[PPT] - Knowledge Representation for the Semantic Web Lecture 2: Description PowerPoint Presentation

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Introduction Syntax of Description Logics

Knowledge Representation for the Semantic Web Lecture 2: Description Logics I

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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Introduction Syntax of Description Logics

Unit Outline

Introduction Syntax of Description Logics

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Introduction Syntax of Description Logics

Logic-based Knowledge Representation

❼ 350 BC: roots of logic-based KR ❼ 17th century: idea to make knowledge explicit by logical

computation

❼ 1930s: disillusion due to results about fundamental limits

for the existence of generic algorithms

❼ adoption of computers and AI as a new area of research

leads to intensified studies

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Introduction Syntax of Description Logics

Propositional and First-order Logic

(1) Aristotel is a man. (2) Socrates is a man.

❼ ❼

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Introduction Syntax of Description Logics

Propositional and First-order Logic

(1) Aristotel is a man. (2) Socrates is a man. In which formalisms can we encode this knowledge?

❼ ❼

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Introduction Syntax of Description Logics

Propositional and First-order Logic

(1) Aristotel is a man. (2) Socrates is a man. In which formalisms can we encode this knowledge?

❼ propositional logic (PL): propositional variables, ¬, ∨, ∧, →

(1) AristotelIsAMan = true; (2) SocratesIsAMan = true

❼

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Introduction Syntax of Description Logics

Propositional and First-order Logic

(1) Aristotel is a man. (2) Socrates is a man. (3) All men are mortal. In which formalisms can we encode this knowledge?

❼ propositional logic (PL): propositional variables, ¬, ∨, ∧, →

(1) AristotelIsAMan = true; (2) SocratesIsAMan = true

❼

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Introduction Syntax of Description Logics

Propositional and First-order Logic

(1) Aristotel is a man. (2) Socrates is a man. (3) All men are mortal. In which formalisms can we encode this knowledge?

❼ propositional logic (PL): propositional variables, ¬, ∨, ∧, →

(1) AristotelIsAMan = true; (2) SocratesIsAMan = true (3) AristotelIsAMan → AristotelIsMortal SocratesIsAMan → SocratesIsMortal; PL is not expressive..

❼

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Introduction Syntax of Description Logics

Propositional and First-order Logic

(1) Aristotel is a man. (2) Socrates is a man. (3) All men are mortal. In which formalisms can we encode this knowledge?

❼ propositional logic (PL): propositional variables, ¬, ∨, ∧, →

(1) AristotelIsAMan = true; (2) SocratesIsAMan = true (3) AristotelIsAMan → AristotelIsMortal SocratesIsAMan → SocratesIsMortal; PL is not expressive..

❼ first order logic (FOL): predicates of arbitrary arity, constants,

variables, function symbols, ¬, ∨, ∧, ∀, ∃, → (1) Man(socrates); (2) Man(aristotel); (3) ∀X(Man(X) → Mortal(X))

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Introduction Syntax of Description Logics

Propositional and First-order Logic

(1) Aristotel is a man. (2) Socrates is a man. (3) All men are mortal. In which formalisms can we encode this knowledge?

❼ propositional logic (PL): propositional variables, ¬, ∨, ∧, →

(1) AristotelIsAMan = true; (2) SocratesIsAMan = true (3) AristotelIsAMan → AristotelIsMortal SocratesIsAMan → SocratesIsMortal; PL is not expressive..

❼ first order logic (FOL): predicates of arbitrary arity, constants,

variables, function symbols, ¬, ∨, ∧, ∀, ∃, → (1) Man(socrates); (2) Man(aristotel); (3) ∀X(Man(X) → Mortal(X)) FOL is expressive but undecidable in general...

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Introduction Syntax of Description Logics

Brief Note on Decidability

Decidability A class of problems is called decidable, if there is an algorithm that given any problem instance from this class as input can output a “yes” or “no” answer to it after finite time. Decidable logics In logic context, the following generic problem is normally studied: Given: a set of statements T and a statement φ, Output: “yes”, iff T logically entails φ and “no” otherwise. In case there is no danger of confusion about the type of problem consid- ered, sometimes the logic itself is called decidable or undecidable.

