An Axiomatic Semantics for RDF, RDF-S, and DAML+OIL

Richard Fikes

Deborah L McGuinness

**Knowledge Systems Laboratory**

**Computer Science Department**

**Stanford University**

March 1, 2001

This document provides an axiomatization for the Resource Description Framework (RDF), RDF Schema (RDF-S), and DAML+OIL by specifying a mapping of a set of descriptions in any one of these languages into a logical theory expressed in first-order predicate calculus. The basic claim of this paper is that the logical theory produced by the mapping specified herein of a set of such descriptions is logically equivalent to the intended meaning of that set of descriptions.

Providing a means of translating RDF, RDF-S, and DAML+OIL descriptions into a first-order predicate calculus logical theory not only specifies the intended meaning of the descriptions, but also produces a representation of the descriptions from which inferences can automatically be made using traditional automatic theorem provers and problem solvers. For example, the DAML+OIL axioms enable a reasoner to infer from the two statements “Class Male and class Female are disjointWith.” and “John is type Male.” that the statement “John is type Female.” is false.

The mapping into predicate calculus consists of a simple
rule for translating RDF (http://www.w3.org/TR/REC-rdf-syntax/)
statements into first-order relational sentences and a set of first-order logic
axioms that restrict the allowable interpretations of the non-logical symbols
(i.e., relations, functions, and constants) in each language. Since RDF-S (http://www.w3.org/TR/rdf-schema/)
and DAML+OIL (http://www.daml.org/2001/03/daml+oil-index.html
) are both vocabularies of non-logical symbols added to RDF, the translation of
RDF statements is sufficient for translating RDF-S and DAML+OIL as well.

The axioms are written in ANSI Knowledge Interchange Format (KIF) (http://logic.stanford.edu/kif/kif.html), which is a proposed ANSI standard. The axioms use standard first-order logic constructs plus KIF-specific relations and functions dealing with lists.[1] Lists as objects in the domain of discourse are needed in order to axiomatize RDF containers and the DAML+OIL properties dealing with cardinality.

The mapping of each of these languages into first-order logic is as follows: A logical theory in KIF that is logically equivalent to a set of RDF, RDF-S, or DAML+OIL descriptions can be produced as follows:

· Translate each RDF statement with predicate P, subject S, and object O into a KIF sentence of the form “(PropertyValue P S O)”.

· Include in the KIF theory the axioms in this document associated with the source language (RDF, RDF-S, or DAML+OIL).

Note that it is not necessary to specify a translation for
every construct in RDF since any set of RDF descriptions can be translated into
an equivalent set of RDF statements.
Thus, the one translation rule above suffices to translate *all* of RDF and therefore *all* of the other two languages as well.

This axiomatization is designed to place minimal constraints on the interpretation of the non-logical symbols in the resulting logical theory. In particular, the axioms do not require use of a set theory, that classes be considered to be sets or to be unary relations, nor do they require that properties be considered to be mappings or binary relations. Such constraints could be added to the resulting logical theory if desired, but they are not needed to express the intended meaning of the RDF, RDF-S, or DAML+OIL descriptions being translated.

Comments are welcomed posted to the www-rdf-logic@w3.org distribution list.[2]

We define the following KIF relations for use in representing RDF, RDF-S, and DAML+OIL descriptions in KIF.

“PropertyValue” is a ternary relation such that each RDF statement with any predicate P, any subject S, and any object O is translated into the KIF relational sentence “(PropertyValue P S O)”.

“Type” is a binary relation that provides a shorthand notation for RDF statements whose predicate is the property “type”.

** **

**%% Relation “Type”
holds for objects S and O if and only if relation “PropertyValue” holds for
objects “type”, S, and O.**

**Ax1.
****(<=> (Type ?s
?o) (PropertyValue type[3]
?s ?o))[4]**

This relation is used in the axiomatization of properties “cardinality”, “minCardinality”, and “maxCardinality”.

** **

**%% A
“no-repeats-list” is a list for which no item occurs more than once.**

**Ax2.
****(<=>
(no-repeats-list ?l) **

** (or (= ?l nil) **

** (exists (?x) (= ?l (listof
?x))) **

** (and (not (item (rest ?l)
(first ?l))) **

** (no-repeats-list (rest
?l)))))[5]**

As stated in the introduction, each RDF statement with predicate P, subject S, and object O is translated into a KIF sentence of the form “(PropertyValue P S O)”.

The default namespace for the classes and properties defined in this section is http://www.w3.org/1999/02/22-rdf-syntax-ns#.

This section axiomatizes the classes that are included in RDF and defines one additional class (i.e., FunctionalProperty) that is useful in axiomatizing other classes and properties.

All things being described by RDF expressions are called resources.

** **

**%% “Resource” is
type “Class”.**

**Ax3.
****(Type Resource
Class)**

** **

**%% If an object P
is type “Property” then P is also type “Resource”. (I.e., properties are resources.)**

**Ax4.
**** (=> (Type ?p Property) (Type ?p Resource))[6][7]**

** **

**%% The first
argument of relation “PropertyValue” is type “Property”.**

**Ax5.
****(=>
(PropertyValue ?p ?s ?o) (Type ?p Property))**

This class is prefixed with “rdfs” to indicate that it is from the namespace http://www.w3.org/2000/01/rdf-schema# and to distinguish it from “Class” as defined in DAML+OIL.

** **

**%% If an object C
is type “rdfs:Class” then C is also type “Resource” (i.e., RDF-S classes are
resources), and no object can be both type “Property” and type “rdfs:Class”
(i.e., properties and RDF-S classes are disjoint).**

**Ax6.
****(=> (Type ?c
rdfs:Class) (Type ?c Resource))**

**Ax7.
****(not (and (Type ?x
Property) (Type ?x rdfs:Class)))**

The class “FunctionalProperty” is not in RDF or RDF-S. We define “FunctionalProperty” only for convenience in axiomatizing other classes and properties.

