# Re: new model theory for DAML+OIL

From: Peter F. Patel-Schneider (pfps@research.bell-labs.com)
Date: 10/10/01

```I didn't account for blank nodes in the model theory in a way that is easy
to define a decent notion of entailment, so I put together a revision of
the model theory.   The development of interpretations and models is now
similar to yours.

I believe that entailment has

G |= G'

for G' isomorphic to G, which would have been somewhat difficult to achieve
in the previous model theory.

peter

Alternative Model Theory for RDF and RDF Schema plus Datatypes

Here is an alternative model theory for RDF and RDF Schema.   The attempt
here is to have a model theory for RDF and RDF Schema that can be extended
to datatypes, as I have proposed them.  This is a VERY terse document,
missing most of what would correspond with Pat Hayes's excellent commentary
on his model theory for RDF and RDF Schema.

Note that this model theory has not gone through any screening.  I have
tried to make it as error-free as possible, but there are undoubtably
errors remaining.  In particular, there may be (little) pieces missing
concerning some of the RDF and RDF Schema vocabulary.

Note also that this is an unofficial and draft model theory.

Peter F. Patel-Schneider

1/ RDF Graph Syntax

I'm sticking with graphs, even though they have some problems for RDF.
Triples have their own problems with respect to RDF.

URI is a collection of URI names.

URI may be the collection of all URI names, but this is not required.
This model theory ignores all aspects of the structure of URIs.

L is the collection of literals, disjoint from URI.

Literals form the lexical space, in XML Schema datatype terms, not the
value space!

An untidy RDF graph, R, is a three-tuple (that can be considered to be a
partially node labeled, directed triple-graph)
< N, E, LN >
where N is the set of nodes in the graph
LN :(partial) N -> URI u L gives labels for nodes
E <= N' x N'' x N is the set of edges in the graph
where N' = { n : LN(n) is undefined or LN(n) in URI }
where N'' = { n : LN(n) is defined and LN(n) in URI }

This accounts for literals not being allowed as ``labels'' of edges, nor
as the labels of nodes that are heads of edges, but does not account for
edge ``labels'' being properties.

An untidy RDF graph is ground if LN is a total function on N.

A tidy RDF graph (also called an RDF graph) is an untidy RDF graph where LN
is injective on URI (but not necessarily total).

Tidy graphs here do not have to be tidy on literals, which is change from
Pat Hayes's model theory.

I have tried to keep as much of the terminology from Pat Hayes's model
theory as possible.

The union of two untidy RDF graphs,
R = < N , E , LN > and R' = < N', E', LN' >
is the untidy RDF graph
R u R' = < N u N', E u E', LN u LN' >
provided that LN and LN' agree where they are both defined.

Note that N and N' need not be disjoint.

The tidy version of an untidy RDF graph R = < N, E, LN >
is the graph t(R) = < N/eLN, E/eLN, LN/eLN >
where eLN is the equivalence relation induced by LN range restricted to URI
i.e., n = n'  iff  LN(n) and LN(n') both defined, LN(n) in URI,
and LN(n) = LN(n')
and / is point congruence.

[Is this the right terminology?]

2/ Literal Values

LV is some collection of literal values.

Literal values form the value space, in XML Schema datatype terms, not
the lexical space!

XLS : L -> powerset ( LV ), maps literals into the set of literal values
that they might have.

Here is the first substantive difference from Pat Hayes's model theory.
The XLS mapping does not provide a definitive answer for the meaning of a
literal.  The reason for not pinning down the mapping for literals is to
allow different mappings for different datatypes.  For example, a node
with literal label 05 might be mapped into the integer 5 or the string "05".

3/ Models and Entailment

An interpretation I is a four-tuple
< IR, IP, IEXT, IS >
where IR is a non-empty set, called resources
IP <= IR, called properties
IEXT : IP -> powerset ( IR x (IR u LV) )
IS : URI -> IR

An interpretation I = < IR, IP, IEXT, IS>
is a model for an untidy RDF graph R = < N, E, LN >
if there is a mapping
M : N -> IR u LV
such that for n, s, p, o in N
1. if LN(n) defined and LN(n) in URI then M(n) = IS(LN(n))
2. if LN(n) defined and LN(n) in L then M(n) in XLS(LN(n))
3. if <s,p,o> is in E then M(p) in IP and <M(s),M(o)> in IEXT(M(p))

This works for both ground and non-ground untidy RDF graphs.

