1. Knowledge Representation and Reasoning University "Politehnica" of Bucharest Department of Computer Science Fall 2011 Adina Magda Florea http://turing.cs.pub.ro/krr_11 curs.cs.pub.ro Master of Science in Artificial Intelligence, 2011-2013

2. 2 Lecture 5 Description Logic Lecture outline  About DL  DL language  Terminologies  World descriptions  Reasoning services  Reasoning algorithms

3. 3 1. About Description Logics  Description Logics (DLs) - knowledge representation formalisms that represent the knowledge of an application domain (the world) by defining the relevant concepts of the domain (its terminology) and then using these concepts to specify properties of individuals occurring in the domain (the world description).  Inferences: classification of concepts - subconcept/superconcept relationships (subsumption) individuals - whether a given individual is always an instance of a concept

4. 4 Description Logics  Semantic networks: IS-A, AKO  KL-ONE – formal semantic  Subsumption  Unlike IS-A links, subsumption relationships and instance relationships are inferred from the definition of the concepts and the properties of the individuals  DLs are based on a formal, logic-based semantics

5. 5 A DL system TBox ABox Description Language Reasoning KB Application Programs Rules TBox – the terminology, i.e., the vocabulary of an application domain = concepts (set of individuals) + roles ABox – assertions about named individuals in terms of this vocabulary

6. 6 Inference services Typical reasoning tasks:  Whether a TBox description is satisfiable (i.e., non- contradictory)  Whether one description subsumes another one in a TBox - organize the concepts of a terminology into a hierarchy according to their generality  Find out whether the set of assertions in a ABox is consistent (has a model)  Whether the assertions in the ABox entail that a particular individual is an instance of a given concept description  A concept description can also be conceived as a query - retrieve the individuals that satisfy the query.

7. 7 2. DL Language  Elementary descriptions: atomic concepts (A,B) atomic roles (R)  Complex descriptions of concepts (C,D) – concept constructors, or of roles – role constructors  Basic language – Attribute Language AL  Description languages are distinguished by the constructors they provide (syntactic rules of concept formation)  Different extensions of AL

8. 8 Syntax C, D  A | (atomic concept) T | (universal concept)  | (bottom concept) A | (atomic negation) C D | (intersection)  R.C | (value restriction)  R.T | (limited existential quantification) C D | (union)  R.C | (full existential quantification) C | (negation of concepts)  n R | (number restrictions)  n R AL ALU ALE ALC ALN AL[U][E][N][C]

9. 9 Examples Atomic concepts: Pers, Fem Atomic role: hasChild Pers FemPers Fem Pers  hasChild.TPers  hasChild.Fem Pers  hasChild.  Pers Fem  hasChild.T  hasChild.Fem Pers Fem  3 hasChild

10. 10 Semantics  Interpretation I  The domain of the interpretation Δ I  An interpretation function  assigns to every atomic concept A a set A I  Δ I  assigns to every atomic role R a binary relation R I  Δ I x Δ I I = (Δ I, · I )

11. 11 Semantics  The interpretation function is extended to concept definitions by the following inductive definitions: Pers  hasChild.Fem Pers  hasChild.T

12. 12 Semantics Pers  hasChild.Fem Fem  3 hasChild Pers Fem

13. 13 Link with FOPL  Atomic concepts - unary predicates  Atomic roles – binary predicates  Concept C  p c (x), for every I the set of elements that satisfies  I is C I

14. 14 3. Terminologies (TBox)  Terminological axioms - make statements about how concepts or roles are related to each other  C D (R S) - inclusion axiom  C  D (R  S) - equality axiom  Semantics of axioms C D if C I  D I C  D if C I  D I  If T is a set of axioms, then I satisfies T i ff I satisfies each element of T – I is a model of T

15. 15 TBox  Definition = an equality whose left-hand side is an atomic concept Introduces symbolic names for complex descriptions  Name symbols N T (defined concepts) - occur on the left-hand side of some axiom  Base symbols B T (primitive concepts) - occur only on the right-hand side of axioms