**%% An object FP is
type “FunctionalProperty” if and only if FP is type “Property” and if objects
V1 and V2 are both values of FP for some object, then V1 is equal to V2. (I.e., functional properties are those
properties that are functional in their values.)**

**Ax8.
****(<=> (Type
?fp FunctionalProperty) **

** (and (Type ?fp Property)**

** (=> (and (PropertyValue ?fp
?s ?v1) **

** (PropertyValue ?fp ?s
?v2))**

** (= ?v1 ?v2))))**

**%% “Literal” is
type “rdfs:Class”.**

**Ax9.
****(Type Literal rdfs:Class)**

** **

**%% If an object S
is type “Statement”, then S is also type “Resource”. (I.e., statements are resources.)**

**Ax10. **** (=> (Type ?s Statement) (Type ?s Resource))**

**%% If an object ST
is type “Statement”, then there exists a value of “Predicate” for ST, a value
of “Subject” for ST, and a value of “Object” for ST. (I.e., every statement has
a predicate, subject, and object.)**

**Ax11. ****(=> (Type Statement ?st)**

** (exists (?p ?r ?o) **

** (and (PropertyValue
Predicate ?st ?p)**

** (PropertyValue Subject ?st ?r)**

** (PropertyValue Object
?st ?o))))**

** **

**%% If an object C
is type “Container”, then C is also type “Resource”. (I.e., containers are resources.)**

**Ax12. **** (=> (Type ?c Container) (Type ?c Resource))**

** **

**%% If an object C
is type “Container”, then C is a KIF list.
(I.e., a container is considered to be a list as defined in KIF.)**

**Ax13. ****(=> (Type ?c Container) (List ?c))**

** **

**%% An object C is
type “Container” if and only if C is type “Bag” or type “Seq” or type
“Alt”. (I.e., a container is a bag, a
sequence, or an alternative.)**

**Ax14. ****(<=> (Type ?c Container) **

** (or (Type ?c Bag) **

** (Type ?c Seq) **

** (Type ?c Alt))) **

** **

**%% If an object C
is type “ContainerMembershipProperty”, then C is also type “Property”. (I.e., container membership properties are
properties.)**

**Ax15. ****(=> (Type ?c
ContainerMembershipProperty) (Type ?c Property))**

This section axiomatizes the properties that are included in RDF.

“type” is used to indicate that a resource is a member of a class.

** **

**%% “type” is type
“Property”. (I.e., “type” is a
property.)**

**Ax16. ****(Type type Property)**

** **

**%% The first
argument of “Type” is a resource and the second argument of “Type” is a class.**

**Ax17. ****(=> (Type ?r ?c) (and (Type ?r Resource)
(Type ?c rdfs:Class)))[8]**

** **

**%% “Subject” is
type “FunctionalProperty”, and if an object SB is the value of “Subject” for an
object ST, then ST is type “Statement” and SB is type “Resource”. (I.e., every statement has exactly one
subject, and that subject is a resource.)**

**Ax18. ****(Type Subject FunctionalProperty)**

**Ax19. ****(=> (PropertyValue Subject ?st ?sb) **

** (and (Type ?st Statement) (Type ?sb
Resource))) **

** **

**%% “Predicate” is
type “FunctionalProperty”, and if an object P is the value of “Predicate” for
an object ST, then P is type “Property” and ST is type “Statement”. (I.e., every statement has exactly one
predicate, and that predicate is a property.)**

**Ax20. ****(Type Predicate FunctionalProperty)**

**Ax21. ****(=> (PropertyValue Predicate ?st ?p) **

** (and (Type ?st Statement) (Type ?p
Property))) **

** **

**%% “Object” is
type “FunctionalProperty”, and if an object O is the value of “Object” for an
object ST, then O is either type “Resource” or type “Literal” and ST is type
“Statement”. (I.e., every statement has
exactly one object, and that object is either a resource or a literal.)**

**Ax22. ****(Type Object FunctionalProperty) **

**Ax23. ****(=> (PropertyValue Object ?st ?o)**

** (and (Type ?st Statement)**

** (or (Type ?o Resource) (Type ?o
Literal))))**

** **

**%% “value” is type
“Property”, and if an object V is a value of “value” for an object SV, then SV
is type “Resource” and V is either type “Resource” or type “Literal”.**

**Ax24. ****(Type value Property)**

**Ax25. ****(=> (PropertyValue value ?sv ?v) **

** (and (Type ?sv Resource) **

** (or (Type ?v Resource) (Type ?v
Literal))))**

** **

**%% For each
positive integer N, “_N” is type “FunctionalProperty”, and if an object O is
the value of “_N” for an object C, then C is type “Collection”. (The _N of a collection is intended to be
the Nth element of that collection.)**

**Ax26. ****(Type FunctionalProperty _1)**

**Ax27. ****(=> (PropertyValue _1 ?c ?o) (Type
Collection ?c))**

** **

**and similarly for
_2, _3, etc.**

RDF-S is a collection of classes and properties that is added to RDF. Statements in RDF-S are RDF statements.

The default namespace for the classes and properties defined in this section is http://www.w3.org/2000/01/rdf-schema#.

This section axiomatizes the classes that are added to RDF in RDF-S.

** **

**%% “Resource” is a
value of “subClassOf” for “ConstraintResource”. (I.e., constraint resources are resources.)**

**Ax28. ****(PropertyValue subClassOf
ConstraintResource Resource)**

** **

**%% An object CP is
type “ConstraintProperty” if and only if it is type “ConstraintResource” and
type “Property”. (I.e., constraint
properties are exactly those constraint resources that are also properties.)**

**Ax29. ****(<=> (Type ?cp ConstraintProperty)**

** (and (Type ?cp ConstraintResource)**

** (Type ?cp Property)))**

**%% An object of
type “NonNegativeInteger” is an integer.**

**Ax30. ****(=> (Type NonNegativeInteger ?n)
(Integer ?n))**

This section axiomatizes the properties that are added to RDF in RDF-S.