One minor difference between this model theory and Pat Hayes's is that
unnamed nodes can denote resources or literal values, unless they appear
in the subject position of an edge.  This could easily be changed to
require that unnamed nodes only denote resources.

An untidy RDF graph R entails another untidy RDF graph R' (R |= R')  iff
R u R' is defined and every model for R is also a model for R u R'.

This definition of entailment admits the possiblity that the two graphs
share nodes.  Will this have interesting consequences?

4/ Core RDF

By core RDF I mean RDF without reification or containers.

A core RDF graph is an untidy RDF graph that contains nodes with the
following labels:

rdf:type
rdf:Property

and an edge <t,t,P> where t is a node with label rdf:type
and P is a node with label rdf:Property

Throughout this development untidy graphs are allowed.  Theorems about
entailment will probably have to use tidy graphs, however.

A core RDF model for a core RDF graph R is a model I for R with the
following extra conditions

1. x in IP  iff  <x,IS(rdf:Property)> in IEXT(IS(rdf:type))
2. IEXT(IS(rdf:type)) <= IR x IR

A core RDF graph R RDF-entails another core RDF graph R'  iff
R u R' is defined and
every core RDF model for R is also a core RDF model for R u R'.

5/ RDFS

A core RDFS graph is a core RDF graph that contains nodes with the
following labels:

rdf:type			[redundant from RDF]
rdf:Property			[redundant from RDF]
rdfs:Resource
rdfs:Class
rdfs:subClassOf
rdfs:subPropertyOf
rdfs:seeAlso
rdfs:isDefinedBy
rdfs:ConstraintResource
rdfs:ConstraintProperty
rdfs:range
rdfs:domain
rdfs:label
rdfs:comment
rdfs:Literal

and the following edges (actually edges between nodes with the following
labels):

<rdfs:Resource,      rdf:type, rdfs:Class>
<rdf:Property,       rdf:type, rdfs:Class>
<rdfs:Class,	       rdf:type, rdfs:Class>		[redundant]
<rdfs:Literal,       rdf:type, rdfs:Class>

<rdf:type,           rdf:type, rdf:Property>		[redundant from RDF]
<rdfs:subClassOf,    rdf:type, rdf:Property>
<rdfs:subPropertyOf, rdf:type, rdf:Property>
<rdfs:seeAlso,       rdf:type, rdf:Property>
<rdfs:isDefinedBy,   rdf:type, rdf:Property>		[redundant]

<rdfs:range,         rdf:type, rdfs:ConstraintProperty>
<rdfs:domain,        rdf:type, rdfs:ConstraintProperty>

<rdfs:Class,              rdfs:subClassOf, rdfs:Resource>
<rdfs:ConstraintResource, rdfs:subClassOf, rdfs:Resource>
<rdfs:ConstraintProperty, rdfs:subClassOf, rdfs:Resource>	[redundant]
<rdfs:ConstraintProperty, rdfs:subClassOf, rdfs:ConstraintResource>

<rdfs:isDefinedBy,   rdfs:subPropertyOf,   rdfs:seeAlso>

<rdf:type,           rdfs:range,  rdfs:Class>
<rdfs:subClassOf,    rdfs:domain, rdfs:Class>
<rdfs:subClassOf,    rdfs:range,  rdfs:Class>
<rdfs:subPropertyOf, rdfs:domain, rdf:Property>
<rdfs:subPropertyOf, rdfs:range,  rdf:Property>
<rdfs:seeAlso,       rdfs:range,  rdfs:Resource>
<rdfs:isDefinedBy,   rdfs:range,  rdfs:Resource>	[redundant]
<rdfs:range,	       rdfs:domain, rdf:Property>
<rdfs:range,	       rdfs:range,  rdfs:Class>
<rdfs:domain,	       rdfs:domain, rdf:Property>
<rdfs:domain,	       rdfs:range,  rdfs:Class>
<rdfs:label,	       rdfs:domain, rdfs:Resource>	[redundant]
<rdfs:label,	       rdfs:range,  rdfs:Literal>
<rdfs:comment,       rdfs:domain, rdfs:Resource>	[redundant]
<rdfs:comment,       rdfs:range,  rdfs:Literal>

A core RDFS model for a core RDFS graph R is a core RDF model I for R with
the following extra conditions:

x in IR  iff  <x,IS(rdfs:Resource)> in IEXT(IS(rdf:type))
x in IP  iff  <x,IS(rdf:Property)> in IEXT(IS(rdf:type)) [redundant from RDF]

if <x,y> in IEXT(IS(rdf:type)) and <y,z> in IEXT(IS(rdfs:subClassOf))
then <x,z> in IEXT(IS(rdf:type))			[2.3.2]

if <x,y> in IEXT(IS(rdfs:subClassOf)) and <y,z> in IEXT(IS(rdfs:subClassOf))
then <x,z> in IEXT(IS(rdfs:subClassOf))		[2.3.2]

if <x,y> in IEXT(r) and <r,s> in IEXT(IS(rdfs:subPropertyOf))
then <x,y> in IEXT(s)				[2.3.3]

if <x,y> in IEXT(IS(rdfs:subPropertyOf))
and <y,z> in IEXT(IS(rdfs:subPropertyOf))
then <x,z> in IEXT(IS(rdfs:subPropertyOf)		[2.3.3?]