16. 16 Example of TBox Woman ≡ Person Female Man ≡ Person ¬Woman Mother ≡ Woman  hasChild.Person Father ≡ Man  hasChild.Person Parent ≡ Father Mother Grandmother ≡ Mother  hasChild.Parent MotherWithManyChildren ≡ Mother  3 hasChild MotherWithoutDaughter ≡ Mother  hasChild.¬Woman Wife ≡ Woman  hasHusband.Man

17. 17 TBox  A base interpretation J for T is an interpretation that interprets only the base symbols.  An interpretation I that interprets also the name symbols is an extension of J if it has the same domain as J, i.e., Δ I =Δ J, and if it agrees with J for the base symbols.  T is definitorial if every base interpretation has exactly one extension that is a model of T. If we know what the base symbols stand for, and T is definitorial, then the meaning of the name symbols is completely determined.

18. 18 TBox  The question whether a terminology is definitorial or not is related to the question whether or not its definitions are cyclic. Human ≡ Animal  hasParent.Human Let A, B be atomic concepts occurring in T  A directly uses B in T - B appears on the right-hand side of the definition of A  A uses B in T - the transitive closure of the relation directly uses.  T contains a cycle i ff there exists an atomic concept in T that uses itself.  Otherwise, T is acyclic

19. 19 TBox  If a terminology T is acyclic, then it is definitorial  T' the expansion of T (all axioms of the form A=C', C' contains only base symbols) Woman ≡ Person Female Mother ≡ Woman  hasChild.Person MotherWithManyChildren ≡ Mother  3 hasChild MotherWithManyChildren ≡ ((Person Female)  hasChild.Person)  3 hasChild only one extension

20. 20 4. World descriptions (ABox)  ABox introduce individuals, by giving them names, and asserts properties of these individuals  Individuals: a, b, c  C(a) - concept assertions - a belongs to (the interpretation of) C,  R(b,c) - role assertions - c is a filler of the role R for b. MotherWithoutDaughter(mary) Father(peter) hasChild(mary, peter) hasChild(peter, harry) hasChild(mary, paul)

21. 21 ABox Semantics  Semantics of ABoxes – extend interpretations to individual names  An interpretation I = (Δ I, · I ) not only maps atomic concepts and roles to sets and relations, but in addition maps each individual name a to an element a I  Δ I.  The mapping has to respect the unique name assumption (UNA) - that distinct individual names denote distinct objects, i.e., if a  b then a I  b I

22. 22 ABox Semantics  An interpretation I satisfies C(a) if a I  C I  An interpretation I satisfies R(a,b) if (a I, b I )  R I  An interpretation I satisfies an ABox if it satisfies each assertion in the ABox – I is a model of A  An interpretation I satisfies an ABox A with respect to a TBox T if it is a model of T and a model of A

23. 23 Individual names in TBox  Sometimes, it is convenient to allow individual names (also called nominals) not only in the ABox, but also in the description language  The most basic concept constructors employing individuals is the set (or one-of ) constructor {a 1,...,a n }, where a 1,...,a n are individual names.  The semantics of a set concept is defined as: {a 1,...,a n } I = {a 1 I,...,a n I }

24. 24 Individual names in TBox  Another constructor involving individual names is the fills constructor R : a, for a role R  The semantics of this constructor is defined as: (R : a) I = {x  Δ I | (x, a I )  R I } R : a stands for the set of those objects that have a as a filler of the role R.

25. 25 5. Reasoning services in DL  Reasoning tasks for TBox Checking satisfiability of a concept Checking subsumption of a concept by another Checking equivalence of 2 concepts Checking disjointness of 2 concepts

26. 26 Reasoning tasks for TBox  Satisfiability: A concept C is satisfiable with respect to T if there exists a model I of T such that C I is nonempty (I is a model of C)  Subsumption: A concept C is subsumed by a concept D with respect to T if C I  D I for every model I of T C D or T|= C D.  Equivalence: Two concepts C and D are equivalent with respect to T if C I = D I for every model I of T C ≡ T D or T |= C ≡ D  Disjointness: Two concepts C and D are disjoint with respect to T if C I ∩ D I = ∅ for every model I of T

