Description Logic 1: Syntax and Semantics Leif Harald Karlsen Autumn 2015 1/35 Contents Introduction ALC: Syntax and Semantics Extensions and other DLs OWL and the Semantic Web Appendix 2/35 Contents Introduction ALC: Syntax and Semantics Extensions and other DLs OWL and the Semantic Web Appendix 3/35 { Each description logic describes a language, and each language differ in expressibility vs. reasoning complexity, defined by allowing or disallowing different constructs (e.g. conjunction, disjunction, negation, quantifiers, etc.) in their language. Overview { Description logics are formal languages designed for knowledge representation and reasoning, and most of these are decidable fragments of FOL. 4/35 Overview { Description logics are formal languages designed for knowledge representation and reasoning, and most of these are decidable fragments of FOL. { Each description logic describes a language, and each language differ in expressibility vs. reasoning complexity, defined by allowing or disallowing different constructs (e.g. conjunction, disjunction, negation, quantifiers, etc.) in their language. 4/35 { Knowledge Representation (KR) { Application oriented { Represent `knowledge' in some way { `Frames,' like classes, with relations and attributes { Try to add some `semantics' in order to do some `reasoning' { Automated Reasoning, Modal Logic { Had theorems and algorithms { Cross-fertilisation of applications and theory { Today: large impact on Semantic Web (sign up for INF3580/4580!) History and motivation { Description logic comes from a merging of two traditions. 5/35 { Automated Reasoning, Modal Logic { Had theorems and algorithms { Cross-fertilisation of applications and theory { Today: large impact on Semantic Web (sign up for INF3580/4580!) History and motivation { Description logic comes from a merging of two traditions. { Knowledge Representation (KR) { Application oriented { Represent `knowledge' in some way { `Frames,' like classes, with relations and attributes { Try to add some `semantics' in order to do some `reasoning' 5/35 { Cross-fertilisation of applications and theory { Today: large impact on Semantic Web (sign up for INF3580/4580!) History and motivation { Description logic comes from a merging of two traditions. { Knowledge Representation (KR) { Application oriented { Represent `knowledge' in some way { `Frames,' like classes, with relations and attributes { Try to add some `semantics' in order to do some `reasoning' { Automated Reasoning, Modal Logic { Had theorems and algorithms 5/35 { Today: large impact on Semantic Web (sign up for INF3580/4580!) History and motivation { Description logic comes from a merging of two traditions. { Knowledge Representation (KR) { Application oriented { Represent `knowledge' in some way { `Frames,' like classes, with relations and attributes { Try to add some `semantics' in order to do some `reasoning' { Automated Reasoning, Modal Logic { Had theorems and algorithms { Cross-fertilisation of applications and theory 5/35 History and motivation { Description logic comes from a merging of two traditions. { Knowledge Representation (KR) { Application oriented { Represent `knowledge' in some way { `Frames,' like classes, with relations and attributes { Try to add some `semantics' in order to do some `reasoning' { Automated Reasoning, Modal Logic { Had theorems and algorithms { Cross-fertilisation of applications and theory { Today: large impact on Semantic Web (sign up for INF3580/4580!) 5/35 { individuals/constants (e.g. james, sensor1) { concepts/unary relations (e.g. Person, Sensor) { roles/binary relations (e.g. isFatherOf , isConnectedTo) Knowledge is represented as a knowledge base, K = hA; T i where: { A is a set of assertions about named individuals, called the ABox (e.g. Person(james), isFatherOf (james; peter)) { T is a set of terminology definitions (i.e. complex descriptions of concepts or roles), called the TBox (e.g. Human v Mammal, Mother ≡ Parent u Woman) Knowledge bases In description logics one works with three different types of elements: 6/35 { concepts/unary relations (e.g. Person, Sensor) { roles/binary relations (e.g. isFatherOf , isConnectedTo) Knowledge is represented as a knowledge base, K = hA; T i where: { A is a set of assertions about named individuals, called the ABox (e.g. Person(james), isFatherOf (james; peter)) { T is a set of terminology definitions (i.e. complex descriptions of concepts or roles), called the TBox (e.g. Human v Mammal, Mother ≡ Parent u Woman) Knowledge bases In description logics one works with three different types of elements: { individuals/constants (e.g. james, sensor1) 6/35 { roles/binary relations (e.g. isFatherOf , isConnectedTo) Knowledge is represented as a knowledge base, K = hA; T i where: { A is a set of assertions about named individuals, called the ABox (e.g. Person(james), isFatherOf (james; peter)) { T is a set of terminology definitions (i.e. complex descriptions of concepts or roles), called the TBox (e.g. Human v Mammal, Mother ≡ Parent u Woman) Knowledge bases In description logics one works with three different types of elements: { individuals/constants (e.g. james, sensor1) { concepts/unary relations (e.g. Person, Sensor) 6/35 Knowledge is represented as a knowledge base, K = hA; T i where: { A is a set of assertions about named individuals, called the ABox (e.g. Person(james), isFatherOf (james; peter)) { T is a set of terminology definitions (i.e. complex descriptions of concepts or roles), called the TBox (e.g. Human v Mammal, Mother ≡ Parent u Woman) Knowledge bases In description logics one works with three different types of elements: { individuals/constants (e.g. james, sensor1) { concepts/unary relations (e.g. Person, Sensor) { roles/binary relations (e.g. isFatherOf , isConnectedTo) 6/35 { T is a set of terminology definitions (i.e. complex descriptions of concepts or roles), called the TBox (e.g. Human v Mammal, Mother ≡ Parent u Woman) Knowledge bases In description logics one works with three different types of elements: { individuals/constants (e.g. james, sensor1) { concepts/unary relations (e.g. Person, Sensor) { roles/binary relations (e.g. isFatherOf , isConnectedTo) Knowledge is represented as a knowledge base, K = hA; T i where: { A is a set of assertions about named individuals, called the ABox (e.g. Person(james), isFatherOf (james; peter)) 6/35 Knowledge bases In description logics one works with three different types of elements: { individuals/constants (e.g. james, sensor1) { concepts/unary relations (e.g. Person, Sensor) { roles/binary relations (e.g. isFatherOf , isConnectedTo) Knowledge is represented as a knowledge base, K = hA; T i where: { A is a set of assertions about named individuals, called the ABox (e.g. Person(james), isFatherOf (james; peter)) { T is a set of terminology definitions (i.e. complex descriptions of concepts or roles), called the TBox (e.g. Human v Mammal, Mother ≡ Parent u Woman) 6/35 Contents Introduction ALC: Syntax and Semantics Extensions and other DLs OWL and the Semantic Web Appendix 7/35 C; D ! A j (atomic concept) > j (universal concept) ? j (bottom concept) :C j (negation) C t D j (union) C u D j (intersection) 9R:C j (existential restriction) 8R:C j (universal restriction) where A is an atomic concept, C and D are concepts, and R is a role. We allow { ABox assertions: C(a) and R(a; b) for individuals a; b, concepts C and roles R; { TBox axioms: C v D for concepts C and D. ALC: Syntax The description logic ALC (Attribute Language with general Complement) allows the following concepts: 8/35 where A is an atomic concept, C and D are concepts, and R is a role. We allow { ABox assertions: C(a) and R(a; b) for individuals a; b, concepts C and roles R; { TBox axioms: C v D for concepts C and D. ALC: Syntax The description logic ALC (Attribute Language with general Complement) allows the following concepts: C; D ! A j (atomic concept) > j (universal concept) ? j (bottom concept) :C j (negation) C t D j (union) C u D j (intersection) 9R:C j (existential restriction) 8R:C j (universal restriction) 8/35 We allow { ABox assertions: C(a) and R(a; b) for individuals a; b, concepts C and roles R; { TBox axioms: C v D for concepts C and D. ALC: Syntax The description logic ALC (Attribute Language with general Complement) allows the following concepts: C; D ! A j (atomic concept) > j (universal concept) ? j (bottom concept) :C j (negation) C t D j (union) C u D j (intersection) 9R:C j (existential restriction) 8R:C j (universal restriction) where A is an atomic concept, C and D are concepts, and R is a role. 8/35 ALC: Syntax The description logic ALC (Attribute Language with general Complement) allows the following concepts: C; D ! A j (atomic concept) > j (universal concept) ? j (bottom concept) :C j (negation) C t D j (union) C u D j (intersection) 9R:C j (existential restriction) 8R:C j (universal restriction) where A is an atomic concept, C and D are concepts, and R is a role. We allow { ABox assertions: C(a) and R(a; b) for individuals a; b, concepts C and roles R; { TBox axioms: C v D for concepts C and D. 8/35 ALC: Semantics A model M for a knowledge base K consists of { a nonempty set ∆, and { an interpretation function M, such that: { for every constant c, cM 2 ∆, { for every atomic concept A, AM ⊆ ∆, { for every atomic role R, RM ⊆ ∆ × ∆, 9/35 >M = ∆ ?M = ; (:C)M = ∆nC M (C t D)M = C M [ DM (C u D)M = C M \ DM (8R:C)M = a 2 ∆ j 8b 2 ∆ ha; bi 2 RM ! b 2 C M (9R:C)M = a 2 ∆ j 9b 2 ∆ ha; bi 2 RM ^ b 2 C M ALC: Semantics M is extended inductively as 10/35 ALC: Semantics M is extended inductively as >M = ∆ ?M = ; (:C)M = ∆nC M (C t D)M = C M [ DM (C u D)M = C M \ DM (8R:C)M = a 2 ∆ j 8b 2 ∆ ha; bi 2 RM ! b 2 C M (9R:C)M = a 2 ∆ j 9b 2 ∆ ha; bi 2 RM ^ b 2 C M 10/35 M M { C v D, denoted M C v D, iff C ⊆ D ; M M { R v P, denoted M R v P, iff R ⊆ P .