[Begin Ali0]
My comments interspersed throughout and in blue. As in this snippet.
[End Ali0]
On Fri, Jan 16, 2009 at 5:29 PM, Pat Hayes <phayes@xxxxxxx> wrote:
On Jan 16, 2009, at 11:01 AM, Ali Hashemi wrote:
Well, that sounds nice, but until you clarify what you mean, its impossible to evaluate it. What does it mean to link as opposed to combine ontologies? I take it that 'combine' means simply to take two sets of axioms and make them into one set. This is the (admittedly very basic and elementary) sense implied by the OWL 'import' primitive, where one ontology can 'combine another ontology into itself. I have never seen any more nuanced or subtle form of combination or 'linking' ever precisely defined.
[Begin Ali1]
Hehe. Ok, when I say links, I mean that there is a relation (not necessarily first order) that connects the two ontologies together. The ontologies can be equivalent, consistent, inconsistent, one contained in another, or disjoint. (Some of the preceding interact with one another... i.e. disjoint ontologies are trivially consistent). These are what i mean by links. So you can say that O1 is consistent with O2.
Or more precisely and interestingly, we might be able to say relations R1,1-R1,n from O1 are consistent with relation R2,1-R2,k in O2, and some other identified relations are inconsistent. Or variations of the types of links i mentioned passingly above.
[End Ali1]
So to take Pat H's illustration of the 3D vs 4D accounts, while they are in metaphysical opposition, they share patterns of thought, of logic. There _are_ (useful) similarities, just as there are differences.
There are indeed similarities. They are so similar in fact that it is hard for many people to even perceive the differences between them. Intuitively they are indistinguishable. But therein lies the problem: because our reasoning engines have no intuition, and formally they (the theories) are incompatible. Intuitive similarity means very little when it comes to ontologies. An analogy: consider two cars made by different manufacturers. They both have similar overall structure, similar shapes and very similar performance and behavior in the real world. But it would be a grave mistake to assume from this that they had interchangeable parts.
[Begin Ali2]
I'm not that interested in intuitive similarities. I'm interested in what the axioms say.
The point i'm trying to make is that I can prove representation theorems that shows the models of Time1 are isomorphic to a particular class of partial orders and I might be able to do the same for Time2, perhaps for a different class of partial orders..
Or a slightly weaker result, if I include a mapping axiom, then I might be able to show that timepoints in O1 and timepoints in O2 are consistent with classes A and B of partial orders respectively. Admittedly, it might not completely capture timepoint as specified in each ontologiy; however, it enables the next step:
If i've carefully set up my ontology repository (i.e. it has some properties), then consistencies and the ways in which i can use Time1 and Time2 together will fall out. Very broadly, this is what my semantic mapping algorithm does.
[End Ali2]
Moreover, they share a language, and describe the same phenomena. There are links between the two accounts.
In a survey talk a few years ago, I introduced the notion of the 'diamond of confusion'. Take two intelligent, competent people who both know logic, and tell them to use identically the same formalism (the bottom of the diamond) and to formalize the same sets of intuitive concepts, such as time and change and identity (the top of the diamond). They will not produce the same ontologies. There is every likelihood, based on actual experience so far, that they will produce logically incompatible ontologies (the two sides of the diamond). Similarity, or even identity, of the topic and of the formalism, is NOT a guarantee that ontologies will be in any useful way formally similar or even formally compatible.
[Begin Ali3]
Doesn't matter. If i have their axioms (at least first order expressive), given a good enough repository, I can tell you how they're consistent and inconsistent with one another.
[End Ali3]
They might not be exactly primitive in the sense that they are the "atoms" from which everything else is built. But they do reuse basic logical concepts.
But not the same metaphysical concepts. Continuants are simply impossible in a strict 4-d ontology, for example. The repository architecture I propose in my thesis uses this notion. It is similar to the idea of metaphor -- how is the solar system similar to an atom?
It would be great, wonderful, if ontologies could make use of metaphor-like relationships. There has been some preliminary work along these lines, based on a notion of metaphoric structural mappings developed at Northwestern University by Ken Forbus and his students. I think it is fair to say that this has not yet reached the ontology mainstream, though John Sowa may have something to say about this. We are mapping substructures from one concept to another. Is there something special about these patterns?
