This document discusses description logic (DL) as a formal knowledge representation language, detailing its evolution to better handle uncertainty and vagueness in real-world applications. It introduces various extensions such as fuzzy description logics, probabilistic description logics, and rough description logics, alongside innovative reasoning techniques like hyper-tableau calculus for complex knowledge structures. The paper emphasizes the need for enhanced expressive power in DL to support reasoning in contexts such as the semantic web and data integration.

1. Description Logic
Author: K.Balamurugan M.Tech, Pondicherry University, Knowledge engineering specialist
1. Abstract:
Descriptionlogicisaformal logic-basedknowledgerepresentationlanguage which“Description"
aboutthe worldintermsof concepts (classes), roles (properties, relationships) and individuals
(instances).descriptionlogics isafamily of logic-basedknowledge representation languages that
can be usedtorepresentthe terminological knowledge of an applicationdomain in a structured
way.Recentexperience withDLs,however,hasshownthattheirexpressivityisofteninsufficient
to accurately describe structured objects. This paper mainly concerned with various kind of
description logic and their power of describing the many real world aspect. Description logic
evolved over the period which will avoid vagueness, uncertain or imprecise knowledge
representationandreasoningindescriptionlogics.Extendedfuzzy description logic is proposed
to increase expressive power for complex fuzzy information.Compared with the other fuzzy
descriptionlogics,the extendedfuzzydescriptionlogiccanexpressmore widefuzzy information.
Anotherkindof newroughdescriptionlogicRDLAC (roughdescriptionlogicbasedonapproximate
concepts) isproposedbasedonapproximate concepts..Inthispapervariousextensions of fuzzy
description logics over lattices are also discussed.
Keyword:Descriptionlogic;Knowledge representation; approximate concepts;rough set theory
2. Introduction:
Descriptionlogics(DLs) are afamily of
knowledge representation formalisms
suitable for representing the
terminological knowledge ina wide range
of applications.The Tableaux algorithm is
a general technique for deciding
theconcept satisfiability problems in
description logics. Historically, the
tableaux algorithm provides an
algorithmic framework that is parametric
withrespect to languageconstructors and
isuseful forstudyingbothcorrectnessand
complexity of concrete decision
procedures.Description Logics (DLs) are a
class of knowledge representation
formalisms in the tradition of semantic
networks and frames, which can be used
to represent the terminological
knowledge of an application domain in a
structured and formally well-understood
way. DL systems provide their users with
inference services (like computing the
subsumption hierarchy) that deduce
implicit knowledge from the explicitly
represented knowledge. They are
employedinvariousapplication domains,
such as semantic Web, ontologies,
databases, and software engineering.
Because classical DLs can only represent
and reason on certain or precise
knowledge, and cannot represent and
reason on uncertain or imprecise
knowledge, therefore, some researchers
extend classical DLs allowing to express
uncertainor imprecise knowledge. At this
aspect, three kinds of description logics,
i.e.,fuzzyDLs,probabilisticDLs,andrough
DLs, are proposed. In what follows, we
will use several DLs, hence it is necessary
to introduce some notationsof DLsfirstly.
The DL that providesthe Booleanconcept
constructors plus the existential and
universal restrictionconstructors is called

