EQL-Lite: Effective First-Order Query Processing in Description Logics(IJCAI07)

1. EQL-Lite: Effective First-Order Query Processing in Description Logics(IJCAI07) Presented by : Elizabeth Brennan Vidhya Dass

2. Agenda • Problem Statement • Motivation • Introduction • Proposed Approach • Epistemic Query Language(EQL) • EQL-Lite(Q) • Case Studies • Constraints • Critique • Future Research

3. Problem Statement • Knowledge bases are incomplete • There is a need to be able to reason about incompleteness and to have a language that’s expressiveness is similar to FOL and decidable. • These two desires are at odds with each other.

4. Motivation • Description Logic (DL) try model the real world • Answering FOL questions posed to a DL is undecidable

5. Motivation • Currently, DLs can be queried for • Instances • Unions of conjunctive queries • Since a DL can never be complete, FOL logic is not appropriate

6. Introduction • EQL is based on a variant of a FOL modal logic of knowledge/belief • The minimal knowledge operator, K, is used • K means that something is known • This allows users to engage in closed world reasoning

7. Introduction • EQL-Lite (Q) implements the operator, K • Q refers to an embedded query language

8. Introduction • A DL knowledge base (KB) is formed by a set of assertions • TBox- express intentional knowledge • “Candy is yummy” • ABox- express extensional knowledge • “Reese’s cups are candy”

9. Proposed Approach • Assumptions: • Express knowledge in KB in terms of atomic concepts and atomic roles/relations through the DL. • Assume that constructs allowed in DL KB are expressible in FOL. • Assume DL KB assertions can be expressed as FOL sentences.

10. Proposed Approach Semantics Syntax • Query Answering: • Interpret DL KB on interpretations sharing same infinite countable domain . • Assume our language includes infinitely countable set of disjoint constants corresponding to elements of , called standard names • Blurs distinction between such constants( syntactic objects) and elements of  that they denote(semantical objects)

11. Epistemic Query Language(EQL) • EQL is first order modal language with equality and single modal operator K, constructed from concepts, roles/relations and the standard names. • Reason about DL KB according to minimal knowledge semantics(K)

12. Hint:  Follows logically from  Y. Katz and B. Parsia, "Minimal Knowledge and Negation as Failure in Description Logics: Part I." vol. 2007: MINDSWAP, University of Maryland, 2005

13. EQL in • K  is known to hold(by the KB) • c - Constant ĉ - Tuple of constants • x - Variable x - Tuple of variables • , - Arbitrary formulas • cx - Formula where each x is replaced by c • World FOL interpretation over  • Epistemic Interpretation E, i.e. (Possibly infinite) set of worlds, world in E • i.e.  is true in interpretation E,

14. EQL E 1 n False 2 K n true

15. EQL • Objective sentence (What is true) • Subjective sentence (What is known to be true) • Inductively define when sentence  is true in interpretation E,

16. EQL • Let  = DL KB • Mod() = set of all FOL interpretations of models of  • -EQL-interpretation = Epistemic interpretation E, for E=Mod() • if • Objective Formulas : for all denoted by

17. EQL • S5 axioms of modal logic : Proposition 1 What is known is true(accurate) } KB has complete knowledge on what is known and what is not known

18. EQL • Proposition 2 • i.e. Sentence is known or not known in KB

19. EQL • Proposition 3 : For objective sentence  }  is known if it is logically implied

20. EQL • EQL Queries -Open EQL Formula • Certain answers to query

21. EQL • DL KB  constituted by T-Box & A-Box FOL Query : Set of males that do not have female children EQL Query : Known males that are not known to be parents of female EQL Query : Known children who have no known sibling

22. EQL-Lite(Q) • Subset of EQL • Parameterized, embedded query language

23. EQL-Lite(Q) • Uses the syntax: • is a query expressed in Q

24. EQL-Lite(Q) • Each occurring in a query is called an epistemic atom • Allowing atoms outside of K does not impact the expressiveness of the language

25. EQL-LITQ(Q) • This formalizes the idea that restricting atoms to be expressed as beliefs does not impact the expressiveness

26. EQL-Lite(Q) • EQL is able to separate the reasoning needed for answering epistemic atoms from the reasoning needed to answer the whole query

27. EQL-Lite(Q) • This says that certain answers are computed in the embedded query language and then the query is evaluated in FOL

28. Problems Posed by Thm 6 • Problems posed by Theorem 6 • The query posed must be finite to be evaluated • The domain, Δ, is infinite. This means the query cannot directly interact with the domain.