** **

**%% “subClassOf” is
type “Property”.**

**Ax31. ****(Type subClassOf Property)**

** **

**%% An object
CSUPER is a value of “subClassOf” for an object CSUB if and only if CSUPER is
type “rdfs:Class”, CSUB is type “rdfs:Class”, CSUB is not CSUPER, and if an
object X is type CSUB then it is also type CSUPER. (I.e., the arguments of subClassOf are classes, and objects in a
subclass are also in the superclass.)**

**Ax32. ****(<=> (PropertyValue subClassOf ?csub
?csuper) **

** (and (Type ?csub rdfs:Class)**

** (Type ?csuper rdfs:Class)**

** (forall (?x) (=> (Type ?x
?csub) (Type ?x ?csuper)))))**

** **

**%% “subPropertyOf”
is type “Property”.**

**Ax33. ****(Type subPropertyOf Property)**

** **

**%% An object
SUPERP is a value of “subPropertyOf” of an object SUBP if and only if SUBP is
type “Property”, SUPERP is type “Property”, and if an object V is a value of
SUBP for an object O then V is also a value of SUPERP for O. (I.e., if a subProperty has a value V for an
object O, then the superproperty also has value V for O.)**

**Ax34. ****(<=> (PropertyValue subPropertyOf
?subP ?superP) **

** (and (Type ?subP Property)**

** (Type ?superP Property)**

** (forall (?o ?v) (=>
(PropertyValue ?subP ?o ?v) **

**
(PropertyValue ?superP ?o ?v)))))**

** **

**%% “seeAlso” is
type “Property”, and “Resource” is a value of both “domain” and “range” for
“seeAlso. (I.e., “seeAlso” is a
property whose arguments are resources.)**

**Ax35. ****(Type seeAlso Property)**

**Ax36. ****(PropertyValue domain seeAlso Resource)**

**Ax37. ****(PropertyValue range seeAlso Resource)**

** **

**%% “seeAlso” is a
value of “subPropertyOf” for “isDefinedBy”.
(I.e., “isDefinedBy” is a subproperty of “seeAlso”.)**

**Ax38. ****(PropertyValue subPropertyOf isDefinedBy
seeAlso)**

** **

**%% “comment” is
type “Property”, and “Literal” is the value of “range” for “comment”. (I.e., “comment” is a property whose value
is a literal.)**

**Ax39. ****(Type comment Property)**

**Ax40. ****(PropertyValue range comment Literal)**

** **

**%% “label” is type
“Property”, and “Literal” is the value of “range” for “label”. (I.e., “label” is a property whose value is
a literal.)**

**Ax41. ****(Type label Property)**

**Ax42. ****(PropertyValue range label Literal)**

**%% “Property” is a
value of “subClassOf” for “ConstraintProperty”. (I.e., constraint properties are properties.)**

**Ax43. ****(PropertyValue subClassOf
ConstraintProperty Property)**

**%% “range” is type
“ConstraintProperty” and type “FunctionalProperty”.**

**Ax44. ****(Type range ConstraintProperty)**

**Ax45. ****(Type range FunctionalProperty)**

** **

**%% An object R is
the value of “range” for an object P if and only if P is type “Property”, R is
type “rdfs:Class”, and if an object Y is a value of P then Y is type R. (I.e., R is the range of P when P is a
property, R is a class, and every value of P is an R.)**

**Ax46. ****(<=> (PropertyValue range ?p ?r) **

** (and (Type ?p Property)**

** (Type ?r rdfs:Class)**

** (forall (?x ?y) **

** (=> (PropertyValue
?p ?x ?y) (Type ?y ?r)))))**

**%% Note that the
following can be inferred from the axioms regarding “domain” and “range”:**

**Th1.
****(PropertyValue
domain range Property)**

**Th2.
****(PropertyValue
range range rdfs:Class)**

**%% “domain” is
type “ConstraintProperty”.**

**Ax47. ****(Type domain ConstraintProperty)**

** **

**%% If an object D
is a value of “domain” for an object P, then P is type “Property”, D is type “rdfs:Class”,
and if P has a value for an object X then X is type D. (I.e., if D is a domain of P, then P is a
property, D is a class, and every object that has a value of P is a D.)**

**Ax48. ****(<=> (PropertyValue domain ?p ?d) **

** (and (Type ?p Property)**

** (Type ?d rdfs:Class)**

** (forall (?x ?y) **

** (=> (PropertyValue
?p ?x ?y) (Type ?x ?d))))) **

** **

**%% Note that the
following can be inferred from the axioms regarding “domain” and “range”:**

**Th3.
****(PropertyValue
domain domain Property)**

**Th4.
****(PropertyValue
range domain Class)**

DAML+OIL is a collection of classes, properties, and objects that is added to RDF and RDF-S. Statements in DAML+OIL are RDF statements.

The default namespace for the classes and properties defined in this section is http://www.daml.org/2001/03/daml+oil#.

This section axiomatizes the classes that are added to RDF and RDF-S.

