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Datatypes: the Bermuda Triangle, and how to fly over it.

From: Pat Hayes <phayes@ai.uwf.edu>
Date: Thu, 31 Jan 2002 18:44:28 -0600
Message-Id: <p05101050b87e2ed2db4d@[65.212.118.208]>
To: w3c-rdfcore-wg@w3.org
Cc: hendler@cs.umd.edu
(I'm CCing this msg to Jim Hendler. Jim, FYI particularly the last 2 
paragraphs, relevance to 'layering'. More on that tomorrow. )

Sorry this is such a long msg, but I think it might solve a lot of problems.

Tinkering with the TDL MT, and offline conversations with Graham, 
convince me that there is a central problem with several of the 
proposed datatyping schemes. Basically, rdf:type just doesn't handle 
datatypes properly, no matter how much one wriggles. You can push the 
problem around, but I don't think there is any way of getting rid of 
it without doing something different. I have a proposal for what to 
do.

First the central problem. (Much of this is a kind of review of what 
other people have noticed, but it helped me get my head clear.) 
Consider any graph with the following kind of 'triangle' with a blank 
node in it (eg see section 3.1 of 
http://lists.w3.org/Archives/Public/www-archive/2002Jan/att-0133/02-tdl.htm, 
or 'idiom B' and 'idiom P' in 
http://lists.w3.org/Archives/Public/www-archive/2002Jan/att-0133/01-s.htm)

eg:thing---eg:prop--->_:1
_:1 ---AAA---> UUU
_:1 ---rdf:type---> DDD

where AAA is some 'special' name like rdf:namedBy or rdf:value or 
whatever, UUU is a unicode string, ie a literal , and DDD is a 
datatype name. The idea is to state some kind of ingeniously worded 
conditions on interpretations of those three triples which guarantee 
that any satisfying interpretation of such a graph will be one where 
the bnode denotes the value of UUU under the datatype mapping 
associated with the datatype DDD, so that the eg:prop of eg:thing is 
that value. (By 'value' here I mean an element of the value space of 
the datatype, eg for xsd:integer it would be an integer.)

The problem is that no such conditions *on RDF triples* are going to 
work properly. Whatever we do, we can't keep a value in a Bermuda 
triangle. Something will fail and the right value will be lost, never 
to be seen again.

Here's why.  In order for the first triple to mean what it is 
supposed to mean, the blank node has to denote the value of UUU under 
the datatype mapping identified by DDD; but that value does not 
provide enough information to enable one to state semantic conditions 
on the other two triples which would restrict UUU to be interpreted 
by the DDD mapping, since the same value may well be the value of a 
different literal under a different datatype mapping.

TDL fixes the last problem by making the bnode denote a pair; but 
then the first triple doesn't make sense. (But we hope some other 
application software will figure it out. Hmm, I wonder if all 
application software is savvy enough to do that...) In Patrick's 
terminology, what we need is some way to state conditions which 
enable the pairing of UUU and DDD to together impose a value on the 
bnode; and the problem is that if that they are only connected at the 
middle blank node, and if its value is the one that any other triple 
would expect it to be (eg Bill eg:age _:x expects the value to be an 
integer, say) then it can't do the pairing all by itself.

Now, it would be possible to fix this problem by stating appropriate 
semantic conditions on the entire subgraph containing the last two 
triples (details below). That option amounts to implicitly giving 
such 'datyping doublets' a special datyped-RDF status, and assigning 
them a meaning that goes beyond their meanings as two separate 
triples, so that a 'datatype doublet' would be more than just a 
conjunction (in a datatype-sensitive interpretation); it really 
amounts to assigning those graphs a special semantic status that 
stands outside the conventional model theory in a particular way. 
That is a slightly unconventional idea in normal logical model 
theory, but it seems to work (and it might be a neat way to approach 
'layering' on RDF in any case, an idea I am trying to write up for 
the webont WG). I'll come back to this proposal later.

However, even if we were to state such 'doublet conditions', we still 
have another problem. If DDD is an rdfs:subClassOf EEE, then this 
graph entails another triangle:

eg:thing---eg:prop--->_:1
_:1---AAA--->UUU
_:1---rdf:type--->DDD
_:1---rdf:type--->EEE

and now any condition that we can state on AAA, UUU and DDD in order 
to restrict I(UUU) to have the relationship to I(_:1) sanctioned by 
I(DDD) is going to apply just as well to I(EEE).  Even if we state 
conditions on doublets, there are two doublets in this graph.  But if 
EEE is a datatype, there is no guarantee that the lexical-to-value 
mapping associated with EEE is compatible with that associated with 
DDD.