2. ALC, where the Boolean concept
constructors are, apart from concept
disjunction, concept conjunction and
concept negation. In addition to the
Booleans, and existential and universal
restriction constructors, DLs typically
provide concept constructors that form
complex concepts. The basic constructors
of this kind are qualified number
restrictions, unqualified number
restrictions, functional number
restrictions, nominal, concrete domain.
More expressive DLs can be obtained by
extending ALC with new concept
constructors. For example, the logic
obtainedfrom ALC by providing qualified
numberrestrictionsiscalledALCQ. On the
otherhand,addingunqualified,functional
number restrictions, nominal, and
concrete domains to ALC results in the
logics ALCN, ALCF, ALCO, and ACL (D),
respectively. Furthermore, besides
concept constructorsDLs may provide a
set of role constructors such as inversion
and transitive closure operator.The logics
that extend ALC with inversion and
transitive closure operatorare called ALCI
and ALC+, respectively.
3. Preliminaries:
3.1 DL Constructors
Description logic has following basic
constructors:
Concepts(unary predicates/formulaewith
one free variable)
‰E.g.,Person,Father,Mother
Roles(binarypredicates/formulae with
twofree variables)
‰E.g.,hasChild,hasHusband
„ Individualnames(constants)
‰E.g.,Alice,Bob,Cindy
„ Subsumption(relationsbetween
concepts)
‰E.g. Female ⊆Person
„ Operators(forformingconceptsand
roles)
‰And(Π),Or(U),Not(¬)
Universal qualifier(∀),Existent
qualifier(∃)
Number restriction: ≤, ≥, =
Inverse role (-), transitive role (+), Role
hierarchy
Description Logics are characterised by
the constructors that they provide to
buildcomplex class and property
descriptions from atomic ones. For
example, ‘elephantswith their ages
greater than 20’ can be described by the
following DL class description:
Elephant Π∃age.>20
Where Elephant is an atomic class, age is
an atomic data-type property, >20 is a
customised data-type (treated as a unary
data-type predicate) and Π, ∃ are class
constructors.Asshownabove, data-types
and predicates (such as =, >, +) defined
over them can be used in the
constructionsof classdescriptions.Unlike
classes, data-types and data-type
predicates have obvious (fixed)
extensions;e.g.,the extensionof >20 is all
the integers that are greater than 20. Due
to the differences between classes and
data-types, there are two kinds of
properties:
(i) Objectproperties,whichrelate objects
to objects, and (ii) data type properties,
whichrelate objectstodata values,which
are instances of data types
Following few types of role constructors:
(Inverse Role)hasParent = hasChild-
>hasParent(Bob,Alice) ->hasChild(Alice,
Bob)
(Transitive Role)hasBrother
hasBrother(Bob,David),hasBrother(David,
Mack) ->hasBrother(Bob,Mack)

3. (Role Hierarchy)hasMother⊆hasParent
‰hasMother(Bob,Alice) ->hasParent(Bob,
Alice)
„HappyFather⊆ Father Π≥1
hasChild.WomanΠ≥1 hasChild.Man
3.2DL Architecture
Besides the concept descriptions for
describing sets of individuals or objects,
thesecond major representation mechanism
of description logics is the knowledge
base,which consists of a Tbox and an Abox.
Tbox:
The first component of a DL
knowledge base is the Tbox (“T” for
terminological).ATbox can be either a simple
Tbox or a general Tbox. A simple Tbox was
also calleda terminology in the past.
Definition :( Simple T box) The elements of a
simple Tbox are either concept inclusions
(e.g.,A⊆C)orconceptdefinitions(e.g., A ≡ C).
Both inclusions and definitions introduce
symbolic names for complex concept
Descriptions .Ina simple Tbox,atmostone
concept definition is allowed per
conceptname.
The concept inclusions or concept definitions
of a simple Tbox can serve as“rewrite rules”
to expand concept names to their definition
without compromisingsoundness. If concept
namesare notallowedtorefertothemselves,
neitherdirectlynorindirectly,thenwe have an
acyclic simple Tbox. Otherwise, it is called a
cyclicsimple Tbox.
As an example, the concept “mother”, as
interpreted in English as “a woman whohas a
child”, could be introduced by a description
like:Mother⊆ WomanΠ ∃haschild.Person
To state that a “womanisa person”,we could
use a second concept inclusion like:
Woman ⊆ Person
Among these two concept inclusions there is
no cyclic reference relationship,
thereforetogether they are acyclic.
Abox
The second component of the
knowledge base is the Abox (“A” for
assertion). AnAbox describes named
individuals and their relations while possibly
referring to theconcept descriptions in the
Tbox.
Definition (Abox Assertion) Given a set of
individual names N1, an Aboxassertion is of
either of the following two forms:

