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 2013-2014 Description Logics 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 2013-2014 Description Logics 3
Chapter 2 A Basic Description Logic
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 2013-2014 Description Logics 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 2013-2014 Description Logics 6
Section 2.1 The Description Language
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 2013-2014 Description Logics 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 2013-2014 Description Logics 9
Examples of Concepts Hero Female fights.mutant Rich Human fights. Human Mutant fights.( Human sidekick.female) Dresden, WS 2013-2014 Description Logics 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 2013-2014 Description Logics 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 2013-2014 Description Logics 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 2013-2014 Description Logics 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 2013-2014 Description Logics 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 2013-2014 Description Logics 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 2013-2014 Description Logics 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 2013-2014 Description Logics 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 2013-2014 Description Logics 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 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 2013-2014 Description Logics 17
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 2013-2014 Description Logics 18
Section 2.2 Terminological Knowledge
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 2013-2014 Description Logics 20
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 2013-2014 Description Logics 21
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 2013-2014 Description Logics 22
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 2013-2014 Description Logics 23
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 2013-2014 Description Logics 24
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 2013-2014 Description Logics 25
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 2013-2014 Description Logics 26
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 2013-2014 Description Logics 27
Section 2.3 Assertional Knowledge
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 2013-2014 Description Logics 29
Examples of Assertions Rich(batman) Human(superman) fights(superman,bizarro) helps(robin,batman) Dresden, WS 2013-2014 Description Logics 30
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 2013-2014 Description Logics 31
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 2013-2014 Description Logics 32
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 2013-2014 Description Logics 33
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 2013-2014 Description Logics 34
Section 2.4 Reasoning Problems and Services
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 2013-2014 Description Logics 36
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 2013-2014 Description Logics 37
Properties of Subsumption Lemma 2.14 1. 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 2013-2014 Description Logics 38
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 2013-2014 Description Logics 39
Reductions Between Reasoning Problems The following polynomial time reductions between the reasoning problems hold 2. 4. equivalence subsumption satisfiability 1. 3. 7. 5. holds for any DL that has conjunction and negation instance 6. consistency Dresden, WS 2013-2014 Description Logics 40
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 2013-2014 Description Logics 41
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 2013-2014 Description Logics 42
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 2.18 1. 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 2013-2014 Description Logics 42
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 2013-2014 Description Logics 43
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 2013-2014 Description Logics 44
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 2013-2014 Description Logics 45
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 2013-2014 Description Logics 46