x in IP and <x,IS(rdfs:ConstraintResource)> in IEXT(IS(rdf:type))
iff  <x,IS(rdfs:ConstraintProperty)> in IEXT(IS(rdf:type))	[3.1.2]

for y in IR, if <x,y> in IEXT(p) and <p,c> in IEXT(IS(rdfs:range))
then <y,c> in IEXT(IS(rdf:type))			[3.1.3]
if <x,y> in IEXT(p) and <p,IS(rdfs:Literal)> in IEXT(IS(rdfs:range))
then y in LV

Yes, this last is a special case for rdfs:Literal, but so what!

if <x,y> in IEXT(p) and <p,c> in IEXT(IS(rdfs:domain))
then <x,c> in IEXT(IS(rdf:type))			[3.1.4]

A core RDFS graph R RDFS-entails another core RDFS graph R'  iff
R u R' is defined and
every core RDFS model for R is also a core RDFS model for R u R'.

6/ Datatypes (general version)

Datatypes add extra structure to literals and literal values.

A datatype theory is a four-tuple <LV,DT,DTC,DTS>
where LV is a collection of literal values
DT is a collection URIs that are also datatypes
DTC : DT -> powerset ( LV )
DTS : DT -> ( L -> LV ), with DTS(d) potentially partial
and DTS(d)(L) <= DTC(d) for all d

DTC maps a datatype to its extension (or value space).
DTS maps a datatype to a partial map from literals (or lexical space) to
literal values (or value space).

Each datatype provides at most one literal value for each literal via
the DTS mapping.

Given a datatype theory <LV,DT,DTC,DTS>
define XLS(l) = { lv in LV : for some d in DT with DT(d) defined on l
lv = DT(d)(l) }

Given a datatype theory <LV,DT,DTC,DTS>
a datatype RDFS model for a core RDFS graph R is a core RDFS model I for R,
with the following extra conditions:

if <s,p,l> is in E with LN(l) in L
and <M(p),c> in IEXT(IS(rdfs:range))
then for any node nc with LN(nc) in DT and M(nc) = c
M(l) = DTS(LN(nc))

for v in LV, if <x,v> in IEXT(p) and <p,c> in IEXT(IS(rdfs:range))
then for any node nc with LN(nc) in DT and M(nc) = c
v in DTC(LN(nc))

These conditions are rather complicated for semantic conditions, so some
explanation is in orer.  The first condition says that literals (n) that
are objects of statements must denote according to any datatype range for
the predicate (M(p)) of the statement.  The second condition says that
literals values (y) that are in relationships must belong to the value
space (DTC(LN(c))) of any range of the relationship.

A core RDFS graph R datatype-entails another core RDFS graph R'  iff
R u R' is defined and
every datatype RDFS model for R is also a datatype RDFS model for R u R'.

7/ Datatypes (for XML Schema datatypes)

A XML Schema datatype theory is a datatype theory <LV,DT,DTC,DTS>
where LV contains the value spaces of the primitive XML Schema datatypes
DT is the subset of URIs consisting of (built-in or all) XML Schema
datatypes, distinguished either by their special names (e.g.,
xsd:integer) or by ``following'' them and finding an XML Schema
datatype expression
DTC maps each d in DT to its value space
DTS maps each d in DT to its map from lexical space to value space

Given an XML Schema datatype theory T
an XML Schema datatype RDFS model for a core RDFS graph R is
a datatype RDFS model for R over T.

A core RDFS graph R xsd-entails another core RDFS graph R'  iff
R u R' is defined and
every XML Schema datatype RDFS model for R is also an XML Schema datatype
RDFS model for R u R'.
```

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