Description logic

# Description logic

Overview
Description logic is a family of formal knowledge representation
Knowledge representation
Knowledge representation is an area of artificial intelligence research aimed at representing knowledge in symbols to facilitate inferencing from those knowledge elements, creating new elements of knowledge...

languages. It is more expressive than propositional logic but has more efficient decision problem
Decision problem
In computability theory and computational complexity theory, a decision problem is a question in some formal system with a yes-or-no answer, depending on the values of some input parameters. For example, the problem "given two numbers x and y, does x evenly divide y?" is a decision problem...

s than first-order predicate logic.
Discussion
 Ask a question about 'Description logic' Start a new discussion about 'Description logic' Answer questions from other users Full Discussion Forum

Recent Discussions
Encyclopedia
Description logic is a family of formal knowledge representation
Knowledge representation
Knowledge representation is an area of artificial intelligence research aimed at representing knowledge in symbols to facilitate inferencing from those knowledge elements, creating new elements of knowledge...

languages. It is more expressive than propositional logic but has more efficient decision problem
Decision problem
In computability theory and computational complexity theory, a decision problem is a question in some formal system with a yes-or-no answer, depending on the values of some input parameters. For example, the problem "given two numbers x and y, does x evenly divide y?" is a decision problem...

s than first-order predicate logic.

DL is used in artificial intelligence
Artificial intelligence
Artificial intelligence is the intelligence of machines and the branch of computer science that aims to create it. AI textbooks define the field as "the study and design of intelligent agents" where an intelligent agent is a system that perceives its environment and takes actions that maximize its...

for formal reasoning on the concepts of an application domain (known as terminological knowledge). It is of particular importance in providing a logical formalism for ontologies and the Semantic Web
Semantic Web
The Semantic Web is a collaborative movement led by the World Wide Web Consortium that promotes common formats for data on the World Wide Web. By encouraging the inclusion of semantic content in web pages, the Semantic Web aims at converting the current web of unstructured documents into a "web of...

. The most notable application outside information science
Information science
-Introduction:Information science is an interdisciplinary science primarily concerned with the analysis, collection, classification, manipulation, storage, retrieval and dissemination of information...

is in bioinformatics
Bioinformatics
Bioinformatics is the application of computer science and information technology to the field of biology and medicine. Bioinformatics deals with algorithms, databases and information systems, web technologies, artificial intelligence and soft computing, information and computation theory, software...

where DL assists in the codification of medical knowledge.

## Introduction

A Description Logic (DL) models concepts, roles and individuals, and their relationships.

The fundamental modeling concept of a DL is the axiom - a logical statement relating roles and/or concepts. This is a key difference from the frames paradigm where a frame specification declares and completely defines a class.

### Differences from OWL

Synonyms
OWL DL
class concept
property role
object individual

### Naming convention

There are many varieties of Description Logic and there is an informal naming convention, roughly describing the operators allowed. The expressivity
Expressive power
In computer science, the expressive power of a language describes the ideas expressible in that language.For example, the Web Ontology Language expression language profile lacks ideas which can be expressed in OWL2 RL . OWL2 EL may therefore be said to have less expressive power than OWL2 RL...

is encoded in the label for a logic using the following letters:
 Functional properties. Full existential qualification (Existential restrictions that have fillers other than owl:Thing). Concept union. Complex concept negation. An abbreviation for with transitive roles. Role hierarchy (subproperties - rdfs:subPropertyOf). Limited complex role inclusion axioms; reflexivity and irreflexivity; role disjointness. Nominals. (Enumerated classes of object value restrictions - owl:oneOf, owl:hasValue). Inverse properties. Cardinality restrictions (owl:cardinality, owl:maxCardinality). Qualified cardinality restrictions (available in OWL 2, cardinality restrictions that have fillers other than owl:Thing). Use of datatype properties, data values or data types.

#### Examples

As an example, is a centrally important description logic from which comparisons with other varieties can be made. is simply with complement of any concept allowed, not just atomic concepts.

A further example, the description logic is the logic plus extended cardinality restrictions, and transitive and inverse roles. The naming conventions aren't purely systematic so that the logic might be referred to as and abbreviations are made where possible, is used instead of the equivalent .

