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语义网的逻辑基础 Logical Foundation of the Semantic Web

主讲： 黄智生 Zhisheng Huang

Vrije University Amsterdam ， The Netherlands

huang@cs.vu.nl

助教： 胡伟 Wei Hu

Southeast University

whu@seu.edu.cn

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课程时间表 Schedule

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• 描述逻辑是什么？• 描述逻辑基本系统• 描述逻辑种类• 描述逻辑实例

讲座 2 ：描述逻辑导论Lecture 2: Introduction to

Description Logics

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• 描述逻辑是一类知识表示语言表达应用领域的概念定义（有可以被看作为专业术语知识） Description logics (DL) are a family of knowledge representation languages which can be used to represent the concept definitions of an application domain (known as terminological knowledge)

描述逻辑是什么？ What are Description Logics?

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描述逻辑简史A Brief History of Description

Logics• Major focus of KR research in the 80’s• Led by Ron Brachman – (AT&T Labs)

• Grew out of early network-based KR systems like semantic networks and frames.

• Major systems and languages –• 80s: KL-ONE, NIKL, KANDOR, BACK, CLASSIC,

LOOM

• 90s: FACT, RACER,

• 00s: DAML+OIL, OWL

• Used as the basis for the Semantic web languages DAML+OIL and OWL

• Some (one) commercial systems

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概念与本体Concepts and Ontologies

• Philosophical discipline, branch of philosophy that deals with the nature and the organisation of reality.

• Science of Being (Aristotle, Metaphysics, IV,1)

• What is being?

• What are the features common to all beings?

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Vocabulary and Ontology

• Controlled vocabulary (Jernst 2003) : • a list of controlled terms• unambiguous• non-redundant definition

• Ontology: a controlled vocabulary expressed in an ontology representation language (Jernst 2003)

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In computer science …

• An ontology is an explicit specification of a conceptualization. [Gruber93]

• An ontology is a shared understanding of some domain of interest. [Uschold, Gruninger96]

• There are many definitions• a formal specification EXECUTABLE• of a conceptualization of a domain COMMUNITY• of some part of world that is of interest APPLICATION

• Defines• A common vocabulary of terms• Some specification of the meaning of the terms• A shared understanding for people and machines

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Why develop an ontology?

• To make domain assumptions explicit• Easier to change domain assumptions• Easier to understand and update legacy data

• To separate domain knowledge from operational knowledge• Re-use domain and operational knowledge

separately• A community reference for applications• To share a consistent understanding of what

information means.

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本体的主要特征Key features of an

Ontology• 概念层次性 Concept hierarchy,

• 概念包含关系 concept subsumption

•特殊与一般关系 InstanceOf Relation (Instances)

•部分与整体关系 PartOf Relation (property)

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Why not other alternatives

• 一阶谓词逻辑 the first-order predicate logic

• 集合论 set theory

• 程序语言 programming languages

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概念与分类

• 设定存在一个所有个体（ Individual ）的集合

• 一个概念被看成是一个个体的集合（ Set of individuals ）

• 定义一个概念就是确定一个分类• 概念集合与个体集合是不相交的• 个体上的一个二元关系集合被称为一

个性质（ Property/Role)

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复合概念 概念的否定， 交 与并

CDCD

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描述逻辑 Description Logic

Knowledge Base

Tbox (schema) 术语部分

Abox (data) 断言部分

Man ´ Human u Male

Happy-Father ´ Man u 9 has-child Female u …

John : Happy-Father

hJohn, Maryi : has-child Infe

ren

ce S

yste

m

Inte

rface

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描述逻辑 Description Logic

Knowledge Base

Tbox (schema) 术语部分

Abox (data) 断言部分

Infe

ren

ce S

yste

m

Inte

rface

Rbox (data) 关系部分

Has-daughter v has-child

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Basic Description Logic: AL （ Attributive

Language ） • Concept Expressions:• A ( 原子概念 atomic concept)• ( 全概念， universal concept) ( 空概念， bottom concept)• A ( 原子否定， atomic negation)

• C ⊓ D ( 并， intersection) R.C ( 值限制， value restriction) R.T ( 有限存在量化 limited existential

quantification)

where A is a concept name, C and D are concept expressions, and R is a role expression

