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Re: [ontolog-forum] Ontologies and Algebraic Specifications

To: "Horning, Jim" <Jim.Horning@xxxxxxxxxx>
Cc: ontolog-forum@xxxxxxxxxxxxxxxx
From: Pat Hayes <phayes@xxxxxxx>
Date: Tue, 27 Feb 2007 15:57:25 -0600
Message-id: <p06230908c20a51899b8c@[]>
>Thanks, that clarifies a lot.  Now let me see how I can do with the
>"much, much greater than" message format that this list prefers.  :-)
>Jim H.
>>  -----Original Message-----
>>  From: Pat Hayes [mailto:phayes@xxxxxxx]
>>  Sent: Tuesday, February 27, 2007 7:37 AM
>>  To: Horning, Jim
>>  Cc: ontolog-forum@xxxxxxxxxxxxxxxx
>>  Subject: Re: Ontologies and Algebraic Specifications
>>  >- *traits* [modules/units/components] that are
>>  >the concrete elements of discourse.  The body of
>>  >a trait may refer to other traits and include
>>  >*declarations*, *axioms*, and *consequences*.
>>  >Axioms are mostly, but not exclusively,
>>  >equations.
>>  That seems very odd to me, speaking from the ONT
>>  side. There are many things one wants to say that
>>  just can't be put into equational form.
>Yes, there are other things, and LSL has other
>forms of axioms for those things that we found
>to be important in specifying abstract data types
>and functions.    (01)

OK, but ontologies can be about almost any topic under the sun. The 
standard OWL tutorial uses an ontology of wines; one of the largest 
deployed ontologies is about oncology, others are about atmospheric 
phenomena, protein folding, processes involving bulk liquids, power 
plant maintenance, Tibetan weaving, events involving on-line 
purchases, etc..    (02)

>I would like to know what other things it is
>important to say in ontologies, and what forms
>of axioms are best suited to saying them.    (03)

See above. (Actually, abstract data types and functions seem to play 
almost no part in ontology composition.) There is a lot of debate on 
the latter point. Obviously (?) if you can make do with a limited 
range of axiom forms, there is more hope for efficient reasoners. 
OWL-DL is an example of a claimed 'sweet spot' between expressivity 
and tractability    (04)

>(LSL treats universally-quantified first-order
>logical expressions as a special case of equations.)    (05)

? I fail to see how most axioms can be cast as equations. Take for 
example these, from a translation of OWL into CL:    (06)

(forall (p x (n number))(iff
   ((ATLEAST n p) x)
   (exists ((m number)) (and
     (lesseq n m)
     (exists m (y)(p x y))
))    (07)

(forall (p c z)(iff ((ALLARE p c) z)(forall (u)(if (p z u)(c u)) ))    (08)

(forall (p c z)(iff ((SOMEARE p c) z)(exists (u)(and (p z u)(c u)) ))    (09)

>>  >- *units* [modules/units/component] that are the
>>  >concrete elements of discourse.  The body of a
>>  >unit may *import* from, and *export* to, other
>>  >units and will include *type declarations* and
>>  >possibly some *constraints*.
>>  Hmm. Im not sure what you mean by a 'unit' here.
>>  (Common Logic has a 'module' construction, but
>>  that has a special purpose of limiting the
>>  'local' universe of discourse.)
>All I was trying to say is that I believe most
>ontologies are not written indivisibly as one
>[block of text/diagram/formula/file/page], but
>are written and presented in chunks (which might
>be called different things in different ONT
>languages).    (010)

Well, actually no. Most of the ontology languages I know are 
essentially just textual, or maybe thought of as an RDF triple set or 
whatever; but essentially 'flat' at the top level. OWL for example is 
just a set of sentences, so you can cut and paste freely at the top 
level. There is no local scoping or modularity between the published 
document and the sentences. Its not anything remotely like Java.    (011)

