Description LogicAuthor: K.Balamurugan M.Tech, Pondicherry University, Knowledge engineering specialist

1. Abstract:Description logic is a formal logic-based knowledge representation language which “Description" about the world in terms of concepts (classes), roles (properties, relationships) and individuals (instances).description logics is a family of logic-based knowledge representation languages that can be used to represent the terminological knowledge of an applicationdomain in a structured way. Recent experience with DLs, however, has shown that their expressivity is often insufficient 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 representation and reasoning in description logics. Extended fuzzy description logic is proposed to increase expressive power for complex fuzzy information.Compared with the other fuzzy description logics, the extended fuzzy description logic can express more wide fuzzy information. Anotherkind of new rough description logic RDLAC (rough description logic based on approximate concepts) is proposed based on approximate concepts.. In this paper various extensions of fuzzy description logics over lattices are also discussed.Keyword: Description logic; Knowledge representation; approximate concepts;rough set theory

2. Introduction:Description logics (DLs) are a family of

knowledge representation formalisms suitable for representing the terminological knowledge in a 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 with respect to languageconstructors and is useful for studying both correctness and 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 employed in various application 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 uncertain or imprecise knowledge. At this aspect, three kinds of description logics, i.e., fuzzy DLs, probabilistic DLs, and rough DLs, are proposed. In what follows, we will use several DLs, hence it is necessary to introduce some notations of DLs firstly. The DL that provides the Boolean concept

constructors plus the existential and universal restriction constructors is called 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 obtained from ALC by providing qualified number restrictions is calledALCQ. On the other hand, adding unqualified, 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 operator are called ALCI and ALC+, respectively.

3. Preliminaries:3.1 DL Constructors

Description logic has following basic constructors:Concepts (unary predicates/formulae with one free variable)� E.g., Person, Father, MotherRoles (binary predicates/formulae with two free variables)� E.g., hasChild, hasHusband� Individual names (constants)� E.g., Alice, Bob, Cindy

� Subsumption (relations between concepts)� E.g. Female ⊆ Person� Operators (for forming concepts and roles) �And (Π),Or (U), Not (¬)Universal qualifier (∀), Existent qualifier(∃)Number restriction: ≤, ≥, = Inverse role (-), transitive role (+), Role hierarchyDescription 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. As shown above, data-types and predicates (such as =, >, +) defined over them can be used in the constructions of class descriptions. Unlike classes, data-types and data-type predicates have obvious (fixed) extensions; e.g., the extension of >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) Object properties, which relate objects to objects, and (ii) data type properties, which relate objects to data values, which are instances of data typesFollowing few types of role constructors:(Inverse Role)hasParent = hasChild->hasParent(Bob,Alice) ->hasChild(Alice, Bob)

(Transitive Role)hasBrotherhasBrother(Bob,David), hasBrother(David, Mack) ->hasBrother(Bob,Mack)

(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)or concept definitions (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 names are not allowed to refer to themselves, neither directlynor indirectly, then we 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 “woman is a 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:

• a : C

• (a, b): R

where a, b ∈ NI are individual names, C is a concept, and R is a 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 between roles. In addition to Tbox andABox, a role hierarchy is sometimes regarded as a third component of a DL knowledgebase and is called Rbox (“R” for Role). However, there is no recognized reasoningtask on an Rbox 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 implemented a highly optimized 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 .automata based 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 algorithmsThe 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 with respect to languageconstructors and is useful for studying both correctness and complexity of concretedecision procedures.Tableau Algorithm is the de facto standard reasoning algorithm used in DLBasic 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⊆ PersonQuestion: 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=(WifeΠ¬Person)� 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.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 qualified cardinality restrictions

o complex subproperty axioms (between a property and a property chain)

o local reflexivity restrictions o reflexive, irreflexive,

symmetric, and antisymmetric properties

o disjoint properties o user-defined data-types

Consistency checking

Ensures that ontology does not contain any contradictory facts. The OWL Semantics standard provides the formal definition of ontology consistency used by Pellet.

Concept satisfiability

Determines whether it’s possible for a� class to have any instances. If a class is unsatisfiable, then defining an instance of that class will cause the whole ontology to be inconsistent.

