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Knowledge Representation for the Semantic Web Lecture 3: Description - - PowerPoint PPT Presentation

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Semantics of Description Logics DL Nomenclature Equivalences Knowledge Representation for the Semantic Web Lecture 3: Description Logics II Daria Stepanova slides based on Reasoning Web 2011 tutorial Foundations of Description Logics and

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Semantics of Description Logics DL Nomenclature Equivalences

Knowledge Representation for the Semantic Web Lecture 3: Description Logics II

Daria Stepanova

slides based on Reasoning Web 2011 tutorial “Foundations of Description Logics and OWL” by S. Rudolph

Max Planck Institute for Informatics D5: Databases and Information Systems group

WS 2017/18

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Semantics of Description Logics DL Nomenclature Equivalences

Unit Outline

Semantics of Description Logics DL Nomenclature Equivalences

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Semantics of Description Logics DL Nomenclature Equivalences

Semantics of Description Logics

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Semantics of Description Logics DL Nomenclature Equivalences

Interpretations

Semantics for DLs is defined in a model theoretic way, i.e., based on ”abstract possible worlds”, called interpretations. Def.: An interpretation I = (∆I, ·I) consists of:

❼ a nonempty set ∆I, called the interpretation domain (of I) ❼ an interpretation function ·I, which maps

❼ each atomic concept A to a subset AI of ∆I ❼ each atomic role r to a subset rI of ∆I × ∆I.

individual names NI . . . a . . . class names NC . . . C . . . role names NR . . . r . . .

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Semantics of Description Logics DL Nomenclature Equivalences

Interpretations

Semantics for DLs is defined in a model theoretic way, i.e., based on ”abstract possible worlds”, called interpretations. Def.: An interpretation I = (∆I, ·I) consists of:

❼ a nonempty set ∆I, called the interpretation domain (of I) ❼ an interpretation function ·I, which maps

❼ each atomic concept A to a subset AI of ∆I ❼ each atomic role r to a subset rI of ∆I × ∆I.

∆I

individual names NI . . . a . . . class names NC . . . C . . . role names NR . . . r . . .

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Semantics of Description Logics DL Nomenclature Equivalences

Interpretations

Semantics for DLs is defined in a model theoretic way, i.e., based on ”abstract possible worlds”, called interpretations. Def.: An interpretation I = (∆I, ·I) consists of:

❼ a nonempty set ∆I, called the interpretation domain (of I) ❼ an interpretation function ·I, which maps

❼ each atomic concept A to a subset AI of ∆I ❼ each atomic role r to a subset rI of ∆I × ∆I.

∆I aI

individual names NI . . . a . . . class names NC . . . C . . . role names NR . . . r . . .

·I

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Semantics of Description Logics DL Nomenclature Equivalences

Interpretations

Semantics for DLs is defined in a model theoretic way, i.e., based on ”abstract possible worlds”, called interpretations. Def.: An interpretation I = (∆I, ·I) consists of:

❼ a nonempty set ∆I, called the interpretation domain (of I) ❼ an interpretation function ·I, which maps

❼ each atomic concept A to a subset AI of ∆I ❼ each atomic role r to a subset rI of ∆I × ∆I.

∆I aI C I

individual names NI . . . a . . . class names NC . . . C . . . role names NR . . . r . . .

·I ·I

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Semantics of Description Logics DL Nomenclature Equivalences

Interpretations

Semantics for DLs is defined in a model theoretic way, i.e., based on ”abstract possible worlds”, called interpretations. Def.: An interpretation I = (∆I, ·I) consists of:

❼ a nonempty set ∆I, called the interpretation domain (of I) ❼ an interpretation function ·I, which maps

❼ each atomic concept A to a subset AI of ∆I ❼ each atomic role r to a subset rI of ∆I × ∆I.

rI ∆I aI C I

individual names NI . . . a . . . class names NC . . . C . . . role names NR . . . r . . .