In what sense of special? (Psychological? Practical? Engineering? Logical? ...?)
[Begin Ali4]
In the practical sense.
I see a loop between interesting abstract patterns and patterns that match things in reality. Sort of how functions in math exist independently of a physical phenomenon, but may be used to fruitfully describe one.
[End Ali4]
{snip}
I disagree (Pat H). Both these accounts use orderings.
They all use different notions of order, though. As I said earlier, if you seek what is common to them all, what you get is a very minimal notion, that time-points are partially ordered. That alone is not enough for a useful temporal ontology.
[Begin Ali5]
Quite so.
But, I can organize all those different partial orders in such a way to facilitate the ontology alignment process (that's in effect what i'm doing - alignment) .
The algorithm uses a referent ontology (the repository) to first align ontologies to a commonly understood set of conceptual theories. It then provides the user with an alignment of the target ontologies as understood (if applicable) by the ontologies in the repository. The terminology i'm using is as defined by Choi et al (2006) - N, Choi, I-Y. Song, and H. Han, "A Survey of Ontology Mapping." ACM SIGMOD Record vol. 35, iss. 3 pp. 34 – 41, September 2006.
[End Ali5]
{snip}
when pat states:
[begin Pat Hayes2]
But my point is that there is only one entity, in fact. And it is important for an ontology to be able to say this, and clearly draw appropriate conclusions. If I am in a room and nobody else is in it, there is one person in the room. Not two people, one of them 4-d and one of them 3-d.
[end Pat Hayes2]
There is surely only one of Pat, but language affords (at least) two ways of describing him; of creating a coherent account of him.
Sure, of course. But my point was that PatC's suggested strategy of including all these alternatives into one ontology, yields an ontology which says that are two of Pat. This, which might be called the thousand-flower strategy, cannot distinguish between two things and two descriptions of one thing. That was my point.
[Begin Ali6]
I think here is where our terminology is obfuscated. I don't see a benefit in "combining" or "merging" these ontologies, since the resultant ontology would be inconsistent.
I do however see a point in figuring out in what ways the semantics of the two ontologies are equivalent, consistent etc. with one another.
This, if the ontologies are axiomatized, is achievable.
[End Ali6]
We would do well to heed some insights from Marshall McLuhan. "The medium is the message."
Not to put too fine a point on it: bullshit. Or as Korzybsky said, quite a few years before McLuhan: "The map is not the territory".
(As you can no doubt tell, I have very little respect for McLuhan, either as a person or as an intellectual guide. IMO, his writings are vaporware. He believed he was in direct communion with the Virgin Mary and was obsessed with the number three. But in any case, nothing he wrote has the remotest bearing on ontology engineering.)
[Begin Ali7]
Not one to mince your words. I respectfully disagree :D.
[End Ali7]
An ontology is the interplay of a number of media, not least of which is language - generally some formal logic.
This is nonsense. Logic is not a medium in McLuhan's sense, because it is not a language in his sense. Formal logics are not (yet) media for human or societal communication.
[Begin Ali8]
I'm not sure what to make of this. Formal logics seem unequivocally an information medium to me. They both carry content, and shape the way that content is recognized and represented. When did you last read McLuhan? :D
[End Ali8]
Ontologies rendered thusly reflect the expressivity and theories of the language.
First-order logic imposes the following conditions on the worlds it describes:
1. The world contains entities which can be conceptually distinguished from one another, and stand in relations to one another.
2. The world is not empty: it contains at least one entity.
And that is all it imposes. This is about the weakest set of conditions one could possibly imagine for an ontology language. It would be ridiculous to conclude that the use of FOL as an ontology language was subject to vague McLuhanesque critiques of imposing its own agenda on theories written in it.
[Begin Ali9]
No comment. :D
[End Ali9]
Possibly a mundane point, but they should also highlight the ways in which inconsistent theories may share similar structures. Indeed, if one were to construct a _model_ (in the logical sense), of pat using 3-d and 4-d views, there would be much in common between the two.
Not sure what you mean by _model_ here. I have to presume you don't mean as in model theory, so I'm guessing you actually mean something like 'logical ontology'. And they do have a lot in common in some sense, but not in any sense that I know how to make precise. Perhaps you can do better. [Begin Ali10]
I mean model in the Tarski sense. In the sense of generating an
interpretation for axoims and having extensions for relations and
functions etc. Taking a closer look at 3D and 4D ontologies, they generally seem to lack clear axioms, which makes it hard to illustrate what i mean.[End Ali10]
The interesting question is how they relate.
