1. ELW2009, Jenova SWCLOS: A Semantic Web Processor on CLOS Advanced Discussion Seiji Koide National Institute of Informatics

2. ELW2009, Jenova Formal Semantics Platonic idealism (from Wikipedia) –Some contemporary linguistic philosophers construe “ Platonism ” to mean the proposition that universals exist independently of particulars (a universal is anything that can be predicated of a particular). Ontology –Ontology is the philosophical study of the nature of being, existence or reality in general, as well as of the basic categories of being and their relations. Formal … –How to describe things in objectivity –How to enable computation on ontology

3. ELW2009, Jenova Formal Semantics Equality –eq, eql, equal, equalp, = –Two names are lexically equal, then both equal? –Two names are different, then both different? –Two anonymous objects have the same structure and the same value, then both equal? What do you mean by “ same ” ? –Two classes share instances, then both equal?

4. ELW2009, Jenova Denotational and Extensional Semantics of RDF based on RDF Semantics by Patrick Hayes and Brian McBride

5. ELW2009, Jenova Russell&Norvig AIMA

6. ELW2009, Jenova Formal Semantics Sentences Entails Aspects of the Real World Aspects of the Real World follows Representation World

7. ELW2009, Jenova Formal Semantics Universe of Discourse Set theory Sentences Entails Aspects of the Real World Aspects of the Real World follows Representation World

8. ELW2009, Jenova Formal Semantics Universe of Discourse Sentences Entails Aspects of the Real World Aspects of the Real World follows Representation World

9. ELW2009, Jenova Formal Semantics Study of Ontology Study of Formal Semantics Universe of Discourse KB =α α Sentences Entails Aspects of the Real World Aspects of the Real World follows Representation World

10. ELW2009, Jenova Formal Semantics Universe of Discourse Interpretation     Sentences Entails Aspects of the Real World Aspects of the Real World follows Representation World Study of Formal Semantics

11. ELW2009, Jenova Formal Semantics Universe of Discourse 0 ×1 = 0 1 ×1 = 1 0 ＋ 1 = 1 1 ＋ 1 = 2 Sentences Entails Aspects of the Real World Aspects of the Real World follows Representation World Interpretation Study of Formal Semantics

12. ELW2009, Jenova Semantic Web LayerCake RDF Semantics RDFS Semantics OWL Semantics

13. ELW2009, Jenova Semantic Web LayerCake RDFSemantics

14. ELW2009, Jenova RDF Model Theory Directed labeled graph

15. ELW2009, Jenova RDF Model Theory An RDF graph is a set of RDF triples. Two graphs described by two N-triples documents that shares a set of triples, in which there is no difference except only blank node identifiers, are equal. A partial graph of RDF is a partial set of triples of the graph. A graph that have no blank node is called ground graph. http:/…/Proposal/ eg:author _:x1. _:x1 eg:name “Tim Berners Lee”. http:/…/Proposal/ eg:author _:x1. _:x1 eg:name “Tim Berners Lee”. http:/…/Proposal/ eg:author _:x2. _:x2 eg:name “Tim Berners Lee”. http:/…/Proposal/ eg:author _:x2. _:x2 eg:name “Tim Berners Lee”.

16. ELW2009, Jenova RDF Vocabulary rdf:_1 rdf:typerdf:predicate rdf:objectrdfs:comment rdf:first rdf:rest rdf:type rdf:valuerdf:subject rdf:_2 rdf:_3 rdf:Property rdf:nil rdf:List rdf:Statement rdf:Alt rdf:Seq rdf:Bag rdf:XMLLiteral

17. ELW2009, Jenova RDF Simple Interpretation A non-empty set IR of resources, called the domain or universe of I. A set IP, called the set of properties of I. A mapping IEXT from IP into the powerset of IR × IR i.e., the set of sets of pairs with x and y in IR. A mapping IS from URI references in V into (IR ∪ IP). A mapping IL from typed literals in V into IR. A distinguished subset LV of IR, called the set of literal values, which contains all the plain literals in V.

18. ELW2009, Jenova RDF Simple Interpretation ex:a ex:p "xx"^^rdf:XMLLiteral “this is a literal” Representation World Interpretation

19. ELW2009, Jenova IS (ex:a) RDF Simple Interpretation A set of resources; IR ex:a ex:p "xx"^^rdf:XMLLiteral “this is a literal” Representation World Interpretation

20. ELW2009, Jenova IS (ex:p) RDF Simple Interpretation A set of propertes IP ex:p IS (ex:a) A set of resources; IR ex:a "xx"^^rdf:XMLLiteral “this is a literal” Representation World Interpretation

21. ELW2009, Jenova RDF Simple Interpretation URI set IR U IP Mapping IS IS (ex:p) A set of propertes IP ex:p IS (ex:a) A set of resources; IR ex:a "xx"^^rdf:XMLLiteral “this is a literal” Representation World Interpretation

22. ELW2009, Jenova IL ("xx"^^rdf:XMLLiteral) RDF Simple Interpretation URI set IR U IP Mapping IS IS (ex:p) A set of propertes IP ex:p IS (ex:a) A set of resources; IR ex:a "xx"^^rdf:XMLLiteral “this is a literal” Representation World Interpretation

23. ELW2009, Jenova RDF Simple Interpretation Mapping IL Typed literal set IR IL ("xx"^^rdf:XMLLiteral) URI set IR U IP Mapping IS IS (ex:p) A set of propertes IP ex:p IS (ex:a) A set of resources; IR ex:a "xx"^^rdf:XMLLiteral “this is a literal” Representation World Interpretation

24. ELW2009, Jenova Representation Mapping IL Typed literal set IR IL ("xx"^^rdf:XMLLiteral) URI set IR U IP Mapping IS IS (ex:p) A set of propertes IP ex:p IS (ex:a) A set of resources; IR ex:a "xx"^^rdf:XMLLiteral “this is a literal” World Interpretation RDF Simple Interpretation Set of literal values LV Set of plain literals “this is literal”