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Introduction Syntax of Description Logics

Brief Note on Decidability (cont’d)

Decidability of propositional logic Consider propositional logic (PL) and the following statements T and φ: (SocrIsAMan → SocrIsMortal) ∧ SocrIsAMan

T

| =

entails

SocrIsMortal

φ

The following questions in PL are equivalent:

❼ T |

= φ?

❼ T → φ for every valuation of socrIsAMan, socrIsMortal? ❼ T ∧ ¬φ is unsatisfiable, i.e., false for every valuation?

The (un)satisfiability problem in PL is called (UN)SAT. Propositional logic is decidable, since (UN)SAT is decidable (consider 2n truth assignments of n variables in T ∧ ¬φ).

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Introduction Syntax of Description Logics

Description Logics

❼ 1930’s: First order logic for KR (undecidable) ❼

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Introduction Syntax of Description Logics

Description Logics

❼ 1930’s: First order logic for KR (undecidable) ❼ 1970’s: Network-shaped structures for KR

❼ Semantic networks [Quillian, 1968], conceptual graphs, SNePs, NETL ❼ Frames [Minsky, 1974]

❼ ❼

❼ ❼ ❼

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Introduction Syntax of Description Logics

Description Logics

❼ 1930’s: First order logic for KR (undecidable) ❼ 1970’s: Network-shaped structures for KR (no formal semantics)

❼ Semantic networks [Quillian, 1968], conceptual graphs, SNePs, NETL ❼ Frames [Minsky, 1974]

❼ ❼

❼ ❼ ❼

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Introduction Syntax of Description Logics

Description Logics

❼ 1930’s: First order logic for KR (undecidable) ❼ 1970’s: Network-shaped structures for KR (no formal semantics)

❼ Semantic networks [Quillian, 1968], conceptual graphs, SNePs, NETL ❼ Frames [Minsky, 1974]

❼ 1979: Encoding of frames into FOL [Hayes, 1979] ❼

❼ ❼ ❼

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Introduction Syntax of Description Logics

Description Logics

❼ 1930’s: First order logic for KR (undecidable) ❼ 1970’s: Network-shaped structures for KR (no formal semantics)

❼ Semantic networks [Quillian, 1968], conceptual graphs, SNePs, NETL ❼ Frames [Minsky, 1974]

❼ 1979: Encoding of frames into FOL [Hayes, 1979] ❼ 1980’s: Description logics (DL) for KR

❼ Decidable fragments of FOL ❼ Theories encoded in DLs are called ontologies ❼ Many DLs with different expressiveness and computational features

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Introduction Syntax of Description Logics

Description Logics

❼ 1930’s: First order logic for KR (undecidable) ❼ 1970’s: Network-shaped structures for KR (no formal semantics)

❼ Semantic networks [Quillian, 1968], conceptual graphs, SNePs, NETL ❼ Frames [Minsky, 1974]

❼ 1979: Encoding of frames into FOL [Hayes, 1979] ❼ 1980’s: Description logics (DL) for KR

❼ Decidable fragments of FOL ❼ Theories encoded in DLs are called ontologies ❼ Many DLs with different expressiveness and computational features

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Introduction Syntax of Description Logics

Description Logics (cont’d)

❼ Goal: ensure decidable reasoning and formal logic-based semantics ❼ Description logics cater for this goal ❼ They can be seen as decidable fragments of first-order logic, closely

related to modal logics

❼ A significant portion of DL-related research devoted to clarifying the

computational effort of reasoning tasks in terms of their worst-case complexity

❼ Despite high worst-case complexity, even for expressive DLs

ptimized reasoning algorithms exist with good behaviour in

practical relevant settings

❼ cf. SAT Solving: NP-complete in general but works well in practice

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Introduction Syntax of Description Logics

Description Logics (cont’d)

❼ Description logics one of today’s main KR paradigms ❼ influenced standardization of Semantic Web

languages, in particular the web ontology language OWL

❼ comprehensive tool support available

Fact++ Pellet HermiT ELK

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Introduction Syntax of Description Logics

Applications

❼ Semantic Web (OWL) ❼ Enterprise Application Integration (EAI) ❼ Data Modelling (UML) ❼ Knowledge Representation for life sciences: SNOMED Clinical

Terms, Gene ontology, UniProtKB/Swiss-Prot protein sequence database, GALEN medical concepts for e-healthcare

❼ Ontology-Based Data Access (OBDA) ❼ . . .

