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Re: AS & S review: RDFS-compatible OWL semantics

From: Peter F. Patel-Schneider <pfps@research.bell-labs.com>
Date: Fri, 17 Jan 2003 12:22:34 -0500 (EST)
Message-Id: <20030117.122234.22285671.pfps@research.bell-labs.com>
To: herman.ter.horst@philips.com
Cc: www-webont-wg@w3.org

From: herman.ter.horst@philips.com
Subject: AS & S review: RDFS-compatible OWL semantics
Date: Fri, 17 Jan 2003 17:13:06 +0100

> Here is the next part of my review comments on the Semantics 
> document:
> Section 5 RDFS-Compatible Model-Theoretic Semantics
> Version of 16 January
> 
> This part of the document is crucial for the semantic
> layering of OWL on RDF.  It is perhaps surprising that
> after all the work and discussion on this topic, more needs
> to be said about several formal aspects to ensure that the 
> text becomes completely suitable.
> 
> The first point that I want to raise is that many 
> small, additional assumptions need to be made on an 
> OWL interpretation in order to assure that each use that
> is made of the functions EXTI and CEXTI can really be made.
> In order to explain this, note that the current text does not 
> start from a suitable summary of the RDF Semantics. 
> In the second and third paragraph of Section 5.2,
> an RDFS interpretation is first described as a three-tuple,
> and then, in a "datatyped" version, as a four-tuple.
> However, an RDFS interpretation, datatyped or not, is currently
> described as a five-tuple, including a set of properties PI as
> part of the basic definition.  The domain of the function EXTI
> is not all of RI, but only the set PI.  This change was made in 
> November (when the RDF Model Theory was renamed to RDF Semantics;
> the OWL AS & S document still speaks of RDF MT).

I've made the appropriate change, to something very close to what you say
below.

> A related point is that the domain of the function CEXTI consists
> only of the set of classes CI, which may also not be all of RI 
> (this was already the case with the RDF MT version of April).

This is not correct.  ICEXT is still defined for all resources, even for
literals.  This may be a bug in the RDFS semantics.

> In view of this, the given summary of RDF semantics should be 
> replaced by an up to date and somewhat more extensive summary.
> Let me briefly summarize the basic definition of the RDF semantics,
> in order to be able to describe the additional assumptions that
> need to be made in the OWL semantics, and in order to facilitate 
> the replacement of the given summary of the RDF semantics. 
> I use the slight adaptation made by Peter of the original notation 
> of Pat, however without making many final I's a subscript, of 
> course.
> An RDFS interpretation of a vocabulary V is a five-tuple consisting 
> of:
> - a set RI (the universe)
> - a set PI subsetOf RI
> - a function EXTI : PI -> P(RI x RI) 
> - a function SI : V -> RI
> - a function LI : {typed literals} -> RI
> satisfying many special conditions specified in the RDF Semantics.
> (By the way, referring to an earlier part of this review,
> note that the P(X) notation for power set is very convenient here.)
> 
> Given such an RDFS interpretation, the set of classes is defined
> to be
> CI := {x in RI: <x,SI(rdfs:Class)> in EXTI(SI(rdf:type))}.
> This set is defined to be the domain of the function
> CEXTI : CI -> P(RI) 
> CEXTI(c) := {x in RI : <x,c> in EXTI(SI(rdf:type))} (c in CI)

This is not in the RDF Semantics document.

> These are all the definitions that need to be summarized.
> It follows from the complete definition of RDFS interpretation
> (actually, it follows already from the definition of RDF 
> interpretation) that
> (*) CI = CEXTI(SI(rdfs:Class))  and  PI = CEXTI(SI(rdf:Property)).
> (The range that I give above to the function CEXTI does not
> appear explicitly in the RDF Semantics document, but follows
> clearly from what is said there.)

range -> domain ?

> So each table in Section 5.2 needs to be expanded with an 
> assumption
> SI(E) in CI (in case CEXTI(SI(E)) is used) or 
> SI(E) in PI (in case EXTI(SI(E)) is used).
> 
> In the second table of Section 5.2 this is easy: each of 
> the empty cells in the second column can just be assigned 
> the content CI.

Not needed.

> In the later tables it is also possible to incorporate the 
> required additional assumptions, in the bold header texts.

Not needed.

> A simpler and more elegant way to incorporate these additional 
> assumptions could be as follows.
> The OWL vocabulary at the beginning of Section 5.1.1 
> (where it now "appears" with an ellipsis) could be expanded
> explicitly, using two disjoint subsets VOWLC and VOWLP (it is 
> clear which vocabulary members should go where).  Then the 
> required additional assumptions on an OWL interpretation
> can be made in one stroke with 
> SI(VOWLC) subsetOf CI   and SI(VOWLP) subsetOf PI.

I think that this is neither necessary nor desirable.

> The equations (*) above can be used to simplify many entries
> in the tables in Section 5.2, by taking CI or PI instead of 
> the expansions in the right-hand sides of these equations.
> It should be noted that CI and PI are more fundamental in the
> RDF Semantics then these expansions.

CI actually is subordinate to ICEXT(I(rdfs:Class)).
Given that CI is subordinate, I prefer to keep consistency by using ICEXT
throughout. 

> Also, the conditions for an OWL interpretation to be OWL Full
> become simply IOT = RI, IOC = CI, IOP = PI.


> There is a problem with the definition of 
> "sequence of y1,...,yn over C".  I can prove that this
> definition cannot be suitable.  Namely, in the first application
> of this definition, C becomes IOC.  The definition uses
> CEXTI(C), so assumes IOC in CI.  In OWL Full, IOC = CI,
> so we get CI in CI.  This contradicts the axiom of foundation.
> The definition can be repaired by letting yi in C instead
> of yi in CEXTI(C).  This seems to be suitable for each use of 
> the definition, except perhaps the use in connection with 
> distinctMembers: shouldn't it be IOT instead of IOC there?