**%% “Class” is a
subclass of “rdfs:Class”.**

**Ax49. ****(PropertyValue subClassOf Class rdfs:Class)**

**%% “Datatype” is a
subclass of “rdfs:Class”.**

**Ax50. ****(PropertyValue subClassOf Datatype
rdfs:Class)**

**%% “Thing” is type
“Class”.**

**Ax51. ****(Type Thing Class)**

** **

**%% If an object C
is type “class”, then if NC is the value of “complementOf” for C and L is a two
element list containing C and NC, then L is a value of “unionOf” for
“Thing”. (I.e., “Thing” is the union of
any class and its complement.)**

**Ax52. ****(=> (Type ?c Class)**

** (forall (?nc ?l) **

** (=> (and (PropertyValue complementOf ?c ?nc)**

** (Type ?l List)**

** (PropertyValue item
?l ?c)**

** (PropertyValue item
?l ?nc)**

** (forall (?x) (=>
(PropertyValue item ?l ?x) **

**
(or (= ?x ?c) **

**
(= ?x ?nc)))))**

** (PropertyValue unionOf
Thing ?l))))**

** **

**%% “Nothing” is
the value of “complementOf” for “Thing”.
(I.e., “Nothing” is the complement of “Thing”.)**

**Ax53. ****(PropertyValue complementOf Thing Nothing)**

**%% “Restriction”
is type “Class”.**

**Ax54. ****(Type Restriction Class)**

**%% “Restriction”
is a subclass of “Class”.**

**Ax55. ****(PropertyValue subClassOf Restriction Class)**

**%% “AbstractProperty” is type “Class”.**

**Ax56. ****(Type AbstractProperty** **Class)**

**%% “Restriction”
is a subclass of “Class”.**

**Ax57. ****(PropertyValue subClassOf AbstractProperty** **Property)**

**%%
“DatatypeProperty” is type “Class”.**

**Ax58. ****(Type DatatypeProperty Class)**

**%% “Restriction”
is a subclass of “Class”.**

**Ax59. ****(PropertyValue subClassOf DatatypeProperty
Class)**

**%%
“TransitiveProperty” is type “Class”.**

**Ax60. ****(Type TransitiveProperty Class)**

** **

**%% An object P is
type “TransitiveProperty” if and only if P is type “AbstractProperty” and if an
object Y is a value of P for an object X and an object Z is a value of P for Y
then Z is a value of P for X.**

**Ax61. ****(<=> (Type ?p TransitiveProperty)**

** (and (Type ?p AbstractProperty)**

** (forall (?x ?y ?z) **

** (=> (and
(PropertyValue ?p ?x ?y) **

**
(PropertyValue ?p ?y ?z)) **

** (PropertyValue ?p
?x ?z)))))**

** **

**%% Note that the
following can be inferred from the axioms regarding “TransitiveProperty” and
“subClassOf”:**

**Th5.
****(PropertyValue
subClassOf TransitiveProperty AbstractProperty)**

**%%
“UniqueProperty” is type “Class”.**

**Ax62. ****(Type UniqueProperty Class)**

** **

**%% An object P is
type “UniqueProperty” if and only if P is type “Property” and if an object Y is
a value of P for an object X and an object Z is a value of P for X then Y is
equal to Z (i.e., then Y and Z are the same object).**

**Ax63. ****(<=> (Type ?p UniqueProperty) **

** (and (Type ?p Property) **

** (forall (?x ?y ?z) **

** (=> (and
(PropertyValue ?p ?x ?y) **

**
(PropertyValue ?p ?x ?z)) **

** (= ?y ?z)))))**

**%% Note that the
following can be inferred from the axioms regarding “UniqueProperty” and
“subClassOf”:**

**Th6.
****(PropertyValue
subClassOf UniqueProperty Property)**

**%%”UnambiguousProperty”
is type “Class”.**

**Ax64. ****(Type UnambiguousProperty Class)**

** **

**%% An object P is
type “UnambiguousProperty” if and only if P is type “AbstractProperty” and if
an object V is a value of P for an object X and V is a value of P for an object
Y then X is equal to Y (i.e., then X and Y are the same object).**

**Ax65. ****(<=> (Type ?p UnambiguousProperty) **

** (and (Type ?p AbstractProperty)**

** (forall (?x ?y ?v) **

** (=> (and
(PropertyValue ?p ?x ?v) **

**
(PropertyValue ?p ?y ?v)) **

** (= ?x ?y)))))**

**%% Note that the
following can be inferred from the axioms regarding “UnambiguousProperty” and
“subClassOf”:**

**Th7.
****(PropertyValue
subClassOf UnambiguousProperty AbstractProperty)**

**%% “Seq” is a
value of “subClassOf” for “List”.
(I.e., lists are sequences.)**

**Ax66. ****(PropertyValue subClassOf List Seq)**

**%% An ontology is
type “Class”.**

**Ax67. ****(Type Ontology Class)**

This section axiomatizes the properties that are added to RDF and RDF-S.

**%% “equivalentTo”
is type “Property”.**

**Ax68. ****(Type equivalentTo Property)**

** **

**%% An object Y is
a value of “equivalentTo” for an object X if and only if X is equal to Y. (I.e., saying that objects X and Y are
“equivalentTo” is logically equivalent to saying that X and Y denote the same
object.)**

**Ax69. ****(<=> (PropertyValue equivalentTo ?x
?y) (= ?x ?y))**

**%% The values of
“subPropertyOf” for “sameClassAs” include “equivalentTo” and “subClassOf”. (I.e., “sameClassAs” is “equivalentTo” for
classes, and two classes that are the same are also subclasses of each other.)**

**Ax70. ****(PropertyValue subPropertyOf sameClassAs
equivalentTo)**

**Ax71. ****(PropertyValue subPropertyOf sameClassAs
subClassOf)**

**%% A value of
“domain” for “sameClassAs” is “Class”.**

**Ax72. ****(PropertyValue domain sameClassAs Class)**

**%% The value of
“range” for “sameClassAs” is “Class”.**

**Ax73. ****(PropertyValue range sameClassAs Class)**

**%% The values of
“subPropertyOf” for “samePropertyAs” include “equivalentTo” and
“subPropertyOf”. (I.e.,
“samePropertyAs” is “equivalentTo” for properties, and two properties that are
the same are also subproperties of each other.)**

**Ax74. ****(PropertyValue subPropertyOf samePropertyAs
equivalentTo)**

**Ax75. ****(PropertyValue subPropertyOf samePropertyAs
subPropertyOf)**

**%% “disjointWith”
is type “Property”.**

**Ax76. ****(Type disjointWith Property)**

** **

**%% An object C2 is
a value of “disjointWith” for an object C1 if and only if C1 is type “Class”,
C2 is type “Class”, and no object X is both type C1 and type C2.**