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Introduction Syntax of Description Logics

Syntax of Description Logics

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Introduction Syntax of Description Logics

DL Building Blocks

❼ Individual names: john, mary, sun, lalaland

aka: constants (FOL), resources (RDF)

❼ Concept names: Male, Planet, Film, Country

aka: unary predicates (FOL), classes (RDFS)

❼ Role names: married, fatherOf , actedIn

aka: binary predicates (FOL), properties (RDFS) The set of all individual, concept and role names is commonly referred to as signature or vocabulary.

fatherOf

married married

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Introduction Syntax of Description Logics

Constituents of a DL Knowledge Base

RBox R TBox T ABox A

❼ information about individuals

and their concept and role memberships

❼ information about concepts and

their taxonomic dependencies

❼ information about roles and

their dependencies

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Introduction Syntax of Description Logics

Constituents of a DL

A DL is characterized by:

❼ A description language: how to form concept/role expressions

Human ⊓ Male ⊓ ∃hasChild ⊓ ∀hasChild.(Doctor ⊔ Lawyer)

❼ A mechanism to specify knowledge about concepts (i.e., TBox T )

and roles (i.e., RBox R) T = {Father ≡ Human ⊓ Male ⊓ ∃hasChild, HappyFather ⊑ Father ⊓ ∀hasChild.(Doctor ⊔ Lawyer)} R = {hasFather ⊑ hasParent}

❼ A mechanism to specify properties of objects (i.e., an ABox)

A = {HappyFather(john), hasChild(john, mary)}

❼ A set of inference services: how to reason on a given KB

T | = HappyFather ⊓ ∃hasChild.(Doctor ⊓ Lawyer) T ∪ A | = (Doctor ⊓ Lawyer)(mary)

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Introduction Syntax of Description Logics

Concept Expressions

❼ Concept expressions are defined inductively as follows:

❼ every concept name is a concept expression, ❼ ⊤ and ⊥ are concept expressions, ❼ for a1, . . . , an individual names, {a1, . . . , an} is a concept expression, ❼ for C and D concept expressions, ¬C and C ⊓ D and C ⊔ D are

concept expressions,

❼ for r a role and C a concept expression, ∃r.C and ∀r.C are concept

expressions,

❼ for s a simple role, C a concept expression and n a natural number,

∃s.Self and ≤n s.C and ≥n s.C are concept expressions. ❼ Note: we formally define roles and simple roles later (for the

moment, we use role names)

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Introduction Syntax of Description Logics

Examples of Concept Expressions

❼ Conjunction: Singer ⊓ Actor ❼ Disjunction: ∀hasChild.(Doctor ⊔ Lawyer) ❼ Qualified existential restriction: ∃hasChild.Doctor ❼ Full negation: ¬(Doctor ⊔ Lawyer) ❼ Number restrictions: (≥ 2hasChild) ⊓ (≤ 1sibling) ❼ Qualified number restrictions: (≥ 2hasChild.Doctor) ❼ Inverse role: ∀hasChild−.Doctor

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Introduction Syntax of Description Logics

TBox

❼ A general concept inclusion (GCI) has the form

C ⊑ D where C and D are concept expressions.

❼ A TBox consists of a set of GCIs.

N.B.: Definition of TBox presumes already known RBox due to role simplicity constraints. TBox T

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Introduction Syntax of Description Logics

Example Knowledge Base

TBox T Healthy ⊑ ¬Dead ”Healthy beings are not dead.” Cat ⊑ Dead ⊔ Alive ”Every cat is dead or alive.” HappyCatOwner ⊑ ∃owns.Cat ⊓ ∀caresFor.Healthy ”A happy cat owner owns a cat and all beings he cares for are healthy.”

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Introduction Syntax of Description Logics

ABox

❼ An individual assertion can have any of the following forms

❼ C(a), called concept assertion ❼ r(a, b), called role assertion ❼ ¬r(a, b), called negated role assertion ❼ a ≈ b, called equality statement, or ❼ a ≈ b, called inequality statement.

