47--60. Google ScholarDigital Library
- M. Rodriguez-Muro and D. Calvanese. 2012. High performance query answering over DL-Lite ontologies. In Proc. of the 13th Int. Conf. on Principles of Knowledge Representation 8 Reasoning (KR’12). AAAI, 308--318. Google ScholarDigital Library
- M. Rodriguez-Muro, R. Kontchakov, and M. Zakharyaschev. 2013. Ontology-based data access: Ontop of databases. In Proc. of the 12th Int. Semantic Web Conf. (ISWC’13), Part I (LNCS), Vol. 8218. Springer, 558--573. Google ScholarDigital Library
- R. Rosati. 2007. The limits of querying ontologies. In Proc. of the 11th Int. Conf. on Database Theory (ICDT’07) (LNCS), Vol. 4353. Springer, 164--178. Google ScholarDigital Library
- R. Rosati. 2012. Prexto: Query rewriting under extensional constraints in DL-Lite. In Proc. of the 9th Extended Semantic Web Conf. (EWSC’12) (LNCS), Vol. 7295. Springer, 360--374. Google ScholarDigital Library
- R. Rosati and A. Almatelli. 2010. Improving query answering over DL-Lite ontologies. In Proc. of the 12th Int. Conf. on Principles of Knowledge Representation 8 Reasoning (KR’10). AAAI, 290--300. Google ScholarDigital Library
- J. F. Sequeda, M. Arenas, and D. P. Miranker. 2014. OBDA: Query rewriting or materialization? In practice, both! In Proc. of the 13th Int. Semantic Web Conf. (ISWC’14), Part I (LNCS), Vol. 8796. Springer, 535--551. Google ScholarDigital Library
- I. H. Sudborough. 1978. On the tape complexity of deterministic context-free languages. J. ACM 25, 3 (1978), 405--414. Google ScholarDigital Library
- R. Szelepcsényi. 1988. The method of forced enumeration for nondeterministic automata. Acta Inform. 26, 3 (1988), 279--284. Google ScholarDigital Library
- M. Thomazo. 2013. Compact rewritings for existential rules. In Proc. of the 23rd Int. Joint Conf. on Artificial Intelligence (IJCAI’13). IJCAI/AAAI, 1125--1131. Google ScholarDigital Library
- M. Vardi. 1982. The complexity of relational query languages (extended abstract). In Proc. of the 14th ACM SIGACT Symp. on Theory of Computing (STOC’82). ACM, 137--146. Google ScholarDigital Library
- H. Venkateswaran. 1991. Properties that characterize LOGCFL. J. Comput. System Sci. 43, 2 (1991), 380--404. Google ScholarDigital Library
- H. Vollmer. 1999. Introduction to Circuit Complexity: A Uniform Approach. Springer. Google ScholarDigital Library
- M. Yannakakis. 1981. Algorithms for acyclic database schemes. In Proc. of the 7th Int. Conf. on Very Large Data Bases (VLDB’81). IEEE Computer Society, 82--94. Google ScholarDigital Library
- Y. Zhou, B. Cuenca Grau, Y. Nenov, M. Kaminski, and I. Horrocks. 2015. PAGOdA: Pay-as-you-go ontology query answering using a Datalog reasoner. J. Artif. Intell. Res. (JAIR) 54 (2015), 309--367. Google ScholarDigital Library
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- Ontology-Mediated Queries: Combined Complexity and Succinctness of Rewritings via Circuit Complexity
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