Phase 1. Phase 2. Phase 3. History. implementation of systems based on incomplete structural subsumption algorithms

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1 History Phase 1 implementation of systems based on incomplete structural subsumption algorithms Phase 2 tableau-based algorithms and complexity results first tableau-based systems (Kris, Crack) first formal study of optimization methods Phase 3 tableau-based algorithms for very expressive DLs highly-optimized tableau-based systems (FaCT, Racer) relationship to modal logic and FOL Dresden, WS Description Logics 2

2 History Phase 4 Web Ontology Language (OWL) based on DL industrial-strength reasoners and ontology editors light-weight (tractable) DLs Phase 5 non-standard reasoning ontology management semantic extensions Dresden, WS Description Logics 3

3 Chapter 2 A Basic Description Logic

4 A Basic Description Logic ALC attributive language with complement Naming Schema basic language AL additional constructors are denoted by appending a letter C stands for complement ALC is obtained by adding to AL Dresden, WS Description Logics 5

5 Structure of Description Logic Systems TBox Ontology terminology of application domain Language constructors for building complex concepts formal logic-based semantics ABox facts about a world Reasoning implicit knowledge practical algorithms Dresden, WS Description Logics 6

6 Section 2.1 The Description Language

7 Syntax of ALC Definition 2.1 (Syntax of ALC) Let N C and N R two disjoint sets of concept names and role names, respectively. ALC (complex) concepts are defined by induction: if A N C, then A is an ALC concept if C, D are ALC concepts and r N R, then the following are ALC concepts: C D (conjunction) C D (disjunction) C (negation) r.c (existential restriction) r.c (value restriction) Abbreviations := A A (top) := A A (bottom) Dresden, WS Description Logics 8

8 Notation concept names are also called atomic concepts all other concepts are called complex instead of ALC concept, we often say concept A, B stand for concept names C, D for (complex) concepts r, s for role names Dresden, WS Description Logics 9

9 Examples of Concepts Hero Female fights.mutant Rich Human fights. Human Mutant fights.( Human sidekick.female) Dresden, WS Description Logics 10

10 Semantics of ALC Definition 2.2 (Semantics of ALC) An interpretation I = ( I, I) consists of: a non-empty domain I, and an extension mapping I (also called interpretation function): A I I for all A N C (concepts interpreted as sets) r I I I for all r N R (roles interpreted as binary relations) The extension mapping is extended to concept descriptions as follows: (C D) I := C I D I (C D) I := C I D I ( C) I := I \ C I ( r.c) I := {d I there is e I with (d, e) r I and e C I } ( r.c) I := {d I for all e I : (d, e) r I implies e C I } Dresden, WS Description Logics 11

11 Interpretation Example Hero R helps Hero B Hero S Hero,Female W fights fights fights fights fights Rich,Human,Female helps Human Female Rich,Human ( Hero fights.human ) I = {B,S} ( Hero fights.(rich Human) ) I = {R,S,W} ( helps.human ) I = I \ {R} Dresden, WS Description Logics 12

12 DL vs FOL ALC can be seen as a fragment of First-order Logic concept names are unary predicates role names are binary predicates Interpretations can obviously be seen as first-order interpretations for this signature concepts then correspond to FOL formulae with one free variable Let φ(x) be such a formula with free variable x, and I an interpretation. The extension of φ w.r.t. I is given by φ I := {d I I = φ(d)} Goal translate ALC concepts C into FOL formulae τ x(c) such that their extensions coincide Dresden, WS Description Logics 13

13 Translation to FOL Syntactic translation C τ x(c) τ x(a) := A(x) for A N C τ x(c D) := τ x(c) τ x(d) τ x(c D) := τ x(c) τ x(d) τ x( C) := τ x(c) τ x( r.c) := y.(r(x, y) τ y C) τ x( r.c) := y.(r(x, y) τ y C) y new variable different from x Example τ x( r.(a s.b)) = y.(r(x, y) τ y (A s.b)) = y.(r(x, y) (A(y) z.(s(y, z) B(z)))) Lemma 2.3 C and τ x(c) have the same extension; that is, Proof by induction on structure of C C I = {d I I = τ x(c)(d)}. Dresden, WS Description Logics 14