**Ax77. ****(<=> (PropertyValue disjointWith ?c1
?c2)**

** (and (Type ?c1 Class)**

** (Type ?c2 Class)**

** (not (exists (?x) (and (Type
?x ?c1) (Type ?x ?c2)))))) **

** **

**%% Note that the
following can be inferred from the axioms regarding “disjointWith”, “domain”,
and “range”:**

**Th8.
****(PropertyValue
domain disjointWith Class)**

**Th9.
****(PropertyValue
range disjointWith Class)**

**%% “unionOf” is a
property.**

**Ax78. ****(Type unionOf Property)**

** **

**%% An object L is
a value of “unionOf” for an object C1 if and only if C1 is type “Class”, L is
type “List”, every item in list L is type “Class”, and the objects of type C1
are exactly the objects that are of type one or more of the classes in the list
L.**

**Ax79. ****(<=> (PropertyValue unionOf ?c1 ?l)**

** (and (Type ?c1 Class)**

** (Type ?l List)**

** (forall (?x) (=>
(PropertyValue item ?x ?l)**

** (Type ?x
Class)))**

** (forall (?x) **

** (<=> (Type ?x ?c1)
**

** (exists (?c2) **

** (and
(PropertyValue item ?c2 ?l) **

**
(Type ?x ?c2)))))))**

** **

**%% Note that the
following can be inferred from the axioms regarding “unionOf”, “domain”, and
“range”:**

**Th10. ****(PropertyValue domain unionOf Class)**

**Th11. ****(PropertyValue range unionOf List)**

** **

**%%
“disjointUnionOf” is type “Property”.**

**Ax80. ****(Type disjointUnionOf Property)**

** **

**%% An object L is
a value of “disjointUnionOf” for an object C if and only if L is type
“Disjoint” and L is a value of “unionOf” for C. (I.e., an object C is a disjoint union of an object L if and only
if L is a list of pairwise disjoint classes and C is the union of the list L of
classes.)**

**Ax81. ****(<=> (PropertyValue disjointUnionOf
?c ?l) **

** (and (PropertyValue unionOf ?c ?l)
(Type ?l Disjoint)))**

** **

**%% Note that the
following can be inferred from the axioms regarding “disjointUnionOf”,
“unionOf”, “domain”, and “range”:**

**Th12. ****(PropertyValue domain disjointUnionOf
Class)**

**Th13. ****(PropertyValue range disjointUnionOf
Disjoint)**

**%%
“intersectionOf” is type “Property”.**

**Ax82. ****(Type intersectionOf Property)**

** **

**%% An object L is
a value of “intersectionOf” for an object C1 if and only if C1 is type “Class”,
L is type “List”, all of the items in list L are type “Class”, and the objects
that are type C1 are exactly those objects that are type all of the classes in
list L.**

**Ax83. ****(<=> (PropertyValue intersectionOf
?c1 ?l)**

** (and (Type ?c1 Class)**

** (Type ?l List)**

** (forall (?x) (=> (PropertyValue item ?x ?l)**

** (Type ?x
Class)))**

** (forall (?x) **

** (<=> (Type ?x
?c1) **

** (forall (?c2) **

** (=>
(PropertyValue item ?c2 ?l) **

** (Type
?x ?c2 )))))))**

**%% Note that the
following can be inferred from the axioms regarding “intersectionOf”, “domain”,
and “range”:**

**Th14. ****(PropertyValue domain intersectionOf Class)**

**Th15. ****(PropertyValue range intersectionOf List)**

**%% “complementOf”
is type “Property”.**

**Ax84. ****(Type complementOf Property)**

** **

**%% An object C2 is
a value of “complementOf” for an object C1 if and only if C1 and C2 are
disjoint classes and all objects are either type C1 or type C2.**

**Ax85. ****(<=> (PropertyValue complementOf ?c1
?c2)**

** (and (PropertyValue disjointWith
?c1 ?c2) **

** (forall (?x) (or (Type ?x ?c1)
(Type ?x ?c2)))))**

**%% Note that the
following can be inferred from the axioms regarding “complementOf”,
“disjointWith”, “domain”, and “range”:**

**Th16. ****(PropertyValue domain complementOf Class)**

**Th17. ****(PropertyValue range complementOf Class)**

**Th18. ****(PropertyValue complementOf Thing Nothing)**

**%% “oneOf” is type
“Property”.**

**Ax86. ****(Type one of Property)**

** **

**%% An object L is a value of “oneOf” for
an object C if and only if C is type “Class”, L is type “List”, and the objects
that are type C are exactly the objects that are values of “item” for L. (I.e., saying that C is “oneOf” L is saying
that C is a class, L is a list, and the objects of type C are the objects on
the list L.)**

**Ax87. ****(<=> (PropertyValue oneOf ?c ?l) **

** (and (Type ?c Class)**

** (Type ?l List)**

** (forall (?x) (<=> (Type
?x ?c) **

**
(PropertyValue item ?l ?x)))))**

**%% Note that the
following can be inferred from the axioms regarding “oneOf”, “domain”, and
“range”:**

**Th19. ****(PropertyValue domain oneOf Class))**

**Th20. ****(PropertyValue range oneOf List)**

**%% “onProperty” is
type “Property”.**

**Ax88. ****(Type onProperty Property)**

** **

**%% “Ristriction”
is a value of “domain” for “onProperty”.
(I.e., the first argument of onProperty is a restriction.)**

**Ax89. ****(PropertyValue domain onProperty
Restriction)**

** **

**%% “Property” is
the value of “range” for “onProperty”.
(I.e., the second argument of onProperty is a property.)**

**Ax90. ****(PropertyValue range onProperty Property)**

**%% “toClass” is
type “Property”.**

**Ax91. ****(Type toClass Property)**

** **

**%% “Restriction”
is a value of “domain” for “toClass”.
(I.e., the first argument of toClass is a restriction.)**

**Ax92. ****(PropertyValue domain toClass Restriction)**

** **

**%% “rdfs:Class” is
the value of “range” for “toClass”.
(I.e., the second argument of toClass is a class.)**

**Ax93. ****(PropertyValue range toClass rdfs:Class)**

**%% If a property P
is a value of “onProperty” for a restriction R and a class C is a value of
“toClass” for R, then an object I is type R if and only if every value of P for
I is type C. (I.e., a “toClass”
restriction of C on a property P is the class of all objects I such that all
values of P for I are type C.)**