4. • a : C
• (a, b): R
where a, b ∈ NI are individual names, C is a
concept,and R isa role.An Abox is afinite set
of assertions.
Role Hierarchy
A role hierarchy (denoted by H as in
ALCHIQ and SHIQ) is a mechanism for
specifying the subsumption relationships
betweenroles.InadditiontoTbox andABox,a
role hierarchy is sometimes regarded as a
third component of a DL knowledgebase and
iscalledRbox (“R” forRole).However,thereis
no recognizedreasoningtaskon anRbox itself.
4. ExistingClassical Description
logic
There are various implemented DL
systems based on tableaux algorithms,
offeringa palette of description
formalisms with different expressive
power. In the history,the first DL-like
system was KL-ONE. KRIS is one of the
firstdescription logic reasoners that
implementedahighlyoptimized tableaux
algorithm.The name ALC stands for
“Attributive concept Language with
Complements.” It was first introduced in
first naming scheme for DLs was
proposed:starting from a basic DL AL, the
addition of constructors is indicated by
appending a corresponding letter; e.g.,
ALC is obtained from AL by adding the
complement operator (¬) and ALE is
obtained from AL by adding existential
restrictions (∃r.C).
Reasoning is a finding fact that is implicit
in the ontology given explicitly stated
facts. A variety of reasoning techniques
can be used to solve the reasoning
problems these include resolution based
approaches.automatabased approaches,
and structural approaches (for sub-
Boolean DLs. The most widely used
technique,however, is the tableau based
approach first introduced by
SchmidtSchauß and Smolka.
4.1 Tableaux algorithms
The Tableaux algorithm is a general
technique for deciding theconcept
satisfiability problems in description
logics. Historically,the tableauxalgorithm
providesanalgorithmicframework that is
parametric with respect to
languageconstructors and is useful for
studyingbothcorrectnessandcomplexity
of concretedecision procedures.
Tableau Algorithm is the de facto
standard reasoning algorithm used in DL
Basic intuitions
‰1.Reduces a reasoning problem to
concept satisfiability problem
‰2.Finds an interpretation that satisfies
concepts in question.
‰3.The interpretation is incrementally
constructed as a "Tableau"
Example:
„ given:Wife⊆Woman,Woman⊆ Person
Question: if Wife⊆ Person
Reasoning process:
‰Test if there is an individual that is a
Woman but not a Person, i.e. Test the
satisfiability of concept
C0=(WifeЬPerson)
‰C0(x) -> Wife(x), (¬Person)(x)
‰Wife(x)->Woman(x)
‰Woman(x) ->Person(x)
‰Conflict!
C0 is unsatisfiable, therefore Wife⊆
Person is true with the given ontology.
4.1.1 Pellet for Reasoning in Protégé:
Pellet is one of the most
commonreasoning engines used for
reasoning with Protege OWL models.„

5. Pellet supports reasoning with the full
expressivity of OWL-DL (SHOIN(D) in
Description Logic jargon) and has been
extended to support OWL 1.1 (the DL
SROIQ(D)). OWL 1.1 adds the following
language constructs to OWL DL:
o qualifiedcardinality
restrictions
o complex subpropertyaxioms
(betweenapropertyanda
propertychain)
o local reflexivityrestrictions
o reflexive,irreflexive,
symmetric,andantisymmetric
properties
o disjointproperties
o user-defineddata-types
Consistency checking
Ensures that ontology does not
contain any contradictory facts. The OWL
Semantics standard provides the formal
definitionof ontologyconsistencyusedby
Pellet.
Concept satisfiability
„Determines whetherit’spossible for
a class to have any instances. If a class is
unsatisfiable,thendefiningan instance of
that classwill cause the whole ontologyto
be inconsistent.
Classification
„computes the subclass relations
betweenevery named class to create the
complete class hierarchy. The class
hierarchy can be used to answer queries
such as getting all or only the direct
subclasses of a class.
Realization
„finds the most specific classes that
an individual belongs to; i.e., realization
computesthe direct types for each of the
individuals. Realization can only be
performedafterclassification since direct
types are defined with respect to a class
hierarchy. Using the classification
hierarchy, it is also possible to get all the
types for each individual.
4.2 Limitations of Existing work
DL knowledge bases describing
structuredobjectsare therefore usuallyunder
constrained, which precludes the entailment
of certain consequences and causes
performance problems during
reasoning.However, classical description
logicsare less suitable in all those domains
where the information tobe represented
comes along with (quantitative) uncertainty
and/orvagueness (or imprecision). For
example,uncertaininformation maybe of the
form “John is a teacher with the degree of
certainty0.3 anda student with the degree of
certainty0.7” (roughly,Johniseitherateacher
or a student,butmore likelyastudent),while
vague information may be of the form “John
is tall with the degree of truth 0.9”(roughly,
John is quite tall);
Formalisms for dealingwith
uncertainty and vagueness have started to
play an importantrole in research related to
the Web andthe SemanticWeb.Forexample,
the order in which Google returns the
answerstoa websearchqueryiscomputedby
using probabilistic techniques.
Furthermore,formalisms for dealing with
uncertaintyandvaguenessinontologies have
been successfully applied in ontology
matching, data integration, and information
retrievalThe rising popularity of description
logicsandtheiruse,andthe need to deal with
uncertaintyand vagueness, both especiallyin
the Semantic Web, is increasingly attracting
the attention ofmany researchers and
practitionerstowardsdescriptionlogicsableto
cope with uncertainty and vagueness. The

