Datalog DL : Datalog Rules Parameterized by Description Logics

1. Canadian Semantic Web Working Symposium June 6, 2006 Laval University, Quebec City, Canada DatalogDL: Datalog Rules Parameterized by Description Logics Jing Mei, Harold Boley, Jie Li,Virendrakumar C. Bhavsar, Zuoquan Lin

2. Contents • Semantic Web Architectures • Context of DatalogDL • Description Logic (DL) Family • Hybrid Knowledge Bases • Strategies for Reasoning Services • Integration Frameworks • Comparison • Proposal of DatalogDL • Syntax • Semantics • Reasoning • Examples • Selected References 2

3. Semantic WebArchitectures Homogeneous approach Hybrid approach 3

4. Content • Semantic Web Architectures • Context of DatalogDL • Description Logic (DL) Family • Hybrid Knowledge Bases • Strategies for Reasoning Services • Integration Frameworks • Comparison • Proposal of DatalogDL • Syntax • Semantics • Reasoning • Examples • Selected References 4

5. ALC The DL Family Bottom-Up • ALC: C and D are classes, R is a property • S = ALCR+: Transitive properties • SI: Inverse properties • SHI: Property hierarchies • SHIF: Functional restrictions • SHIN: Cardinality (Number) restrictions • SHIQ: Qualified number restrictions • Support for datatype predicates (e.g. string, integer) leads to the concrete domain of D • Using nominals O allows to construct classes from singleton sets, with the so-called one-of operator • OWL Lite = SHIQ(D) • OWL DL = SHOIN(D) [10] 5

6. Hybrid Knowledge Base • Hybrid KB: K = (, ) • : A DL KB • : A Datalog program with DL-queries to  • Hybrid Rules • h(X):-b1(Y1) … bm(Ym) & q1(Z1) … qn(Zn) • h(X), bi(Yi)Datalog atoms (1≤i≤m); X, Yisequences of constants|variables • qj (Zj) DL-queries (1≤j≤n); Zjsequence of constants|variables • Safeness Condition • Weak safeness condition • Variables appearing in the head of a rule must also appear in the body, but not necessarily in the DL body • That is, a variable occurring in X must occur in one of the Yi|Zj's • Strong safeness condition • Each variable appearing in the DL component also appears in the Datalog component, in addition to weak safeness • That is, a variable occurring in X|Zj must occur in one of the Yi's 6

7. Strategies for Reasoning Services • Beyond classical DL tableaux calculus • Based on reduction • Reducing a DL KB to (disjunctive, function-free, negation-free) Datalog rules • Rule engines support for DL reasoning • Based on components • SLD-resolution for rules • Backward chaining, Top-Down • Collecting DL-queries, which are finally evaluated for DL satisfiability • Entailment for DL • Forward chaining, Bottom-Up • Building DL tableaux, whose inferred assertions are fed into rules • Fixpoint Iteration for both DL and rules • Modular reasoning method with separation of reasoning for components • Running DL reasoners and rule engines at the same time • Exchanging information until a fixpoint is reached 7

8. Integration Frameworks • Homogeneous approaches • DLP [1]: Description Logic Programs • SWRL [2]: Semantic Web Rule Language • KAON2 [3]: OWL extended with DL-safe rules • Hybrid approaches • AL-log [4]: ALC DL + Datalog • CARIN [5]: ALCNR DL + Datalog where N means cardinality (number) restrictionsand R means role conjunctions [10] • dl-programs [6]: SHIF(D) | SHOIN(D) DL + Answer Set Programming • r-hybrid KBs [7]: A decidable DL + Datalog 8

9. Comparison Safeness Condition: Safeness Condition: Information Flow Information Flow Reasoning Reasoning Strong Weak Strong Weak between between Datalog Datalog & DL: & DL: Strategy Strategy Uni Uni - - direct. Bi direct. Bi - - direct. direct. Homogeneous Homogeneous DLP DLP X X X X Reduction Reduction Approaches Approaches SWRL SWRL X X X X – – KAON2 KAON2 X X X Reduction Reduction Hybrid Hybrid AL AL - - log log X X X SLD SLD - - resolution resolution Approaches Approaches CARIN CARIN X X X X Entailment Entailment dl dl - - programs programs X X X X Fixpoint Fixpoint iteration iteration r r - - hybrid hybrid KBs KBs X X X X – – Datalog Datalog X X X SLD SLD - - resolution resolution DL DL • Notes: • AL-log: Only concept constraints • CARIN: Recursive CARIN-ALCNR undecidable • dl-programs: Nonmonotonic semantics • r-hybrid KBs: Nonmonotonic semantics • Notes: • DLP: Expressivity restrictions • SWRL: Undecidable • KAON2: DL-safe rules 9