25. ELW2009, Jenova RDF Simple Interpretation 〈 x, y 〉 y Mapping IEXT from IP to 〈 x, y 〉 IS (ex:p) x Representation World Interpretation A set of propertes IP A set of resources; IR

26. ELW2009, Jenova RDF Simple Interpretation 〈 x, y 〉 PowerSet IR × IR y x IS (ex:p) Representation World Interpretation A set of propertes IP A set of resources; IR Mapping IEXT from IP to 〈 x, y 〉

27. ELW2009, Jenova 〈 x, y 〉 PowerSet IR × IR y x IS (ex:p) Representation World Interpretation A set of propertes IP A set of resources; IR Mapping IEXT from IP to 〈 x, y 〉 RDF Simple Interpretation IEXT(IS (ex:p) ) is called a property extension of IS (ex:p). IP may be a partial set of IR. We distinguish IS (ex:p) as an element of pair from denotated property IS (ex:p), e.g., ex:p ex:p y. IEXT(IS (ex:p) ) is called a property extension of IS (ex:p). IP may be a partial set of IR. We distinguish IS (ex:p) as an element of pair from denotated property IS (ex:p), e.g., ex:p ex:p y. IS (ex:p) ∈ IRIS (ex:p) ∈ IP

28. ELW2009, Jenova 〈 x, y 〉 PowerSet IR × IR y x IS (ex:p) Representation World Interpretation A set of propertes IP A set of resources; IR Mapping IEXT from IP to 〈 x, y 〉 IEXT(IS (ex:p) ) is called a property extension of IS (ex:p). IP may be a partial set of IR. We distinguish IS (ex:p) as an element of pair from denotated property IS (ex:p), e.g., ex:p ex:p y. IEXT(IS (ex:p) ) is called a property extension of IS (ex:p). IP may be a partial set of IR. We distinguish IS (ex:p) as an element of pair from denotated property IS (ex:p), e.g., ex:p ex:p y. RDF Simple Interpretation 〈 IS (ex:p), y 〉∈ IEXT(IS (ex:p))

29. ELW2009, Jenova Semantic Conditions for Ground Graphs If E is a plain literal “ aaa ” in V, then I (E) ＝ aaa. If E is a plain literal “ aaa ” @ttt in V, then I (E) ＝ 〈 aaa, ttt 〉. If E is a typed literal in V, then I (E) ＝ IL(E). If E is a URI reference in V, then I (E) ＝ IS(E). If E is a ground triple s p o., then I (E) ＝ true if s, p, and o are in V, I (p) is in IP, and 〈 I (s), I (o) 〉 is in IEXT( I (p)), otherwise I (E) ＝ false. If E is a ground RDF graph, then I (E) ＝ false if I (E') ＝ false for some triple E' in E, otherwise I (E) ＝ true. “soccur@en-US” = “football”@en

30. ELW2009, Jenova Simple Entailment between RDF Graphs I satisfies E if I (E) = true, and a set S of RDF graphs (simply) entails a graph E if every interpretation which satisfies every member of S also satisfies E. If A entails B, then any interpretation that makes A true also makes B true, so that an assertion of A already contains the same “ meaning ” as an assertion of B. Through the notions of satisfaction, entailment and validity, formal semantics gives a rigorous definition to a notion of “ meaning ”. Any process which constructs a graph E from some other graph(s), S is said to be (simply) valid if S entails E in every case, otherwise invalid. S ⊨ ES ⊨ E

31. ELW2009, Jenova Simple Entailment Rules of RDF Rule Name If E contains:then add: se1uuu aaa xxx. uuu aaa _:nnn. where _:nnn identifies a blank node allocated to xxx by rule se1 or se2. allocated se2uuu aaa xxx. _:nnn aaa xxx. where _:nnn identifies a blank node allocated to uuu by rule se1 or se2. allocated.. ex:aex:b ex:c ex:q _:x. ex:b ex:c ex:q _:x.

32. ELW2009, Jenova Literal Generalization/Instantiation Rule Rule Name If E contains:then add: lguuu aaa lll. uuu aaa _:nnn. where _:nnn identifies a blank node allocated to the literal lll by this rule. allocated gl uuu aaa _:nnn. where _:nnn identifies a blank node allocated to the literal lll by rule lg.allocatedrule lg uuu aaa lll. ex:a ex:p “literal1” ex:a ex:p “literal2” ex:q

33. ELW2009, Jenova RDF entailment rules Rule Name If E contains:then add: rdf1uuu aaa yyy.aaa rdf:type rdf:Property. rdf2 uuu aaa lll. where lll is a well-typed XML literal. _:nnn rdf:type rdf:XMLLiteral. where _:nnn identifies a blank node allocated to lll by rule lg. allocated torule lg ex:a ex:p “1”^^xsd:integer ex:a ex:p rdf:XMLLiteral rdf:type

34. ELW2009, Jenova RDF Semantic Condition x ∈ IP ， iff 〈 x, I (rdf:Property) 〉 ∈ IEXT( I (rdf:type)) xxx rdf:type rdf:Property. IEXT( I (rdf:type)) ∧ xxx is a well-typed XML literal ⇒ IL( “ xxx ” ^^rdf:XMLLiteral) is an XML value of xxx, IL( “ xxx ” ^^rdf:XMLLiteral) ∈ LV, 〈 IL( “ xxx ” ^^rdf:XMLLiteral), I (rdf:XMLLiteral) 〉∈ IEXT( I (rdf:type)) “ xxx ” ^^rdf:XMLLiteral rdf:type rdf:XMLLiteral. IEXT( I (rdf:type)) ∧ xxx is ill-typed XML literal ⇒ IL( “ xxx ” ^^rdf:XMLLiteral) ∉ LV ， 〈 IL( “ xxx ” ^^rdf:XMLLiteral), I (rdf:XMLLiteral) 〉 ∉ IEXT( I (rdf:type))

35. ELW2009, Jenova RDF Axiomatic Triples rdf:_1 rdf:typerdf:predicate rdf:objectrdfs:comment rdf:first rdf:rest rdf:type rdf:valuerdf:subject rdf:_2 rdf:_3 rdf:Property rdf:nil rdf:Litst rdf:Statement rdf:Alt rdf:Seq rdf:Bag rdf:XMLLiteral rdf:type rdf:type rdf:Property. rdf:subject rdf:type rdf:Property. rdf:predicate rdf:type rdf:Property. rdf:object rdf:type rdf:Property. rdf:first rdf:type rdf:Property. rdf:rest rdf:type rdf:Property. rdf:value rdf:type rdf:Property. rdf:_1 rdf:type rdf:Property. rdf:_2 rdf:type rdf:Property.... rdf:nil rdf:type rdf:List.

