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《描述逻辑(Description Logic)手册》是一本深入探讨理论、实现和应用的权威著作，它在知识表示和人工智能领域占有重要地位。描述逻辑（Description Logic，简称DL）是一种形式化的逻辑系统，主要用于构建和处理概念化知识模型。下面将详细阐述描述逻辑的主要概念、理论基础、实现方法以及其广泛应用。 1. **描述逻辑的基本概念** 描述逻辑是基于谓词逻辑的一种简化版本，它引入了类（Concepts）、角色（Roles）和个体（Individuals）的概念。类代表对象的集合，角色则表示对象之间的关系，个体则是具体的实体。 2. **描述逻辑的形式语法** 在描述逻辑中，类定义使用类似于面向对象编程的结构，如继承（Subsumption）和并集（Union）、交集（Intersection）、差集（Difference）等操作。角色也有类似的关系，如对称性、反称性、传递性等。 3. **描述逻辑的表达能力** 不同的描述逻辑系统有不同的表达能力，通过不同的符号和运算符组合，可以形成不同复杂度的逻辑。例如，ALC是最基本的描述逻辑系统，而更复杂的系统如SHOIN则包含更多的概念和角色特性。 4. **描述逻辑的推理** 描述逻辑推理包括概念蕴含（Concept Inclusion）、实例查询（Instance Query）、类型查询（Type Query）等。这些推理任务在知识库中寻找满足特定条件的对象或判断两个概念之间的关系。 5. **描述逻辑的语义** 描述逻辑的语义基于模型理论，每个逻辑公式对应一组模型，称为模型集合。模型集合的性质决定了逻辑公式的真值。例如，某个概念在逻辑上是另一个概念的子概念，当且仅当所有该概念的模型都是另一个概念的模型。 6. **描述逻辑的实现** 实现描述逻辑通常涉及算法设计，如ABox推理（处理个体实例）和TBox推理（处理概念和关系）。这些算法效率至关重要，因为大型知识库可能包含数百万个元素。 7. **应用领域** 描述逻辑广泛应用于知识管理系统、本体工程、数据库集成、信息检索、生物医学信息学等领域。例如，OWL（Web Ontology Language）就是基于描述逻辑的一个重要标准，用于构建和共享Web上的本体。 8. **相关资源** 提供的压缩包文件可能包含了描述逻辑的理论章节、实现细节、应用案例等内容，如"dlhb-01.pdf"可能是介绍基础概念的开篇，"dlhb-09.pdf"可能涉及到高级主题，"dlhb-bibliography.pdf"则可能包含了相关研究文献的引用。通过深入学习和理解《描述逻辑手册》中的内容，不仅可以掌握描述逻辑的基本原理，还能了解到如何将其应用于实际问题中，提升知识表示和智能系统的性能。

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Relationships with other Formalisms

Ulrike Sattler

Diego Calvanese

Ralf Molitor

Abstract

In this chapter, we are concerned with the relationship between Description Log-

ics and other formalisms, regardless of whether they were designed for knowledge

representation issues or not. We concentrated on those representation formalisms

that either (1) had or have a strong inﬂuence on Description Logics (e.g., modal

logics), (2) are closely related to Description Logics for historical reasons (e.g., se-

mantic networks and structured inheritance networks), or (3) have similar expressive

power (e.g., semantic data models). There are far more knowledge representation

formalisms than those mentioned in this section. For example, “verb-centered”

graphical formalisms like those introduced by Simmons

[

1973

]

are not mentioned

since we believe that their relationship with Description Logics is too weak.

4.1 AI knowledge representation formalisms

In artiﬁcial intelligence (AI), various “non-logical” knowledge representation for-

malisms were developed, motivated by the belief that classical logic is inadequate

for knowledge representation in AI applications. This belief was mainly based upon

cognitive experiments carried out with human beings and the wish to have repre-

sentational formalisms that are close to the representations in human brains. In this

Section, we will discuss some of these formalisms, namely semantic networks, frame

systems, and conceptual graphs. The ﬁrst two formalisms are mainly presented for

historical reasons since they can be regarded as ancestors of Description Logics. In

contrast, the third formalism can be regarded as a “sibling” of Description Logics

since both have similar ancestors and live in the same time.