·I ·I ·I

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Semantics of Description Logics DL Nomenclature Equivalences

Interpretations: an Example

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Semantics of Description Logics DL Nomenclature Equivalences

Interpretation of Individuals

Unique Name Assumption (UNA) If c1 and c2 are two individuals such that c1 = c2, then cI

1 = cI 2

Note: When the UNA holds, equality and distincntness assertions are

  • meaningless. In DLs one can drop UNA.

Example: absence of UNA Two fathers (f1, f2) and two sons (s1, s2) went to a pizzeria and bought three pizzas for picnic lunch. When they started their lunch, every-

  • ne had a whole pizza. How could

this happen?

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Semantics of Description Logics DL Nomenclature Equivalences

Interpretation of Individuals

Unique Name Assumption (UNA) If c1 and c2 are two individuals such that c1 = c2, then cI

1 = cI 2

Note: When the UNA holds, equality and distincntness assertions are

  • meaningless. In DLs one can drop UNA.

Standard Name Assumption (SNA) The UNA holds, and moreover individuals are interpreted in the same way in all interpretations. Hence, we can assume that ∆I contains the set of individuals, and that for each interpretation I, we have that cI = c (then c is called standard name)

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Semantics of Description Logics DL Nomenclature Equivalences

Interpretation of Concept Expressions

Construct Syntax Example Semantics atomic concept A Doctor AI ⊆ ∆I atomic role r hasChild rI ⊆ ∆I × ∆I atomic negation ¬A ¬Doctor ∆I\AI conjunction C ⊓ D Human ⊓ Male CI ∩ DI

  • unqual. exist. res.1

∃r ∃hasChild {o | ∃o′.(o, o′) ∈ rI} value res. ∀r.C ∀hasChild.Male {o | ∀o′.(o, o′) ∈ rI → o′ ∈ CI} bottom ⊥ ∅

C, D denote arbitrary concepts and r denotes an arbitrary role. The above constructs form the basic language AL

1Unqualified existential restriction 7 / 34

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Semantics of Description Logics DL Nomenclature Equivalences

Interpretation of Concept Expressions, cont’d

Construct AL Syntax Semantics disjunction U C ⊔ D Singer ⊔ Dancer CI ∪ DI

  • qual. exist. res.2

E ∃R.C ∃hasChild.Male {o | ∃o′.(o, o′) ∈ rI ∧ o′ ∈ CI} (full) negation C ¬C ¬(∃hasSibling.Female) ∆I\CI

  • num. res.

N (≥ k r) ≥ 2 hasSister {o | #{o′ | (o, o′) ∈ rI} ≥ k} (≤ k r) ≤ 3 hasBrother {o | #{o′ | (o, o′) ∈ rI} ≤ k}

  • qual. num. res.

Q (≥ k r.C) ≥ 2 hasSibling.Female {o | #{o′ | (o, o′) ∈ rI ∧ o′ ∈ CI} ≥ k (≥ k r.C) ≤ 3 hasSibling.Male {o | #{o′ | (o, o′) ∈ rI ∧ o′ ∈ CI} ≤ k top ⊤ ∆I

Many different DL constructs and their combinations have been investigated. Combining various constructs we obtain a concrete DL fragment, i.e., language (see slide 26 for further details).

2Qualified existential restriction 8 / 34

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Semantics of Description Logics DL Nomenclature Equivalences

Boolean Concept Expressions

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Semantics of Description Logics DL Nomenclature Equivalences

Boolean Concept Expressions

Politician ¬

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Semantics of Description Logics DL Nomenclature Equivalences

Boolean Concept Expressions

Politician ¬ I

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Semantics of Description Logics DL Nomenclature Equivalences

Boolean Concept Expressions

Politician ¬ I I

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Semantics of Description Logics DL Nomenclature Equivalences

Boolean Concept Expressions

Politician ¬ I I

⊓ Politician Actor

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Semantics of Description Logics DL Nomenclature Equivalences

Boolean Concept Expressions

Politician ¬ I I

⊓ Politician Actor I

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Semantics of Description Logics DL Nomenclature Equivalences

Boolean Concept Expressions

Politician ¬ I I

⊓ Politician Actor I I

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Semantics of Description Logics DL Nomenclature Equivalences

Boolean Concept Expressions

Politician ¬ I I

⊓ Politician Actor I I I

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Semantics of Description Logics DL Nomenclature Equivalences

