Knowledge Repn. & Reasoning Lec #11+13: Frame Systems and Description Logics

1. Knowledge Repn. & ReasoningLec #11+13: Frame Systems and Description Logics UIUC CS 498: Section EA Professor: Eyal AmirFall Semester 2004

2. Today • Restricting expressivity of FOL: DLs • Description Logics (DLs) • Language • Semantics • Inference

3. Description Logics (DLs) • Originate in semantic networks (NLP), and Frame Systems (KR) • Hold information about concepts, objects, and simple relationships between them • Hierarchical information • Many DLs, differing in their expressive power

4. Person Man Woman Frame Systems Concept frames Object frames Jane

5. age age 26 Person child child Jill,John Man Woman Frame Systems Roles Object frames Jane

6. Differences from DBs • Hierarchical structure (?) • Many times no closed-world assumption • Values may be missing • More expressive (?) • Semantic structure between concepts and roles • Typical reasoning tasks (satisfiability, generality/classification)

7. Description Logics: Language • Formal language that can be analyzed • Describe frame systems with attention to the expressive power needed • Definitions are stated in a terminological part of the KB (TBox) • Assertions are made at an assertional part of the KB (Abox)

8. Description Logics: Language . • Definitions are stated in a terminological part of the KB (TBox) • Assertions are made at an assertional part of the KB (Abox) TBox Description Language Reasoning ABox

9. Description Logics: Language . • Example definition: C = AпB • Example assertion: C(John), CпD = AпB TBox Description Language Reasoning ABox

10. AL Description Logic: Language • AL: C,D  A | (atomic concept) T | (universal concept)  | (bottom concept) A | (atomic negation) CпD | (intersection) R.C | (value restrict.) R.T | (limited existential quantific.)

11. AL Description Logic: Language • AL: C,D  A | (atomic concept) A  Person | Female • An atomic concept corresponds to a unary predicate symbol in FOL • Extensionally, a set of world elements

12. AL Description Logic: Language • AL: C,D  A | (atomic concept) T | (universal concept) • Intuition: The universal concept corresponds in FOL to λx.TRUE(x), the unary predicate that holds for every object

13. AL Description Logic: Language • AL: C,D  A | (atomic concept) T | (universal concept)  | (bottom concept) • Intuition: The bottom concept corresponds in FOL to λx.FALSE(x), the unary predicate that holds for no object

14. AL Description Logic: Language • AL: C,D  A | (atomic concept) T | (universal concept)  | (bottom concept) A | (atomic negation) • The negation of A is the concept that is the complement of A, i.e., contains all elements that A does not • Female, Person

15. AL Description Logic: Language • AL: C,D  A | (atomic concept) T | (universal concept)  | (bottom concept) A | (atomic negation) CпD | (intersection) • Intersection of concepts corresponds to set intersection of their elements • Person п Female

16. AL Description Logic: Language • AL: C,D  A | (atomic concept) T, | (universal, bottom) A | (atomic negation) CпD | (intersection) R.C | (value restrict.) • All elements whose R is filled only by C-elements • hasChild.Female

17. AL Description Logic: Language • AL: C,D  A | (atomic concept) T, | (universal, bottom) A, CпD R.C | (value restrict.) R.T | (limited existential quantific.) • The concept including all elements that have role R filled by some element • hasChild.T

18. AL DL: FOL Semantics • Interpretation I maps Δ to nonempty set ΔI and, • Every atomic concept A is mapped to AI  ΔI • TI = ΔI • I = Ø • (A)I = ΔI \ AI • (CпD)I = CI пDI • (R.C)I = {a  ΔI |  b. (a,b)RI b  CI } • (R.T)I = {a  ΔI |  b. (a,b)RI}

19. DLs that Extend AL • R.C – full existential quantification • (≥n R) - number restrictions • C – negation of arbitrary concepts • CUD – union of concepts • Trigger rules – CLASSIC (configuration of systems), LOOM

20. TBox: Terminological Axioms • C D – The left-hand side is a symbol • R S – same • C D – same • RS – same • Mother  Woman п hasChild.Person • Parent  Mother U Father • Grandmother Mother п hasChild.Mother п п

