Cactus Language • Preliminaries 8

Defining the basic operations of concatenation and surcatenation on arbitrary strings gives them operational meaning for the all‑inclusive language $\mathfrak{L} = \mathfrak{A}^*.$ With that in hand it is time to adjoin the notion of a more discriminating grammaticality, in other words, a more properly restrictive concept of a sentence.

If $\mathfrak{L}$ is an arbitrary formal language over an alphabet of the type we have been discussing, that is, an alphabet of the form $\mathfrak{A} = \mathfrak{M} \cup \mathfrak{P},$ then there are a number of basic structural relations which can be defined on the strings of $\mathfrak{L}.$

Concatenation: $s$ is the concatenation of $s_1$ and $s_2$ in $\mathfrak{L}$
if and only if
$s_1$ is a sentence of $\mathfrak{L},$ $s_2$ is a sentence of $\mathfrak{L},$
and
$s = s_1 \cdot s_2$; $s$ is the concatenation of the $k$ strings $s_1, \ldots, s_k$ in $\mathfrak{L}$
if and only if
$s_j$ is a sentence of $\mathfrak{L}$ for all $j = 1 \ldots k$
and
$s = \mathrm{Conc}_{j=1}^k s_j = s_1 \cdot \ldots \cdot s_k$

Discatenation: $s$ is the discatenation of $s_1$ by $t$
if and only if
$s_1$ is a sentence of $\mathfrak{L},$ $t$ is an element of $\mathfrak{A},$
and
$s_1 = s \cdot t$
in which case we more commonly write
$s = s_1 \cdot t^{-1}$

Subclause: $s$ is a subclause of $\mathfrak{L}$
if and only if
$s$ is a sentence of $\mathfrak{L}$
and
$s$ ends with a $``\text{)}"$

Subcatenation: $s$ is the subcatenation of $s_1$ by $s_2$
if and only if
$s_1$ is a subclause of $\mathfrak{L},$ $s_2$ is a sentence of $\mathfrak{L},$
and
$s = s_1 \cdot (``\text{)}")^{-1} \cdot ``\text{,}" \cdot s_2 \cdot ``\text{)}"$

Surcatenation: $s$ is the surcatenation of the $k$ strings $s_1, \ldots, s_k$ in $\mathfrak{L}$
if and only if
$s_j$ is a sentence of $\mathfrak{L}$ for all ${j = 1 \ldots k}$
and
$s = \mathrm{Surc}_{j=1}^k s_j = ``\text{(}" \cdot s_1 \cdot ``\text{,}" \cdot \ldots \cdot ``\text{,}" \cdot s_k \cdot ``\text{)}"$

The converses of the above decomposition relations amount to the corresponding composition operations. As complementary forms of analysis and synthesis they make it possible to articulate the structures of strings and sentences in two directions.

Resources

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

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