Eigenclass model: Difference between revisions

Revision as of 19:25, 9 November 2012

Revision as of 19:15, 15 November 2012

Line 385:

<i>member-of-instance-of</i> relation with the

<i>instance-of-instance-of</i> relation.

=== The primary structure ===

The restriction of a canonical structure of object membership

to primary objects can be expressed as a structure

<span style="white-space:nowrap"><i>(<u>O</u>.c</i>, &#1013;, &le;<i>)</i></span>

where

<ul>

<li>

<i><u>O</u>.c</i> is the set of (primary) objects,

<li>

&#1013; is the <em>instance-of</em> relation between (primary) objects,

<li>

&le; is the <em>inheritance</em> relation between (primary) objects.

</ul>

<em>Helix classes</em> are (primary) objects <i>x</i>

such that <i>x</i> &#1013; <i>x</i>.

<em>Classes</em> are (primary) inheritance descendants of helix classes.

<em>Terminals</em> are the remaining (primary) objects.

<em>Metaclasses</em> are the [[Upper and lower bounds|lower bounds]] of helix classes, i.e. they are objects <i>x</i> such that

all helix classes are among ancestors of <i>x</i>.

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The structure is subject to the following axioms:

<ol style="list-style-type:none; margin-left: 2ex;">

<li style="text-indent:-3ex; margin-left:3ex;">(p~1)

(&#1013;) &#9675; (&le;) = (&#1013;) = (&le;) &#9675; (&#1013;).

<li>(p~2) Inheritance &le; is a partial order.

<li>(p~3)

Terminals have neither strict descendants nor instances.

<li style="text-indent:-3ex; margin-left:3ex;">(p~4)

Helix classes are totally ordered by inheritance, and at least two in number.

<li>(p~5) Metaclasses cannot have terminal instances.

<li>(p~6) Every object <i>x</i> has a least (primary) container,

<i>x.class</i>.

<li>(p~7) Cycles in &#1013; only occur between helix classes.

<li>(p~8) The set of primary objects is finite.

</ol>

=== Eigenclass completion ===

Axiom [[#Axioms|(5)]] can be considered in the following stronger form

since it is satisfied in all mentioned programming languages (and presumably all relevant ones in general):

Line 398 ⟶ 446:

</ol>

Assuming (5'), the object membership structure,

<span style="white-space:nowrap"><i>(<u>O</u></i>, &#1013;<i>)</i></span>,

is uniquely determined by the inheritanceprimary andstructure, instance-of relations

between primary objects, up to isomorphism

<span style="white-space:nowrap"><i>(<u>O</u>.c</i>, &#1013;, &le;<i>)</i></span>,

(so that the structure is also determined by inheritance and the class map).

up to isomorphism.

(Therefore,

<span style="white-space:nowrap"><i>(<u>O</u></i>, &#1013;<i>)</i></span> is also determined by

<span style="white-space:nowrap"><i>(<u>O</u>.c</i>, <i>.class</i>, &le;<i>)</i></span>.)

Given a structure