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

1. 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

2. 课程时间表Schedule

3. 讲座2：描述逻辑导论Lecture 2: Introduction to Description Logics • 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

4. 描述逻辑是什么？ What are Description Logics? • 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)

5. 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

6. 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?

7. 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)

8. 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

9. 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.

10. 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)

11. 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

12. 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)

13. 复合概念 • 概念的否定， 交 与并 C D CD

14. 描述逻辑 Description Logic Knowledge Base Tbox (schema)术语部分 Man ´ Human u Male Happy-Father ´ Man u9 has-child Female u … Interface Inference System Abox (data)断言部分 John : Happy-Father hJohn, Maryi : has-child

15. 描述逻辑 Description Logic Knowledge Base Tbox (schema)术语部分 Interface Abox (data)断言部分 Inference System Rbox (data)关系部分 Has-daughter v has-child

16. 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

17. 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 EN =AL + [E]+ [N] • Smallest propositionally closed DL is ALC (equiv modal K(m)) • Concepts constructed using u, t, :, 9 and 8

18. Examples woman ≡ person ⊓ female man ≡ person ⊓woman mother ≡ woman ⊓hasChild.person father ≡ man ⊓hasChild.person

19. 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}.

20. 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

21. 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。

22. 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

23. 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

24. DL Semantics • Semantics defined by interpretations • An interpretation I = (DI, ¢I), where • DI is the domain (a non-empty set) • ¢I is an interpretation function that maps: • Concept (class) name A! subset AI of DI • Role (property) name R! binary relation RI over DI • Individual name i!iI element of DI

25. DL Semantics (cont.) • Interpretation function ¢I extends to concept (and role) expressionsin the obvious way, e.g.:

26. 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。

27. 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 vD而且 DvC • 概念不相交（Disjointness）: C  ⊓ D  

28. General Concept Inclusion Expressivity with GCIs • Disjointness : C ⊓ D v • Identity: {a} v {b} • Distinctiveness : {a} ⊓ {b} v

29. DL Knowledge Base • A DL Knowledge baseK 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: • CvD, C´D, RvS, R´S and R+vR • where C,D concepts, R, S roles, and R+ set of transitive roles • Abox axioms are of the form: • x:D, hx,yi:R • where x,y are individual names, D a concept and R a role

30. 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

31. SHOIN • SHION: • S: ALC + role transitivity • H: role hiersrchies • O: nominals • I: Inverse roles • N: cardinality restriction • SHOIN(D) = OWL-DL • D: datatypes

32. Knowledge Base Semantics • An interpretationI satisfies (models) a Tbox axiom A (I²A): • I²CvD iff CIµDII²C´D iff CI = DI • I²RvS iff RIµSII²R´S iff RI = SI • I²R+vR iff (RI)+µRI • Isatisfiesa TboxT (I²T ) iff I satisfies every axiom A in T • An interpretationI satisfies (models) an Abox axiom A (I²A): • I²x:D iff xI2DII²hx,yi:R iff (xI,yI) 2RI • Isatisfies an AboxA (I²A) iff I satisfies every axiom A in A • Isatisfies an KBK (I²K) iff I satisfies both T and A

33. Reasoning Tasks for Concept

34. Reduction to Subsumption

35. Reduction to Unsatisfiability

36. Reducing Unsatisfiability • The followings are equivalent:

37. 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 6n R, >n R) • Q for qualified number restrictions (of form 6 n R.C, > n R.C) • …

38. 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

39. 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

40. 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)

41. 想一想： DL-Lite能不能表达下列描述? • A1 tA2v C • B v C1⊓C2 • Disjoint（A,B) • A1 ⊓A2v C • B v C1tC2

42. 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？ 为什么？如何对其进行形式化描述？ • 今天下午：描述逻辑与知识表示的专题讨论

43. 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”

44. 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”

45. 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) t9 R.T.

46. 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。

47. 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。。。

48. Questions and Discussions