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Family of AL-Language

• U: C t D ( 交 union)• E: R.C ( 完全存在量化 full existential

quantification)• N: ( 数量限制 Number restrictions)

• ( n R) ( 至少限制 at least restriction)• ( n R) ( 最多限制 at most restriction)

• C: (Negation): : C

• AL E N =AL + [E]+ [N] • Smallest propositionally closed DL is ALC (equiv modal

K(m))• Concepts constructed using u, t, :, 9 and 8

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Examples

woman ≡ person ⊓ female

man ≡ person ⊓ woman

mother ≡ woman ⊓ hasChild.person

father ≡ man ⊓ hasChild.person

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一个实例 Example

• whitehorse ≡ horse ⊓ white.

• color(white).

• whitehorse ≡ horse ⊓ hasColor. {white}.

这里 white 是一个列名（ nominal ）

• whitehorse ≡ horse ⊓ hasColor. {white} ⊓ hasColor. {white}.

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AL 句法规则

， AL• p AtomicConcept => p AL ，• p AtomicConcept => p AL ，• C ， D AL =>C ⊓ D AL ，• C AL , R Role=> R.C AL, • R Role => R.T AL

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描述逻辑的语义模型

一个描述逻辑语言 DL 上的一个语义模型M=(S, {R1, R2,… Rn}, V)

这里 S 是所有可能个体（ Individual) 的集合Ri SXS 是一个 S 上的二元关系V ： P -> PowerSet(S) 是一个赋值函数，它

给一个原子概念赋予 S 的一个子集。

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对照：模态逻辑的语义模型

• 命题模态逻辑语言 L 上的一个语义模型M=(S, A, V)

这里 S 是可能世界的集合A SXS 是一个可达世界的关系V ： P -> PowerSet(S) 是一个赋值函数，它

给一个原始命题赋予一个可能世界子集。

所以说，一个描述逻辑实质上就是一个多模态逻辑

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描述逻辑 AL 的真值条件

• M ， s |= p if s V(p)

• M ， s |= p iff M, s |=\= p

• M ， s |= C ⊓ D iff M, s |= C and M, s |= D

• M ， s |= R.C iff M, s’ |=C for all s’ such that <s,s’> R

想想看： ， ， R.T 对应的真值条件是什么

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DL Semantics• Semantics defined by interpretations

• An interpretation I = (I, ¢I), where• I is the domain (a non-empty set)

•¢I is an interpretation function that maps:• Concept (class) name A ! subset AI of I

• Role (property) name R ! binary relation RI over I

• Individual name i ! iI element of I

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DL Semantics (cont.)

• Interpretation function ¢I extends to concept (and role) expressions in the obvious way, e.g.:

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规范的 AL 语义

• {x}I = {xI}

• {p }I =S/ pI

• {C ⊓ D }I = {C} I {D}I

• {R.C}I = {x |y(<x,y> RI =>y CI}

这里 I 被称作一个解释（ Interpretation ）， 实质上就是一个模型。

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Axioms define relations between concepts

• 概念包含（ Subsumption ） : C v D iff CI DI

定义：• 概念相等 (Equivalence): C D iff C

v D 而且 D v C

• 概念不相交（ Disjointness ） : C ⊓ D

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General Concept Inclusion

Expressivity with GCIs

• Disjointness : C ⊓ D v •Identity: {a} v {b}

•Distinctiveness : {a} ⊓ {b} v

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DL Knowledge Base• A DL Knowledge base K is a pair hT ,Ai

where• T is a set of

“terminological” axioms (the Tbox)• A is a set of “assertional” axioms (the Abox)

• Tbox axioms are of the form:C v D, C ´ D, R v S, R ´ S and R+ v Rwhere C, D concepts, R, S roles, and R+ set of transitive roles

• Abox axioms are of the form:x:D, hx,yi:Rwhere x,y are individual names, D a concept and R a role

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More about Family of AL Language

• Additional letters indicate other extension, e.g.:• H for role inclusion axioms (role hierarchy)• O for nominals (singleton classes, written {x})• I for inverse roles• Q for qualified number restrictions (of form