6. goal of this paper is toprovide extended
description logic that deals with uncertainty
and vagueness in description logics for
theSemantic Web, integration, and
information retrieval. Vagueness and
imprecisionalso abound in multimedia
information processing and retrieval.
5. ProposedWorks:
Following papers proposed that
provides several type of Extended
Description logic for dealing with
uncertainty and vagueness. For example
some researchers extend classical DLs
allowing expressing uncertain or
imprecise knowledge.Atthisaspect,three
kinds of description logics:
1. FuzzyDLs,
2. ProbabilisticDLs,and
3. Rough DLs, are proposed.
5.1. Representing ontologies using
description logics, description graphs, and
rules:
 DLs are often used to describe
structuredobjects-objectswhoseparts
are interconnectedinarbitrary,rather
than tree-like ways.
 DL knowledge bases describing
structured objects are therefore
usually under-constrained, which
precludes the entailment of certain
consequences and causes
performance problems during
reasoning.
 To addressthisproblem, anextension
of DL languages with description
graphs—knowledge modelling
construct thatcan accuratelydescribe
objects with parts connected in
arbitrary ways.
 To enable modelling the conditional
aspectsof structured objects, we also
extend DLs with rules.
 Hyper-tableau-based reasoning
algorithm that decides the
satisfiabilityprobleminthe decidable
cases,and that acts as a semi decision
procedure for some undecidable
ones.
 The algorithm(hypertableaucalculus)
first pre-processes (T, A) into a set of
rules ΞT (T) and an ABox A ∪ ΞA (T).
 Hyper tableau calculus consists of
three steps. First, transitivity axioms
are eliminated from T by encoding
them using general concept
inclusions;similar encodings are well
known in the context of various
description and modal logics
 Second, axioms are normalized and
complex concepts are replaced with
atomic ones in a way similar to the
structural transformation
 Third, the normalized axioms are
translated into rules by using the
correspondencesbetweendescription
and first-order logic
5.2. Reasoning with rough description
logics: An approximate concepts
approach
 Existing problems of uncertain or
imprecise knowledge representation
and reasoningindescriptionlogicsare
analysed. Approximate concepts are
introducedtodescriptionlogicsbased
on rough set theory (RDLAC). The
theoryof rough set is an extension of
set theory, in which a subset of a
universe is described by a pair of
ordinary sets called the lower and
upper approximations.
 DL thatprovidesthe Boolean concept
constructors plus the existential and
universal restriction constructors is
called ALC