That has been worked out in some detail in this particular case. Not that a merge would simply render the resultant account incoherent. But in what ways are they similar and how to they differ?
****
{snip}
how serendipitous, my ontology design algorithm (chapter) partially addresses, and at the very least alleviates a lot of the problems here. it allows ontology designers to define a relation semantically, with little need to navigate the syntax and grammar of a particular formal logic.
To my mind, that is saying that you can define it semantically without semantics (since any precise semantics must be attached to a formal grammar of some kind, if only a very 'abstract' one.)
[Begin Ali11]
They are being defined not syntactically but model theoretically. Details are too cumbersome to go into in this email. I have a chapter devoted to this.
[End Ali11]
***
[begin P hayes3 ]They are all "atomic". (Well, virtually all: all the 'natural kind' concepts, as opposed to artificial combinations such as "French women between the ages of 25 and 50".) Just look at a first-order theory: all the names in it are on an equal footing; none are more 'primitive' than another, and they have no internal conceptual structure which would permit their being decomposed into something simpler or more elemental.
...
But surely the time catalog is a clear case of this non-uniqueness. I fail to understand how you cannot see this, it seems so obvious. What would you reduce all of it to?
[end P hayes3]
I would reduce it to theories it is reusing. In the case of time, orderings.
I suggest that you actually read the catalog before proceeding. There are some re-use cases in there, where one theory extends or builds on another. But there are also a number of clear incompatibilities between different alternative building blocks. To take the first: are times dense or not? Some time-orderings assume a discrete ordering in which each time has a unique 'next' time. These correspond to the time "theories" incorporated into for example the UNIX operating system, where all times and dates are defined as a number of milliseconds since the start of 1970. Other temporal ontologies require time to be continuous, or at least dense. These are incompatible: in fact, one gets an early version of the first by explicitly denying a basic axiom of the second. Others still use other, more complex, order structures. These are not re-uses of one theory: they are different, incompatible, basic theories which have ramifications throughout the subsequent development of more complex theories. For example, some of them make the familiar algebra of 13 interval relations incoherent, others reduce it to fewer relations, and others still allow 16 relations (if time is allowed to be circular).
[Begin Ali12]
Hehe funny you should say that, since your catalog was partial inspiration for the identification of the problem and the resultant repository architecture :D.
I am agnostic as to what time is.
I am very interested in what theories of math descriptions of time use. Some are dense, some are discrete, some branch, some are intervals. They are not always necessarily consistent. That's fine. I can collect them in a repository and show exactly how they are consistent and inconsistent.
I only care to know how the different concepts of time are related to one another and which parts they reuse. I'm not trying to collect them in one single ontology. I'm trying to collect them in an organized repository that will maximize reuse :D. (and let me have my two algorithms...)
[End Ali12]
That sounds interesting. Please don't take this the wrong way, but it sounds almost too interesting to be a master's thesis. But I look forward to reading it. This idea isn't an attack on UO's, but rather a complement. The different views, and biases as captured by UO's are valuable. It's simply that they are not exclusively "true". The repository described in my thesis unifies these within the language of _expression_.
It also consequently enables to pretty useful algorithms. One for semantic mapping and another for ontology design.
The semantic mapping algorithm uses a centralized repository to conduct ontology alignment between n or more ontologies.
The ontology design algorithm intelligently navigates the repository to match an agent's intended semantics to sets of axioms.
Sorry for the tease, but next week my thesis will be bound and in the libraries, so i'll be able to link the document then.
I look forward to seeing it. Until then, let us all hope you are onto something. I have to say, anyone who can bring together Common Logic and Marshall McLuhan and make something meaningful from this collision will have my respect. :-)
PatH
[Begin Ali13]
Lol, the Marshall McLuhan aspects are, for practical reasons, curtailed to focus on the (hopefully) interesting engineering results.
[End Ali13]
(PS if someone has an interesting project, i'd likely be interested now that my Master's is complete ;) .)
Ali Hashemi
MASc (waiting to officially convocate)
Semantic Technologies Laboratory
University of Toronto
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