Description Logics
Dr. Alexandra I. Cristea
Description Logics
Description Logics allow formal concept definitions that
can be reasoned about to be expressed
Not a single logic, but a family of KR logics
Subsets of first-order logic (FOL)
Advantages:
– Well-defined model theory
– Known computational complexity
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Why Description Logics in OWL?
OWL exploits results of 20+ years of DL research
– Well defined (model theoretic) semantics
– Formal properties well understood, e.g., computational complexity
“I can’t find an efficient algorithm,
but neither can all these famous people.”
Garey & Johnson. Computers and Intractability:
A Guide to the Theory of NP-Completeness. 1979.
– Known reasoning algorithms
– Implemented systems (highly optimised): FaCT++, Racer, Pellet, etc.
Foundational research was crucial to the design of OWL
– “Why not extend the language with feature x, which is clearly harmless?”
– “Adding x would lead to undecidability — see proof in [. . . ]”
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Why ontology reasoning?
Given key role of ontologies in many applications, it is essential to
provide tools and services to help users:
– Design and maintain high quality ontologies, e.g.:
– Meaningful — all named classes can have instances (remember ‘mad cows’?)
– Correct — captures intuitions of domain experts
– Minimally redundant — no unintended synonyms
– Answer queries, e.g.:
– Find more general/specific classes
– Retrieve individuals/tuples matching a given query
– But what is (formal) reasoning?
The laws of correct reasoning are the subject of mathematical logic
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Exercise: give examples of (principles of) ‘correct reasoning’
All humans are mortal.
Socrates is a human.
Ergo Socrates is mortal.
Hamlet: “To be or not to be”
David Hume: “The sun has risen in the east every morning up
until now. Ergo the sun will also rise in the east tomorrow.”
Aristotle:
If a drink is made with boiling water, it will be hot.
This drink was not made with boiling water.
Ergo this drink is not hot.
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Example concept definitions in DL
.
Woman = Person ⊓ Female
.
Man = Person ⊓ ¬Woman
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Reasoning in DL
A classifier (a reasoning engine) can construct
class hierarchy from ontology concepts defs
Concept definitions composed from primitives
>> ontology is more maintainable
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Example Reasoning
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Example Reasoning
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Example Reasoning
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DL Terminology
DL separates assertions and concept definitions:
A Box: Assertions
– e.g. hasChild(john, mary)
– This is the knowledge base
• T Box: Terminology
– The definitions of concepts in the ontology
– Example axioms for definitions
• C ⊑ D [C is a subclass of D, D subsumes C]
• C ≡ D [C is defined by the expression D]
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DL architecture
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DL Terminology (2)
Concept: class, category or type
Role: binary relation
– Attributes are functional roles
Subsumption:
– D subsumes C if C is a subclass of D : i.e. all Cs are Ds
Unfoldable terminologies:
– The defined concept does not occur in the defining expression:
– C ≡ D where C does not occur in the expression D
Language families
– AL: Attributive Language
– ALC: adds full negation to AL
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Language elements for concept expressions
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Universal restriction
Universal (value) restriction: ∀R.C
The set {x|∀y, R(x, y) → y∈C}
The set of things x such that for all y where x and y are related by R, y is in C.
e.g., ∀hasChild.Parent
Set of things x so that for all y where x and y are related by hasChild, y will be in class
Parent; everything in set x is a child, everything in set y is a parent.
That is, anything that is the object of the relation hasChild must be in class Parent,
regardless of what the subject is.
This is a local statement: this is true for every statement in your
dataset.
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Existential restriction
Existential restriction - also called exists restriction: ∃R.C
The set {x|∃y, R(x, y) ⋀ y∈C}
The set of things x such that there exists a y where x and y are related via R and y is
in class C.
e.g., ∃hasChild.Doctor
The set of all x’s such that x is related to y via hasChild and y is in class Doctor.
the set of all children which have at least one parent who is a doctor
This is a local statement: this is true for at least one statement in your
dataset.