**Ax94. ****(=> (and (PropertyValue onProperty ?r
?p)**

** (PropertyValue toClass ?r ?c)) **

** (forall (?i) (<=> (Type ?i ?r)**

** (forall (?j)
(=> (PropertyValue ?p ?i ?j) **

**
(Type ?j ?c))))))**

**%% “hasValue” is
type “Property”.**

**Ax95. ****(Type hasValue Property)**

** **

**%% “Restriction”
is a value of “domain” for “hasValue”.
(I.e., the first argument of hasValue is a restriction.)**

**Ax96. ****(PropertyValue domain hasValue Restriction)**

**%% If a property P
is a value of “onProperty” for a restriction R and an object V is a value for
“hasValue” for R, then an object I is type R if and only if V is a value of P
for I. (I.e., a “hasValue” restriction
of V on a property P is the class of all objects that have V as a value of P.)**

**Ax97. ****(=> (and (PropertyValue onProperty ?r
?p) **

** (PropertyValue hasValue ?r ?v))
**

** (forall (?i) (<=> (Type ?i ?r)
(PropertyValue ?p ?i ?v))))**

**%% “hasClass” is
type “Property”.**

**Ax98. ****(Type hasClass Property)**

** **

**%% “Restriction”
is a value of “domain” for “hasClass”.
(I.e., the first argument of hasClass is a restriction.)**

**Ax99. ****(PropertyValue domain hasClass Restriction)**

** **

**%% “rdfs:Class” is
the value of “range” for “hasClass”.
(I.e., the second argument of hasClass is a class.)**

**Ax100.****(PropertyValue range hasClass rdfs:Class)**

**%% If a property P
is a value of “onProperty” for a restriction R and a class C is a value of
“hasClass” for R, then an object I is type R if and only if there exists a
value of P for I that is type C. (I.e.,
a “hasClass” restriction of C on a property P is the class of all objects I
which have a value of P that is type C.)**

**Ax101.****(=> (and (PropertyValue onProperty ?r
?p)**

** (PropertyValue hasClass
?r ?c)) **

** (forall (?i) (<=> (Type
?i ?r)**

** (exists (?j) **

** (and
(PropertyValue ?p ?i ?j) **

**
(Type ?j ?c))))))**

**%%
“minCardinality” is type “Property”.**

**Ax102.****(Type minCardinality Property)**

** **

**%% “Restriction”
is a value of “domain” for “minCardinality”.
(I.e., the first argument of minCardinality is a restriction.)**

**Ax103.****(PropertyValue domain minCardinality
Restriction)**

** **

**%%
“NonNegativeInteger” is the value of “range” for “minCardinality”. (I.e., the second argument of minCardinality
is a non-negative integer.)**

**Ax104.****(PropertyValue range minCardinality
NonNegativeInteger)**

** **

**%% If a property P
is a value of “onProperty” for a restriction R and a non-negative integer N is
a value of “minCardinality” for R, then an object I is type R if and only if
there is a “no-repeats-list” all of whose items are values of P for I and whose
length is greater than or equal to N.
(I.e., a “minCardinality” restriction of N on a property P is the class
of all objects I which have at least N values of P.)**

**Ax105.****(=> (and (PropertyValue onProperty ?r
?p)**

** (PropertyValue
minCardinality ?r ?n))**

** (forall (?i) **

** (<=> (Type ?i
?r) **

** (exists (?vl) **

** (and (no-repeats-list ?vl) **

**
(forall (?v) **

**
(=> (PropertyValue **

** item ?vl
?v) **

**
(PropertyValue**

** ?p ?i ?v)))
**

**
(>= (length ?vl) ?n))))))[9]**

**%%
“maxCardinality” is type “Property”.**

**Ax106.****(Type maxCardinality Property)**

** **

**%% “Restriction”
is a value of “domain” for “maxCardinality”.
(I.e., the first argument of maxCardinality is a restriction.)**

**Ax107.****(PropertyValue domain maxCardinality
Restriction)**

** **

**%%
“NonNegativeInteger” is the value of “range” for “maxCardinality”. (I.e., the second argument of maxCardinality
is a non-negative integer.)**

**Ax108.****(PropertyValue range maxCardinality
NonNegativeInteger)**

** **

**%% If a property P
is a value of “onProperty” for a restriction R and a non-negative integer N is
a value of “maxCardinality” for R, then an object I is type R if and only if
all “no-repeats-lists” whose items are exactly the values of P have length less
than or equal to N. (I.e., a
“maxCardinality” restriction of N on a property P is the class of all objects I
which have at most N values of P.)**

**Ax109.****(=> (and (PropertyValue onProperty ?r
?p)**

** (PropertyValue
maxCardinality ?r ?n))**

** (forall (?i) **

** (<=> (Type ?i
?r)**

** (forall (?vl) **

** (=>
(and (no-repeats-list ?vl) **

**
(forall **

**
(?v) **

**
(<=> (PropertyValue **

** item ?vl ?v)
**

**
(PropertyValue **

** ?p ?i ?v))))**

**
(=< (length ?vl) ?n))))))[10]**

**%% “cardinality”
is type “Property”.**

**Ax110.****(Type cardinality Property)**

** **

**%% “Restriction”
is a value of “domain” for “cardinality”.
(I.e., the first argument of cardinality is a restriction.)**

