The present patch of discussion is concerned with describing a family of formal languages whose typical representative is the painted cactus language 
The standard devices of formal grammars and formal language theory are adequate to describe the language of interest from an external point of view but the ultimate desire is for something more exacting, to turn the tables on the order of description and enter on a process of eversion which evolves to the point of asking: To what extent can the language capture the essential features and laws of its own grammar and describe the active principles of its own generation? In other words: How well can the language be described by using the language itself to do so?
To address the above questions, we have to express what a grammar says about a language in terms of what a language can say on its own. In effect, it is necessary to analyze the kinds of meaningful statements grammars are capable of making about languages in general and to relate them to the kinds of meaningful statements the sentences of the cactus language might be interpreted as making about the same topics. So far in the present discussion, the sentences of the cactus language make no meaningful statements at all, much less any meaningful statements about languages and their constitutions. As of yet, those sentences subsist in the form of purely abstract, formal, and uninterpreted combinatorial constructions.
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is an ordered pair 
of sentential forms in 
by writing 
immediately derives 
by saying that 
is immediately derived from 
and also by writing the following form.
of sentential forms over 
of sentential forms in the sequence is an immediate derivation in 
derives 
in 
generated by the formal grammar 
is the set of strings over the terminal alphabet 
derivable from the initial symbol 
by way of the intermediate symbols in 
according to the characterizations in 
All that is summed up in the following form.
is a finite set of characterizations, variously described as covering rules, formations, productions, rewrite rules, subsumptions, transformations, or typing rules.
form the augmented alphabet of 
are the non‑terminal symbols of 
are the non‑initial symbols of 
are the augmented strings for 
are the sentential forms for 
taking the following form.![\begin{matrix} S_1 & = & Q_1 \cdot q \cdot Q_2 \\[8pt] S_2 & = & Q_1 \cdot W \cdot Q_2 \end{matrix}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Bmatrix%7D++S_1+%26+%3D+%26+Q_1+%5Ccdot+q+%5Ccdot+Q_2++%5C%5C%5B8pt%5D++S_2+%26+%3D+%26+Q_1+%5Ccdot+W+%5Ccdot+Q_2++%5Cend%7Bmatrix%7D&bg=ffffff&fg=000000&s=0&c=20201002)
and 
are augmented strings for 
is a non‑terminal symbol, in other words, 
The elements 
and 
The language 
is then said to be governed, licensed, or regulated by the grammar 
in the following forms.![\begin{matrix} S_1 & :> & S_2 \\[8pt] Q_1 \cdot q \cdot Q_2 & :> & Q_1 \cdot W \cdot Q_2 \end{matrix}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Bmatrix%7D++S_1+%26+%3A%3E+%26+S_2++%5C%5C%5B8pt%5D++Q_1+%5Ccdot+q+%5Ccdot+Q_2+%26+%3A%3E+%26+Q_1+%5Ccdot+W+%5Ccdot+Q_2++%5Cend%7Bmatrix%7D&bg=ffffff&fg=000000&s=0&c=20201002)
amounts to a grammatical license to transform a string of the form 
into a string of the form 
in effect, replacing the non‑terminal symbol 
In that application the notation 
is a finite set of intermediate symbols, all distinct from 
is a finite set of terminal symbols, also known as the alphabet of 
Depending on the particular conception of the language 
with the set 
shows how an initially effective and succinct definition of a formal language can be terse to the point of forcing its interpreters to spend exorbitant amounts of time developing its consequences, but that it can be converted to a form more efficient from the operational point of view, however ungainly in regard to its elegance.
in which all of the covering productions have one of the following two forms.![\begin{array}{ccll} S & :> & \varepsilon & \\[6pt] q & :> & W, & \text{with}~ q \in \{ ``S" \} \cup \mathfrak{Q} ~\text{and}~ W \in (\mathfrak{Q} \cup \mathfrak{A})^+ \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Bccll%7D++S+%26+%3A%3E+%26+%5Cvarepsilon+%26++%5C%5C%5B6pt%5D++q+%26+%3A%3E+%26+W%2C+%26+%5Ctext%7Bwith%7D%7E+q+%5Cin+%5C%7B+%60%60S%22+%5C%7D+%5Ccup+%5Cmathfrak%7BQ%7D+%7E%5Ctext%7Band%7D%7E+W+%5Cin+%28%5Cmathfrak%7BQ%7D+%5Ccup+%5Cmathfrak%7BA%7D%29%5E%2B++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=0&c=20201002)
always replaces 
so the lengths of the augmented strings or sentential forms which follow one another in an ordinary derivation never decrease at any stage of the process, up to and including the terminal string finally generated by the grammar.
are non‑contracting.
and the set 
as 
in parallel fashion with the division of its overlying type as 
That is an option we will close off for now but leave open to consider at a later point, noting only the issues involved in choosing between grammars, and then moving on to the next alternative.
and sentential forms 
on the collection of non‑terminal symbols 
for all 
ordains the empty string to be strictly less than every other sentential form. That amounts to an axiom of the form 
for every non‑empty sentential form 
Inquiry Into Inquiry 