The Protégé ontology editor supports . Three major bioinformatic terminology bases, Snomed, Galen, and GO, are expressible in (with additional role properties).

OWL 2 provides the expressiveness of , OWL-DL is based on , and for OWL-Lite it is .

#### Exceptions

Some canonical DLs that do not exactly fit this convention are: >
 Attributive language. This is the base language which allows: Atomic negation (negation of concepts that do not appear on the left hand side of axioms) Concept intersection Universal restrictions Limited existential quantification A sub-language of , which is obtained by disallowing atomic negation. A sub-language of , which is obtained by disallowing limited existential quantification. Intersection and full existential restriction.

## History

Description logic (DL) was given its current name in the 1980s. Previous to this it was called (chronologically): terminological systems, and concept languages.

### Knowledge representation

Frames and semantic network
Semantic network
A semantic network is a network which represents semantic relations among concepts. This is often used as a form of knowledge representation. It is a directed or undirected graph consisting of vertices, which represent concepts, and edges.- History :...

s lack formal (logic-based) semantics. DL was first introduced into Knowledge Representation
Knowledge representation
Knowledge representation is an area of artificial intelligence research aimed at representing knowledge in symbols to facilitate inferencing from those knowledge elements, creating new elements of knowledge...

(KR) systems to overcome this deficiency.

The first DL-based KR system was KL-ONE
KL-ONE
KL-ONE is a well known knowledge representation system in the tradition of semantic networks and frames; that is, it is a frame language. The system is an attempt to overcome semantic indistinctness in semantic network representations and to explicitly represent conceptual information as a...

(by Ronald J. Brachman
Ronald J. Brachman
Ronald J. "Ron" Brachman is currently Vice President of Yahoo! Labs and Research Operations at Yahoo! Labs. He is also the Head of Yahoo!'s Academic Relations organization. Prior to working at Yahoo!, he worked at DARPA as a the Director of the Information Processing Technology Office , one of...

and Schmolze, 1985). During the '80s other DL-based systems using structural subsumption algorithms were developed including KRYPTON (1983), LOOM
LOOM (ontology)
Loom or LOOM is a knowledge representation language developed by researchers in the Artificial Intelligence research group at the University of Southern California's Information Sciences Institute...

(1987), BACK (1988), K-REP (1991) and CLASSIC (1991). This approach featured DL with limited expressiveness but relatively efficient (polynomial time) reasoning.

In the early '90s, the introduction of a new tableau based algorithm paradigm allowed efficient reasoning on more expressive DL. DL-based systems using these algorithms - such as KRIS (1991) - show acceptable reasoning performance on typical inference problems even though the worst case complexity is no longer polynomial.

From the mid '90s, reasoners were created with good practical performance on very expressive DL with high worst case complexity. Examples from this period include FaCT, RACER (2001), CEL (2005), and KAON 2
KAON
KAON is an ontology infrastructure developed by the University of Karlsruhe and the Research Center for Information Technologies in Karlsruhe....

(2005).

DL reasoners, such as FaCT, FaCT++, RACER, DLP and Pellet, implement the analytic tableau method. KAON2 is implemented by algorithms which reduce a SHIQ(D) knowledge base to a disjunctive datalog
Datalog
Datalog is a query and rule language for deductive databases that syntactically is a subset of Prolog. Its origins date back to the beginning of logic programming, but it became prominent as a separate area around 1977 when Hervé Gallaire and Jack Minker organized a workshop on logic and databases...

program.

### Semantic Web

The DARPA Agent Markup Language (DAML) and Ontology Inference Layer
Ontology Inference Layer
OIL can be regarded as an Ontology infrastructure for the Semantic Web. OIL is based on concepts developed in Description Logic and frame-based systems and is compatible with RDFS....

(OIL) ontology languages for the semantic web
Semantic Web
The Semantic Web is a collaborative movement led by the World Wide Web Consortium that promotes common formats for data on the World Wide Web. By encouraging the inclusion of semantic content in web pages, the Semantic Web aims at converting the current web of unstructured documents into a "web of...

can be viewed as
syntactic
Syntax (logic)
In logic, syntax is anything having to do with formal languages or formal systems without regard to any interpretation or meaning given to them...

variants of DL. In particular, the formal semantics and reasoning in OIL use the DL. The DAML+OIL DL was developed as a submission to - and formed the starting point of - the World Wide Web Consortium
World Wide Web Consortium
The World Wide Web Consortium is the main international standards organization for the World Wide Web .Founded and headed by Tim Berners-Lee, the consortium is made up of member organizations which maintain full-time staff for the purpose of working together in the development of standards for the...