**Ax111.****(PropertyValue domain cardinality Restriction)**

** **

**%%
“NonNegativeInteger” is the value of “range” for “cardinality”. (I.e., the second argument of cardinality is
a non-negative integer.)**

**Ax112.****(PropertyValue range cardinality
NonNegativeInteger)**

** **

**%% If a property P
is a value of “onProperty” for a restriction R and a non-negative integer N is
a value of “cardinality” for R, then an object I is type R if and only if all
“no-repeats-lists” whose items are exactly the values of P have length N. (I.e., a “cardinality” restriction of N on a
property P is the class of all objects I which have exactly N values of P.)**

**Ax113.****(=> (and (PropertyValue onProperty ?r
?p)**

** (PropertyValue
cardinality ?r ?n))**

** (forall (?i) **

** (<=> (Type ?i
?r)**

** (forall (?vl) **

** (=>
(and (no-repeats-list ?vl) **

**
(forall **

**
(?v) **

**
(<=> (PropertyValue **

** item ?vl ?v)
**

**
(PropertyValue **

** ?p ?i ?v))))**

** (= (length ?vl) ?n))))))**

**%% “hasClassQ” is
type “Property”.**

**Ax114.****(Type hasClassQ Property)**

** **

**%% “Restriction”
is a value of “domain” for “hasClassQ”.
(I.e., the first argument of hasClassQ is a restriction.)**

**Ax115.****(PropertyValue domain hasClassQ Restriction)**

** **

**%% “rdfs:Class” is
the value of “range” for “hasClassQ”.
(I.e., the second argument of hasClassQ is a class.)**

**Ax116.****(PropertyValue range hasClassQ rdfs:Class)**

**%%
“minCardinalityQ” is type “Property”.**

**Ax117.****(Type minCardinalityQ Property)**

** **

**%% “Restriction”
is a value of “domain” for “minCardinalityQ”.
(I.e., the first argument of minCardinalityQ is a restriction.)**

**Ax118.****(PropertyValue domain minCardinalityQ
Restriction)**

** **

**%%
“NonNegativeInteger” is the value of “range” for “minCardinalityQ”. (I.e., the second argument of
minCardinalityQ is a non-negative integer.)**

**Ax119.****(PropertyValue range minCardinalityQ
NonNegativeInteger)**

** **

**%% If a property P
is a value of “onProperty” for a restriction R, a non-negative integer N is a
value of “minCardinalityQ” for R, and a class C is a value of “hasClassQ” for
R, then an object I is type R if and only if there is a “no-repeats-list” all
of whose items are type C and values of P for I and whose length is greater
than or equal to N. (I.e., a
“minCardinalityQ” restriction of N on a property P is the class of all objects
I which have at least N values of P that are type C.)**

**Ax120.****(=> (and (PropertyValue onProperty ?r
?p)**

** (PropertyValue
minCardinalityQ ?r ?n)**

** (PropertyValue hasClassQ
?r ?c))**

** (forall **

** (?i) **

** (<=> (Type ?i ?r) **

** (exists **

** (?vl) **

** (and (no-repeats-list
?vl) **

** (forall **

** (?v) **

** (=>
(PropertyValue item ?vl ?v)**

** (and
(PropertyValue ?p ?i ?v) **

** (Type
?v ?c))) **

** (>= (length
?vl) ?n))))))**

**%%
“maxCardinalityQ” is type “Property”.**

**Ax121.****(Type maxCardinalityQ Property)**

** **

**%% “Restriction”
is a value of “domain” for “maxCardinalityQ”.
(I.e., the first argument of maxCardinalityQ is a restriction.)**

**Ax122.****(PropertyValue domain maxCardinalityQ
Restriction)**

** **

**%%
“NonNegativeInteger” is the value of “range” for “maxCardinalityQ”. (I.e., the second argument of
maxCardinalityQ is a non-negative integer.)**

**Ax123.****(PropertyValue range maxCardinalityQ
NonNegativeInteger)**

** **

**%% If a property P
is a value of “onProperty” for a restriction R, a non-negative integer N is a
value of “maxCardinalityQ” for R, and a class C is a value of “hasClassQ” for
R, then an object I is type R if and only if all “no-repeats-lists” whose items
are exactly the values of P that are type C have length less than or equal to
N. (I.e., a “maxCardinalityQ”
restriction of N on a property P is the class of all objects I which have at
most N values of P that are type C.)**

**Ax124.****(=> (and (PropertyValue onProperty ?r
?p)**

** (PropertyValue maxCardinalityQ ?r ?n)**

** (PropertyValue hasClassQ
?r ?c))**

** (forall **

** (?i) **

** (<=> (Type ?i ?r)**

** (forall **

** (?vl) **

** (=> (and (no-repeats-list ?vl) **

** (forall **

** (?v) **

** (<=>
(PropertyValue item ?vl ?v) **

** (and
(PropertyValue ?p ?i ?v)**

**
(Type ?v ?c)))))**

** (=< (length
?vl) ?n))))))**

**%% “cardinalityQ”
is type “Property”.**

**Ax125.****(Type cardinalityQ Property)**

** **

**%% “Restriction”
is a value of “domain” for “cardinalityQ”.
(I.e., the first argument of cardinalityQ is a restriction.)**

**Ax126.****(PropertyValue domain cardinalityQ
Restriction)**

** **

**%%
“NonNegativeInteger” is the value of “range” for “cardinalityQ”. (I.e., the second argument of cardinalityQ
is a non-negative integer.)**

**Ax127.****(PropertyValue range cardinalityQ
NonNegativeInteger)**

** **

**%% If a property P
is a value of “onProperty” for a restriction R, a non-negative integer N is a
value of “cardinalityQ” for R and a class C is a value of “hasClassQ” for R,
then an object I is type R if and only if all “no-repeats-lists” whose items
are exactly the values of P that are type C have length N. (I.e., a “cardinalityQ” restriction of N on
a property P is the class of all objects I which have exactly N values of P
that are type C.)**