14 Decidable Fragments of FOL ALC can be seen as a fragment of FOL: each concept C yields a formula τ x(c) with one free variable Decidability These formulae belong to known decidable subclasses of FOL: two variable fragment guarded fragment τ x( r.(a s.b)) = y.(r(x, y) τ y (A s.b)) = y.(r(x, y) (A(y) x.(s(y, x) B(x)))) Dresden, WS Description Logics 15

15 DL vs Modal Logic ALC is a syntactic variant of multimodal K concept names are propositional variables role names are transition relations multimodal K: several pairs of boxes and diamonds Concept descriptions C yield modal formulas Θ(C) Θ(A) := A for A N C Θ(C D) := Θ(C) Θ(D) Θ(C D) := Θ(C) Θ(D) Θ( C) := Θ(C) Θ( r.c) := [r]θ(c) Θ( r.c) := r Θ(C) C and Θ(C) have the same semantics: I is a Kripke structure C I is the set of worlds that make Θ(C) true in I. Dresden, WS Description Logics 16

16 More Expressivity ALC is only one example of many description logics that have been studied many other constructors exist and can be used for KR Qualified Number Restrictions (Q) ( n r.c) I := {d I {e (d, e) r I, e C I } n} at least n r-successors that belong to C ( n r.c) I := {d I {e (d, e) r I, e C I } n} at most n r-successors that belong to C A hero that fights at least two villains, of which at most one is a sidekick Hero ( 2 fights.villain) ( 1 fights.( helps.villain)) Dresden, WS Description Logics 17

17 More Expressivity ALC is only one example of many description logics that have been studied many other constructors exist and can be used for KR Number Restrictions (N ) ( n r) I := ( n r. ) I = {d I {e (d, e) r I, e C I } n} at least n r-successors ( n r) I := ( n r. ) I = {d I {e (d, e) r I, e C I } n} at most n r-successors Dresden, WS Description Logics 17

18 More Expressivity ALC is only one example of many description logics that have been studied many other constructors exist and can be used for KR Inverse Roles (I) for a role name r, r 1 denotes the inverse: (r 1 ) I := {(e, d) (d, e) r I } A hero that only fights villains with female sidekicks Hero fights.(villain helps 1.Female) Dresden, WS Description Logics 17

19 Structure of Description Logic Systems Ontology TBox terminology of application domain Language constructors for building complex concepts formal logic-based semantics ABox facts about a world Reasoning implicit knowledge practical algorithms Dresden, WS Description Logics 18

20 Section 2.2 Terminological Knowledge

21 GCIs and TBoxes Definition 2.4 (GCIs and TBoxes) A general concept inclusion (GCI) is of the form C D, where C, D are concepts A TBox is a finite set of GCIs The interpretation I satisfies the GCI C D iff C I D I I is a model of the TBox T iff it satisfies all GCIs in T Note: this definition is not specific of ALC; applies to any description language Example Hero Villain Hero haspower. Rich hassidekick 1.Rich Two TBoxes are equivalent if they have the same models Dresden, WS Description Logics 20

22 Concept Definitions Definition 2.5 A concept definition is of the form A C where A is a concept name C is a concept description The interpretation I satisfies the concept definition A C if A I = C I A C abbreviates the two GCIs A C, C A Dresden, WS Description Logics 21

23 Acyclic TBoxes Definition 2.5 (continued) An acyclic TBox is a finite set of concept definitions that does not contain multiple definitions A C A D does not contain cyclic definitions (directly or indirectly) A r.b B C C s.a The interpretation I is a model of the acyclic TBox T is it satisfies all concept definitions in T A concept name A occurring in T is a defined concept iff there is C such that A C T ; primitive concept otherwise Dresden, WS Description Logics 22

24 Example Heroine Hero Female Sidekick helps. Criminal fights.hero MutantCriminal Criminal fights.mutant Superhero Hero (Rich Human haspower.superpower) Overlord ( 3 helps 1.Criminal) fights.superhero Dresden, WS Description Logics 23