One reaction is to just stop this happening by some kind of fiat: 
forbid subClassOf to be applied to datatypes ....
(objection: in order to make use of RDFS reasoning on datatypes, we 
presumably want to be able to say that one datatype is a sub-datatype 
of another, and it seems natural to do that by saying that one of 
them is an rdfs:subClassOf the other. In fact, its more than natural: 
its the *only* way we have of doing it, if datatypes are classes. 
Another objection: it might follow from normal class reasoning that 
one datatype value space is a subset of another. After all, they are, 
as a matter of fact.)
.....or declare that we will only allow the use of rationally 
constructed datatype schemes....
(objection: XSD isn't rational enough.)
But none of these seem satisfactory as a solution.

Now, the key point about this second problem is not the details of 
the semantic conditions or the use of bnodes or whatever: it is RDFS 
class inheritance. That is what causes all this trouble.  And that 
arises from our using rdf:type in the triangle in the first place. 
Even if we allowed literals as subjects, so that we could put the 
literal and the datatype name in a single triple (which is about as 
'local' as one could get in a triples notation with datatype class 
names) :

UUU rdf:type DDD .

the problem would still be there.

So here is my proposal to fix this second problem: DON'T use rdf:type 
to associate datatypes with nodes. Instead use a new property, which 
I will call rdf:dtype. This can be required to be an 
rdfs:subPropertyOf rdf:type, so we can do a fair amount of 
class-membership reasoning using it. But now the bad inferences are 
blocked, because

foo rdf:dtype DDD .
DDD rdfs:subClassOf EEE .

does not entail

foo rdf:dtype EEE .

The utility of this is that it enables us to carefully limit the 
extent to which this datatyping property gets inherited. For example, 
we might want to impose a 'ranging condition':

If:
foo rdf:Range DDD
xxx foo yyy
then:
yyy rdf:dtype DDD

whenever I(DDD) is a datatype. Then we can use ordinary subclass 
reasoning on the ranges, using rdfs:subClass, but that will only give 
us assertions using rdf:type, not rdf:dtype, so they will not get the 
datatyping constraints confused.  (Recall that foo rdf:Range baz and 
baz rdfs:subClassOf bar does *not* entail foo rdf:Range bar, even 
though rdf:type is inherited by superclasses.)

I think that is the only change this would require to the current MT 
would be to add

rdf:dtype rdfs:subPropertyOf rdf:type .

to #1 in the RDFS closure.

It would be perfectly *permissible* to use rdf:type in a doublet, of 
course, and in 'simple' cases it would have the same effect (which I 
am willing to bet would cover all the legacy uses)  but it is 
potentially dangerous, since it might lead to inconsistencies  with 
more complicated datatype hierarchies. (Notice the it would be quite 
harmless to conclude that DDD was a subclass of EEE as long as EEE 
wasn't a datatype class, that would have no effect an anything.) So 
we can still allow it, but deprecate it in favor of rdf:dtype. The 
'simple' cases would be those in which rdf:type rdfs:subPropertyOf 
rdf:dtype, so that the two were equivalent in meaning; and if someone 
were willing to assert that, then they could use rdf:type freely to 
attach datatypes to literals and it would work just fine from normal 
rdfs reasoning, as long as there were no datatype mapping inheritance 
inconsistencies.  (And even then, all that would happen is that some 
particularly dumb RDF graphs might give silly or inconsistent 
results; the reasoners wouldn't actually break.) Of course, a DPH 
could achieve the same effect, and run the same risks, just by 
ignoring the difference between rdf:type and rdf:dtype, and we could 
even point this out to said DPH, with our blessing.

Now, with this change, we can solve the Bermuda triangle property by 
stating semantic conditions on doublets as follows. (This is an 
adaptation of an idea from Graham):

A dt-interpretation I of E   - with respect to some externally 
defined set D of datatypes, where each datatype d in D has an 
associated lexical-to-value mapping LV(d)  - is an 
rdfs-interpretation of E where for every doublet

S rdf:value "uuu"
S rdf:dtype ddd

in E, if I(ddd) is in D, then I(S)=LV(I(ddd))(uuu)

Notice that this really is a condition on the doublet because it 
mentions both ddd and uuu. It is consistent with I("uuu") being the 
string uuu, so we could make graphs tidy on literals (keeps Sergey 
and Dan C happy and simplifies the graph syntax.)  It is a 
restriction on an rdfs-interpretation, so every dt-interpretation is 
an rdfs-interpretation, so dt-entailment is a strengthening of 
rdfs-entailment, as we would want.

BTW, the easiest way to understand rdf:value here would be that it is 
a union (disjunction) of inverses of canonical submappings of the 
datatype mappings in D, ie IEXT(I(rdf:value)) = {<x,y> : for some 
datatype d in D, LV(d)(y)=x} . But this really doesn't matter all 
that much (as long as it can be given some consistent 
interpretation), since the only significant role of the rdf:value 
triple is to be part of the datatyping doublet. Also, we could call 
that property anything we like, as long as it is reserved for this 
special use.

Notice that uses exactly the same RDF graph as the TDL proposal 
(apart from the rdf:dtype change). Its just a different MT.  It works 
pretty much the same way for rdfs:Range as well.