(W3C) Web Ontology Working Group. In 2004, the Web Ontology Working Group completed its work by issuing the OWL
Web Ontology Language
The Web Ontology Language is a family of knowledge representation languages for authoring ontologies.The languages are characterised by formal semantics and RDF/XML-based serializations for the Semantic Web...

recommendation. The design of OWL is based on the family of DL with OWL DL and OWL Lite based on and respectively.

The W3C OWL Working Group began work in 2007 on a refinement of - and extension to - OWL. In 2009, this was completed by the issuance of the OWL2 recommendation. OWL2 is based on the description logic . Practical experience demonstrated that OWL DL lacked several key features necessary to model complex domains.

## Modeling

In DL, a distinction is drawn between the so-called TBox
Tbox
In Computer Science, a TBox is a "terminological component"—a conceptualization associated with a set of facts, known as an ABox.The terms ABox and TBox are used to describe two different types of statements in ontologies. TBox statements describe a conceptualization, a set of concepts and...

(terminological box) and the ABox
Abox
In Computer Science, an ABox is an "assertion component"—a fact associated with a terminological vocabulary within a knowledge base.The terms ABox and TBox are used to describe two different types of statements in ontologies. TBox statements describe a system in terms of controlled vocabularies,...

(assertional box). In general, the TBox contains sentences describing concept hierarchies (i.e., relations between concept
Concept
The word concept is used in ordinary language as well as in almost all academic disciplines. Particularly in philosophy, psychology and cognitive sciences the term is much used and much discussed. WordNet defines concept: "conception, construct ". However, the meaning of the term concept is much...

s) while the ABox contains ground sentences stating where in the hierarchy individuals belong (i.e., relations between individuals and concepts). For example, the statement:

(1) Every employee is a person

belongs in the TBox, while the statement:

(2) Bob is an employee

belongs in the ABox.

Note that the TBox/ABox distinction is not significant, in the same sense that the two "kinds" of sentences are not treated differently in first-order logic (which subsumes most DL). When translated into first-order logic, a subsumption axiom
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...

like (1) is simply a conditional restriction to unary
Unary operation
In mathematics, a unary operation is an operation with only one operand, i.e. a single input. Specifically, it is a functionf:\ A\to Awhere A is a set. In this case f is called a unary operation on A....

predicates (concepts) with only variables appearing in it. Clearly, a sentence of this form is not privileged or special over sentences in which only constants ("grounded" values) appear like (2).

So why was the distinction introduced? The primary reason is that the separation can be useful when describing and formulating decision-procedures for various DL. For example, a reasoner might process the TBox and ABox separately, in part because certain key inference problems are tied to one but not the other one ('classification' is related to the TBox, 'instance checking' to the ABox). Another example is that the complexity of the TBox can greatly affect the performance of a given decision-procedure for a certain DL, independently of the ABox. Thus, it is useful to have a way to talk about that specific part of the knowledge base
Knowledge base
A knowledge base is a special kind of database for knowledge management. A Knowledge Base provides a means for information to be collected, organised, shared, searched and utilised.-Types:...

.

The secondary reason is that the distinction can make sense from the knowledge base modeler's perspective. It is plausible to distinguish between our conception of terms/concepts in the world (class axioms in the TBox) and particular manifestations of those terms/concepts (instance assertions in the ABox). In the above example: when the hierarchy within a company is the same in every branch but the assignment to employees is different in every department (because there are other people working there), this distinction makes sense to reuse the TBox for different branches.

There are two features of Description Logic that are not shared by most other data description formalisms: DL does not make the Unique Name Assumption
Unique name assumption
The Unique Name Assumption is a concept from ontology languages and description logics. In logics with the unique name assumption, different names always refer to different entities in the world...

(UNA) or the Closed World Assumption
Closed world assumption
The closed world assumption is the presumption that what is not currently known to be true, is false. The same name also refers to a logical formalization of this assumption by Raymond Reiter. The opposite of the closed world assumption is the open world assumption , stating that lack of knowledge...