21. Definitional / Nondefinitional • Base interpretation for atomic concepts • The TBox is definitional if every base interpretation has only one extension • Observation: If the TBox has no cycles then it is definitional

22. ABox: Assertions About Elements • Father(Peter) C(a) • Grandmother(Mary) C(a) • hasChild(Mary,Peter) R(b,c) • hasChild(Mary,Paul) R(b,c) • hasChild(Peter,Harry) R(b,c) • C(a) – concept assertions • R(b,c) – role assertions

23. ABox: Assertions About Elements • UNA – Unique Names Assumption • Interpretation I maps object names to elements in ΔI • Some languages allow other statements, within a fragment of FOL. • TBox,Abox equivalent to a set of axioms in FOL (with two variables, without functions)

24. Take a Breath • So far: Language + Semantics • From here: • Reasoning Tasks • Algorithms • Later: NLP using Description Logics

25. TBox Reasoning Tasks • Satisfiability of C: • A model I of T such that CI is nonempty • Subsumption of C by D • For every model I of T, CI DI • Equivalence of C and D • Disjointness of C and D п

26. Reductions to Subsumption • C is unsatisfiable iff C  • C,D equivalent iff C D, D C • C,D disjoint iff CпD  • With an empty or nonempty TBox • Assuming we have the needed operations п п п п

27. Reductions to Unsatisfiability • C D iff CпD unsatisfiable • C,D equivalent iff CпD , CпD unsatisfiable • C,D disjoint iff CпD unsatisfiable • With an empty or nonempty TBox • Assuming we have the needed operations п

28. Systems vs Reasoning • CLASSIC, LOOM : Subsumption • KRIS, CRACK, FACT, DLP, RACE: Satisfiability • Subsumption is most general and therefore most expensive computationally

29. Eliminating the TBox • Converting definitional TBox problems to concept problems T={ Woman  Person п Female Man  Person п Woman } C = Woman п Man C’= Person п Female п Person п (Person п Female)

30. ABox Queries • Consistency • Instance check – A C(a) • “a” is an instance name • Reduces to concept satisfiability if “set” and “fill” constructors are allowed • Retrieval of all individuals satisfying C • Find most specific concept for individual a ╨

31. Structural Subsumption • Language: FL0 • Concept conjunction C п D • Value restriction R.C • Normal form of concepts in FL0 C  A1п… пAmпR1.C1п… пRn.Cn D  B1п… пBkпS1.D1п… пSl.Dl • C D iff i≤k j≤m s.t. Bi = Aj i≤l j≤n s.t. Si = Rj , Ci Dj • Proof? п п • Proof?

32. Structural Subsumption Algorithm for FL0 1. Convert concepts to normal form C  A1п… пAmпR1.C1п… пRn.Cn D  B1п… пBkпS1.D1п… пSl.Dl 2. Check recursively: i≤k j≤m s.t. Bi = Aj i≤l j≤n s.t. Si = Rj , Ci Dj п

33. Extending FL0 • Language: FL0 • Concept conjunction C п D • Value restriction R.C • Language: ALN • AL (C п D, R.C , T, , A, R.T) • Number restrictions (≥nR, ≤nR)

34. Structural Subsumption for ALN • Language: ALN • AL (C п D, R.C , T, , A, R.T) • Number restrictions (≥nR, ≤nR) • Normal form for ALN C  L1п… пLmпR1.C1п… пRn.Cn or C  , • Li atomic concepts, their negation, or ≥nR,≤nR • Ci in normal form, Ri, Ai distinct

35. Computing Normal Form for ALN • C п D, R.C , T, , A, R.T, ≥nR, ≤nR C L1п…пLmпR1.C1п…пRn.Cnor C • Look at outermost connective • , T, , ≥nR, ≤nR, R.T : return concept • R.C : C’ = recurse on C; return R.C’ • C п D – recurse on C,D, generating C’,D’; • If top level of C’ п D’ includes conflict (A,A; ; ≥nR,≤mR (n<m); ≥nR,R.), return  • Return C’ п D’

