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, 
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 
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 
In particular, 
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.
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coming with the territory of the cactus language 
it specifies 
as listed in the following display.
covers any concatenation of sentences in 
continuing with a sentence, possibly empty, following with a finite number of phrases of the form 
and closing with a 
one can imagine but there are a couple of problems commonly felt to afflict its style of presentation and to make it less than completely acceptable. Briefly stated, the problems turn on the following properties of the formulation.
and 
where the symbols 
may be taken to signify either the tokens or the subtypes of their covering types.
as the augmented alphabet of the candidate formal grammar for 
as the augmented strings of the grammar for 
from all other cases of augmented strings. In those situations the strings in the disjoint union 
are known as the sentential forms of the associated grammar.
where 
and 
are augmented strings or sentential forms of specified types which depend on the style of the grammar being sought, are variously known as characterizations, covering rules, productions, rewrite rules, subsumptions, transformations, or typing rules. Statements of that form are collected together in a set 
has the converse form 
which may be read as 
is covered by 
![\begin{array}{l} S ~\mathrm{covers}~ T \\[2pt] S ~\mathrm{governs}~ T \\[2pt] S ~\mathrm{rules}~ T \\[2pt] S ~\mathrm{subsumes}~ T \\[2pt] S ~\mathrm{types~over}~ T \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Bl%7D++S+%7E%5Cmathrm%7Bcovers%7D%7E+T++%5C%5C%5B2pt%5D++S+%7E%5Cmathrm%7Bgoverns%7D%7E+T++%5C%5C%5B2pt%5D++S+%7E%5Cmathrm%7Brules%7D%7E+T++%5C%5C%5B2pt%5D++S+%7E%5Cmathrm%7Bsubsumes%7D%7E+T++%5C%5C%5B2pt%5D++S+%7E%5Cmathrm%7Btypes%7Eover%7D%7E+T++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=0&c=20201002)
The letter 
signifying other types of strings, will be found necessary for a reasonable account or rational reconstruction of the sentences in 
whose sole member is the initial symbol with the set 
of intermediate symbols results in the set 
of non‑terminal symbols. Even though 
as a typical variable ranging over the full set of non‑terminal symbols, 
of the language 
is the formal language 
defined as follows.
and 
for all 
as a member of the cactus language 
a number of frequently occurring sublanguages and commonly invoked operations on their expressions can now be given succinct definition.
A bare Parce is a sentence in the bare cactus language, variously notated as follows.
A bare cactus expression is commonly encountered in practice when one has occasion to start with an arbitrary Parce and then finds reason to delete or erase all its paints.
Inquiry Into Inquiry 