36. ELW2009, Jenova Semantic Web LayerCake RDFS Semantics

37. ELW2009, Jenova RDFS Vocabulary rdf:_1 rdfs:subClassOfrdfs:subPropertyOfrdf:typerdf:predicate rdfs:domainrdfs:rangerdfs:labelrdf:objectrdfs:comment rdf:first rdf:rest rdfs:subClassOf rdfs:subPropertyOf rdf:type class instance rdfs:seeAlsordfs:isDefinedByrdf:valuerdf:subjectrdfs:member rdf:_2 rdf:_3 rdf:Propertyrdfs:ContainerMembershipProperty rdf:nil rdfs:Resource rdf:List rdf:Statement rdfs:Containerrdf:Alt rdf:Seq rdf:Bag rdfs:Literal rdf:XMLLiteral rdfs:Classrdfs:Datatype

38. ELW2009, Jenova RDFS Semantic Condition Universe of Discourse IR

39. ELW2009, Jenova RDFS Semantic Condition IR = ICEXT(IS(rdfs:Resource)) IS(rdfs:Resource) x ∈ ICEXT(c) ， iff 〈 x,c 〉∈ IEXT(IS(rdf:type))

40. ELW2009, Jenova RDFS Semantic Condition IR = ICEXT(IS(rdfs:Resource)) IS(rdfs:Resource) x ∈ ICEXT(c) ， iff 〈 x,c 〉∈ IEXT(IS(rdf:type)) A set of propertes IP

41. ELW2009, Jenova RDFS Semantic Condition IR = ICEXT(IS(rdfs:Resource)) IS(rdfs:Resource) IS(rdf:Property) x ∈ ICEXT(c) ， iff 〈 x,c 〉∈ IEXT(IS(rdf:type)) IP = ICEXT(IS(rdf:Property))

42. ELW2009, Jenova RDFS Semantic Condition IS(rdfs:Resource) IS(rdfs:Literal) IS(rdf:Property) IP = ICEXT(IS(rdf:Property)) IR = ICEXT(IS(rdfs:Resource)) x ∈ ICEXT(c) ， iff 〈 x,c 〉∈ IEXT(IS(rdf:type)) LV = ICEXT(IS(rdfs:Literal))

43. ELW2009, Jenova RDFS Semantic Condition IS(rdfs:Resource) IS(rdfs:Literal) IS(rdf:Property) IP = ICEXT(IS(rdf:Property)) IR = ICEXT(IS(rdfs:Resource)) x ∈ ICEXT(c) ， iff 〈 x,c 〉∈ IEXT(IS(rdf:type)) LV = ICEXT(IS(rdfs:Literal))

44. ELW2009, Jenova RDFS Semantic Condition IS(rdf:Property) IS(rdfs:Literal) IS(rdfs:Resource) IP = ICEXT(IS(rdf:Property)) IR = ICEXT(IS(rdfs:Resource)) x ∈ ICEXT(c) ， iff 〈 x,c 〉∈ IEXT(IS(rdf:type)) LV = ICEXT(IS(rdfs:Literal))

45. ELW2009, Jenova RDFS Semantic Condition IC = ICEXT(IS(rdfs:Class)) IS(rdfs:Class) IP = ICEXT(IS(rdf:Property)) IR = ICEXT(IS(rdfs:Resource)) x ∈ ICEXT(c) ， iff 〈 x,c 〉∈ IEXT(IS(rdf:type)) LV = ICEXT(IS(rdfs:Literal))

46. ELW2009, Jenova RDFS Semantic Condition IS(rdfs:Class) IS(rdfs:Literal) IS(rdfs:Resource) IP = ICEXT(IS(rdf:Property)) IR = ICEXT(IS(rdfs:Resource)) x ∈ ICEXT(c) ， iff 〈 x,c 〉∈ IEXT(IS(rdf:type)) LV = ICEXT(IS(rdfs:Literal))

47. ELW2009, Jenova RDFS Axiomatic Triples rdf:_1 rdfs:subClassOfrdfs:subPropertyOfrdf:typerdf:predicate rdfs:domainrdfs:rangerdfs:labelrdf:objectrdfs:comment rdf:first rdf:rest rdfs:subClassOf rdfs:subPropertyOf rdf:type class instance rdfs:seeAlsordfs:isDefinedByrdf:valuerdf:subjectrdfs:member rdf:_2 rdf:_3 rdf:Propertyrdfs:ContainerMembershipProperty rdf:nil rdfs:Resource rdf:Litst rdf:Statement rdfs:Containerrdf:Alt rdf:Seq rdf:Bag rdfs:Literal rdf:XMLLiteral rdfs:Classrdfs:Datatype rdfs:Class ∈ ICEXT(rdfs:Class) ， iff 〈 rdfs:Class,rdfs:Class 〉∈ IEXT(IS(rdf:type))