27. 27 Properties Reduction to Subsumption  C is unsatisfiable  C is subsumed by   C and D are equivalent  C is subsumed by D and D is subsumed by C  C and D are disjoint  C D is subsumed by  Reduction to Unsatisfiability  C is subsumed by D  C ¬D is unsatisfiable  C and D are equivalent  both (C ¬D) and (¬C D) are unsatisfiable  C and D are disjoint  C D is unsatisfiable

28. 28 Reasoning tasks for ABox  Consistency of an ABox  Instance check  Realization  Retrieval  An ABox A is consistent with respect to a TBox T, if  I that is a model of both A and T  A is consistent if it is consistent with respect to the empty TBox. { Mother(mary), Father(mary) }

29. 29 Reasoning tasks for ABox  The expansion of A with respect to T - the ABox A' that is obtained from A by replacing each concept assertion C(a) in A with the assertion C'(a), where C ' is the expansion of C with respect to T  An ABox A is consistent with respect to a TBox iff the expansion A' is consistent

30. 30 Reasoning tasks for ABox  Instance check  An assertion C(a) is entailed by A if  I that satisfies A also satisfies C(a) (A|= C(a)) A|= C(a) i ff A  {¬C(a)} is inconsistent  Retrieval problem  Given an ABox A and a concept C, find all individuals a such that A|= C(a)  Realization problem  Given an individual a and a set of concepts, find the most specific concepts C from the set such that A|= C(a) The most specific concepts are those that are minimal with respect to the subsumption ordering

31. 31 Assumptions  Closed-world vs. open-world semantics

32. 32 6. Reasoning algorithms  Structural subsumption algorithms = algorithms that compare the syntactic structure of concept descriptions Very efficient Complete for only rather simple languages For ALNU subsumption algorithms are not complete  Tableau-based algorithms = satisfiability of concepts, TBox, etc.

33. 33 Tableau algorithms  Negation normal form of a concept – negation occurs only in front of concept names  Example of tableau algorithm to check subsumption (reduced to unsatisfiability)

34. 34 Tableau algorithms  Test satisfiability of C 0  Let C 0 a concept (ALCN) in negation normal form  A 0 ={C 0 (x 0 )}  Apply transformation rules (preserve consistency if any) until no more rule apply or a clash is obtained  ABox – transformed in many new {ABox'} by some rules  ABox is consistent iff one of the {ABox'} is consistent

35. 35 Transformation rules for satisfiability algorithms

36. 36 Transformation rules for satisfiability algorithms

37. 37 Tableau algorithms  An ABox is complete iff none of the transformation rules apply to it  Consistency of a set of complete ABoxes can be decided by looking for clashes (contradictions).  An ABox A contains a contradiction if: {  (x)}  A for some individual x {A(x), ¬A(x)}  A for some individual x and some concept A {(  n R)(x)}  {R(x,y i ) | 1  i  n+1}  {y i  y j | 1  i<j  n+1}  A for individuals x, y 1, …, y n+1, a nonnegative integer n, and a role R

38. 38 Tableau algorithms Tr. rule ABoxset of complete {ABoxes}  if one ABox is clash-free, then {C 0 (x 0 )} is consistent and C 0 is satisfiable  if all {ABoxes} contain a clash, then {C 0 (x 0 )} is inconsistent and C 0 is unsatisfiable

39. 39 Theorems T1 (termination) Let C 0 be an ALCN concept description in negation normal form. There cannot be an infinite sequence of rule applications {C 0 (x 0 )}  S1  S2  … T2 (decidability) It is decidable whether or not an ALCN-concept is satisfiable

40. 40 Credits Slides based on the book  The Description Logic Handbook: Theory, Implementation, and Applications, Edited by Franz Baader, Diego Calvanese, Deborah L. McGuinness, Daniele Nardi, Peter F. Patel-Schneider, Cambridge University Press, 2007, 2nd Edition