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DL naming
Attributive language - basic language which allows:
–
–
–
–
atomic negation
concept intersection
universal restrictions
limited existential quantification
Frame based description language, allows:
–
–
–
–
concept intersection
universal restrictions
limited existential quantification
role restriction
allows:
– concept intersection
– existential restrictions (of full existential quantification)
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DL naming: Extensions
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DL naming: Exceptions
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Common DLs
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Homework
Explain, based on the naming conventions
you’ve just seen, what do the DL types for
OWL2, Protégé, OWL-Lite, and OWL-DL
stand for.
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Example concept expressions
Parent ≡ “Persons who have (amongst other things) some children”
Person ⨅ ∃hasChild.⊤
ParentOfBoys ≡ “Persons who have some children, and only have
children that are male”
Person ⨅ (∃hasChild. ⊤) ⨅(∀hasChild.Male)
ScottishParent ≡ “Persons who only have children that drink (amongst
other things) some IrnBru”
Person ⨅ (∀hasChild. (∃drink.IrnBru))
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Value and exists restrictions
{a,b,c,d,e,f} are instances; Plant and Animal are classes
⊓
Plant ⊓ Animal ⊑ ⊥
(disjointness)
⊤ ⊑ Plant Animal
(partition)
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Value and exists restrictions
{a,b,c,d,e,f} are instances; Plant and Animal are classes
∃eats.Animal = {c,d,e}
∃eats.Animal ⊓ ∀eats.Animal = {c,e}
∀eats.Animal = {a,b,c,e,f}
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Model theory
Δ1 universal domain of individuals, let Δ1 ={a,b,c,d,e,f}
eats1 set of pairs for the relation eats, let
eats1 = {<d,a>,<d,e>,<e,d>,<e,f>,<c,f>}
For all concepts C:
i) C1 ⊆ Δ 1
ii) C1≠ ∅
Let Animal1 = {d,e,f}
∴ (¬Animal)1 = {a,b,c}
∴ (∀eats.Animal)1={a,b,c,e,f}
∴ (∃eats. Animal)1 = {c,d,e}
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Inference
MeatEater ≡ ∀eats. Animal = {a,b,c,e,f}
Vegetarian ≡ ∀eats. ¬Animal = {a,b,f}
Omnivore ≡ ∃eats. Animal = {c,d,e}
Inference:
From the above classes we can see that:
• MeatEater subsumes Vegetarian
• Vegetarian is disjoint from Omnivore
in this model, with these definitions.
The problem is to prove this for ALL models.
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DL Inference
Inference can be expressed in terms of the model
– Satisfiability of C: C1 is non-empty
– Subsumption C ⊑ D iff C1 ⊆ D1 (“C is subsumed
by D”)
– Equivalence C ≡ D iff C1 = D1
– Disjointness(C ⨅ D) ⊑ iff C1 ∩ D1 ≡ ∅
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• Tractable/terminating inference algorithms exist
Last time:
– Started DL
– DL terminology, some reasoning/ inference,
models, architecture
Next:
– More on DL inference
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Value and exists restrictions
{a,b,c,d,e,f} are instances; Plant and Animal are classes
Vegetarian = {a,b,f}
Omnivore = {c,d,e}
disjoint?
MeatEater= {a,b,c,e,f}
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DL Inference (2)
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DL Inference (2)
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DL Inference (2)
Yes
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DL Inference (2)
Yes
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DL Inference (2)
Yes
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DL Inference (2)
Yes
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DL Inference (2)
Yes
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DL Inference (3)
Inference has 2 equivalent notions - so implementing one lets us
prove all 4 properties
Reduction to subsumption ⊑:
– Unsatisfiability of C: C ⊑ ⊥
– Equivalence C≡D iff C ⊑ D and D ⊑ C
– Disjointness (C ⨅ D) ⊑ ⊥
Reduction to unsatisfiability CI = ∅ :
– Subsumption C ⊑ D iff (C ⨅ ¬D) is unsatisfiable
– Equivalence C≡D iff (C ⨅ ¬D) and (D ⨅ ¬C) are unsatisfiable
– Disjointness (C ⨅ D) is unsatisfiable
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DL Summary
DLs: family of languages based on subsets of FOL.
Expressivity depends on language attributes.
Attributes: indicated by letters; DL language names: series
of letters => expressivity of DL language in its name.
DLs allow complex expressions of how concepts relate to
one another.
Many algorithms (e.g., Tableaux Algorithms) allowing
efficient reasoning over DLs.
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Additional Slides DL
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