Boolean Concept Expressions

Politician ¬ I I

⊓ Politician Actor I I I ⊔ Politician Actor

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Semantics of Description Logics DL Nomenclature Equivalences

Boolean Concept Expressions

Politician ¬ I I

⊓ Politician Actor I I I ⊔ Politician Actor I

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Semantics of Description Logics DL Nomenclature Equivalences

Boolean Concept Expressions

Politician ¬ I I

⊓ Politician Actor I I I ⊔ Politician Actor I I

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Semantics of Description Logics DL Nomenclature Equivalences

Boolean Concept Expressions

Politician ¬ I I

⊓ Politician Actor I I I ⊔ Politician Actor I I I

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Semantics of Description Logics DL Nomenclature Equivalences

Existential Role Restrictions

parentOf. ∃ Male

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Semantics of Description Logics DL Nomenclature Equivalences

Universal Role Restrictions

parentOf. ∀ Male

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Semantics of Description Logics DL Nomenclature Equivalences

Qualified Number Restrictions

parentOf. ≥2 Male

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Semantics of Description Logics DL Nomenclature Equivalences

Self-Restrictions

killed. ∃ Self

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Semantics of Description Logics DL Nomenclature Equivalences

Interpretation of Role Expressions

Construct Syntax Example Semantics atomic role r hasChild rI ⊆ ∆I × ∆I role negation ¬r ¬hasSister ∆I × ∆I\{(o, o′) ∈ rI} inverse role r− hasParent− {(o, o′) | (o′, o) ∈ rI} transitivity r ◦ r′ hasChild ◦ hasParent {(o, o′) | (o, o′′) ∈ rI, (o′′, o′) ∈ r′I}

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Semantics of Description Logics DL Nomenclature Equivalences

Inverse Role

childOf − = parentOf

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Semantics of Description Logics DL Nomenclature Equivalences

Role Chain

childOf ◦ parentOf

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Semantics of Description Logics DL Nomenclature Equivalences

Semantics of Axioms

Given a way to determine a semantic counterpart for all expressions, we now define the criteria for checking whether an interpretation I satisfies an axiom alpha α (written: I | = α).

❼ I |

= r1 ◦ . . . ◦ rn ⊑ r

❼ I |

= Dis(s1, s2)

❼ I |

= C ⊑ D

❼ I |

= C(a)

❼ I |

= r(a, b)

❼ I |

= ¬r(a, b)

❼ I |

= a ≈ b

❼ I |

= a ≈ b if (r1 ◦ . . . ◦ rn)I ⊆ rI if sI

1 ∩ sI 2 = {}

if CI ⊆ DI if aI ∈ DI if (aI, bI) ∈ rI if (aI, bI) ∈ rI if aI = bI if aI = bI

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Semantics of Description Logics DL Nomenclature Equivalences

Concept and Role Membership

Male(nicolas)

  • married(carla, nicolas)

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Semantics of Description Logics DL Nomenclature Equivalences

General Inclusion Axioms

ExPresident ⊑ Politician I I

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Semantics of Description Logics DL Nomenclature Equivalences

Role Inclusion Axioms

childOf ◦ parentOf ⊑ siblingOf

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Semantics of Description Logics DL Nomenclature Equivalences

DL to First Order Logic

Syntax FOL formalization A1 ⊑ A2 ∀x(A1(x) → A2(x)) R1 ⊑ R2 ∀x, y(R1(x, y) → R2(x, y)) A1 ⊑ ¬A2 ∀x(A1(x) → ¬A2(X)) R1 ⊑ ¬R2 ∀x, y(R1(x, y) → ¬R2(x, y)) ∃R ⊑ A ∀x(∃y(R(x, y)) → A(x)) ∃R− ⊑ A ∀x(∃y(R(y, x)) → A(x)) A ⊑ ∃R ∀x(A(x) → ∃y(R(x, y))) funct(R) ∀x, y, y′(R(x, y) ∧ R(x, y′) → y = y′) A1 ⊓ A2 ⊑ A3 ∀xA1(x) ∧ A2(x) → A3(x) ∃R.A1 ⊑ A2 ∀x(∃y(R(x, y) ∧ A1(y)) → A2(x) A1 ⊑ ∃R.A2 ∀x(A(x) → ∃y(R(x, y) ∧ A2(y))) . . . . . .