**Ax128.****(=> (and (PropertyValue onProperty ?r
?p)**

** (PropertyValue
cardinalityQ ?r ?n)**

** (PropertyValue hasClassQ
?r ?c))**

** (forall **

** (?i) **

** (<=> (Type ?i ?r)**

** (forall **

** (?vl) **

** (=> (and
(no-repeats-list ?vl) **

** (forall **

** (?v) **

** (<=>
(PropertyValue item ?vl ?v) **

** (and
(PropertyValue ?p ?i ?v)**

**
(Type ?v ?c)))))**

** (= (length ?vl)
?n))))))**

**%% “inverseOf” is
type “Property”.**

**Ax129.****(Type inverseOf Property)**

** **

**%% An object P2 is
a value of “inverseOf” for an object P1 if and only if P1 is type
“AbstractProperty”, P2 is type “AbstractProperty”, and an object X2 is a value
of P1 for an object X1 if and only X1 is a value of P2 for X2.**

**Ax130.****(<=> (PropertyValue inverseOf ?p1
?p2) **

** (and (Type ?p1
AbstractProperty)**

** (Type ?p2
AbstractProperty)**

** (forall (?x1 ?x2) **

** (<=>
(PropertyValue ?p1 ?x1 ?x2) **

** (PropertyValue ?p2 ?x2 ?x1)))))**

**%% Note that the
following can be inferred from the axioms regarding “inverseOf”, “domain”, and
“range”:**

**Th21. ****(PropertyValue domain inverseOf
AbstractProperty)**

**Th22. ****(PropertyValue range inverseOf
AbstractProperty)**

**%% “first” is type
“FunctionalProperty”.**

**Ax131.****(Type first FunctionalProperty)**

** **

**%% An object X is
the value of “first” for an object L if and only if L is type “List” and the
value of “_1” for L is X. (I.e.,
“first” is “_1” for lists.)**

**Ax132.****(<=> (PropertyValue first ?l ?x) **

** (and (Type ?l List)
(PropertyValue _1 ?l ?x)))**

**%% “rest” is type
“FunctionalProperty”.**

**Ax133.****(Type rest FunctionalProperty)**

** **

**%% An object R is
the value of “rest” for an object L if and only if L is type “List”, R is type
“List”, and L has the same items in the same order as list R with one
additional object as its first item.**

**Ax134.****(<=> (PropertyValue rest ?l ?r)**

** (and (Type ?l List)**

** (Type ?r List)**

** (exists (?x) (= ?l (cons
?x ?r)))))[11]**

**%% Note that the
following can be inferred from the axioms regarding “oneOf”, “domain”, and
“range”:**

**Th23. ****(PropertyValue domain rest List))**

**Th24. ****(PropertyValue range rest List)**

**%% “item” is type
“Property”.**

**Ax135.****(Type item Property)**

** **

**%% An object X is
a value of “item” for an object L if and only if L is type “List” and either X
is the value of “first” for L or X is a value of “item” for the value of “rest”
of L. (I.e., Saying that X is an item
of L is saying that X is the first item in list L or there is a list R that is
the rest of list L and X is an item in the list R.)**

**Ax136.****(<=> (PropertyValue item ?l ?x)**

** (and (Type ?l List)**

** (or (PropertyValue first
?l ?x) **

** (exists (?r) **

** (and (PropertyValue rest ?l ?r) **

**
(PropertyValue item ?r ?x))))))**

**%% Note that the
following can be inferred from the axioms regarding “item”, and “domain”:**

**Th25. ****(PropertyValue domain item List))**

**%% “versionInfo”
is a property.**

**Ax137.****(Type versionInfo Property)**

**%% “imports” is a
property.**

**Ax138.****(Type imports Property)**

**%% “nil” is a list
for which there are no values of “item”.
(I.e., “nil” is the empty list.)**

**Ax139.****(Type nil List)**

**Ax140.****(not (PropertyValue item nil ?x))**

[1] Previous versions of this document used additional KIF constructs such as “holds” and sequence variables not found in other first-order logic languages. Those constructs have been eliminated from the axiomatization in this version of the document.

[2] The authors would like to thank Pat Hayes and Peter Patel-Schneider for their generous help in debugging earlier versions of this document. We would also like to acknowledge constructive comments from many on the RDF mailing list including Alex Borgida, Dan Connolly, and Ora Lassila.

[3] For more on type see section 3.2.1.

[4] KIF note: “<=>” means “if and only if”. Relational sentences in KIF have the form “ (<relation name> <argument>*)”. Names whose first character is ``?'' are variables. If no explicit quantifier is specified, variables are assumed to be universally quantified.

[5] KIF notes: “nil” denotes an empty list. “(listof x)” is a term that denotes the list of length 1 whose first (and only) item is x. “=” means “denotes the same object in the domain of discourse”. “item” is a binary relation that holds when the object denoted by its second argument is an element of the list denoted by its first argument. “first” is a unary function whose value is the first element of the list denoted by its argument. “rest” is a unary function which has a value when its argument is a non-empty list and has as a value the list consisting of all but the first element of the list denoted by the argument.

[6] KIF note: “=>” means “implies”.

[7] We do not use the property “subClassOf” in the RDF axioms since “subClassOf” is a property defined in RDF-S.

[8] We do not use the properties “domain” and “range” in the RDF axioms since those properties are defined in RDF-S.

[9] KIF note: “length” is a KIF function that denotes the number of items in the list denoted by the argument of “length”, and “>=” is the KIF relation for “greater than or equal”.

[10] KIF note: “=<” is the KIF relation for “less than or equal”.

[11] KIF note: “cons” is a KIF function that takes an object and a list as arguments and whose value is the list produced by adding the object to the beginning of the list so that the object is the first item on the list that is the function’s value.