7.  Expressive DLs can be obtained by
extending ALC with new concept
constructors. reasoning algorithms of
more expressive rough description
logics including approximate
concepts, number restrictions,
nominal, inverse roles and role
hierarchies are provided by the
(RDLAC)
 Reasoning algorithms of rough
description logics including number
restrictions, nominal, inverse roles
and rolehierarchies, and an
integration between approximate
concepts and fuzzy DLs or
probabilisticDLsbased onfuzzy rough
set theory or probabilistic rough set
theory respectively. Furthermore,
additionalresearch effort can be
focused on the investigation of the
construction of approximate
ontologies (or rough TBoxes) using
formal concept analysis.
5.3 Reasoning within extended fuzzy
description logic:
 FALC is a fuzzy extension of ALC
by adopting fuzzy interpretation
to redefine the semantics of ALC
syntax and extending the axiom
forms in TBox and ABox.
 FALC just offers limited but not
sufficient expressive power of
complex fuzzy information
 FALC only focuses on three
reasoning tasks with respect to
empty TBox. They are
1. Fuzzy entailment
2. Value bound
3. Fuzzy subsumption
 An extended fuzzy description
logic is proposed to increase
expressive power for complex
fuzzy information.
 sets of the fuzzy concepts and
fuzzy roles as atomic concepts
and atomicroles,anduse concept
constructors of the description
logics to support a new logic
system for fuzzy knowledge
representation
 Knowledge base ∑E (TE, AE) of
EFALC not only offers the fuzzy
information, but also supports
several reasoningtasks.Similar to
FALC,we will definethe reasoning
tasks of EFALC with purely
assertional knowledge base.
Satisfiability and consistency are
usually considered as
representative reasoning tasks in
classical DLs.
 Consistencyisaproblemof telling
whetheran ABox AE is consistent.
It can be converted into sat-
domain. Now we define process
of tableau algorithm for
consistency in following steps
(1) Let AE be an EFALC ABox and
any cut concept in AE is in NNF.
The tableau algorithm starts with
a set of a single alterable ABox.
S0={(AE,{n0})}. For any x : C[n1...nk]
or (x;y): R[n1]in AE, we consider ni
as a constant function fi(n).
Therefore we can extend AE as an
alterable ABox {(AE,{n0})} and
apply translation rules to it.
Obviously,AE is consistent iff S0 is
n0-satisfiable.
(2) Apply translation rules
excluding £ exhaustively to
currentS0. So there isa chain of Si
by applying rules:
S0 -> S1 ->…-> Si. In classical ALC,
the similar process is called ‘‘pre-

8. completion” step. And after this
step, all role assertions can be
ignored, because they cannot be
applied by any translation rule.
The similar conclusion also
holds for EFALC. And S0 is n0-
satisfiable iff Si is n0-satisfiable.
The size of any ABox in Si is
polynomial inthe size of (AE,{n0}).
(3) Apply T-rules to Si, check
whether any T-subsequence of Si
is n0-T satisfiable.
5.4 Inconsistency-tolerant description
logic.
 As a matter of fact, the
information we have about the
world, however, is often
inconsistent.The mergingof data,
for example, may lead to
contradictory pieces of
information.
 three systems of inconsistency-
tolerant constructive description
logic: CALCC (a constructive
version of the basic description
logic ALC with a semantics
induced by a translation into
classical predicate logic), CALCN4
(a versionof ALCwitha semantics
induced by a translation into the
constructive predicate logicQN4),
and CALCN4d
(a versionof ALCwith
a semantics induced by another
translationintoQN4guaranteeing
the duality of existential and
universal role restrictions)
 The best-known system of
constructive logic is intuitionistic
logic. One attractive feature of
intuitionistic logic is that its
relational possible-worlds
semantics admits of an
‘informational’ interpretation
according to which the possible
worldsare informationstates and
the assumedaccessibilityrelation
between worlds is a relation of
possible expansion of information
states.
 The tableau algorithms used to
test for satisfiability. If classical
negation ￢ is present, and
subsumption is defined in terms
of material implication ￢C Π D,
subsumption testing can be
reducedtosatisfiability testing: C
⊆D iff C Π ￢D is unsatisfiable.
 Tableau algorithm allows it to
make the observation that in the
constructionof a completiontree,
extending R-subtrees and
extending subtrees are rather
independent of each other. One
can say that the work with
operators of classical description
logic and the work with
constructive implication go in
orthogonal directions.So,one can
expectthatthe expressive power
of CALCC can be enlarged along
familiar lines.
5.5 A formal framework for description
logics with uncertainty
 Uncertaintyisa form of deficiency
or imperfection commonly found
in real-world information/data.
Framework for knowledge bases
withuncertaintyexpressed in the
description logic ALCU, which is a
propositionally complete
representation language
providing conjunction,
disjunction, existential and
universal quantifications, and full
negation.
 Framework is equipped with a
constraint-based reasoning

9. procedure that derives a
collection of assertions as well as
a set of linear/nonlinear
constraints that encode the
semantics of the uncertainty
knowledge base.
6. References
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