(CWA). Not having UNA means that two concepts with different names may be allowed by some inference to be shown to be equivalent. Not having CWA, or rather having the Open World Assumption
Open World Assumption
In formal logic, the open world assumption is the assumption that the truth-value of a statement is independent of whether or not it is known by any single observer or agent to be true. It is the opposite of the closed world assumption, which holds that any statement that is not known to be true is...

(OWA) means that lack of knowledge of a fact does not immediately imply knowledge of the negation of a fact.

## Formal description

Like first order logic (FOL), a syntax
Syntax (logic)
In logic, syntax is anything having to do with formal languages or formal systems without regard to any interpretation or meaning given to them...

defines which collections of symbols are legal expressions in a Description Logic (DL), and semantics
Semantics
Semantics is the study of meaning. It focuses on the relation between signifiers, such as words, phrases, signs and symbols, and what they stand for, their denotata....

determine meaning. Unlike FOL, a DL may have several well known syntactic variants.

### Syntax

The syntax of a member of the description logic family is characterized by its recursive definition, in which the constructors that can be used to form concept terms are stated. Some constructors are related to logical constructors in first-order logic
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...

(FOL) such as intersection
Intersection (set theory)
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....

or conjunction
Logical conjunction
In logic and mathematics, a two-place logical operator and, also known as logical conjunction, results in true if both of its operands are true, otherwise the value of false....

of concepts, union
Union (set theory)
In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...

or disjunction of concepts, negation
Negation
In logic and mathematics, negation, also called logical complement, is an operation on propositions, truth values, or semantic values more generally. Intuitively, the negation of a proposition is true when that proposition is false, and vice versa. In classical logic negation is normally identified...

or complement
Complement (set theory)
In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...

of concepts, universal restriction and existential restriction. Other constructors have no corresponding construction in FOL including restrictions on roles for example, inverse, transitivity
Transitive relation
In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c....

and functionality.

#### Notation

Let C and D be concepts, a and b be individuals, and R be a role.
Conventional Notation
Symbol Description Example Read
all concept names top
empty
Empty set
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...

concept
bottom
intersection
Intersection (set theory)
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....

or conjunction
Logical conjunction
In logic and mathematics, a two-place logical operator and, also known as logical conjunction, results in true if both of its operands are true, otherwise the value of false....

of concepts
C and D
union
Union (set theory)
In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...

or disjunction of concepts
C or D
negation
Negation
In logic and mathematics, negation, also called logical complement, is an operation on propositions, truth values, or semantic values more generally. Intuitively, the negation of a proposition is true when that proposition is false, and vice versa. In classical logic negation is normally identified...

or complement
Complement (set theory)
In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...

of concepts
not C
universal restriction all R-successors are in C
existential restriction an R-successor exists in C
Concept inclusion all C are D
Concept equivalence C is equivalent to D
Concept definition C is defined to be equal to D
Concept assertion a is a C
Role assertion a is R-related to b

#### The description logic ALC

The prototypical DL Attributive Concept Language with Complements () was introduced by Manfred Schmidt-Schauß and Gert Smolka in 1991, and is the basis of many more expressive DL. The following definitions follow the treatment in Baader et al.

Let , and be (respectively) sets of concept names (also known as atomic concepts), role names and individual names (also known as individuals, nominals or objects). Then the ordered triple (, , ) is the signature.
##### Concepts

The set of concepts is the smallest set such that:
• The following are concepts:
• (top is a concept)
• (bottom is a concept)
• Every (all atomic concepts are concepts)
• If and are concepts and then the following are concepts:
• (the intersection of two concepts is a concept)
• (the union of two concepts is a concept)
• (the complement of a concept is a concept)
• (the universal restriction of a concept by a role is a concept)
• (the existential restriction of a concept by a role is a concept)

##### Terminological axioms

A general concept inclusion (GCI) has the form where and are concepts. Write when and . A TBox is any finite set of GCIs.
##### Assertional axioms
• A concept assertion is a statement of the form where and C is a concept.
• A role assertion is a statement of the form where and R is a role.