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Semantics of Description Logics DL Nomenclature Equivalences

Semantics via Translation into FOL

As (common) DLs can be seen as fragments of FOL, one can also define the semantics by providing a translation of DL axioms into FOL formulas.

❼ τR(r, x, y): produce for r(x, y) a formula with free variables x, y ❼ τC(C, x): produce for C(x) a formula with free variable x ❼ define transformations recursively

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Semantics of Description Logics DL Nomenclature Equivalences

Semantics via Translation into FOL

As (common) DLs can be seen as fragments of FOL, one can also define the semantics by providing a translation of DL axioms into FOL formulas.

❼ τR(r, x, y): produce for r(x, y) a formula with free variables x, y ❼ τC(C, x): produce for C(x) a formula with free variable x ❼ define transformations recursively

Bottom rewriting: τR(u, xi, xj) = true τR(r, xi, xj) = r(xi, xj) τR(r−, xi, xj) = r(xj, xi)

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Semantics of Description Logics DL Nomenclature Equivalences

Semantics via Translation into FOL

As (common) DLs can be seen as fragments of FOL, one can also define the semantics by providing a translation of DL axioms into FOL formulas.

❼ τR(r, x, y): produce for r(x, y) a formula with free variables x, y ❼ τC(C, x): produce for C(x) a formula with free variable x ❼ define transformations recursively

Bottom rewriting: τR(u, xi, xj) = true τR(r, xi, xj) = r(xi, xj) τR(r−, xi, xj) = r(xj, xi) τC(A, xi) = A(xi) τC(⊤, xi) = true τC(⊥, xi) = false τC({a1, . . . , an}, xi) =

n

  • j=1

xi = aj

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Semantics of Description Logics DL Nomenclature Equivalences

Semantics via Translation into FOL (ctd.)

Complex concepts: τC(C ⊓ D, xi) = τC(C, xi) ∧ τC(D, xi) τC(C ⊔ D, xi) = τC(C, xi) ∨ τC(D, xi) τC(¬C, xi) = ¬τC(C, xi) τC(∃r.C, xi) = ∃xi+1.(τR(r, xi, xi+1) ∧ τC(C, xi+1)) τC(∀r.C, xi) = ∀xi+1.(τR(r, xi, xi+1) → τC(C, xi+1)) τC(∃r.Self, xi) = τR(r, xi, xi) τC(≥ nr.C, xi) = ∃xi+1 . . . xi+n.(

i+n

  • j=i+1

i+n

  • k=j+1

(xj = xk) ∧

i+n

  • j=i+1

i+n

  • k=j+1

(τR(r, xi, xj) ∧ τC(C, xj)) τC(≤ nr.C, xi) = ¬τC(≥ (n + 1)r.C, xi)

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Semantics of Description Logics DL Nomenclature Equivalences

Semantics via Translation into FOL (ctd.)

Axioms: τ(C ⊑ D) = ∀x0(τC(C, x0) → τC(D, x0)) τ(r1 ◦ . . . ◦ rn ⊑ r) = ∀x0 . . . xn(n

i=1 τR(ri, xi−1, xi)) → τR(r, x0, xn)

τ(Dis(r, r′)) = ∀x0, x1(τR(r, x0, x1) → ¬τR(r′, x0, x1) τ(Ref (r, r′)) = ∀x τR(r, x, x) τ(Asym(r)) = ∀x0, x1.(τR(r, x0, x1) → ¬τR(r, x1, x0))

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Semantics of Description Logics DL Nomenclature Equivalences

Semantics via Translation into FOL (ctd.)