25 ABox Expansion Proposition 2.6 For every acyclic TBox T, we can effectively construct an equivalent acyclic TBox T such that the right-hand sides of concept definitions in T contain only primitive concepts. Proof blackboard We call T the expanded version of T Dresden, WS Description Logics 24

26 Primitive Interpretations Given an acyclic TBox T, a primitive interpretation J for T together with an extension mapping J that maps primitive concepts P to sets P J J role names r to binary relations r J J J consists of a non-empty set J The interpretation I is an extension of the primitive interpretation J iff J = I, P J = P I for all primitive concepts P r J = r I for all role names r Corollary 2.7 Let T be an acyclic TBox. Any primitive interpretation J has a unique extension to a model of T Proof blackboard Dresden, WS Description Logics 25

27 Translation to FOL Any ALC-TBox can be translated into FOL: τ(t ) := x.(τ x(c) τ x(d)) C D T Lemma 2.8 Let T a TBox and τ(t ) its translation into FOL. Then T and τ(t ) have the same models Proof blackboard Dresden, WS Description Logics 26

28 Structure of Description Logic Systems TBox Ontology terminology of application domain Language constructors for building complex concepts formal logic-based semantics ABox facts about a world Reasoning implicit knowledge practical algorithms Dresden, WS Description Logics 27

29 Section 2.3 Assertional Knowledge

30 Assertions and ABoxes Definition 2.9 (Assertions and ABoxes) An assertion is of the form C(a) (concept assertion) or r(a, b) (role assertion) where C is a concept, r a role, and a, b are individual names from a set N I (disjoint with N C,N R ) An ABox is a finite set of assertions Interpretations I are extended to assign elements a I I to individual names a N I I is a model of the ABox A if it satisfies all its assertions: a I C I for all C(a) A (a I, b I ) r I for all r(a, b) A Dresden, WS Description Logics 29

31 Examples of Assertions Rich(batman) Human(superman) fights(superman,bizarro) helps(robin,batman) Dresden, WS Description Logics 30

32 Translation to FOL Any ALC-ABox can be translated into FOL: τ(a) := C(a) A (individual names are viewed as constants) τ x(c)(a) Lemma 2.10 Let A an ABox and τ(a) its translation into FOL. Then A and τ(a) have the same models Proof easy (Exercise!) r(a,b) A r(a, b) Dresden, WS Description Logics 31

33 Ontologies Definition 2.11 An ontology O = (T, A) consists of a TBox T and an ABox A The interpretation I is a model of the ontology O = (T, A) iff it is a model of T and a model of A FOL translation: τ(o) := τ(t ) τ(a) Lemma 2.12 Let O be an ontology and τ(o) its FOL translation. O and τ(o) have the same models Proof Immediate from Lemmas 2.8 and 2.10 Dresden, WS Description Logics 32

34 Nominals We can increase the expressive power of the description language by using individual names as concept constructors They are interpreted as singleton sets containing only the extension of the individual name Nominals (O) Syntax: {a} for a N I Semantics: {a} I := {a I } Using nominals, ABox assertions can be expressed through GCIs: C(a) is expressed by {a} C r(a, b) is expressed by {a} r.{b} Dresden, WS Description Logics 33

35 Structure of Description Logic Systems TBox Ontology terminology of application domain Language constructors for building complex concepts formal logic-based semantics ABox facts about a world Reasoning implicit knowledge practical algorithms Dresden, WS Description Logics 34

36 Section 2.4 Reasoning Problems and Services

37 Reasoning Problems Goal: to make implicitly represented knowledge explicit Definition 2.13 (Terminological Reasoning) Let T be a TBox. Terminological reasoning refers to deciding the following problems Satisfiability: C is satisfiable w.r.t. T iff C I for some model I of T Subsumption: C is subsumed by D w.r.t. T (C T D) iff C I D I for all models I of T Equivalence: C is equivalent to D w.r.t. T (C T D) iff C I = D I for all models I of T If T = we simply remove the w.r.t T from the name Dresden, WS Description Logics 36

38 Examples A A and r.a r. A are not satisfiable (unsatisfiable) A A and r.a r. A are equivalent A B is subsumed by A and by B r.(a B) is subsumed by r.a and by r.b r.(a B) is equivalent to r.a r.b r.a r.b is subsumed by r.(a B) Dresden, WS Description Logics 37

