graph representation of knowledge

Saturday, October 16, 2004

Algorithm development using Orbs

Comment from the Founding Committee

The Adi letter semantics has a three-decade-old development period and is associated with a 1986 patent on organizing data by collapsing categories into representations using a single letter.

(See previous note from Tom Adi at [62])

(additional extended comments by Prueitt [66])

The Readware Framework of Knowledge

By Tom Adi, October 14, 2004

Footnotes by Founding Committee

1. Introduction

In 1985, Adi developed a theory of the Arabic language by looking at an old Arabic book and asking a single question thousands of times: "Do the many contexts in which a word is used point to an invariant abstract parallel between the word structure and the structure of the physical context to which the word refers?"

In Arabic, there is one-to-one correspondence between letters and sounds. Arabic has 28 letters and four vowels. Most Arabic word stems are verbs and most of them consist of three consonants. Vowels are added to create words out of stems. Vowels do not change the basic meaning.

First, Adi examined the contextual usage of prepositions, pronouns and articles that consisted of a single consonant and a vowel. Then he turned to stems that had two soft consonants (ya, hamza, waw, ha). Soft consonants are related to vowels and are known to add much less meaning to stems than other consonants. Then, he examined stems with one soft consonant, and finally stems without soft consonants. Adi started by examining stems that denote simple processes that are easy to visualize. The Readware framework of knowledge is based on the results of Adi's 1983 analyses.

2: The abstract concept of a framework

The notion of a framework is central to at least three scholar’s works, Zachman, Sowa and Ballard. An Internet reference to these scholar’s work is given by Prueitt [1]

Formal definition of the abstract framework: Let S₍₁₎, S₍₂₎, , … , S_(n) be n framework aspect sets. These sets may have any allowable member, and are considered here to be finite and in fact having only a few members. The dimensions of the framework match the integer that is the number of framework aspect sets. The size of the framework is the integer that is the product of the size of all framework aspects sets.

F = S₍₁₎ x S₍₂₎ x … x S_(n)

An abstract framework is the cross product of the set of framework aspect sets.

Example, Let A, B, C be three sets, with the size(A) = m, size(B) = n, and the size(C) = o. Then the size(F) = m*n*o, when

F = A x B x C

If A = (aspect1, aspect2) and B = (aspect3, aspect4, aspect5), then the size(F) = 6.

F is often represented as a matrix, where the cells contain the elements of the cross product. In this case we have a 2x3 Matrix

F = [ f(i,j) | I = 1,2 and j = 1,2,3 ]

The cell f(1,1) contains the ordered pair (aspect1, aspect5), and so forth.

The dimensions of a framework are selected by a human mind as a coherent domain of inquiry with a specific set of descriptive elements like words or phrases. These should be selected as if the axioms of the foundational elements of a specific geometry, i.e., each aspect of the dimensional description should be as independent as possible from the other aspects. The set of aspects should “cover” a coherent domain of inquiry. The set should be minimal in number, only having a sufficient number of elements to serve the purpose of description and conversation about the domain. The independence and the sufficiency constraints will create a situation where each element has the same “status”, in terms of importance. Each is important and necessary.

Coherence is related to completeness and consistency. This is a deep discussion with some simple consequences. The process of separating the dimensions of an inquiry is quite natural and occurring in all children’s minds as they make sense of the world. Coherence is not possible across all of the experiences that we humans have. So the selection of a coherent figure in a figure ground relationship in the perception of reality is natural. We see this in the development of the Adi Notational System as applied to the structural ontology related to the Arabic language. For example, the first dimension is a cross product itself of two sets that must lie at the base of any organization of data into a structured ontology. These two underlying sets are P and G:

P = {closed, open}

G = {self, engaged}

G is the set of elementary engagements and P the set of elementary polarities. In the next section, the reader will see a separation of the four elements of the cross product of P and G. This indicates a cross-scale transformation operating in which at first a bag [2] is created.

The bag contains the elements of the cross product, and then this bag is ordered in a different way, i.e. having now only one dimension rather than two. The cross scale transform imposes an order. The transform also imposes a categorical collapse of elements in the bag into a partition [3]of the bag’s elements. These processes are formal constructions that create a specific ordering of a specific set of elements. These elements are then laid down into a new framework aspect set, following the principles of categoricalAbstraction (Prueitt, 2001).

By making this cross scale transformation, a measurement of the elements occurring. By taking the cross scale transform of P x G Adi discovered the dimension of the elementary frame types underlying the use and practice of the Arabic language.

As defined below, let S be the set of elementary essences:

S = {element, domain, order}

Adi discovered that the power set of S is the elements of a dimension of the realization of the four elementary frame types.

2. The Readware Substructural Ontology

Let F = { f(i) | i = 1 to 4 } be the set of elementary frame types. For convenience we may write f1 for f(1), f2 for f(2), f3 for f(3) and f4 for f(4). We have

f1 = (closed)

f2 = (open)

f3 = (engaged)

f4 = (separate)

Define a frame as a unity, indicated by "(-)", filled with a list -, so (a b c) is a frame.

Define list((-)) = - , where "(-)" is a frame, so list((a b c)) = a b c.

Define list(X) = list of elements of X, where X is a set, so list({a, b, c}) = a b c.

Define frame(-) = (-), where "-" is a list, so frame(a b c) = (a b c).

Define set(-) = {elements of the list - }, so set(a b c) = {a, b, c}.

Let P =

{ p(i) | i = 1, 2, 3 } = {assignment, manifestation, containment}

be the set of elementary process types. For convenience we may write p1 for p(1), p2 for p(2) and p3 for p(3).