10. Content • Semantic Web Architectures • Context of DatalogDL • Description Logic (DL) Family • Hybrid Knowledge Bases • Strategies for Reasoning Services • Integration Frameworks • Comparison • Proposal of DatalogDL • Syntax • Semantics • Reasoning • Examples • Selected References 10

11. A Hybrid Approach: DatalogDL • DatalogDL:Combining (sublanguage of) SHIQ DL and Datalog rules • The rule component: (Disjunctive, Function-free, Negation-free) Datalog with terms consisting of variables and constants • The DL Component: Any specific decidable DL language ranging from ALC to SHIQ • Safeness: Weak safeness condition • Requirement: Independent properties • Reasoning Strategy • SLD-resolution for rules: Extending a rule engine (OO jDREW) to incorporate hybrid rules • Tableaux algorithm for DL queries: Using an external DL reasoner (Racer) to check ALC to SHIQ satisfiability 11

12. Syntax • An alphabet of predicates A = AT AP with AT AP =  • ADatalogLKB: K = (, ) • : An L-based DL KB with predicates in AT where L ranges from ALC to SHIQ •  : A Datalog program with DL-queries to , s.t. each hybrid rule r is [r]h(X):-b1(Y1) … bm(Ym) & q1(Z1) … qn(Zn) where • X, Y1, ..., Ym are n-ary sequences of terms (constants|variables) • Z1, ..., Zn are unary/binary sequences of terms • h(X),bi(Yi) (1≤i≤m) are Datalog atoms withpredicates in AP • Each qj(Zj) (1≤j≤n) is aDL-query with predicate in AT • Notes: • “DL body” means: “q1(Z1) … qn(Zn)” • “Datalog body” means: “b1(Y1) … bm(Ym)” • “Datalog rule” means: hybrid rule after deletion of “& DL body” 12

13. Decidability Issues • It has been pointed out in CARIN • Recursive Datalog rules + cyclic TBox with only DL constructor P.C • Reducing the halting problem of a Turing machine (known to be undecidable) to the entailment problem of a KB containing • DL ABox: integer(1) • DL TBox: integer  succ.integer • rule-primitive: lessThan(x, y) :- & succ(x, y). • rule-recursive: lessThan(x, y) :- lessThan(z, y) & succ(x, z). • Remark: Strong safeness condition would demand that “x” occur in “lessThan(z, y)” in the above KB example • Re-obtaining decidability • AL-log: Disallowing DL property queries like “succ(x, y)“ • CARIN: A (maximal) decidable sublanguage namely CARIN-MARC • DLP: Disallowing the existential DL constructor P.C to occur on the right hand side of “” subsumptions • DatalogDL: By means of constrained SLD-resolution, provided by hybrid rules with independent properties 13

14. Features of DatalogDL • Pure-DL Variables • A pure-DL variable in a rule r is a variable that only occurs in one of the Zj's • Pure-DL variables lead to the violation of the strong safeness condition in cases where the weak safeness condition is obeyed • According to the classical SLD-resolution with rules, non-pure-DL variables will be bound to ground values, still leaving pure-DL variables free • Folding • Classical DL algorithms: Reducing DL queries to KB unsatisfiability, e.g. by transforming the query into a negated assertion, but the negation of properties is not supported by most DLs • DL-query of C(x) is reduced to checking whether C is non-empty, where x is a pure-DL variable • DL-query of P(u, x)∧C(x) becomes folding result P.C(u), where x is a pure-DL variable • DL-query of P(x, u)∧C(x) becomes folding result P-.C(u), where x is a pure-DL variable and P- is the inverse of P 14

15. Features of DatalogDL (cont’d) • Independent Properties • Folding cannot be applied to query parts that contain cycles (e.g. P(x, y)∧Q(y, z) ∧R(z, x)), or where more than one property arc enters a node that corresponds to a variable (e.g. P(u, x)∧Q(y, x)) • Tree-shaped DL queries: Adding rules to DLs, in a unrestricted manner, causes the loss of any form of tree model property • A property P is independent in a rule r, if no P occurrence shares any pure-DL variables with other property occurrences (including other P occurrences) • Correspondence: For hybrid rules with independent properties, the folding results are equivalent to the original DL-queries 15