As stated, this only works for 'local typing', ie where the (rdfs 
closure of the) graph contains an actual triple with "rdf:dtype" in 
it.  What about ranges and so on? Well, we can introduce yet a 
further semantic extension, which would be a range-dt-interpretation, 
which also satisfies the ranging condition described above, ie that 
if both

fff rdf:Range ddd
xxx fff yyy

are true then

yyy rdf:dtype ddd

must be true also. Obviously this has a corresponding closure rule 
that would add the appropriate rdf:dtype assertions to a graph, and 
then we can characterize 'remote' datatyping by saying that it 
amounts to doing local datatyping in the range-dt-closure of the 
graph. Then we would have an entailment lemma for each kind of 
datatyping as an extension of the previous kind, extending the 
'layers' that we already have for simple/rdf/rdfs/...entailment. 
Expressed more operationally, that is just saying that in order to 
check 'remote' datatyping, we need to do some extra inferencing of a 
particular kind. Most of it will be normal rdfs inference, but it 
needs to pay special attention to rdf:dtype.

We could even express something like a preference for local over 
global, by saying that one is supposed to check dt-entailment before 
checking for range-dt-entailment. That makes the distinction 
rigorously clear in MT terms without actually including barbaric 
things like defaults into the MT itself :-) Notice that something can 
be range-dt-inconsistent but still be dt-consistent; that would be a 
range d.t. contradicting a local d.t.. But if we decide to stick with 
simple dt-entailment and only worry about that, and never check 
range-dt-entailment, then we will never notice the higher-level 
inconsistency, and indeed the graph will be consistent as far as we 
are concerned. And all these graphs are perfectly fine seen as simple 
RDFS, though they aren't saying much about the values of the bnodes 
in the triangles when looked at from that low a level.

<datatyping proposal />
  <more general issue>

Now, all this business of defining more and more different notions of 
interpretation and associated kinds of entailment might seem kind of 
a crock: after all, RDF is supposed to be a single language, not a 
whole series of languages, right? But in fact, I would argue that it 
is appropriate, since RDF(S) is supposed to be a 'ground layer' for 
the whole SWeb, so one would expect that it should support a whole 
lot of different kinds of interpretation, as one puts more and more 
things into it that are supposed to have extra meanings over and 
above the basic RDF meaning. In a sense, it *is* a series of 
languages, but which are all encoded in the same RDF triples-graph 
syntax, so we have to define a whole range of different kinds of 
entailment for the same syntax.  And this is all OK, as long as those 
special 'extra' meanings are indeed genuinely extra, ie if you ignore 
the extra constraints, then the RDF triples that are there can be 
understood as normal RDF triples at some 'lower' level of the layer 
cake, and all their conclusions will be valid there (and will not 
screw up any higher levels). Which they are in this case. (Though not 
when we get to DAML, which is a problem for the webont WG; but that's 
another story.) Then each 'layer' can be seen as a licence to use 
some extra kind of inferencing or processing power on the same graph, 
rather than an extension to the notation/syntax of the language. 
Presumably the licence to use such extra power would come from some 
kind of enclosing tags, like the <daml> tags that DAML uses; or, we 
could introduce the new vocabulary with a new Qname prefix, and write 
rdfd:type and rdfd:value to indicate that they constitute a new 
name-space with special meanings, maybe. (Can we, the RDF WG, do 
that?).

The overall picture one gets is something like Tim's famous 
layer-cake, but with finer structure in the layers. (But then we 
ought to expect that when we get closer, we can see finer structure, 
right?) Like this:

Full datatype entailment (where datatypes on rdf:ranges of properties 
can fix meanings of literals)
---------------------------------
Local datatype entailment (where only the explicit typing triples fix 
meanings of literals)
---------------------------------
RDFS entailment (where the rdf: and rdfs: vocabulary is properly 
interpreted but literals are just strings)
---------------------------------
RDF entailment (where rdf:type and rdf:Property is properly 
interpreted: it might not be worth distinguishing this layer, in 
practice.)
---------------------------------
Simple RDF entailment(just from the raw triples)
---------------------------------
(I guess XML is under here somewhere :-)

Its the same syntax everywhere: sets of RDF triples. But each layer 
adds some inferential power to the layer below, expressed in the MT 
as a new set of semantic constraints on allowable interpretations 
imposed on some part of the vocabulary. When those restrictions have 
to mention more than one triple (as here) then in a sense we could 
say that this amounts to encoding a different, higher-level, syntax 
in the RDF graph, but we aren't *obliged* to talk about it that way.

BTW, since the top two are relative to D, this is really a tree, if 
there are several datatyping schemes in use. But RDFS is all in the 
trunk.

Pat

PS. More later on the S idioms and how they can be included into this 
picture. They don't need the use of rdf:dtype to avoid inappropriate 
inheritance.


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Received on Thursday, 31 January 2002 19:44:03 EST

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