>  This subdivision might or might
>not have semantic significance in a particular
>ONT language (e.g., scoping of names and
>quantifiers).  These chunks have to be named
>or identified somehow in other chunks; their
>relations constitute a "gross anatomy" of
>the presentation.    (012)

Well, yes, importing requires that pieces of OWL (or whatever) are 
given names or identifiers, but that really isn't much beyond giving 
a URI to an XML resource.    (013)

>  > This seems to miss the central point, which is
>>  that an ontology is, at base, a theory; that is,
>>  a collection of sentences. Plus, maybe, some
>  > bells and whistles, but in many cases even those
>>  are re-understood as kinds of sentence (e.g.
>>  'imports'). Some languages have types which may
>>  in some cases require declarations, but these are
>>  not an essential part of the mix.
>No, IMHO this is one of the points that I have
>not missed.  However, I fear, some in this
>community have overlooked it or gotten it wrong:
>The presentation is not the same thing as what the
>presentation denotes.    (014)

True, but I fail to see how that is relevant here. The ontology IS a 
set (bag, document containing, graph structure encoding, whatever) of 
sentences, which denote truthvalues.    (015)

>  What I write in an ONT
>language is not the same as the associated theory.    (016)

Depends how you understand 'theory'. If that means a set of sentence, 
an ontology is a theory. If it means a logically closed set of 
sentences, then the closure of the ontology is a theory.    (017)

>I don't care which one you call "the ontology,"
>but you also need to have a name for the other one,
>and any reader needs to know which you mean where.
>>  Importing is a
>>  Web idea, and there is no such thing as
>>  "exporting" (AFAIK).
>My mistake.  I thought the discussion I had seen
>here of import and export related to ontologies.
>E.g., John F. Sowa "I also added two special
>kinds of statements:  'Import' for those types
>that are specified elsewhere (I won't say
>'defined'), and 'Export' for those types
>specified in this file that may be used elsewhere."    (018)

Well, that is John's idea, but I havn't seen it given flesh in any 
working system. And the whole idea of a Web architecture is inimical 
to the notion of 'exporting'. "Publishing" would be better.    (019)

>If "ontology" means theory, then I agree that
>it is not very useful to import and export theories,
>and that import and export statements probably
>don't make sense as part of theories.    (020)

Importing does, it allows you to include another theory in your own. 
At the very least, that can save a lot of typing.    (021)

>  That does
>not mean that these constructs are not valuable
>constructs in ONT languages, with or without the
>Web.  Without them, the languages just won't scale.    (022)

Importing is useful. I fail to see what the point would be of 
exporting. Maybe I just don't fully understand what y'all mean by 
this.    (023)

>>  Also, one of the most central and salient aspects
>>  of an ONT language is how it handles quantifiers,
>>  and what it can quantify over. You don't mention
>>  this anywhere, which I also find odd. Does Larch
>>  have quantifiers? (It is possible to treat
>>  quantification in terms of implicit universal
>>  quantification of variables and handling
>>  existentials by the deft use of functions,
>>  avoiding any need for an explicit quantifier
>>  scoping mechanism in the syntax. I'll make a
>>  guess that this is how Larch does it also.)
>Absolutely correct.  This caused relatively few
>problems in our domains.  How bad does it make
>things for ontologies?    (024)

Well, this only works for certain quantifier patterns. It may be that 
these will suffice for a lot of cases. I'm happier with a more 
general language to work with, myself, with explicit quantification. 
I find that Skolem function names often get very confusing.    (025)

>>  >- *types* [classes/kinds/classes].
>>  Be careful of terminology. "types" often is
>>  understood to imply that type membership is
>>  syntactically checkable or used to determine
>>  wellformedness. This is quite unusual in ontology
>>  languages. If you really do just mean classes,
>>  then yes, almost all ontology languages have this.
>I was trying to indicate that I realized that there
>were various notions of type and class and that each
>ONT language probably picked just one of them.    (026)

OK, fair enough. And yes, they tend to do that.    (027)

Pat    (028)

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