Cactus Language • Preliminaries 7

The array of syntactic operators may be put in more organized form by making a few additional conventions and auxiliary definitions.

Concatenation

The conception of concatenation permits extension to its natural prequel, the corresponding operator on zero operands.

$\mathrm{Conc}^0 = ``" = \text{the empty string.}$

From that beginning the operation of concatenation may be broken into stages by means of the following conceptions.

The precatenation $\mathrm{Prec}(s_1, s_2)$ of two strings $s_1, s_2$ is defined as follows.

$\mathrm{Prec} (s_1, s_2) = s_1 \cdot s_2.$

The concatenation of $n$ strings $s_1, \ldots, s_n$ may now be given a new definition as the iterated precatenation of $n+1$ strings beginning with $s_0 = \mathrm{Conc}^0 = ``"$ and continuing through the remaining $n$ strings.

$\text{For}~ n = 0, ~\mathrm{Conc}_{k=0}^n s_k = \mathrm{Conc}^0 = ``".$

$\text{For}~ n > 0, ~\mathrm{Conc}_{k=1}^n s_k = \mathrm{Prec}(\mathrm{Conc}_{k=0}^{n-1} s_k, s_n).$

Surcatenation

The conception of surcatenation permits extension to its natural prequel, the corresponding operator on zero operands.

$\mathrm{Surc}^0 = ``()".$

From that beginning the operation of surcatenation may be broken into stages by means of the following conceptions.

A subclause in $\mathfrak{A}^*$ is a string ending with $``)".$

The subcatenation $\mathrm{Subc} (s_1, s_2)$ of a subclause $s_1$ by a string $s_2$ is defined as follows.

$\mathrm{Subc} (s_1, s_2) = s_1 \cdot (``)")^{-1} \cdot ``," \cdot s_2 \cdot ``)".$

The surcatenation of $n$ strings $s_1, \ldots, s_n$ may now be given a new definition as the iterated subcatenation of $n+1$ strings beginning with $s_0 = \mathrm{Surc}^0 = ``()"$ and continuing through the remaining $n$ strings.

$\text{For}~ n = 0, ~\mathrm{Surc}_{k=0}^n s_k = \mathrm{Surc}^0 = ``()".$

$\text{For}~ n > 0, ~\mathrm{Surc}_{k=1}^n s_k = \mathrm{Subc}(\mathrm{Surc}_{k=0}^{n-1} s_k, s_n).$

Notice that the expressions $\mathrm{Conc}_{k=0}^0 s_k$ and $\mathrm{Surc}_{k=0}^0 s_k$ are defined in such a way that the respective operators $\mathrm{Conc}^0$ and $\mathrm{Surc}^0$ simply ignore, in the manner of constants, whatever sequences of strings $s_k$ may be listed as their ostensible arguments.

Resources

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