142

Relationships with other Formalisms 143

4.1.1 Semantic networks

Semantic networks originate in Quillian’s

semantic memory models

[

Quillian, 1967

]

a graphical formalism designed to represent “word concepts” in a deﬁnitorial way,

i.e., similar to the one that can be found in an encyclopedia deﬁnition. This for-

malism is based on labelled graphs with diﬀerent kinds of edges and nodes. Besides

others, Quillian’s networks allow for subclass/superclass edges, for and- and or

edges, and for

subject/object edges between nodes.

Following Quillian’s memory models, a great variety of

semantic network

for-

malisms were proposed; an overview of their history can be found in

[

Brachman,

1979

]

. In general, semantic networks distinguish between concepts (denoted by

generic nodes

) and individuals (denoted by

individual nodes

), and between sub-

class/superclass edges and property edges. Using subclass/superclass links, concepts

can be organised in a specialisation hierarchy. Using property edges, properties can

be associated to concepts, that is, to the individuals belonging to the concept the

properties are associated with. Figure 4.1 contains a hierarchy of animals, birds,

ﬁshes, etc. Interestingly, the cognitive adequacy of this approach was proven em-

pirically

[

Collins and Quillian, 1970

]

The two kinds of edges interact with each other: A property is

inherited along

subclass/superclass edges—if not modiﬁed in a more speciﬁc class. For example,

birds are equipped with skin because animals are equipped with skin, and birds

Canar Shark

Bird Fish

Animal

- has skin

- can move around

- eats

- breathes

- has feathers

- has wings

- can fly

- has fins

- can swim

- has gills

- can sing

- is yellow

- can bite

- is dangerous

Ostrich

- has long,

thin legs

- is tall

- can’t fly

Salmon

- is pink

- is edible

- swims upstreams

to lay eggs

ljbdfg

Fig. 4.1. A semantic network describing animals.

144 U. Sattler, D. Calvanese, R. Molitor

inherit this property because of the subclass/superclass edge between birds and

animals. In contrast, although ostriches are birds, they do not inherit the property

“can ﬂy” from birds because this property is “modiﬁed” for ostriches.

Intuitively, it should be possible to translate subclass/superclass edges into con-

cept deﬁnitions, for example,

Shark ≡

Fish u

CanBite

u IsDangerous.

According to Brachman

[

1985

]

, the above translation is not always intended. Sub-

class/superclass edges can also be read as

primitive

concept deﬁnitions, that is,

they impose only necessary properties but not suﬃcient ones. Hence the above

translation might better be

Shark v Fish u CanBite u

IsDangerous

Due to the lack of a precise semantics, there are even more readings of subclass/su-

perclass edges which are discussed in Woods

[

1975

]

[

1977b; 1985

]

. A prominent

reading is the one of inheritance by default

, which can be speciﬁed in diﬀerent ways,

thus leading to misunderstandings and to the question which of these speciﬁcations

is the “right” one (see also Chapter 6).

As a consequence of this ambiguity, new formalisms mainly evolved along two

lines: (1) To capture inheritance by default, various non-monotonic inheritance sys-

tems, respectively various ways of reasoning in non-monotonic inheritance systems,

were investigated

[

Touretzky et al.

, 1987; 1991; Selman and Levesque, 1993

]

. (2) To

capture the monotonic aspects of semantic networks, a new graphical formalism,

structured inheritance networks

, was introduced and implemented in the system

Kl-One

[

Brachman, 1979; Brachman and Schmolze, 1985

]

. It was designed to cover

the declarative, monotonic aspects of semantic networks, and hence did not specify

the way in which (non-monotonic multiple) inheritance was supposed to function

in conﬂicting situations. Brachman and Schmolze

[

1985

]

argue that Kl-One does

not allow for cancellation or inheritance by default because such mechanisms would

make taxonomies meaningless. Indeed, all properties of a given concept could be

cancelled, so that it would ﬁt everywhere in the taxonomy. Their proposition is to

make a strict separation of default assertions and conceptual descriptions.

Brachman and Schmolze

[

1985

]

, b esides pointing out the computation of the

taxonomy as a core system service, describe the meaning of various concept con-

structors that were implemented in

Kl-One, for example conjunction, universal

value restrictions, role hierarchies, role-value-maps, etc. Moreover, we ﬁnd a clear

distinction between individuals and concepts, and between a terminological and an

assertional formalism.

In the following, we use standard Description Logics as deﬁned in Chapter 2.