An ABox is a finite set of assertional axioms.
##### Knowledge base

A knowledge base (KB) is an ordered pair for TBox and ABox .

### Semantics

The semantics
Semantics
Semantics is the study of meaning. It focuses on the relation between signifiers, such as words, phrases, signs and symbols, and what they stand for, their denotata....

of description logic are defined by interpreting concepts as sets of individuals and roles as sets of pairs of individuals. Those individuals are typically assumed from a given domain. The semantics of non atomic concepts and roles is then defined in terms of atomic concepts and roles. This is done by using a recursive definition similar to the syntax.

#### The description logic ALC

The following definitions follow the treatment in Baader et al.

A terminological interpretation over a signature consists of
• a non-empty set called the domain
Domain of discourse
In the formal sciences, the domain of discourse, also called the universe of discourse , is the set of entities over which certain variables of interest in some formal treatment may range...

• a interpretation function that maps:
• every individual to an element
• every concept to a subset of
• every role name to a subset of

such that
• (union
Union (set theory)
In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...

means disjunction)
• (intersection
Intersection (set theory)
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....

means conjunction
Logical conjunction
In logic and mathematics, a two-place logical operator and, also known as logical conjunction, results in true if both of its operands are true, otherwise the value of false....

)
• (complement
Complement (set theory)
In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...

means negation
Negation
In logic and mathematics, negation, also called logical complement, is an operation on propositions, truth values, or semantic values more generally. Intuitively, the negation of a proposition is true when that proposition is false, and vice versa. In classical logic negation is normally identified...

)

Define (read I models) as follows
##### TBox
• if and only if
• if and only if for every

##### ABox
• if and only if
• if and only if
• if and only if for every

##### Knowledge base

Let be a knowledge base.
• if and only if and

### Decision problems

In addition to the ability to describe concepts formally, one also would like to employ the description of a set of concepts to ask questions about the concepts and instances described. The most common decision problems are basic database-query-like questions like instance checking (is a particular instance (member of an A-box) a member of a given concept) and relation checking (does a relation/role hold between two instances, in other words does a have property b), and the more global-database-questions like subsumption (is a concept a subset of another concept), and concept consistency (is there no contradiction among the definitions or chain of definitions). The more operators one includes in a logic and the more complicated the T-box (having cycles, allowing non-atomic concepts to include each other), usually the higher the computational complexity is for each of these problems (see Navigator on Description Logic Complexity for examples).

### First order logic

Many Description Logic models (DLs) are decidable fragments of first order logic (FOL). Some DLs now include operations (for example, transitive closure of roles) that allow efficient inference but cannot be expressed in FOL.

### Fuzzy description logic

Fuzzy description logic combines fuzzy logic
Fuzzy logic
Fuzzy logic is a form of many-valued logic; it deals with reasoning that is approximate rather than fixed and exact. In contrast with traditional logic theory, where binary sets have two-valued logic: true or false, fuzzy logic variables may have a truth value that ranges in degree between 0 and 1...

with DLs. Since many concepts that are needed for intelligent systems
Intelligent Systems
is a Japanese first-party video game developer and internal team of Nintendo Co., Ltd. It has its headquarters in the Nintendo Kyoto Research Center in Higashiyama-ku, Kyoto, Kyoto Prefecture....

lack well defined boundaries, or precisely defined criteria of membership, fuzzy logic is needed to deal with notions of vagueness and imprecision. This offers a motivation for a generalization of description logic towards dealing with imprecise and vague concepts.

What people should also think about for intelligent systems is multiple viewpoints of the data. This will lead to subjective (as opposed to objective) intelligent systems.

### Modal logic

Description Logic is related to — but developed independently of — modal logic
Modal logic
Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...

(ML). Many - but not all - DL are syntactic variants of ML.

#### Examples

Syntactic Variants
DL ML
K

### Temporal description logic

Temporal description logic represents - and allow reasoning about - time dependent concepts and many different approaches to this problem exist. For example, a description logic might be combined with a modal
Modal logic
Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...

temporal logic
Temporal logic
In logic, the term temporal logic is used to describe any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of time. In a temporal logic we can then express statements like "I am always hungry", "I will eventually be hungry", or "I will be hungry...

such as Linear temporal logic
Linear temporal logic
In logic, Linear temporal logic is a modal temporal logic with modalities referring to time. In LTL, one can encode formulae about the future of paths such as that a condition will eventually be true, that a condition will be true until another fact becomes true, etc. It is a fragment of the more...

