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From: scottw@advsysres.com  (Scott A. Whitmire)
Subject: Re: mathematics (algebra?) of object orientation
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References: <49c627$bgn@sparcserver.lrz-muenchen.de> <49d23s$6bk@Starbase.NeoSoft.COM> <MUSTO.95Nov28125302@hermes.informatik.unibw-muenchen.de>
Date: Tue, 5 Dec 1995 04:37:41 GMT
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Xref: glinda.oz.cs.cmu.edu comp.object:42054 comp.object.logic:608 comp.theory:14841 sci.math:127470

In <MUSTO.95Nov28125302@hermes.informatik.unibw-muenchen.de>, musto@informatik.unibw-muenchen.de (Alexandra Musto) writes:
>In article <49d23s$6bk@Starbase.NeoSoft.COM> timd@Starbase.NeoSoft.COM (Zombie) writes:
>
>>Helmut Richter <Helmut.Richter@lrz-muenchen.de> wrote:
>>>[Crossposted, follow-ups directed to group comp.object]
>>>
>
>>>Now when I'm trying to translate things back, I
>>>get soon stuck because the similarity to mathematical objects is
>>>limited. Especially the concept of "class" is hard to express in
>>>mathematical terms; [...]
>>
>>I'm sure there is a lot of theory out there regarding OO systems.
>>
>>Let's start with this one:  A class is a set of entities.  A sub-
>>class is a subset.  Where do you want to go from here?
>
>Actually, math. theory is rather poorly developed for
>OO systems.
>
>E.g. if you consider a class a set, which is defined
>as the product of the domains of the attributes
>(e.g. person = name x date_of_birth), then the set
>defined by a subclass that has more attributes
>(e.g employee (subset of person) = name x date_of_birth x employer)
>is bigger than the first. How can it be a subset?
>

Well, no, that's not quite right. A class is not the product of the domains
of its attributes, it is merely the set of attributes. Its ordinal is the number
of attributes it contains. The product of the attribute domains is actually
the universe of potential objects. The attributes of a class have no values,
only types. The attributes take on values (one each) when you create an
object of the class.

A subclass is not a subset of its superset. In fact, it's the other way around.
The attributes of the superclass form a subset of the subclass.

One other thing: there are several models of classes and objects, but no
theories. None of the models are general enough to be extended to a true
theory. I have a feeling that a theory of objects will be a generalization of a
blending of several of the existing models. I'll have more on this in a few weeks.

Scott A. Whitmire             scottw@advsysres.com
Advanced Systems Research     
25238 127th Avenue SE         tel:(206)631-7868
Kent Washington 98031         fax:(206)630-2238

Consultants in object-oriented development and software metrics.