6nR.C, >nR.C)• S often used for ALC with transitive roles

(R+)• SHIQ: ALC + R+ + role hierarchy +

inverse roles + Q

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SHOIN

• SHION:S: ALC + role transitivity

H: role hiersrchies

O: nominals

I: Inverse roles

N: cardinality restriction

SHOIN(D) = OWL-DLD: datatypes

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Knowledge Base Semantics• An interpretation I satisfies (models) a Tbox axiom

A (I ² A):I ² C v D iff CI µ DI I ² C ´ D iff CI = DI

I ² R v S iff RI µ SI I ² R ´ S iff RI = SI

I ² R+ v R iff (RI)+ µ RI

• I satisfies a Tbox T (I ² T ) iff I satisfies every axiom A in T

• An interpretation I satisfies (models) an Abox axiom A (I ² A):

I ² x:D iff xI 2 DI I ² hx,yi:R iff (xI,yI) 2 RI

• I satisfies an Abox A (I ² A) iff I satisfies every axiom A in A

• I satisfies an KB K (I ² K) iff I satisfies both T and A

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Reasoning Tasks for Concept

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Reduction to Subsumption

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Reduction to Unsatisfiability

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Reducing Unsatisfiability

• The followings are equivalent:

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描述逻辑系统命名规则 DL Naming

• Basic description logic is ALC (equiv modal K(m))• Concepts constructed using u, t, :, 9 and 8

• S often used for ALC with transitive roles• Additional letters indicate other extension, e.g.:

• H for role inclusion axioms (role hierarchy)• O for nominals (singleton classes, written {x})• I for inverse roles• N for number restrictions (of form 6 n R, > n R)• Q for qualified number restrictions (of form 6 n R.C,

> n R.C)• …

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DL-Lite: A “Scalable” DL Family

• R AtomicRole =>R, R- BasicRole

• A AtomicConcept =>A BasicConcept,

• R AtomicRole => R BasicConcept

• C BasicConcept=> C, C GeneralConcept

• R BasicRole=> R, R GeneralRole

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Simplified DL-Lite Syntax Rules

• R ->P|P-

• B ->A| R

• C->B| B

• E ->R| R

Where R is basic role, P is atomic role, A is atomic concept, B is basic concept, C is general concept, E is general role

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DL-Lite Family

• DL-Litecore Tbox is a set of inclusion axioms of the from B v C

• DL-LiteR = DL-Litecore + role inclusion axioms of the from R v E

• DL-LiteF = DL-Litecore + functionality on role or on their inverse with the form (funct R)

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想一想： DL-Lite 能不能表达下列描述 ?

• A1 t A2 v C

• B v C1 ⊓ C2

• Disjoint （ A,B)

• A1 ⊓ A2 v C

• B v C1 t C2

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思考：描述逻辑的下一步扩展？

想一想，如果让你对现有的描述逻辑的表达功能进行扩展，你将会添加一些什么？

•今天下午：描述逻辑与知识表示的专题讨论

为什么？如何对其进行形式化描述？

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练习题

思考题 2.1 ：如何用描述逻辑来表示下列一些概念 ?

• “ 至少有一个女孩的女人”• “ 没有女孩的女人”• “ 所有女孩都上学的女人”• “ 有一个女孩不上学的女人”

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练习题

思考题 2.2 用 ALC 语言能不能来表示下列一些概念 ? 为什么？• “ 有 3 个女孩的女人”• “ 最大的孩子是女孩的女人”• “ 非女青年人”• “ 所有的女儿都没上学的女人”

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练习题思考题 2.3 判断下列一些公式哪些是满足的，哪些是不可满足的，哪些是永真的？• (8R. :C) u C

• (8R. :C) u 9 R. C

• (8R. :C) t 9 R. C

• (8R. C) t 9 R.T.

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练习题思考题 2.4. 证明 DL-LiteF 不满足有限模型性(Finite Model Property),

并分析其问题特征。

一个逻辑具有有限模型性 则表明其任何一个公式集如果是可满足的，那么必存在它的一个有限模型。

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语义网逻辑基础演义

第二回：本体为纲描述逻辑展头角 概念当先形式方法显风采

欲知后事如何，请听下回分解。。。

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Questions and Discussions