Axioms: τ(C ⊑ D) = ∀x0(τC(C, x0) → τC(D, x0)) τ(r1 ◦ . . . ◦ rn ⊑ r) = ∀x0 . . . xn(n

i=1 τR(ri, xi−1, xi)) → τR(r, x0, xn)

τ(Dis(r, r′)) = ∀x0, x1(τR(r, x0, x1) → ¬τR(r′, x0, x1) τ(Ref (r, r′)) = ∀x τR(r, x, x) τ(Asym(r)) = ∀x0, x1.(τR(r, x0, x1) → ¬τR(r, x1, x0)) Assertions: τ(C(a)) = τC(C, x0)[x0/a] τ(r(a, b)) = τR(r, x0, x1)[x0/a][x1/b] τ(¬r(a, b)) = ¬τ(r(a, b)) τ(a ≈ b) = a = b τ(a ≈ b) = ¬(a = b)

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Semantics of Description Logics DL Nomenclature Equivalences

Description Logics Nomenclature

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Semantics of Description Logics DL Nomenclature Equivalences

Naming Scheme for Expressive DLs

((ALC | S)[H] | SR)[O][I][F | N | Q]

❼ S stands for ALC + role transitivity ❼ H stands for role hierarchies ❼ O stands for nominals, i.e., closed classes {o} such as {john, mary, tom} ❼ I stands for inverse roles (seen soon) ❼ F stands for role functionality (⊤ ⊑≤ 1.r) ❼ N (Q) stands for arbitray (qualified) cardinality restrictions ❼ R stands for role box with all kinds of role axioms plus self concepts Note: ❼ S subsumes ALC, SR subsumes (ALC | S)[H] ALCH ❼ SROIQ subsumes all the other description logics in this scheme. ❼ N makes F obsolete ❼ Q makes N (and F) obsolete

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Semantics of Description Logics DL Nomenclature Equivalences

DL Syntax - Overview

Concepts ALC Atomic A, B Not ¬C And C ⊓ D Or C ⊔ D Exists ∃r.C For all ∀r.C Q (N) At least ≥n r.C (≥n r) At most ≤n r.C (≤n r) O Closed class {i1, . . . , in} R Self ∃r.Self Roles I Atomic r Inverse r−

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Semantics of Description Logics DL Nomenclature Equivalences

DL Syntax - Overview ctd.

Ontology (=Knowledge Base) Concept Axioms TBox Subclass C ⊑ D Equivalent C ≡ D Assertional Axioms ABox Instance C(a) Role r(a, b) Same a ≈ b Different a ≈ b Role Axioms RBox H Subrole r ⊑ s S Transitivity Trans(r) SR Role Chain r ◦ r′ ⊑ s Role Disjointness Disj(s, r)

❼ Transitivity and Disjointness are role characteristics. ❼ Further characteristics in SROIQ are asymmetry, Asym(r), and

reflexivity, Ref (r).

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Semantics of Description Logics DL Nomenclature Equivalences

Concept Equivalences

C ≡ D

Two concept expressions C and D are called equivalent (written: C ≡ D), if for every interpretation I holds CI = DI.

❼ Commutativity, Associativity, Idempotence: C ⊓ D ≡ D ⊓ C (C ⊓ D) ⊓ E ≡ C ⊓ (D ⊓ E) C ⊓ C ≡ C C ⊔ D ≡ D ⊔ C (C ⊔ D) ⊔ E ≡ C ⊔ (D ⊔ E) C ⊔ C ≡ C ❼ Double Negation: ¬¬C ≡ C ❼ Complement, de Morgan laws: ¬⊤ ≡ ⊥ C ⊓ ¬C ≡ ⊥ ¬(C ⊓ D) ≡ ¬D ⊔ ¬C ¬⊥ ≡ ⊤ C ⊔ ¬C ≡ ⊤ ¬(C ⊔ D) ≡ ¬D ⊓ ¬C

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Semantics of Description Logics DL Nomenclature Equivalences

Concept Equivalences ctd.