48. ELW2009, Jenova RDFS Axiomatic Triples rdf:_1 rdfs:subClassOfrdfs:subPropertyOfrdf:typerdf:predicate rdfs:domainrdfs:rangerdfs:labelrdf:objectrdfs:comment rdf:first rdf:rest rdfs:subClassOf rdfs:subPropertyOf rdf:type class instance rdfs:seeAlsordfs:isDefinedByrdf:valuerdf:subjectrdfs:member rdf:_2 rdf:_3 rdf:Propertyrdfs:ContainerMembershipProperty rdf:nil rdfs:Resource rdf:Litst rdf:Statement rdfs:Containerrdf:Alt rdf:Seq rdf:Bag rdfs:Literal rdf:XMLLiteral rdfs:Classrdfs:Datatype rdf:type rdfs:domain rdfs:Resource. rdfs:domain rdfs:domain rdf:Property. rdfs:range rdfs:domain rdf:Property. rdfs:subPropertyOf rdfs:domain rdf:Property. rdfs:subClassOf rdfs:domain rdfs:Class. rdf:subject rdfs:domain rdf:Statement. rdf:predicate rdfs:domain rdf:Statement. rdf:object rdfs:domain rdf:Statement. rdfs:member rdfs:domain rdfs:Resource. rdf:first rdfs:domain rdf:List. rdf:rest rdfs:domain rdf:List. rdfs:seeAlso rdfs:domain rdfs:Resource. rdfs:isDefinedBy rdfs:domain rdfs:Resource. rdfs:comment rdfs:domain rdfs:Resource. rdfs:label rdfs:domain rdfs:Resource. rdf:value rdfs:domain rdfs:Resource. rdf:type rdfs:range rdfs:Class. rdfs:domain rdfs:range rdfs:Class. rdfs:range rdfs:range rdfs:Class. rdfs:subPropertyOf rdfs:range rdf:Property. rdfs:subClassOf rdfs:range rdfs:Class. rdf:subject rdfs:range rdfs:Resource. rdf:predicate rdfs:range rdfs:Resource. rdf:object rdfs:range rdfs:Resource. rdfs:member rdfs:range rdfs:Resource. rdf:first rdfs:range rdfs:Resource. rdf:rest rdfs:range rdf:List. rdfs:seeAlso rdfs:range rdfs:Resource. rdfs:isDefinedBy rdfs:range rdfs:Resource. rdfs:comment rdfs:range rdfs:Literal. rdfs:label rdfs:range rdfs:Literal. rdf:value rdfs:range rdfs:Resource.

49. ELW2009, Jenova RDFS Entailment Rules Rule NameIf E contains:then add: rdfs1 uuu aaa lll. where lll is a plain literal (with or without a language tag). _:nnn rdf:type rdfs:Literal. where _:nnn identifies a blank node allocated to lll by rule rule lg. allocated torule lg rdfs2 aaa rdfs:domain xxx. uuu aaa yyy. uuu rdf:type xxx. rdfs3 aaa rdfs:range xxx. uuu aaa vvv. vvv rdf:type xxx. rdfs4auuu aaa xxx.uuu rdf:type rdfs:Resource. rdfs4buuu aaa vvv.vvv rdf:type rdfs:Resource. rdfs5 uuu rdfs:subPropertyOf vvv. vvv rdfs:subPropertyOf xxx. uuu rdfs:subPropertyOf xxx. rdfs6uuu rdf:type rdf:Property.uuu rdfs:subPropertyOf uuu.

50. ELW2009, Jenova RDFS Entailment Rules Rule NameIf E contains:then add: rdfs7 aaa rdfs:subPropertyOf bbb. uuu aaa yyy. uuu bbb yyy. rdfs8uuu rdf:type rdfs:Class.uuu rdfs:subClassOf rdfs:Resource. rdfs9 uuu rdfs:subClassOf xxx. vvv rdf:type uuu. vvv rdf:type xxx. rdfs10uuu rdf:type rdfs:Class.uuu rdfs:subClassOf uuu. rdfs11 uuu rdfs:subClassOf vvv. vvv rdfs:subClassOf xxx. uuu rdfs:subClassOf xxx. rdfs12 uuu rdf:type rdfs:ContainerMembershipProp erty. uuu rdfs:subPropertyOf rdfs:member. rdfs13uuu rdf:type rdfs:Datatype.uuu rdfs:subClassOf rdfs:Literal.

51. ELW2009, Jenova Hierarchical Structure of RDFS rdf:_1 rdfs:subClassOfrdfs:subPropertyOfrdf:typerdf:predicate rdfs:domainrdfs:rangerdfs:labelrdf:objectrdfs:comment rdf:first rdf:rest rdfs:subClassOf rdfs:subPropertyOf rdf:type class instance rdfs:seeAlsordfs:isDefinedByrdf:valuerdf:subjectrdfs:member rdf:_2 rdf:_3 rdf:Propertyrdfs:ContainerMembershipProperty rdf:nil rdfs:Resource rdf:List rdf:Statement rdfs:Containerrdf:Alt rdf:Seq rdf:Bag rdfs:Literal rdf:XMLLiteral rdfs:Classrdfs:Datatype Instance Meta-class Class

52. ELW2009, Jenova Transitivity Rule implies … rdf:_1 rdfs:subClassOfrdfs:subPropertyOfrdf:typerdf:predicate rdfs:domainrdfs:rangerdfs:labelrdf:objectrdfs:comment rdf:first rdf:rest rdfs:subClassOf rdfs:subPropertyOf rdf:type class instance rdfs:seeAlsordfs:isDefinedByrdf:valuerdf:subjectrdfs:member rdf:_2 rdf:_3 rdf:Propertyrdfs:ContainerMembershipProperty rdf:nil rdfs:Resource rdf:List rdf:Statement rdfs:Containerrdf:Alt rdf:Seq rdf:Bag rdfs:Literal rdf:XMLLiteral rdfs:Classrdfs:Datatype rdf:Propertyrdfs:ContainerMembershipProperty rdfs:Resourcerdf:Statement rdfs:Containerrdf:Alt rdf:Seq rdf:Bag rdfs:Literal rdf:XMLLiteral rdf:List rdfs:Classrdfs:Datatype rdfs:Resource is a superclass of other classes rdfs:Resource is a superclass of other classes