Yes, this is a stupid bug in my definition.  Thanks for picking it up.

> This seems to be suitable for each use of 
> the definition, except perhaps the use in connection with 
> distinctMembers: shouldn't it be IOT instead of IOC there?

Yes.

> In my view, the formal definition of an OWL interpretation should
> include, in addition to an RDFS interpretation <RI,PI,EXTI,SI,LI>,
> the distinguished subsets IOC, IOP, IOT, IOR, IOOP, IODP, IDC,
> IAD, and IL of RI.  Otherwise, these sets "fall out of the air".
> Each of these 9 subset relationships is implied by the second
> table of Section 5.2, except for IAD subsetOf RI, which should
> be added to this table.

I think that this is neither needed nor desirable.

> The second table does not include the inclusion LV subsetOf RI,
> which can be assumed by the RDF Semantics, and which would
> be appropriate to recall in the table.

This is now incorporated in the text, but putting it in the table is a good
idea.

> The first table in Section 5.2 uses sets IOP, IOC etc. whose 
> meaning is not yet clear.  Therefore I propose to move this table
> to the third position.  Then, moreover, we get three more coherent 
> "groups of tables" in a row:
> 1. universe/syntactic categories; classes/datatypes/properties
> 2. the "iff tables": domains/ranges; equivalence
> 3. the "DL tables": Boolean combinations; restrictions; 
>    comprehension principles

I don't think that this reordering is helpful.

> I feel that Section 5.2 could use more text to motivate these
> different kinds of tables.  For example, can it be 'explained' 
> why is there an iff for owl:sameClassAs and owl:disjointWith
> but not for owl:complementOf?

There is very little motivation now.  However, I think that a useful amount
of motivation would turn out to be a large amount of motivation.  I've been
too busy with RDF to attend to it.


> In my view, the first condition on oneOf is an unsuitable integration
> of dissimilar conditions.  In fact, the next table, which is
> completely devoted to oneOf, could be omitted by slightly 
> extending the condition in the previous table, as follows:
>   ( x in IOC and l is a sequence of y1,...,yn over IOT
>   or x in CI and l is a sequence of y1,...,yn over LV )
>   and CEXTI(X) = {y1,...,yn}

The condition for oneOf is similar to the ones for unionOf and
intersectionOf.  I thus think that it should stay.  The change you suggest
would cause problems for OWL DL.

> The first table where oneOf currently appears has two header
> lines with bold text.  The first of these lines should be
> omitted.

first -> second?  Yes, I see how this can be done.

> Section 5.1 starts with the following sentence:
> "All of the OWL vocabulary is defined on the 'OWL universe', 
> which is a collection of RDFS classes that are intended to 
> circumscribe the domain of application of the OWL vocabulary: 
> owl:Thing, owl:Class and owl:Property."
> I read here that 
> OWL universe = {owl:Thing, owl:Class and owl:Property}.
> However, with the RDF semantics it is inherited that
> the set RI is called the universe of the interpretation,
> as is also mentioned in the beginning of Section 5.2.
> As the word universe is used here in two different ways,
> I feel that the wording of the cited sentence should be
> adapted to incorporate the connection with RDF semantics.

Changed to

<p>All of the OWL vocabulary is defined on the 'OWL universe', which is a
   division of part of the RDFS universe into three parts, namely the class
   extensions of owl:Thing, owl:Class and owl:Property. 
  The  class extension of owl:Thing comprises the individuals of the
  OWL universe.  The  class extension of owl:Class comprises the
  classes of the OWL universe.  The  class extension of
  owl:Property comprises the properties of the OWL universe.
</p>

> 
> The table on the semantics of the cardinality restrictions 
> does not yet include the corrections which I believe you 
> confirmed earlier.
> It should be, three times:
>    card{v in IOT union LV : <u,v> in EXTI(p)}
> (In our earlier discussion I missed the LV part.
> In this way, both object properties and datatype properties
> are covered, in the correct, intended way.)
> Without this addition, formally, there is no set, so
> no cardinality can be taken.  Instead, formally, there is 
> only a class, not in the sense of OO or RDF or OWL, but 
> in the sense of Zermelo-Fraenkel set theory.

I disagree.  I believe that the definitions are fine as they are.

> I find it confusing, in the definition of separated OWL
> vocabulary in Section 5.3.2, to identify a vocabulary
> with a partition of it.  I am in favor of omitting the 
> = sign, and of speaking of a vocabulary V' with partition
> <...>.  This would also affect (improve) the next paragraphs, 
> including the statement of Theorem 1.

I should have used V = VI + VC + VD + VOP + VDP

> As to the next definition, of OWL abstract ontology with
> separated names, it is not consistent with the first sentence
> of Section 2.1, to speak of a set of axioms and facts
> in the abstract syntax: it should become a sequence.

I think that this is a suitable abuse of terminology.

> In Sections 5.3.1/2, three RDF Graphs should become RDF graphs.

Technical terms can (and often should) be capitalized, I believe.

> In the definition of OWL DL interpretation, it is awkward to
> use the letter n for a vocabulary element, in the third line 
> of the table (since n suggests integers).

Changed to v

> As to notation, I prefer the standard notation for empty set
> instead of {} (this also appeared in earlier sections).

I prefer {}, as it uses the same notation as non-empty sets.

peter
Received on Friday, 17 January 2003 12:22:46 GMT

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