❼ Distributivity, Absorption: (C ⊔ D) ⊓ E ≡ (C ⊓ E) ⊔ (D ⊓ E) (C ⊓ D) ⊔ E ≡ (C ⊔ E) ⊓ (D ⊔ E) C ⊔ (C ⊓ D) ≡ C (C ⊔ D) ⊓ C ≡ C (C ⊓ D) ⊔ C ≡ C C ⊔ (C ⊓ D) ≡ C ❼ Quantifiers and Counting: ¬∃r.C ≡ ∀r.¬C ¬∀r.C ≡ ∃r.¬C ¬ ≤ nr.C ≡ ≥ (n + 1)r.C ¬ ≥ (n + 1)r.C ≡ ≤ nr.C ≥ 0r.C ≡ ⊤ ≥ 1r.C ≡ ∃r.C ≤ 0r.C ≡ ∀r.¬C

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Semantics of Description Logics DL Nomenclature Equivalences

Axiom and KB Equivalences

❼ Lloyd-Topor equivalences

{A ⊔ B ⊑ C} ⇐ ⇒ {A ⊑ C, B ⊑ C} {A ⊑ B ⊓ C} ⇐ ⇒ {A ⊑ B, A ⊑ C}

❼ Turning GCIs into universally valid concept descriptions

C ⊑ D ⇐ ⇒ ⊤ ⊑ ¬C ⊔ D

❼ Internalisation of ABox into TBox

C(a) ⇐ ⇒ {a} ⊑ C r(a, b) ⇐ ⇒ {a} ⊑ ∃r.{b} ¬r(a, b) ⇐ ⇒ {a} ⊑ ¬∃r.{b} a ≈ b ⇐ ⇒ {a} ⊑ {b} a ≈ b ⇐ ⇒ {a} ⊑ ¬{b}

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Semantics of Description Logics DL Nomenclature Equivalences

(Non-)Concept Equivalences

Exercise: Show that the following equivalences are not valid. ∃r.(C ⊓ D) ≡ ∃r.C ⊓ ∃r.D (1) C ⊓ (D ⊔ E) ≡ (C ⊓ D) ⊔ E (2) ∃r.{a} ⊓ ∃r.{b} ≡ ≥ 2 r.{a, b} (3) ∃r.⊤ ⊓ ∃s.⊤ ≡ ∃r.∃r−.∃s.⊤ (4) Exercise: Show that the following equivalences are valid. ∃r−.C ⊑ D ≡ C ⊑ ∀r.D (5) C ⊑ ∀r−.D ≡ ∃r.C ⊑ D (6)

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Semantics of Description Logics DL Nomenclature Equivalences

Concept Subsumption

C ⊑ D

A concept expression C is subsumed by a concept expression D (written: C ⊆ D), if for every interpretation I holds CI ⊆ DI. Some elementary properties:

❼ C ⊑ D ⇐

⇒ C ≡ C ⊔ D

❼ C ≡ D ⇐

⇒ C ⊑ D and D ⊑ C

❼ C ⊑ D and D ⊑ E implies C ⊑ E (transitivity) ❼ C ⊑ D ⇐

⇒ ¬D ⊑ ¬C

❼ C ⊑ D implies C ⊓ E ⊑ D ❼ C ≡ D implies C ⊓ E ≡ D ⊓ E

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Semantics of Description Logics DL Nomenclature Equivalences

Summary

  • 1. Semantics of DLs

❼ Interpretation of ❼ individuals ❼ concept expressions ❼ role expressions ❼ Semantics of axioms ❼ DLs to FOL

  • 2. DL nomenclature

❼ Naming schema ❼ DL syntax overview

  • 3. Concept equivalences

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References I

Franz Baader, Diego Calvanese, Deborah McGuinness, Daniele Nardi, and Peter Patel-Schneider, editors. The Description Logic Handbook: Theory, Implementation and Applications. Cambridge University Press, 2007. Pascal Hitzler, Markus Kr¨

  • tzsch, and Sebastian Rudolph.

Foundations of Semantic Web Technologies. Chapman and Hall, 2010. Sebastian Rudolph. Foundations of description logics. In Axel Polleres, Claudia d’Amato, Marcelo Arenas, Siegfried Handschuh, Paula Kroner, Sascha Ossowski, and Peter Patel-Schneider, editors, Reasoning Web. Semantic Technologies for the Web of Data, volume 6848

  • f Lecture Notes in Computer Science, pages 76–136. Springer Berlin /

Heidelberg, 2011.