Cactus Language • Preliminaries 13

Consider what effects that might conceivably
have practical bearings you conceive the
objects of your conception to have. Then,
your conception of those effects is the
whole of your conception of the object.

Charles S. Peirce • Issues of Pragmaticism

We have before us what appears to be a maximally concise description of our subject matter.

The painted cactus language with paints in the set $\mathfrak{P} = \{ p_j : j \in J \}$ is the formal language $\mathfrak{L} = \mathfrak{C} (\mathfrak{P}) \subseteq \mathfrak{A}^* = (\mathfrak{M} \cup \mathfrak{P})^*$ defined as follows.

$\begin{array}{ll} \text{PC 1.} & \text{The blank symbol}~ m_1 ~\text{is a sentence.} \\ \text{PC 2.} & \text{The paint}~ p_j ~\text{is a sentence for each}~ j ~\text{in}~ J. \\ \text{PC 3.} & \mathrm{Conc}^0 ~\text{and}~ \mathrm{Surc}^0 ~\text{are sentences.} \\ \text{PC 4.} & \text{For each positive integer}~ n, \\ & \text{if}~ s_1, \ldots, s_n ~\text{are sentences} \\ & \text{then}~ \mathrm{Conc}_{k=1}^n s_k ~\text{is a sentence} \\ & \text{and}~ \mathrm{Surc}_{k=1}^n s_k ~\text{is a sentence.} \end{array}$

Here we encounter a problem. The very conciseness of that description presents an obstacle to understanding, glossing over infinities and divinities of detail which must be comprehended in effectively finite form, especially if we have in mind developing a fully computational parser.

A start in that direction, taking steps toward an effective description of cactus languages, a finitary conception of their membership conditions, and a bounded characterization of a typical sentence of that form, can be made by recasting the above description of cactus expressions into the pattern of what is called, more or less roughly, a formal grammar.

Resources

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

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