39 Properties of Subsumption Lemma The subsumption relation T is a pre-order on concepts: C T C (reflexivity) if C T D and D T E, then C T E (transitivity) 2. Existential restrictions and value restrictions are monotonic w.r.t. subsumption if C T D, then r.c T r.d and r.c T r.d Proof blackboard Note T is not a partial order since it is not antisymmetric: C T D and D T C does not imply that C = D (no syntactic equivalence, just semantical) Dresden, WS Description Logics 38

40 Reasoning Problems Definition 2.15 (Assertional Reasoning) Let O = (T, A) be an ontology. The following are assertional reasoning problems Consistency: O is consistent iff there exists a model of O Instance: a is an instance of C w.r.t. O iff a I C I for all models I of O Lemma 2.16 Let O = (T, A) be an ontology. If a is an instance of C w.r.t. O and C T D, then a is an instance of D w.r.t. O Exercise! Dresden, WS Description Logics 39

41 Reductions Between Reasoning Problems The following polynomial time reductions between the reasoning problems hold equivalence subsumption satisfiability holds for any DL that has conjunction and negation instance 6. consistency Dresden, WS Description Logics 40

42 Reductions Theorem 2.17 Let O = (T, A) be an ontology, C, D concepts and a N I. 1. C T D iff C T D and D T C 2. C T D iff C T C D 3. C T D iff C D is unsatisfiable w.r.t. T 4. C is satisfiable w.r.t. T iff C T 5. C is satisfiable w.r.t. T iff (T, {C(a)}) is consistent 6. a is an instance of C w.r.t. O iff (T, A { C(a)}) is inconsistent 7. O is consistent iff a is not an instance of w.r.t. O Proof blackboard Dresden, WS Description Logics 41

43 Expansions Let O = (T, A) be an ontology where T is acyclic, and C a concept The expanded versions Ĉ and  of C and A w.r.t. T are obtained by replacing all defined concepts occurring in C and A by their defintions from T T : Woman Person Female Criminal fights.hero Minion Person helps.criminal C: Woman Minion Ĉ = Person Female Person helps. fights.hero Dresden, WS Description Logics 42

44 Expansions Let O = (T, A) be an ontology where T is acyclic, and C a concept The expanded versions Ĉ and  of C and A w.r.t. T are obtained by replacing all defined concepts occurring in C and A by their defintions from T Proposition C is satisfiable w.r.t. T iff Ĉ is satisfiable 2. O = (T, A) is consistent iff (, Â) is consistent Proof blackboard similar reductions exist for other reasoning problems Dresden, WS Description Logics 42

45 Not a Polynomial Reduction The expansion of concepts and ABoxes is in general not polynomial The acyclic TBox T A 0 r.a 1 s.a 1 A 1 r.a 2 s.a 2. A n 1 r.a n s.a n has n axioms, all of the same size; i.e., the size of T is linear in n The expanded version Â0 of A 0 contains the name A n 2 n times! induction on n Dresden, WS Description Logics 43

46 Relationship with FOL We can translate ALC reasoning into FOL reasoning Lemma 2.19 Let O = (T, A) be an ontology, C, D be concepts, and a N I 1. C T D iff τ(t ) = x.(τ x(c)(x) τ x(d)(x)) 2. O is consistent iff τ(o) is consistent 3. a is an instance of C w.r.t. O iff τ(o) = τ x(c)(a) Proof blackboard Dresden, WS Description Logics 44

47 Classification Computing the subsumption relations between all concept names in T Heroine Hero Female MaleHero Hero Female MutantHeroine Heroine Mutant Elite Rich Human Superhero Hero Elite Elite Hero Female Mutant Rich SuperHero MaleHero Heroin MutantHeroin Human Dresden, WS Description Logics 45

48 Realization Computing the most specific concept names to which an individual belongs Heroine Hero Female MaleHero Hero Female MutantHeroine Heroine Mutant Elite Rich Human Superhero Hero Elite Hero(Superman) Superman is an instance of Hero, MaleHero, Elite, Superhero Dresden, WS Description Logics 46