16. Two Other Transformations • Making weakly safe rules strongly safe • Referring to DL-safe rules in KAON2 [3] • A special predicate symbol OAP • For each pure variable w in a rule r, add an atom O(w) to the Datalog body of r • For each constant c occurring in K = (, ), add a fact O(c) to  • Rolling-up to eliminate DL property queries • Referring to a conjunctive query language for DL ABox [8] • Similar to folding in our setting • Exploiting the DL tree model feature for queries containing cycles • Simulating the one-of operator by substituting each individual a with a representative concept Pa of the individual a 16

17. Semantics • A first-order interpretation I = (△, I) of DatalogL • △: The non-empty domain of I • I: The interpretation function of I • A model of the DatalogL KB K=(, ) • The interpretation I is a model of  • The interpretation I satisfies each hybrid rule r in , i.e. [r] h(X):-b1(Y1) … bm(Ym) & q1(Z1) … qn(Zn) s.t. • If Tr(Yi)  biI and Tr(Zj)  qjI (1≤i≤m, 1≤j≤n) for every atom in the body of r, then Tr(X)  hI for the head of r, where Tr is a term assignment w.r.t I for constants and variables in r. • Notes: • The interpretation of constants is according to the standard names assumption and to the unique name assumption • Without negation-as-failure, this first-order semantics gives a platform for DL-and-Datalog combination, both of which are first-order fragments 17

18. Reasoning • A kind of constrained SLD-resolution • Grounding variables in hybrid rules, but pure-DL variables still left free • Folding (independent) properties, to eliminate pure-DL variables • DL satisfiability • DL queries without variables • Building a disjunctive DL class for the collection of DL queries from hybrid rules sharing the same head 18

19. Referring to AL-log [4], a query to mayDoThesis(paul, mary): The final ground queries after constrained SLD-resolution without folding  expert(mary, lp), exam(paul, ai), subject(ai, lp) & St(paul), Tp(lp), AC(ai), Example of AL-log 19

20. Referring CARIN [5], a query to price(a, usa high): The final ground queries after constrained SLD-resolution plus folding  made-by(a, b), monopoly(b, a, usa) & Example of CARIN 20

21. Referring to RuleML FOAF [9], a query to possiblyKnows(Laura, Ben): The final ground queries after constrained SLD-resolution plus folding  & Use Case of RuleML FOAF 21

22. Selected References [1] Benjamin N. Grosof, Ian Horrocks, Raphael Volz, and Stefan Decker.Description Logic Programs: Combining Logic Programs with Description Logic.In WWW 2003, pages 48–57, 2003. [2] Ian Horrocks, Peter F. Patel-Schneider, Harold Boley, Said Tabet, Benjamin Grosof, and Mike Dean.Semantic Web Rule Language (SWRL). W3C Member Submission.http://www.w3.org/Submission/2004/SUBM-SWRL-20040521/, May 2004. [3] Boris Motik, Ulrike Sattler, and Rudi Studer.Query Answering for OWL-DL with Rules. Journal of Web Semantics, 3(1):41–60, 2005. [4] Francesco M. Donini, Maurizio Lenzerini, Daniele Nardi, and Andrea Schaerf.AL-log: Integrating Datalog and Description Logics.Journal of Intelligent Information Systems (JIIS), 10(3):227–252, 1998. [5] Alon Y. Levy and Marie-Christine Rousset.CARIN: A Representation Language Combining Horn Rules and Description Logics.In ECAI-96, pages 323–327, 1996. [6] Thomas Eiter, Thomas Lukasiewicz, Roman Schindlauer, and Hans Tompits.Combining Answer Set Programming with Description Logics for the Semantic Web.In KR 2004, pages 141–151, 2004. [7] Riccardo Rosati.On the decidability and complexity of integrating ontologies and rules.Journal of Web Semantics, 3(1):61–73, 2005. [8] Ian Horrocks and Sergio Tessaris.Querying the Semantic Web: a Formal Approach.In Workshop on Principles and Practice of Semantic Web Reasoning, pages 177—191, 2005. [9] Jie Li, Harold Boley, Virendrakumar C. Bhavsar, and Jing Mei. Expert Finding for eCollaboration Using FOAF with RuleML Rules. In: The Montreal Conference on eTechnologies, May 2006. [10] Franz Baader, Diego Calvanese, Deborah McGuinness, Daniele Nardi, and Peter F. Patel-Schneider. The Description Logic Handbook: Theory, Implementation and Applications. Cambridge University Press, 2003. 22

23. Thanks Questions? 23