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

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

3 Semantic Web Architectures Homogeneous approach Hybrid approach

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

5 The DL Family Bottom-Up ALC: C and D are classes, R is a property S = ALC R+ : 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] ALC

6 Hybrid Knowledge Base Hybrid KB: K = ( ,  )   : A DL KB   : A Datalog program with DL-queries to  Hybrid Rules  h(X) :- b 1 (Y 1 )  …  b m (Y m ) & q 1 (Z 1 )  …  q n (Z n )  h(X), b i (Y i ) Datalog atoms (1≤i≤m); X, Y i sequences of constants|variables  q j (Z j ) DL-queries (1≤j≤n); Z j sequence 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 Y i |Z j '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|Z j must occur in one of the Y i 's

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

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) restrictions and R means role conjunctions [10]  dl-programs [6]: SHIF(D) | SHOIN(D) DL + Answer Set Programming  r-hybrid KBs [7]: A decidable DL + Datalog 

9 Comparison Notes: 1.DLP: Expressivity restrictions 2.SWRL: Undecidable 3.KAON2: DL-safe rules Notes: 1.AL-log: Only concept constraints 2.CARIN: Recursive CARIN-ALCNR undecidable 3.dl-programs: Nonmonotonic semantics 4.r-hybrid KBs: Nonmonotonic semantics SLD-resolution Entailment Fixpointiteration – SLD-resolution X X X X X X X X X X AL-log CARIN dl-programs r-hybridKBs Datalog DL Hybrid Approaches Reduction – X X X X X X DLP SWRL KAON2 Homogeneous Approaches Reasoning Strategy Information Flow betweenDatalog& DL: Uni-direct. Bi-direct. Safeness Condition: Strong Weak SLD-resolution Entailment Fixpointiteration – SLD-resolution X X X X X X X X AL-log CARIN dl-programs r-hybridKBs Datalog DL Hybrid Approaches Reduction – X X X X X DLP SWRL KAON2 Homogeneous Approaches Reasoning Strategy Information Flow betweenDatalog& DL: Uni-direct. Bi-direct. Safeness Condition: Strong Weak

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

11 A Hybrid Approach: Datalog DL Datalog DL : 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

12 Syntax An alphabet of predicates A = A T  A P with A T  A P =  A Datalog L KB: K = ( ,  )   : An L-based DL KB with predicates in A T where L ranges from ALC to SHIQ   : A Datalog program with DL-queries to , s.t. each hybrid rule r is [r] h(X) :- b 1 (Y 1 )  …  b m (Y m ) & q 1 (Z 1 )  …  q n (Z n ) where X, Y 1,..., Y m are n-ary sequences of terms (constants|variables) Z1,..., Zn are unary/binary sequences of terms h(X), b i (Y i ) (1≤i≤m) are Datalog atoms with predicates in A P Each q j (Z j ) (1≤j≤n) is a DL-query with predicate in A T Notes: 1.“DL body” means: “q 1 (Z 1 )  …  q n (Z n )” 2.“Datalog body” means: “b 1 (Y 1 )  …  b m (Y m )” 3.“Datalog rule” means: hybrid rule after deletion of “& DL body”

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  Datalog DL : By means of constrained SLD-resolution, provided by hybrid rules with independent properties

14 Features of Datalog DL Pure-DL Variables  A pure-DL variable in a rule r is a variable that only occurs in one of the Z j '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

15 Features of Datalog DL (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

16 Two Other Transformations Making weakly safe rules strongly safe  Referring to DL-safe rules in KAON2 [3]  A special predicate symbol O  A P  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 P a of the individual a

17 Semantics A first-order interpretation I = ( △,  I ) of Datalog L  △ : The non-empty domain of I   I : The interpretation function of I A model of the Datalog L KB K=( ,  )  The interpretation I is a model of   The interpretation I satisfies each hybrid rule r in , i.e. [r] h(X) :- b 1 (Y 1 )  …  b m (Y m ) & q 1 (Z 1 )  …  q n (Z n ) s.t. If T r (Y i )  b i I and T r (Z j )  q j I (1≤i≤m, 1≤j≤n) for every atom in the body of r, then T r (X)  h I for the head of r, where T r is a term assignment w.r.t I for constants and variables in r. Notes: 1.The interpretation of constants is according to the standard names assumption and to the unique name assumption 2.Without negation-as-failure, this first-order semantics gives a platform for DL-and-Datalog combination, both of which are first-order fragments

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

19 Example of AL-log 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),

20 Example of CARIN 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) &

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

22 [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, [2] Ian Horrocks, Peter F. Patel-Schneider, Harold Boley, Said Tabet, Benjamin Grosof, and Mike Dean. Semantic Web Rule Language (SWRL). W3C Member Submission. May [3] Boris Motik, Ulrike Sattler, and Rudi Studer. Query Answering for OWL-DL with Rules. Journal of Web Semantics, 3(1):41–60, [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, [5] Alon Y. Levy and Marie-Christine Rousset. CARIN: A Representation Language Combining Horn Rules and Description Logics. In ECAI-96, pages 323–327, [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, [7] Riccardo Rosati. On the decidability and complexity of integrating ontologies and rules. Journal of Web Semantics, 3(1):61–73, [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, [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 [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, Selected References

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