29. EQL-Lite(Q) • A FOL query, q, is domain independent if for Ι1 and I2 whose respective domains are a subset of Δ, all atomic relations will have the same evaluations

30. EQL-Lite(Q) • If a query evaluates to a finite set of tuples, it is said to be ∑-range-restricted • A query is ∑-range-restricted if each of its epistemic atoms involves a ∑-range-restricted query

31. EQL-Lite(Q) • This says that if a constant that is not in ∑ occurs in the solution set, then such a constant can be replaced with any other constant not in ∑ and still get a tuple in the solution set • This would mean the solution set is infinite

32. EQL-Lite(Q) • We want to make sure that our solution set is range-restricted. • A range-restricted and domain independent DL will guarantee that we can compute certain answers

33. EQL-Lite(Q)

34. EQL-Lite(Q) • This says that ∑-range-restricted queries can be solved in logarithmic time.

35. Case Studies • The DL was considered • Q is a ∑-range-restricted concept and role expressions • Conjunctive queries with the existential quantification are performed over named items in ABox • Instance checking is coNP-complete • Queries can be performed in logarithmic amount of time

36. Case Studies • Unions of conjunctive queries are performed • Most expressive subset of FOL for querying KBs • This problem is coNP-complete • Queries can be answered in logarithmic time

37. Case Studies • A PTime complete DL and a ∑-range-restricted Q with roles and concepts is considered • Queries were solved in polynomial time

38. Case Studies • A KB formed of ABox expressions with the embedded query language consisting of roles and concepts and extended with the operator K is considered • Instance checking is performed in polynomial time

39. Case Studies • DL-Lite KB and queries with embedded UCQ • DL-Lite KBs deal with large ABox(data) • Query in DL-Lite tractable due to controlled expressive power of DL-Lite • Answering UCQ(union conjunctive query) in DL-Lite is in LOGSPACE with respect to data complexity

40. Case Studies • Due to LOGSPACE data complexity of answering UCQ in DL-Lite and tightly controlled expressiveness

41. Case Studies • Due to Theorem 8 and data complexity being in LOGSPACE - Can use EQL(UCQ) instead of UCQ in DL-Lite

42. Case Studies • FOL Reducibility : • Query answering FOL reducible  FOL queries over Abox FOL interpretation (infinite to finite FOL interpretation) • Interpretation of ABox (Check ABox)

43. Case Studies • Domain independent queries can be expressed in relational algebra(SQL). Use DBMS optimization

44. Constraints • A Boolean EQL-Lite(Q) query can be viewed as an integrity constraint • A DL KB ∑=(T,A) where T is a TBox and A is an ABox satisfies an integrity constraint γ expressed as a Boolean EQL-Lite(Q) query if ans(γ,∑)= true • ∑ satisfies a set of integrity constraints, C, if for all γC, ans(γ,∑)=true

45. Constraints • The definition of a DL KB can be modified to include the idea of constraints • (T,A,C) • This can be interpreted as • If Mod(∑) is non-empty, the DL KB with constraints is satisfiable • The ideas discussed in this paper can easily be extended to DL KBs with constraints

46. Critique • Strengths : • Great approach to balance expressiveness and complexity of queries in DL KB that models open world. • Good propositions and theorems.

47. Critique • Weakness : • Introduction was lacking. • Repetitive sentences. • Flow was choppy in paper. • Lack of concept elaboration. • Use of non standard symbols (PTIME,LOGSPACE, eval, ans, )

48. Future Research • Extending the concept to DL-Lite KBs. • Implementing this within an existing DL reasoning system. • Possibly try to make it work on DL’s with second order constructs.

49. Thank you