❼ An ABox consists of a set of individual

assertions. ABox A

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Introduction Syntax of Description Logics

Example Knowledge Base

TBox T Healthy ⊑ ¬Dead ”Healthy beings are not dead.” Cat ⊑ Dead ⊔ Alive ”Every cat is dead or alive.” HappyCatOwner ⊑ ∃owns.Cat ⊓ ∀caresFor.Healthy ”A happy cat owner owns a cat and all beings he cares for are healthy.” ABox A HappyCatOwner(schroedinger) ”Schr¨

dinger is a happy cat owner.”

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Introduction Syntax of Description Logics

Role Incusion Axioms

❼ A role can be

❼ a role name r or ❼ an inverted role name r− (intuitively, reversed participants) or ❼ the universal role u.

❼ A role inclusion axiom (RIA) is a statement of the form

r1 ◦ · · · ◦ rn ⊑ r where r1, . . . , rn, r are roles.

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Introduction Syntax of Description Logics

Role Simplicity

❼ Given RIAs, roles are divided into simple and non-simple roles. ❼ Roughly, roles are non-simple if they may occur on the rhs of a

complex RIA.

❼ More precisely,

❼ for any RIA r1 ◦ r2 ◦ . . . ◦ rn ⊑ r with n > 1, r is non-simple, ❼ for any RIA s ⊑ r with s non-simple, r is non-simple, and ❼ all other properties are simple.

Example

q ◦ p ⊑ r r ◦ p ⊑ r r ⊑ s p ⊑ r q ⊑ s

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Introduction Syntax of Description Logics

Role Simplicity

❼ Given RIAs, roles are divided into simple and non-simple roles. ❼ Roughly, roles are non-simple if they may occur on the rhs of a

complex RIA.

❼ More precisely,

❼ for any RIA r1 ◦ r2 ◦ . . . ◦ rn ⊑ r with n > 1, r is non-simple, ❼ for any RIA s ⊑ r with s non-simple, r is non-simple, and ❼ all other properties are simple.

Example

q ◦ p ⊑ r r ◦ p ⊑ r r ⊑ s p ⊑ r q ⊑ s non-simple: r, s

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Introduction Syntax of Description Logics

Role Simplicity

❼ Given RIAs, roles are divided into simple and non-simple roles. ❼ Roughly, roles are non-simple if they may occur on the rhs of a

complex RIA.

❼ More precisely,

❼ for any RIA r1 ◦ r2 ◦ . . . ◦ rn ⊑ r with n > 1, r is non-simple, ❼ for any RIA s ⊑ r with s non-simple, r is non-simple, and ❼ all other properties are simple.

Example

q ◦ p ⊑ r r ◦ p ⊑ r r ⊑ s p ⊑ r q ⊑ s non-simple: r, s simple: p, q

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Introduction Syntax of Description Logics

RBox

❼ A role disjointness statement has the form

Dis(s1, s2) where s1 and s2 are simple roles.

❼ An RBox consists of regular1 set of RIAs and a set of role

disjointness statements.

❼ In expressive Description Logics, R might

contain further axioms, such as Asym(r) (asymmetry) and Ref (r) (reflexivity). RBox R

1Syntactic conditions put on the usage of non-simple roles (see [Rudolph, 2011]) 23 / 25

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Introduction Syntax of Description Logics

Example Knowledge Base

RBox R

wns

⊑ caresFor ”If somebody owns something, s/he cares for it.” TBox T Healthy ⊑ ¬Dead ”Healthy beings are not dead.” Cat ⊑ Dead ⊔ Alive ”Every cat is dead or alive.” HappyCatOwner ⊑ ∃owns.Cat ⊓ ∀caresFor.Healthy ”A happy cat owner owns a cat and all beings he cares for are healthy.” ABox A HappyCatOwner(schroedinger) ”Schr¨

dinger is a happy cat owner.”

Exercise: try to compute all facts that follow from the KB yourself!24 / 25

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Introduction Syntax of Description Logics

Summary

1. Introduction and background

❼ Brief recap on propositional and first order logic ❼ Decidability of logics ❼ History of DLs

2. Syntax of DLs

❼ DL building blocks ❼ Concept expressions ❼ TBox ❼ ABox ❼ RBox

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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 /