36. Structural Subsumption Algorithm for ALN 1. Convert concepts to normal form C  L1п… пLmпR1.C1п… пRn.Cn D  N1п… пNkпS1.D1п… пSl.Dl 2. Check recursively: i≤k j≤m s.t. Bi = Aj i≤l j≤n s.t. Si = Rj , Ci Dj with ≥nR ≥mR iff n≥m п п

37. Example • C=Person пFemaleпhasChild.TпhasChild.PersonпhasChild.FemaleпhasChild.hasChild.Female пhasChild.hasChild.Female • D=Person п≥1.hasChild ON BOARD

38. Extending ALN • Language: ALCN • ALN: CпD, R.C , T, , A, R.T, ≥nR, ≤nR • Arbitrary negation (complement) C • Overall algorithm for satisfiability • Convert to negation normal form (negation in front of atoms only) • Use tableau theorem proving to find model

39. Principles of Tableau Reasoning • Apply rules and build tree (defines model): • When a branch of the tree is contradictory to itself (e.g., has A,A), we backtrack p  (~q  ~p) Tableau for Propositional logic: Rules for , p (~q  ~p) ~q ~p

40. Tableau-based Satisfiability Algorithm for ALCN • Want to show that C0 (in NNF) is satisfiable • We look for a model of Abox A = {C0(x0)}, with x0 a new constant symbol • Apply (consistency preserving) transformation rules • If at some point a “complete” ABox is generated, then C0 is satisfiable • If no complete ABox found, C0 unSAT

41. Tableau-based Satisfiability Algorithm for ALCN • п-rule: • Condition: A contains (C1 п C2)(x), but neither C1(x),C2(x) • Action: A’=A{C1(x),C2(x)} • U-rule: • Condition: A contains (C1 U C2)(x), but neither C1(x),C2(x) • Action (nondeterministically choose): A’=A{C1(x)}, A’’=A{C2(x)}

42. Tableau-based Satisfiability Algorithm for ALCN • -rule: • Condition: A contains (R.C)(x), but there is no individual name z s.t. C(z) and R(x,z) in A • Action: A’=A{C(y),R(x,y)} for y an individual name not occuring in A • -rule: • Condition: A contains (R.C)(x) and R(x,y), but C(y) is not in A • Action: A’=A{C(y)}

43. Tableau-based Satisfiability Algorithm for ALCN • ≥-rule: • Condition: A contains (≥nR)(x), but no individual names z1,…, zn s.t. R(x,zj) (i≤n) and zj≠zj (i<j≤n) • Action: A’=A{R(x,yj)| i≤n}{yi≠yj| i<j≤n}, and y1,…,yn distinct individual names not in A • ≤-rule: • Condition: A contains distinct individual names y1,…,yn+1 s.t. (≤nR)(x) and R(x,yi) (i≤n) in A, but yi≠yj not in A for some i≠j • Action (nondeterministically choose j<i≤n with yi≠yj): A’=A[yi/yj]

44. Example ? • (R.A) п (R.B) R.(A п B) п

45. Example 2 ? • (R.A) п (R.B) п (≤1R) R.(A п B) п

46. Computational Properties • Satisfiability (and subsumption) in ALCN is PSpace-complete • This tableau algorithm takes time O(22^n) • Small improvement gives a nondeterministic PSpace tableau algorithm which takes time O(22n) • n = length of concept/s

47. Related to DL • Natural language processing • Semantic web • Complexity of reasoning and decidable first-order languages • Conceptual modeling • CYC

48. Summary So Far • Description Logics provide expressivity / tractability tradeoff • ALN reasoning in polynomial time • ALCN reasoning in PSpace • Next: Medical informatics

49. Application: Medical Informatics • GALEN: A terminological knowledge base (TBox) of human anatomy • Hierarchical display • Multiple axes • Simple combinations of concepts • Automatic-dynamic classification of new concepts • Aid in creating new concepts

50. Application: Medical Informatics • Example: classification • Leg which • hasLeftRightSelector leftSelection • Leg пleftRightSelector.leftSelection, or • Leg пleftRightSelector.{leftSelection} • The language does not include negation • If have time – show demo