Classification

computes the subclass relations� between every 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 computes the direct types for each of the individuals. Realization can only be performed after classification 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 structured objects are therefore usually under 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, uncertain information maybe of the form “John is a teacher with the degree of certainty 0.3 anda student with the degree of certainty 0.7” (roughly, John is either ateacher or a student, but more likely a student), 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 and the Semantic Web. For example, the order in which Google returns the answers to a web searchquery is computed by using probabilistic techniques. Furthermore,formalisms for dealing with uncertainty and vagueness in ontologies have been successfully applied in ontology matching, data integration, and information retrievalThe rising popularity of description logics and their use, andthe need to deal with uncertainty and vagueness, both especiallyin the Semantic Web, is increasingly attracting the attention ofmany researchers and practitioners towards description logics ableto cope with uncertainty and vagueness. The 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. Proposed Works: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. At this aspect, three kinds of description logics:1. Fuzzy DLs,2. Probabilistic DLs, and3. Rough DLs, are proposed.

5.1. Representing ontologies using description logics, description graphs, and rules:

DLs are often used to describe structured objects-objects whoseparts are interconnected in arbitrary, 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 address this problem, an extension of DL languages with description graphs—knowledge modelling construct that can accurately describe objects with parts connected in arbitrary ways.

To enable modelling the conditional aspects of structured objects, we also extend DLs with rules.

Hyper-tableau-based reasoning algorithm that decides the satisfiability problem in the decidable cases, and that acts as a semi decision procedure for some undecidable ones.

The algorithm (hyper tableau calculus) 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 correspondences between description and first-order logic

5.2. Reasoning with rough description logics: An approximate concepts approach

Existing problems of uncertain or imprecise knowledge representation and reasoning in description logics are analysed. Approximate concepts are introduced to description logics based on rough set theory (RDLAC). The theory of 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 that provides the Boolean concept constructors plus the existential and universal restriction constructors is called ALC

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 probabilistic DLs based on fuzzy 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 are1. Fuzzy entailment2. Value bound3. 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 atomic roles, and use 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 reasoning tasks. Similar toFALC, we will define the reasoning tasks of EFALC with purely assertional knowledge base. Satisfiability and consistency are usually considered as representative reasoning tasks in classical DLs.

Consistency is a problem of telling whether an 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 isn0-satisfiable.(2) Apply translation rules excluding £ exhaustively to current S0. So there is a chain of Si by applying rules:S0 -> S1 ->…-> Si. In classical ALC, the similar process is called ‘‘pre-completion” step. And after this step, all role assertions can be ignored, because they cannot beapplied by any translation rule. The similar conclusion alsoholds for EFALC. And S0 is n0-satisfiable iff Si is n0-satisfiable.The size of any ABox in Si is polynomial in the 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 merging of 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 version of ALC with a semantics induced by a translation into the constructive predicate logic QN4), and CALCN4d (a version of ALC with a semantics induced by another translation into QN4 guaranteeing 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 worlds are information states and the assumed accessibility relation 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 reduced to satisfiability testing: C

D iff C Π ⊆ ￢D is unsatisfiable. Tableau algorithm allows it to

make the observation that in the construction of a completion tree, 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 expect that the expressive power of CALCC can be enlarged along familiar lines.

5.5 A formal framework for description logics with uncertainty

Uncertainty is a form of deficiency or imperfection commonly found in real-world information/data. Framework for knowledge bases with uncertainty expressed 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 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

1. Haarslev, V., Pai, H., & Shiri, N. (2009). International Journal of Approximate Reasoning A formal framework for description logics with uncertainty q. International Journal of Approximate Reasoning, 50(9),

1399–1415. doi:10.1016/j.ijar.2009.04.009

2. Jiang, Y., Tang, Y., Wang, J., Deng, P., & Tang, S. (2010). Knowledge-Based Systems Expressive fuzzy description logics over lattices. Knowledge-Based Systems, 23(2), 150–161. doi:10.1016/j.knosys.2009.11.002

3. Jiang, Y., Wang, J., Tang, S., & Xiao, B. (2009). Reasoning with rough description logics : An approximate concepts approach. Information Sciences, 179(5), 600–612. doi:10.1016/j.ins.2008.10.021

4. Lu, J., Li, Y., Zhou, B., & Kang, D. (2009). Knowledge-Based Systems Reasoning within extended fuzzy description logic. Knowledge-Based Systems, 22(1), 28–37. doi:10.1016/j.knosys.2008.04.010

5. Motik, B., Cuenca, B., Horrocks, I., & Sattler, U. (2009). Representing ontologies using description logics , description graphs ,. Artificial Intelligence, 173(14), 1275–1309. doi:10.1016/j.artint.2009.06.003

6. Odintsov, S. P., & Wansing, H. (2008). Inconsistency-tolerant description logic . Part II : A tableau algorithm for CALC C, 6, 343–360. doi:10.1016/j.jal.2007.06.001