Grammar 1
Grammar 1 is something of a misnomer. It is nowhere near exemplifying any kind of a standard form and it’s put forth only as a starting point for the initiation of more respectable grammars. Such as it is, it uses the terminal alphabet 
The last two rules of Grammar 1 dictate the following typings.
- The concept of a sentence in
covers any concatenation of sentences in
that is, any finite number of freely chosen sentences available to be concatenated one after another.
- The concept of a sentence in
continuing with a sentence, possibly empty, following with a finite number of phrases of the form
and closing with a
The above appears to be just about the most concise description of the cactus language 
- The invocation of the kleene star operation is not reduced to a manifestly finitary form.
- The type
indicative of a sentence is allowed to cover not only itself but also the empty string.
We’ll discuss those issues at first in general, and especially in regard to how the two features interact with one another, and then we’ll return to address in further detail the questions they engender on their individual bases.
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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 
Depending on the context, 
![\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.
more briefly as a Parce on 
is a sentence.
is not a sentence.
Inquiry Into Inquiry 