We enumerate the power set of P

P* = {s(i) | i = 1 to 8}

= { null set,

{p1}, {p2}, {p3},

{p1, p2}, {p1, p3}, {p2, p3} ,

{p1, p2, p3} }

Remember that size(P*) = 8.

We now define the "puts" operator on F x P*

puts(s(i), f(j)) = frame( list( set( list( f(j) ) ) union s(i) ) )

which "puts" up to three elementary process types in one elementary frame type. We call the resulting frame a substructural ontological frame type.

The ontology-generating function B is defined from F x P* to the 4x8 matrix Q

Q = [ q(i, j) | i = 1 to 4 and j = 1 to 8 ]

where q(i, j) = puts(s(i), f(j))

	(closed)	(open)	(engaged)	(separate)
	(closed assignment)	(open assignment)	(engaged assignment)	(separate assignment)
Q = [	(closed manifestation)	(open manifestation)	(engaged manifestation)	(separate manifestation)	]
	(closed containment)	(open containment)	(engaged containment)	(separate containment)
	(closed assignment manifestation)	(open assignment manifestation)	(engaged assignment manifestation)	(separate assignment manifestation)
	(closed assignment containment)	(open assignment containment)	(engaged assignment containment)	(separate assignment containment)
	(closed manifestation containment)	(open manifestation containment)	(engaged manifestation containment)	(separate manifestation containment)
	(closed assignment manifestation containment)	(open assignment manifestation containment)	(engaged assignment manifestation containment)	(separate assignment manifestation containment)

The substructural ontology Q contains all possible substructural ontological frame types. We find that if we arrange the 4 vowels and 28 consonants of the Arabic language in a certain order in a 4x8 matrix A

A = [ a(i, j) | i = 1 to 8 and j = 1 to 4 ]

	vowel_i	vowel_a	vowel_u	sukoon
	ya	hamza	waw	ha
A = [	meem	fa	dal	thal	]
	'ain	noon	qaf	ghain
	ra	lam	ba	ta
	seen	zay	ssad	tha
	kaf	ddad	tta	kha
	hha	sheen	geem	zza

We conjecture that an ontological correspondence exists between A and Q.

There is a one-to-one ontological correspondence between the vowels and consonants of the Arabic language and the frame types of our substructural ontology.

3. Consonants as Substructural Ontological Frame Types

Let us first look at the consonants of the second row of A (ya, hamza, waw, ha) which correspond to the ontological assignment process in different frames (second row of Q).

The Arabic conjunctive "wa" means "and," "while" or "I swear an oath by..." "wa" is spelled as the single consonant "waw" which corresponds to the ontological element q(2, 3) "engaged assignment." This seems fitting for all three meanings. "And" engages (connects) two things. The "while" sense is as in "eating while talking" which expresses engagement into something or connects two events. An oath is an engagement (a commitment), too.

We find the consonant "ha" (pronounced like "h") at the beginning of all third person pronouns "hoowa" (he, spelled ha waw) "heeya" (she, spelled ha ya) etc. This seems fitting, too, as "ha" corresponds to q(2, 4) "separate assignment" and the third person is basically a separate entity.

The "hamza" is a question article similar to "is this...?" and it happens to correspond to q(2, 2) "open assignment." Asking a question is equivalent to saying that the matter is open.

The address article "ya" (spelled ya and a stretched vowel_a) is used to tell someone that the message is "for you only." It is a "closed assignment" (q(2, 1)).

4. Object View of Processes and Facets of Frame Types

We find that each elementary process type has an object view (Table 1). Object views propagate to every aspect of the substructural ontology and the replacement of "process" with "object" creates new types of substructural frame types in the rows 5 to 8 of Q.

Process Type	Object View
assignment	element
manifestation	domain
containment	order

Table 1. Object views of elementary process types

We also find that frame types ((closed), (open), (engaged), (separate)) have additional facets. Table 2 lists the frame type as the first facet. Similar facets are grouped together. In a meaningful abstract sense, "(closed)" has the facets "(defined)", "(backward)" etc.

Facet 1	Facet 2	Facet 3	Facet 4	. . .
closed	defined, positive	backward, inward,	returning, recursive
open	undefined, negative	forward outward	propagating procursive
engaged	valid, complete	dual, multiple	universal
separate	invalid, non-thing, empty	third, new	unusual, singular

Table 2. Facets of elementary frame types

Consider the third row of the substructural ontology Q and the corresponding third row of A (the Arabic consonants meem, fa, dal, thal). We have here the manifestation process and its object view "domains" as well as the different facets of the frame types.

The consonant "meem" (q(3, 1)) is used as a prefix in Arabic words to denote "defined domain" (place of action) and "defined manifestation" (item of action, manner of action, actor, instrument of action). The word "ma" (spelled meem stretched vowel_a) means "not" (closed manifestation) as well as "what" (defined manifestation).

The conjunctive "fa" (spelled fa) means "then" as in "first this, then that." This constitutes a "forward manifestation" (q(3, 2)). "foo" (spelled fa waw) means "mouth" (open domain).

The consonant "thal" (q(3, 4)) is used to express a separate manifestation (special attribute, this, that, the one who, that which, etc.).

To be continued. ….

[2] A bag is like a set except duplicates are not collapsed into a single element, i.e. the bag [ 3,3 ] has size 2. The elements of the bag have no categorical collapses, nor any notion of order.

[3] In this case the categories are each having only one member, but in other cases the formation of categories will organize the lower set in a cross scale transform into a small number of categories