53. ELW2009, Jenova Transitivity Rule implies … rdf:_1 rdfs:subClassOfrdfs:subPropertyOfrdf:typerdf:predicate rdfs:domainrdfs:rangerdfs:labelrdf:objectrdfs:comment rdf:first rdf:rest rdfs:subClassOf rdfs:subPropertyOf rdf:type class instance rdfs:seeAlsordfs:isDefinedByrdf:valuerdf:subjectrdfs:member rdf:_2 rdf:_3 rdf:Propertyrdfs:ContainerMembershipProperty rdf:nil rdfs:Resource rdf:List rdf:Statement rdfs:Containerrdf:Alt rdf:Seq rdf:Bag rdfs:Literal rdf:XMLLiteral rdfs:Classrdfs:Datatype rdf:Propertyrdfs:ContainerMembershipProperty rdfs:Resourcerdf:Statement rdfs:Containerrdf:Alt rdf:Seq rdf:Bag rdfs:Literal rdf:XMLLiteral rdf:List rdfs:Classrdfs:Datatype rdfs:Resource is a superclass of other classes and meta-classes rdfs:Resource is a superclass of other classes and meta-classes

54. ELW2009, Jenova Subsumption Rule implies … rdf:_1 rdfs:subClassOfrdfs:subPropertyOfrdf:typerdf:predicate rdfs:domainrdfs:rangerdfs:labelrdf:objectrdfs:comment rdf:first rdf:rest rdfs:subClassOf rdfs:subPropertyOf rdf:type class instance rdfs:seeAlsordfs:isDefinedByrdf:valuerdf:subjectrdfs:member rdf:_2 rdf:_3 rdf:Propertyrdfs:ContainerMembershipProperty rdf:nil rdfs:Resource rdf:List rdf:Statement rdfs:Containerrdf:Alt rdf:Seq rdf:Bag rdfs:Literal rdf:XMLLiteral rdfs:Classrdfs:Datatype rdf:_1 rdfs:subClassOfrdfs:subPropertyOfrdf:typerdf:predicate rdfs:domainrdfs:rangerdfs:labelrdf:objectrdfs:comment rdf:first rdf:rest rdfs:seeAlsordfs:isDefinedByrdf:valuerdf:subjectrdfs:member rdf:_2 rdf:_3 rdf:Propertyrdfs:ContainerMembershipProperty rdf:Propertyrdfs:ContainerMembershipProperty rdfs:Resource rdf:nil rdf:List Every instance is an instance of rdfs:Resource Every instance is an instance of rdfs:Resource

55. ELW2009, Jenova Subsumption Rule implies … rdf:_1 rdfs:subClassOfrdfs:subPropertyOfrdf:typerdf:predicate rdfs:domainrdfs:rangerdfs:labelrdf:objectrdfs:comment rdf:first rdf:rest rdfs:subClassOf rdfs:subPropertyOf rdf:type class instance rdfs:seeAlsordfs:isDefinedByrdf:valuerdf:subjectrdfs:member rdf:_2 rdf:_3 rdf:Propertyrdfs:ContainerMembershipProperty rdf:nil rdfs:Resource rdf:List rdf:Statement rdfs:Containerrdf:Alt rdf:Seq rdf:Bag rdfs:Literal rdf:XMLLiteral rdfs:Classrdfs:Datatype Every class is an instance of rdfs:Resource Every class is an instance of rdfs:Resource rdf:Propertyrdfs:ContainerMembershipProperty rdfs:Resource rdf:List rdf:Statement rdfs:Containerrdf:Alt rdf:Seq rdf:Bag rdfs:Literal rdf:XMLLiteral rdfs:Classrdfs:Datatype rdfs:Classrdfs:Datatype

56. ELW2009, Jenova Subsumption Rule implies … rdf:_1 rdfs:subClassOfrdfs:subPropertyOfrdf:typerdf:predicate rdfs:domainrdfs:rangerdfs:labelrdf:objectrdfs:comment rdf:first rdf:rest rdfs:subClassOf rdfs:subPropertyOf rdf:type class instance rdfs:seeAlsordfs:isDefinedByrdf:valuerdf:subjectrdfs:member rdf:_2 rdf:_3 rdf:Propertyrdfs:ContainerMembershipProperty rdf:nil rdfs:Resource rdf:List rdf:Statement rdfs:Containerrdf:Alt rdf:Seq rdf:Bag rdfs:Literal rdf:XMLLiteral rdfs:Classrdfs:Datatype rdfs:Datatype and rdfs:Class are also instances of rdfs:Resource rdfs:Datatype and rdfs:Class are also instances of rdfs:Resource rdfs:Resource rdfs:Classrdfs:Datatype

57. ELW2009, Jenova Semantic Web LayerCake OWL Semantics

58. ELW2009, Jenova Three sub languages of OWL OWL Lite –supports users primarily needing a classification hierarchy and simple constraints. It should be simpler to provide tool support for OWL Lite than the other sublanguages. OWL DL –supports users who want the maximum expressiveness while retaining computational completeness (all conclusions are guaranteed to be computable) and decidability (all computations will finish in finite time). OWL Full –is meant for users who want maximum expressiveness and the syntactic freedom of RDF with no computational guarantees. For example, in OWL Full a class can be treated simultaneously as a collection of individuals and as an individual in its own right.

59. ELW2009, Jenova Conditions concerning the parts of the OWL universe and syntactic categories If E isthen IS(E) ∈ ICEXT(IS(E))=and owl:ClassICIOC IOC ⊆ IC owl:ThingIOCIOT IOT ⊆ IR and IOT ≠ ∅ owl:RestrictionICIOR IOR ⊆ IOC owl:ObjectPropertyICIOOP IOOP ⊆ IP owl:DatatypePropert y ICIODP IODP ⊆ IP owl:AnnotationPrope rty ICIOAP IOAP ⊆ IP

