Grammar 1 (cont.)

The degree of intermediate organization in a grammar is measured by the number of its intermediate symbols and the complexity of their mutual interplay within the frame of the grammar’s productions.

Grammar 1 has no intermediate symbols at all, and so remains at a trivial degree of intermediate organization.  Some additions to the list of intermediate symbols are practically obligatory in order to arrive at any reasonable grammar at all.  Other inclusions have a more optional character, though useful from the standpoints of clarity and ease of comprehension.

One of the troubles perceived to affect Grammar 1 is the way it wastes so much of the available potential for efficient description in recounting over and over again the simple fact that the empty string is present in the language.  The problem arises partly from the covering relation which has the following implications.

There is nothing wrong with the more expansive pan of the covered equation, since it follows straightforwardly from the definition of the kleene star operation.  But the statement is not a very productive piece of information, in the sense of telling us much about the language falling under the type of a sentence

In particular, implies   Since for any formal language the empty string is counted over and over in every term of the union, and every non‑empty sentence under appears again and again in every term of the union following the initial appearance of   As a result, the overall style of characterization has to be classified as true but not very informative.

If at all possible, one prefers to partition the language of interest into a disjoint union of subsets, thereby accounting for each sentence under its proper term, and one whose place under the sum serves as a useful parameter of its character or its complexity.  Such an ideal form of description is not always possible to achieve but it is usually worth the trouble to actualize it whenever one can.

Suppose one tries to deal with the problem by eliminating each use of the kleene star operation, by reducing it to a purely finitary set of steps, or by finding another way to cover the sublanguage it is used to generate.  That amounts, in effect, to recognizing a type, a complex process involving the following steps.

• Noticing a category of strings which is generated by iteration or recursion.
• Acknowledging the fact that it needs to be covered by a non‑terminal symbol.
• Making a note of it by instituting an explicitly‑named grammatical category.

In sum, one introduces a non‑terminal symbol for each type of sentence and each part of speech or sentential component generated by means of iteration or recursion under the ruling constraints of the grammar.  To do that one needs to analyze the iteration of each grammatical operation in a way which is analogous to a mathematically inductive definition but further in a way which is not forced explicitly to recognize a distinct and separate type of expression merely to account for and recount every increment in the parameter of iteration.

Resources

• Cactus Language • Syntax
• Survey of Animated Logical Graphs
• Survey of Theme One Program

cc: Academia.edu • BlueSky • Laws of Form • Ma^thstodon • Research Gate
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science