=Paper=
{{Paper
|id=None
|storemode=property
|title=Finite Models in RDF(S), with Datatypes
|pdfUrl=https://ceur-ws.org/Vol-1035/iswc2013_poster_14.pdf
|volume=Vol-1035
|dblpUrl=https://dblp.org/rec/conf/semweb/Patel-SchneiderH13
}}
==Finite Models in RDF(S), with Datatypes==
https://ceur-ws.org/Vol-1035/iswc2013_poster_14.pdf
Finite Models in RDF(S), with datatypes
Peter F. Patel-Schneider1 and Pat Hayes2
1
Nuance Communications, pfpschneider@gmail.com
2
IHMC, phayes@ihmc.us
The details of reasoning in RDF [2] and RDFS [1] are generally well known.
There is a model-theoretic semantics for RDF [3, 4] and there are sound and com-
plete proof theories for RDF without datatypes [6]. However, the model-theoretic
characteristics of RDF3 have been less studied, particularly when datatypes are
added. We show that RDF reasoning can be performed by only considering finite
models or pre-models, and sometimes only very small models need be considered.
Ter Horst [6] does define Herbrand models for RDF and RDFS, providing the
basis for some model-theoretic characteristics of RDF and RDFS, but he does
not provide a full analysis of RDF datatypes, analyzing instead an incomplete
semantics for datatypes that is easier to reason in. As well, the recent minor
modifications to the semantics of RDF [4], while cleaning up some aspects of en-
tailment in RDF, do make some technical changes that might appear to interfere
with finite model-based reasoning in RDF. An analysis of finite models for RDF
shows that the modified semantics does not introduce any unintended changes
to reasoning in RDF. As well, it provides insights into the modeling strength of
RDF, particularly when blank nodes are not present, and illustrates how finite
datatypes interact with the rest of RDF.
As shown by ter Horst, sound and complete reasoning in RDF and RDFS
without datatypes is decidable, even though RDF and RDFS have an infinite
number of axioms. However, the decidability of RDF reasoning does not neces-
sarily mean that RDF reasoning can be done by considering only finite models.
For example, OWL [5] has decidable reasoning, but nonetheless requires infinite
models, for example by encoding the number line using inverse properties and
number restrictions.
In the new semantics for RDF [4], all IRIs are given denotations in all in-
terpretations. This means that the standard construction for Herbrand models
will result in an infinite model. Infinite datatypes (e.g., xsd:integer) also produce
infinite models, so finite model reasoning in RDF with datatypes is technically
concerned with pre-models, semantic structures that are finite and can be triv-
ially extended to real models. Nonetheless finite model reasoning should be pos-
sible in RDF as RDF does not have inverses, counting, or even equality and
inequality.
Given an RDF graph (or set of RDF graphs), we define the set of identifiers
for the graph as the nodes and predicates of the graph plus the IRIs used in
the RDF (and RDFS, if considering RDFS entailment) semantic conditions and
3
In this paper, RDF by itself will generally be used to indicate both RDF and RDFS,
both with and without datatypes, unless otherwise specified.
axioms, except that no container membership property not occurring in the
graph is an identifier for the graph.
We then build some models for the RDF graph as follows.4 We start with the
data values for the recognized datatypes. For every identifier that is not a literal
with a recognized datatype we nondeterministically either nondeterministically
choose some data value as its denotation or add a new domain element as its
denotation. This results in denotation functions where identifiers that do not
denote data values all denote different domain elements. For IRIs and literals
with unrecognized datatypes that are not identifiers of the graph we add two
extra domain elements, one being the denotation of the container membership
properties that are not identifiers for the graph and one being the denotation of
all other identifiers. We then build up the rest of the semantic structure using
the graph and the axioms and rules of inference from ter Horst augmented with
axioms for datatypes and co-denoting identifiers, resulting in a structure like a
datatype-aware Herbrand interpretation except that some non-literal identifiers
might denote data values.
Some of these denotation functions might fail to produce an interpretation
because some datatype domain or range restriction requires a domain element
to be a data value for a particular datatype when it is not. However, if there
is a model for the RDF graph, then this construction will produce at least one
model, because there are no semantic conditions in RDF that require a partic-
ular denotation for an identifier or require or prohibit co-denotation between
identifiers except those related to data values and we have not constrained data
value denotations here except for non-identifiers.
So for every satisfiable RDF graph we have ended up with a set of models.
We now need to show that any model of the graph is at least as strong as one
of these models. For a particular set of denotations the inference rules produce
the weakest possible model. Now consider the extra domain element added for
unmentioned container membership properties. Replicating this domain element
and splitting denotations produces a model that has the same strength as the
original model because the RDF semantic conditions treat all these domain ele-
ments the same and identity cannot be detected. Similarly replicating the other
additional domain element produces a model of the same strength. Because there
is no inequality in RDF, identifying any two domain elements always produces
model that is at least as strong, if it produces a model at all. Thus the restric-
tion that denotations that are not data values be unique produces the weakest
possible models.
