lecture 16: description logic
- basic formalisms, TBox, ABox
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description logic
basic formalism
DL provides facilities to set up knowledge bases, to
reason about their content, and to manipulate them
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description logic
basic formalism
TBox
Description
Language
Reasoning
ABox
application
program
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rules
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description logic
basic formalism
vocabulary consists of concepts, that denote sets of
individuals, and roles, that denote binary
relationships between individuals
in addition, DL systems allow their users to build
complex descriptions of concepts and roles (Tbox can
be used to assign names to them)
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description logic
basic formalism
statements in the TBox and ABox can be identified
with formulae in first-order logic or, in some cases, a
slight extension of it
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description logic
basic formalism
typical reasoning tasks (the previous lecture):
determining if a description is satisfiable
or
whether one description is more general than
the other – subsumption
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description logic
basic formalism
important problems for ABox (the previous lecture):
to find out if its set of assertions is consistent
(has a model, and if individuals are instances of
concept descriptions)
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description logic
basic formalism
satisfiability checks descriptions and consistency
checks assertions – all to determine whether a
knowledge base is meaningful at all
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description logic
basic formalism
rules – an extension of the logical core formalism –
allow for adding assertions to a knowledge base
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description logic
description language
atomic concepts – A, B
atomic roles – R
concept descriptions – C, D
description languages are distinguished by the
constructors they provide – we look at the family of
AL-languages (=attributive language)
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description logic
description language – AL
(atomic concept)
(universal concept)
(bottom concept)
(atomic negation)
(intersection)
(value restriction)
(limited exist. quant.)
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description logic
description language – AL
examples:
atomic concepts – Person and Female
Person Female
Ø Female
Person
$ hasChild.
Person
"hasChild.Female
Person
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description logic
semantics of AL–language
to define semantics of AL-concepts – we consider
interpretations I that consist of a non-empty set
and an interpretation function for every atomic
concept A:
and a binary relation:
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description logic
semantics of AL–language
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description logic
semantics of AL–language
examples:
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description logic
family of AL–languages
to obtain more expressive languages – we add
further constructor to AL:
union (U):
I
I
I
(C D) = C È D
full existential quantifier (E):
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description logic
family of AL–languages
number restrictions (N)
negation (C)
AL{U}{E}{N}{C}
and others …
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description logic
AL–language and predicate logic
the semantics of concepts identifiers are fragments of
first-order predicate logic – atomic concepts and roles
are unary and binary predicates
any concept C and be translated into a predicate logic
formula FC(x), with one free variable x such that
elements of satisfying FC(x) is exactly CI
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description logic
AL–language and predicate logic
examples:
f (y) = $x. R(y,x)Ù f (x)
$R .C
C
variable free syntax of DLs is much more concise
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description logic
terminology
forming complex descriptions of concepts:
terminological axioms – make statements
about how concepts or roles are related to each
other
definitions – specific axioms
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description logic
terminology - axioms
in the most general case
inclusion: C D (R S)
equality: C º D (R º S)
C, D are concepts, R, S are roles
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description logic
terminology - axioms
an interpretation I satisfies an inclusion C
, and
it satisfies an equality C º D if C I = D I
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D
if
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description logic
terminology - axioms
if T is a set of axioms then I satisfies T
iff I satisfies each element of T,
if I satisfies a set of axiom then we say that it is a
model of this set of axioms
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description logic
definitions
an equality whose left-hand side is an atomic concept
is a definition
definitions are use to introduce symbolic names for
complex descriptions
Mother º Woman
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$hasChild.Person
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description logic
TBox …
a finite set of definitions T is called
a terminology or TBox
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description logic
TBox …
Woman º Person
Man º Person
Female
ØWoman
Mother º Woman $ hasChild.Person
Father º Man $ hasChild.Person
Parent º Father
Grandmother º Mother
MotherWithManyChildren º Mother
MotherWithoutDaughter º Mother
Wife º Woman
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Mother
$ hasChild.Parent
³ 3 hasChild
" hasChild.ØWoman
$hasHusband.Man
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description logic
TBox …
name symbols – symbols that appear on the lefthand side of axioms (also called defined concepts)
base symbols – symbols that appear only on the
right-hand side of axioms (also called primitive
concepts)
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description logic
TBox …
if we know what the base symbols stand for – then
the meaning of of the name symbols is completely
determined*
*additionally terminology has to be definitorial
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description logic
TBox …
an important (and related to the issue of being
definitorial) question – are definitions of a
terminology cyclic or not?
Human’ º Animal
"hasParent.Human’
(… is cyclic iff there exists an atomic concept that
uses itself)
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description logic
TBox …
Woman º Person
Man º Person
Mother º (Person
Father º Person
Female
Ø(Person
Female)
Female)
Ø(Person
$ hasChild.Person
Female)
$ hasChild.Person
Parent º ...
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description logic
TBox …
there are terminologies with cycles that are
definitorial …
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description logic
TBox …with inclusion axioms
for some concepts we will not be able to define them
completely – we will state necessary conditions for
concept membership using an inclusion
Woman
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Person
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description logic
TBox …with inclusion axioms
if we do this – the terminology loses its definitorial
impact, even if it is acyclic, then …
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description logic
ABox …
the second component of a knowledge base
a world description or ABox
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description logic
ABox …
in ABox we describe a specific state of affairs of a
given application domain
we introduce individuals, by giving them names, and
we asserts properties of these individuals
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description logic
ABox …
we denote names as a, b, c
using concepts C and roles R, we can make
assertions of the following two kinds in the ABox
C(a)
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R(b,c)
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description logic
ABox …
C(a) – this concept assertion used to stat that a
belongs to (the interpretation of C)
R(b,c) – this role assertion states that c is a filler of
the role R for b
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description logic
ABox …
Father(PETER) – means that PETER is a father
hasChild(MARY,PAUL) – means that Paul is a child
of Mary
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description logic
ABox …
examples:
MotherWithoutDaughter(MARY)
hasChild(MARY, PETER)
hasChild(MARY, PAUL)
Father(PETER)
hasChild(PETER, HARRY)
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description logic
ABox …
the semantic of ABox is “open-world semantics” – we
cannot assume that the knowledge in the knowledge
base is complete (on the contrary to “closed-world”
semantics of classical databases)
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description logic
ABox …
the semantic is given to ABoxs by extending
interpretations to individual names
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description logic
ABox …
now, an interpretation I not only maps atomic
concepts and roles to sets and relations, but in
addition it maps each individual name a to and
element
we assume that distinct individual names denote
distinct objects
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description logic
ABox …
the interpretation I satisfies the concept assertion
C(a) if a I Î C I, and it satisfies the role assertion
I
I
I
(a
,
b
)
Î
R
R(a, b) if
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description logic
ABox …
an interpretation satisfies the ABox A if it satisfies
each assertion in A
in this case we say that I is a model of the assertion
or the ABox the concept assertion
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description logic
ABox …
I satisfies an ABox A with respect to a TBox T
if in addition to being a model of A,
it is a model of T
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description logic
inference …
and more ….
… to be continued (sometime)
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