.

• Formal concept analysis
Formal concept analysis
Formal concept analysis is a principled way of automatically deriving an ontology from a collection of objects and their properties. The term was introduced by Rudolf Wille in 1984, and builds on applied lattice and order theory that was developed by Birkhoff and others in the 1930s.-Intuitive...

• Lattice (order)
Lattice (order)
In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...

• Semantic parameterization
Semantic parameterization
Semantic parameterization is a conceptual modeling process for expressing natural language descriptions of a domain in first-order predicate logic...

• Semantic reasoner
Semantic reasoner
A semantic reasoner, reasoning engine, rules engine, or simply a reasoner, is a piece of software able to infer logical consequences from a set of asserted facts or axioms. The notion of a semantic reasoner generalizes that of an inference engine, by providing a richer set of mechanisms to work with...

• SWRL
SWRL
SWRL is a proposal for a Semantic Web rules-language, combining sublanguages of the OWL Web Ontology Language with those of the Rule Markup Language ....

### Tools

Reasoners

There are some reasoners that deal with OWL and Description Logic. These are some of the most popular:
• CEL is a free (for non-commercial use) LISP-based reasoner
• Cerebra Engine was a commercial C++-based reasoner, acquired in 2006 by webMethods.
• FaCT++ is a free open-source C++-based reasoner.
• KAON
KAON
KAON is an ontology infrastructure developed by the University of Karlsruhe and the Research Center for Information Technologies in Karlsruhe....

2 is a free (free for non-commercial usage) Java reasoner.
• MSPASS is a free open-source C reasoner for numerous description logic models.
• Pellet is a dual-licensed (AGPL and proprietary) commercial, Java-based reasoner.
• RacerPro of Racer Systems is a commercial (free trials and research licenses are available) lisp-based reasoner.
• Sim-DL is a free open-source Java-based reasoner for the language ALCHQ. It also provides a similarity measurement functionality between concepts. To access this functionality a Protégé plugin can be used.
• HermiT is an open source
Open source
The term open source describes practices in production and development that promote access to the end product's source materials. Some consider open source a philosophy, others consider it a pragmatic methodology...

reasoner based on the hypertableaux calculus. It is developed by the University of Oxford
University of Oxford
The University of Oxford is a university located in Oxford, United Kingdom. It is the second-oldest surviving university in the world and the oldest in the English-speaking world. Although its exact date of foundation is unclear, there is evidence of teaching as far back as 1096...

.

Editors
• Protégé
Protege (software)
Protégé is a free, open source ontology editor and a knowledge acquisition system. Like Eclipse, Protégé is a framework for which various other projects suggest plugins. This application is written in Java and heavily uses Swing to create the rather complex user interface...

is a free, open source ontology editor and knowledge-base framework, which can use DL reasoners which offer a DIG interface as backends for consistency checks.
• SWOOP is an open source
Open source
The term open source describes practices in production and development that promote access to the end product's source materials. Some consider open source a philosophy, others consider it a pragmatic methodology...

ontology editor originally developed at the University of Maryland
University of Maryland
When the term "University of Maryland" is used without any qualification, it generally refers to the University of Maryland, College Park.University of Maryland may refer to the following:...

.

Interfaces
• DIG Implementation. DIG is an XML interface to DL systems, recommended by the DL Implementation Group. DIG 2.0 is an ongoing effort for a new DIG interface standard.
• OWL API is an open source
Open source
The term open source describes practices in production and development that promote access to the end product's source materials. Some consider open source a philosophy, others consider it a pragmatic methodology...

Java
Java (programming language)
Java is a programming language originally developed by James Gosling at Sun Microsystems and released in 1995 as a core component of Sun Microsystems' Java platform. The language derives much of its syntax from C and C++ but has a simpler object model and fewer low-level facilities...

interface to the W3C Web Ontology Language OWL
Web Ontology Language
The Web Ontology Language is a family of knowledge representation languages for authoring ontologies.The languages are characterised by formal semantics and RDF/XML-based serializations for the Semantic Web...

developed by the University Of Manchester