In these models all the data values in a datatype that are not the denotation
of some identifier have exactly the same characteristics. A pre-model can thus be
constructed that collapses all these data values into one, finally resulting in finite
model reasoning for RDF. (The result is, of course, not generally a model because
it violates the semantic conditions on the denotation of literals.) A completely
finite semantic structure can be constructed by simply ignoring these denotation
mappings and the mappings for other non-identifiers.
4
For purposes of space some shortcuts in notation will be taken throughout this paper.
In the absence of recognized datatypes, the above construction results in
unique Herbrand models just like the ones in ter Horst. In the presence of rec-
ognized datatypes this construction is different from that in ter Horst, as it
captures the full meaning of datatypes, including the requirement to consider
several models. For example, consider a datatype with only two data values, say
ex:two, and the RDF graph
ex:p rdf:range ex:two.
ex:a ex:p ex:u, ex:v. ex:b ex:p ex:u, ex:w. ex:c ex:p ex:v, ex:w.
This RDF graph entails :x ex:p ex:u, ex:v, ex:w. To determine entailment in
these situations more than one model must be considered, hence the choice of
values in the graph above.
If datatypes have sufficient data values of the right kind, however, then it is
possible to only consider models that are more like Herbrand models. Given a
finite set of recognized datatypes D and E a subset of D, let the unconstrained
portion of E be the elements of the intersection of the data spaces for each e in E
that are not in any other datatype in E that is not a superset of the intersection.
Consider two RDF graphs A and B and a set of recognized datatypes D. If the
unconstrained portion of every E, a non-empty subset of D, is of size greater
than the number of data values in it denoted by literals in A and B plus the
number of identifiers in A that are not literals with recognized datatypes plus one
then it is possible to always choose unconstrained elements when picking data
values for identifiers that are not literals with recognized datatypes. Then all
such identifiers will have different denotations, and different denotations from
all literals. This in turn permits the determination of the datatypes that the
denotation of an identifier must belong to by using the D* rules of ter Horst.
Then when determining the denotation of an identifier, if this set is empty add a
new domain element and otherwise pick an unconstrained value for this set. This
results in a single, finite model that can be used for reasoning. Note, however,
that the presence of even a single too-small unconstrained portion may require
examining multiple models.
So we have shown that RDF reasoning can be done by considering only
models of the size of the RDF graph. Is it possible to consider only very small
models? (Datatypes make these considerations even more complex, so this sec-
tion of the paper will ignore datatypes.) If RDF had disjunction, then it would
not be possible to significantly shrink the minimum size of considered models.
For example, consider RDF graphs containing n triples of the form
Si S1 Si . for 1 <= i <= n.
In any model with less than n domain elements, there is some 1 ≤ i 6= j ≤ n
such that Si and Sj have the same denotation. In this model Si S1 Sj . is true
and so in any such model the disjunction of all these triples is true, which is not
a valid entailment.
Even with RDF lacking disjunctions, it is possible to show that very small
models are not adequate. Consider RDF graphs containing triples of the form
Si S1 Sj . for 1 ≤ i 6= j ≤ n.
In any model with less than n domain elements, there is some 1 ≤ i 6= j ≤ n such
that Si and Sj have the same denotation, which is then related to the denotation
of each of the Si by the denotation of S1 , so the graph containing
:x S1 Sj . for 1 <= i <= n,
is true in each of these models, but this graph is not entailed. Therefore consid-
ering only models of this size or smaller is not sufficient.
If we only consider entailments with no blank nodes in the entailed graph
then smaller models suffice. Consider an interpretation I (for RDF or RDFS
without any recognized datatypes) containing two domain elements e1 and e2
that are neither properties nor classes (call these domain elements ordinary).
Form I 0 from I by simply replacing e1 and e2 with a single domain element e
throughout. Then I 0 is an interpretation, which can be determined by examining
all the appropriate semantic conditions.
So for B1 and B1 identifiers whose denotation in I are neither e1 nor e2 , I 0
supports any triple of the form B1 P B2 ., if and only if I supports the triple.
For any particular such triple this process can be repeated until only three
ordinary domain elements remain. Considering all such shrunken interpretations
is adequate to rule out any invalid entailments, so we need only consider models
with three ordinary domain elements, but of course we need to consider many
interpretations. The ability to have such small models and to then only consider
them shows how weak RDF is as a logic.
We have argued that RDF reasoning can be done by only considering finite
models, even in the presence of datatypes. The exact size and number of the
models that need to be considered depends on a number of factors, including
which recognized datatypes are involved, but generally models of size at least the
number of identifiers in the graph must be considered. If there are no datatypes
or the datatypes are of sufficient size, then a single Herbrand-like model is all
that need be considered. If there are no blank nodes in the entailed graph, then
much smaller models suffice, although multiple models must then be considered.
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