60. ELW2009, Jenova RDF-Compatible Model-Theoretic Semantics IR IOT

61. ELW2009, Jenova RDF-Compatible Model-Theoretic Semantics IP IR IOT IOOP IODP IOAP

62. ELW2009, Jenova RDF-Compatible Model-Theoretic Semantics IP IR IC IOC IOT IOOP IODP IOAP

63. ELW2009, Jenova RDF-Compatible Model-Theoretic Semantics IP IR IC IOC IOT IOR IOOP IODP IOAP

64. ELW2009, Jenova RDF-Compatible Model-Theoretic Semantics IP IR IC IOC IOT IOR IOOP IODP IOAP OWL Full IOT = IR IOOP = IP IOC = IC IOT = IR IOOP = IP IOC = IC OWL DL LVI, IOT, IOC, IDC, IOOP, IODP, IOAP, IOXP, IL, and IX are all pairwise disjoint. For v in the disallowed vocabulary, IS(v) ∈ IR － (LV ∪ IOT ∪ IOC ∪ IDC ∪ IOOP ∪ IODP ∪ IOAP ∪ IOXP ∪ IL ∪ IX). LVI, IOT, IOC, IDC, IOOP, IODP, IOAP, IOXP, IL, and IX are all pairwise disjoint. For v in the disallowed vocabulary, IS(v) ∈ IR － (LV ∪ IOT ∪ IOC ∪ IDC ∪ IOOP ∪ IODP ∪ IOAP ∪ IOXP ∪ IL ∪ IX).

65. ELW2009, Jenova RDF-Compatible Model-Theoretic Semantics IP IR IC IOC IOT IOR IOOP IODP IOAP OWL Full IOT ⊆ IR IOOP ⊆ IP IOC ⊆ IC IOT ⊆ IR IOOP ⊆ IP IOC ⊆ IC OWL DL LVI, IOT, IOC, IDC, IOOP, IODP, IOAP, IOXP, IL, and IX are all pairwise disjoint. For v in the disallowed vocabulary, IS(v) ∈ IR － (LV ∪ IOT ∪ IOC ∪ IDC ∪ IOOP ∪ IODP ∪ IOAP ∪ IOXP ∪ IL ∪ IX). LVI, IOT, IOC, IDC, IOOP, IODP, IOAP, IOXP, IL, and IX are all pairwise disjoint. For v in the disallowed vocabulary, IS(v) ∈ IR － (LV ∪ IOT ∪ IOC ∪ IDC ∪ IOOP ∪ IODP ∪ IOAP ∪ IOXP ∪ IL ∪ IX).

66. ELW2009, Jenova OWL-Full Style Connection rdfs:subClassOf rdfs:subPropertyOf rdf:type class instance rdf:Property rdfs:Resource rdfs:Class

67. ELW2009, Jenova rdfs:Resource OWL-Full Style Connection rdfs:subClassOf rdfs:subPropertyOf rdf:type class instance rdf:Property rdfs:Class owl:Restrictionowl:Class IOC ⊆ IC IOR ⊆ IOC

68. ELW2009, Jenova rdfs:Resource vin:Wine OWL-Full Style Connection rdfs:subClassOf rdfs:subPropertyOf rdf:type class instance rdf:Property rdfs:Class owl:Restrictionowl:Class IOC ⊆ IC IOR ⊆ IOC

69. ELW2009, Jenova rdfs:Resource vin:Wine OWL-Full Style Connection All classes under owl:Class are in the domain of rdfs:Resource. rdfs:subClassOf rdfs:subPropertyOf rdf:type class instance rdf:Property rdfs:Class owl:Restriction rdfs:Class owl:Class IOC ⊆ IC ⊆ IR IOR ⊆ IOC

70. ELW2009, Jenova rdfs:Resource hasColor.Red OWL-Full Style Connection All restrictions under owl:Restriction are in the domain of rdfs:Resource. rdfs:subClassOf rdfs:subPropertyOf rdf:type class instance rdf:Property rdfs:Class owl:Restriction rdfs:Class owl:Class IOC ⊆ IC ⊆ IR IOR ⊆ IOC ⊆ IC ⊆ IR owl:Restriction

71. ELW2009, Jenova rdfs:Resource vin:Wine OWL-Full Style Connection rdfs:subClassOf rdfs:subPropertyOf rdf:type class instance rdf:Property rdfs:Class owl:Classowl:Restriction owl:Thing

72. ELW2009, Jenova OWL-Full Style Connection rdfs:subClassOf rdfs:subPropertyOf rdf:type class instance rdf:Property rdfs:Resource rdfs:Class owl:Classowl:Restriction owl:Thing vin:Wine vin:Zinfandel vin:ElyseZinfandel

73. ELW2009, Jenova OWL-Full Style Connection We need the new axiom to make OWL individuals be in RDFS universe. –owl:Thing rdfs:subClassOf rdfs:Resource. rdfs:subClassOf rdfs:subPropertyOf rdf:type class instance rdf:Property rdfs:Resource rdfs:Class owl:Classowl:Restriction owl:Thing vin:Wine vin:Zinfandel vin:ElyseZinfandel rdfs:Resource owl:Thing vin:Wine vin:Zinfandel vin:ElyseZinfandel IOT ⊆ IR

74. ELW2009, Jenova OWL-Full Style Connection rdfs:subClassOf rdfs:subPropertyOf rdf:type class instance rdf:Property rdfs:Resource rdfs:Class owl:Classowl:Restriction owl:Thing vin:Wine vin:Zinfandel owl:Thing vin:Wine vin:Zinfandel owl:Class

75. ELW2009, Jenova OWL-Full Style Connection rdfs:subClassOf rdfs:subPropertyOf rdf:type class instance rdf:Property rdfs:Resource rdfs:Class owl:Classowl:Restriction owl:Thing vin:Wine vin:Zinfandel rdfs:Resource owl:Thing vin:Wine vin:Zinfandel owl:Class We need the new axiom to make OWL classes be in OWL universe. –owl:Class rdfs:subClassOf owl:Thing. IOC ⊆ IOT

76. ELW2009, Jenova Conditions concerning the parts of the OWL universe and syntactic categories If E isthen IS(E) ∈ ICEXT(IS(E))=and owl:ClassICIOC IOC ⊆ IC owl:ThingIOCIOT IOT ⊆ IR and IOT ≠ ∅ owl:RestrictionICIOR IOR ⊆ IOC owl:ObjectPropertyICIOOP IOOP ⊆ IP owl:DatatypePropert y ICIODP IODP ⊆ IP owl:AnnotationPrope rty ICIOAP IOAP ⊆ IP