Relationships with other Formalisms 145

Later

[

Levesque and Brachman, 1987

]

, Kl-One

was provided with a well-deﬁned

“Tarski-style” semantics which ﬁxed the precise meaning of its graphical constructs

and led to the deﬁnition of the ﬁrst Description Logic

[

Levesque and Brachman,

1987

]

, at that time also called terminological languages, concept languages, or Kl-

One

based languages. Besides giving a precise meaning to semantic networks, this

formalisation allowed the investigation of inference algorithms with respect to their

soundness, completeness, and computational complexity. For example, it turned

out that subsumption in Kl-One is undecidable, mainly due to role-value-maps

[

Schmidt-Schauß, 1989

]

4.1.2 Frame systems

Minsky

[

1981

]

introduced frame systems as an alternative to logic-oriented ap-

proaches to knowledge representation, which he thought were not adequate to “sim-

ulate common sense thinking” for various reasons. His system provides record-like

data structures to represent prototypical knowledge concerning situations and ob-

jects and includes defaults, multiple perspectives, and analogies. Nowadays, se-

mantic networks and frame systems are often viewed as the same family of for-

malisms. However, in standard semantic networks, properties are restricted to

primitive, atomic ones, whereas, in general, properties in frame systems can be

complex concepts described by frames.

One goal of the frame approach was to gather all relevant knowledge about a situa-

tion (e.g., entering a restaurant) in one object instead of distributing this knowledge

across various axioms. Roughly spoken, a situation (or an object) is described in one

frame. Similar to entries in a record, a frame contains

slots to represent properties

of the situation described by the frame. Reasoning comes in two shapes: (1) Using

a “partial matching”, more speciﬁc frames are embedded into more general ones,

thus giving, for example, meaning to a new situation or classifying an object as a

kind of, say, bird. (2) Searching for slot ﬁllers to collect more information con-

cerning a speciﬁc situation. A variety of expert systems

[

Fikes and Kehler, 1985;

Christaller et al., 1992; Gen, 1995; Flex, 1999

]

are based on a frame-based formalism

and are further enhanced with rules, triggers, daemons, etc.

Despite the fact that frame systems were designed as an alternative to logic, the

monotonic, declarative part of this formalism could be shown to be captured us-

ing ﬁrst-order predicate logic

[

Hayes, 1977; 1979

]

. To our knowledge, no precise

semantics could be given for the non-declarative, non-logic, or non-monotonic as-

pects of frame systems. Hence neither their expressive power nor the quality of

the corresponding reasoning algorithms and services can be compared with other

formalisms.

In the remainder of this section, we show how the monotonic part of a frame-

146 U. Sattler, D. Calvanese, R. Molitor

based knowledge base can be translated into an

ALUN TBox

[

Calvanese et al.,

1994

]

Since there is no standard syntax for frame systems, we have chosen to use

basically the notation adopted by Fikes and Kehler

[

1985

]

, which is used also in the

Kee

system.

A frame deﬁnition is of the form

Frame : F in KB F E

, where F is a

frame

name

and E

is a frame expression, i.e., an expression formed according to the

following syntax:

−→ SuperClasses

, . . . , F

MemberSlot : S

ValueClass : H

Cardinality.Min

: m

Cardinality.Max

· · ·

MemberSlot :

ValueClass

Cardinality.Min : m

Cardinality.Max

: n

denotes a frame name, S

denotes a slot name,

and

denote positive integers,

and

denotes slot constraints. A

slot constraint can be speciﬁed as follows:

H −→

F |

(INTERSECTION H

)

(UNION

) |

(NOT

)

A frame knowledge base

is a set of frame deﬁnitions.

For example, Figure 4.2 shows a simple

Kee

knowledge base describing courses in

a university. Cardinality restrictions are used to impose a minimum and maximum

number of students that may be enrolled in a course, and to express that each

course is taught by exactly one individual. The frame AdvCourse

represents courses

which enroll only graduate students, i.e., students who already have a degree. Basic

courses, on the other hand, may be taught only by professors.

Hayes

[

1979

]

gives a semantics to frame deﬁnitions by translating them to ﬁrst-

order formulae in which frame names are translated to unary predicates, and slots

are translated to binary predicates.

In order to translate frame knowledge bases to ALUN knowledge bases, we ﬁrst

deﬁne the function Ψ that maps each frame expression into an ALUN concept ex-

pression as follows: Each frame name

is mapped onto an atomic concept Ψ(F ),

Not only the translation but also the example are by Calvanese et al.

[

1994

]

Kee

is a trademark of Intellicorp. Note that a Kee user does not directly specify her knowledge base in

this notation, but is allowed to deﬁne frames interactively via the graphical system interface.

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