77. ELW2009, Jenova Conditions concerning the parts of the OWL universe and syntactic categories If E isthen IS(E) ∈ ICEXT(IS(E))=and owl:ClassICIOC IOC ⊆ IOT ⊆ IC owl:ThingIOCIOT IOT ⊆ IR and IOT ≠ ∅ owl:RestrictionICIOR IOR ⊆ IOC owl:ObjectPropertyICIOOP IOOP ⊆ IP owl:DatatypePropert y ICIODP IODP ⊆ IP owl:AnnotationPrope rty ICIOAP IOAP ⊆ IP

78. ELW2009, Jenova OWL-Full Style Connection Classes may be treated as individual. rdfs:subClassOf rdfs:subPropertyOf rdf:type class instance rdf:Property rdfs:Resource rdfs:Class owl:Classowl:Restriction owl:Thing

79. ELW2009, Jenova OWL-Full Style Connection Classes may be treated as individual. –vin:Wine owl:sameAs food:Wine. rdfs:subClassOf rdfs:subPropertyOf rdf:type class instance rdf:Property rdfs:Resource rdfs:Class owl:Classowl:Restriction owl:Thing vin:Wine food:Wine

80. ELW2009, Jenova Equality in OWL Equivalency as Class –owl:equivalentClass, owl:intersectionOf, owl:unionOf, owl:complementOf, owl:oneOf –owl:disjointWith, owl:complementOf, Equivalency as Individual –owl:sameAs, owl:FunctionalProperty, owl:InverseFunctionalProperty –owl:differenceFrom, owl:distinctMembers –owl:oneOf Equivalency as Property –owl:equivalentProperty http://www.w3.org/TR/owl-ref/#DescriptionAxiom

81. ELW2009, Jenova Equivalent Property http://www.w3.org/TR/owl-ref/#equivalentProperty-def Property equivalence is not the same as property equality. Equivalent properties have the same "values" (i.e., the same property extension), but may have different intensional meaning (i.e., denote different concepts). Property equality should be expressed with the owl:sameAs construct. As this requires that properties are treated as individuals, such axioms are only allowed in OWL Full. owl:sameAs

82. ELW2009, Jenova Equality in RDF Graph http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-graph-equality M maps blank nodes to blank nodes. M(lit)=lit for all RDF literals lit which are nodes of G.RDF literals M(uri)=uri for all RDF URI references uri which are nodes of G.RDF URI references The triple ( s, p, o ) is in G if and only if the triple ( M(s), p, M(o) ) is in G'. Two RDF graphs G and G' are equivalent if there is a bijection M between the sets of nodes of the two graphs, such that: With this definition, M shows how each blank node in G can be replaced with a new blank node to give G'.

83. ELW2009, Jenova Non-Unique Name Assumption and Equality In Semantic Webs, two or more different names or URIs may denote same object. Negative identity of two or more graphs and nodes does not mean the difference.

84. ELW2009, Jenova Graph Equality in RDF http://www.w3.org/TR/rdf-mt/#graphdefs Any instance of a graph in which a blank node is mapped to a new blank node not in the original graph is an instance of the original and also has it as an instance, and this process can be iterated so that any 1:1 mapping between blank nodes defines an instance of a graph which has the original graph as an instance. Two such graphs, each an instance of the other but neither a proper instance, which differ only in the identity of their blank nodes, are considered to be equivalent. We will treat such equivalent graphs as identical; this allows us to ignore some issues which arise from 're-naming' nodeIDs, and is in conformance with the convention that blank nodes have no label. Equivalent graphs are mutual instances with an invertible instance mapping. ex:a ex:p ex:a ex:p _:x. _:y.

85. ELW2009, Jenova Non-Unique Name Assumption and Equality ex:a and ex:b may be identical (denoted by different URIs) in non-UNA. Two graphs that show different appearance may be identical. ex:a ex:p ex:b ex:p 1. 11

86. ELW2009, Jenova Non-Unique Name Assumption and Equality equal?different? ex:a ex:p ex:b ex:p ex:a ex:p ex:a ex:p Yes.No. ex:a ex:p ex:a ex:p Yes.No. ex:z ex:a ex:p ex:b ex:p ex:z ex:a ex:p ex:a ex:p ex:yex:z ex:p Yes.No. ex:p ex:yex:z in UNA No.Yes. No.Yes. No.Yes. No.Yes. ex:p ex:z Yes.No.

87. ELW2009, Jenova Non-Unique Name Assumption and Equality equal?different? ex:a ex:p ex:b ex:p ?? ex:a ex:p ex:a ex:p Yes.No. ex:a ex:p ex:a ex:p Yes.No. ex:z ex:a ex:p ex:b ex:p ?? ex:z ex:a ex:p ex:a ex:p ?? ex:yex:z ex:p Yes.No. ex:p ?? ex:yex:z in non-UNA ex:p ex:z Yes.No.

88. ELW2009, Jenova Non-Unique Name Assumption and Equality equal?different? ex:a ex:p ex:b ex:p ?? ex:a ex:p ex:a ex:p Yes.No. ex:a ex:p ex:a ex:p Yes.No. ex:z ex:a ex:p ex:b ex:p ?? ex:z ex:a ex:p ex:a ex:p ex:yex:z ex:p Yes.No. ex:p ex:yex:z Atomic UNA in non-UNA No.Yes. ex:p ex:z Yes.No. Yes.

89. ELW2009, Jenova Non-Unique Name Assumption and Equality equal?different? ex:a ex:p ex:b ex:p ?? ex:a ex:p ex:b ex:p ?? ex:z ex:a ex:p ex:a ex:p ex:yex:z ex:p ex:yex:z Atomic UNA in non-UNA No.Yes. No.Yes. No. ex:yex:z No.Yes. Different names without slots Same names with different slots Different names with same slots

90. ELW2009, Jenova Non-Unique Name Assumption and Equality Two atomic objects must be different if the two have different names, because they are basics of composite objects. If slots of two objects are different, then different. –intensional logics If slots of two objects are same, then same in non-UNA?

91. ELW2009, Jenova Auto Epistemic Local Closed World Open World is troublesome. –owl:someValuesFrom restriction entails very few entailments. It may be satisfiable somewhere you do not know. After ontology building, agents want to infer the world which they know. User-Settable Closed World –owl:intersectionOf, owl:unionOf, owl:one of –owl:allValesFrom + owl:maxCardinality

92. ELW2009, Jenova Russell&Norvig AIMA

93. ELW2009, Jenova Formal Semantics Sentences Entails Aspects of the Real World Aspects of the Real World follows Representation World

94. ELW2009, Jenova Formal Semantics triplesa triple Entails RDF graph?a graph? follows Representation World

95. ELW2009, Jenova Formal Semantics Lisp SymbolsA Lisp Symbol Entails CLOS Objects?a CLOS object? follows Representation World

96. ELW2009, Jenova Semantics in Lisp and RDF based on Brian Cantwell Smith 1983

97. ELW2009, Jenova A Simple Semantic Interpretation Function Syntactic Domain S Semantic Domain D Mapping Φ Interpretation function symbol s sign s significance d interpretation of s 1.The objects and events in the world in which a computational process is embedded, including both real-world objects like cars and caviar, and set- theoretic abstractions like numbers and functions. 2.The internal elements, structures, or processes inside the computer, including data structures, program representations, execution sequences and so forth. 3.The notational or communicational expressions, in some externally observable and consensually established medium of interaction, such as strings of characters, streams of words, or sequences of display images on a computer terminal. 1.The objects and events in the world in which a computational process is embedded, including both real-world objects like cars and caviar, and set- theoretic abstractions like numbers and functions. 2.The internal elements, structures, or processes inside the computer, including data structures, program representations, execution sequences and so forth. 3.The notational or communicational expressions, in some externally observable and consensually established medium of interaction, such as strings of characters, streams of words, or sequences of display images on a computer terminal.

98. ELW2009, Jenova Semantic Relationships in a Computational Process “ … ” Structural Field O Ψ Φ Φ(O(“T.S.Eliot”) = T.S.Eliot

99. ELW2009, Jenova A Framework for Computational Semantics Structure S 1 Designation D 1 Φ Notation N 1 O Structure S 2 Designation D 2 Φ Notation N 2 Ψ O－1O－1

100. ELW2009, Jenova Russell&Norvig AIMA Sentences Aspects of the real world Φ Notation N 1 O Sentences Aspects of the real world Φ Notation N 2 O－1O－1 Entails Follows

101. ELW2009, Jenova LISP Evaluation vs. Designation 3 Three Φ Ψ (+ 1 2) Three Φ Ψ 3 Φ T Truth Φ Ψ 'X Φ Ψ X (EQ 'A 'B) Falsity Φ Ψ NIL Φ (CAR '(A. B)) Φ Ψ A CDR the CDR function Φ ???

102. ELW2009, Jenova LISP ’ s “De-reference If You Can” Evaluation Protocol Φ Ψ Φ Ψ … edge of the machine … Internal Structures External World Φ Ψ de-reference

103. ELW2009, Jenova LISP Evaluation vs. Designation 3 Three Φ Ψ (+ 1 2) Three Φ Ψ 3 Φ T Truth Φ Ψ 'X Φ Ψ X (EQ 'A 'B) Falsity Φ Ψ NIL Φ (CAR ‘(A. B)) Φ Ψ A CDR the CDR function Φ ???

104. ELW2009, Jenova Model Theoretic Semantics of RDF

105. ELW2009, Jenova Model Theoretic Semantics of RDF rdfs:Resource, rdf:Property, rdfs:subClassOf, rdfs:domain, rdfs:range eg:Agent, eg:Work, eg:author, eg:Person, eg:Documents eg:author rdfs:Resource rdf:Property rdfs:subClassOf rdfs:domain rdfs:range Information Management: A Proposal “Information Management: A Proposal”, “Tim Berners-Lee” Tim Berners-Lee eg:Agent eg:Work eg:Person eg:Documents

106. ELW2009, Jenova Model Theoretic Semantics of RDF rdfs:Resource, rdf:Property, rdfs:subClassOf, rdfs:domain, rdfs:range eg:Agent, eg:Work, eg:author, eg:Person, eg:Documents eg:author rdfs:Resource rdf:Property rdfs:subClassOf rdfs:domain rdfs:range Information Management: A Proposal “Information Management: A Proposal”, “Tim Berners-Lee” Tim Berners-Lee URI References Plane Literals eg:Agent eg:Work eg:Person eg:Documents PIPI RIRI Vocabulary V

107. ELW2009, Jenova Model Theoretic Semantics of RDF rdfs:Resource, rdf:Property, rdfs:subClassOf, rdfs:domain, rdfs:range eg:Agent, eg:Work, eg:author, eg:Person, eg:Documents eg:author rdfs:Resource rdf:Property rdfs:subClassOf rdfs:domain rdfs:range Information Management: A Proposal “Information Management: A Proposal”, “Tim Berners-Lee” Tim Berners-Lee SISI LV eg:Agent eg:Work eg:Person eg:Documents PIPI RIRI Vocabulary V

108. ELW2009, Jenova A Framework for Computational Semantics for RDF ComputationalObject 1 Designation D 1 Φ URI_Reference 1 O ComputationalObject 2 Designation D 2 Φ URI_Reference 2 Ψ O－1O－1

109. ELW2009, Jenova SWCLOS ’ s “De-reference If You Can” Evaluation Protocol Φ Ψ Φ Ψ … edge of the machine … CLOS Objects External World Φ Ψ de-reference

110. ELW2009, Jenova Evaluation vs. Designation in SWCLOS eg:Person a Class Person Φ Ψ # Φ (typep eg:Person rdfs:Class) Truth Φ Ψ t Φ (slot-value a:Proposal 'dc:title) “Information Management: A Proposal ” Φ Ψ 3 Three Φ Ψ # Tim Berners-Lee Φ

111. ELW2009, Jenova Continue to the paper