Cactus Language • Syntax 11

Grammar 5

With the foregoing array of considerations in mind, one is gradually led to a grammar for $\mathfrak{L} = \mathfrak{C} (\mathfrak{P})$ in which all of the covering productions have one of the following two forms.

$\begin{array}{ccll} S & :> & \varepsilon & \\[6pt] q & :> & W, & \text{with}~ q \in \{ ``S" \} \cup \mathfrak{Q} ~\text{and}~ W \in (\mathfrak{Q} \cup \mathfrak{A})^+ \end{array}$

A grammar fitting that description is called a context‑free grammar. The first type of rewrite rule is called a special production while the second type of rewrite rule is called an ordinary production.

An ordinary derivation is one composed solely of ordinary productions. An ordinary production $q :> W$ always replaces $q$ with a non‑empty string $W,$ 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.

The feature just described is known as the non‑contracting property of productions, derivations, and grammars. A grammar is said to have the property if all of its covering productions, with the possible exception of $S :> \varepsilon,$ are non‑contracting.

In particular, context‑free grammars are special cases of non‑contracting grammars. The presence of the non‑contracting property in a grammar makes the length of the augmented string available as a parameter to figure into mathematical induction and motivate recursive proofs. A handle like that on the generative process makes it possible to establish results about the generated language which are not easy to achieve in more general cases, nor even by other means in the context‑free case.

Grammar 5 is a context‑free grammar for the painted cactus language $\mathfrak{L} = \mathfrak{C} (\mathfrak{P})$ with the intermediate alphabet $\mathfrak{Q} = \{ ``S'", ``T" \}$ and the set $\mathfrak{K}$ of covering rules listed in the next display.

Finally, it is worth trying to bring together the separate advantages of diverse styles of grammar, to the extent they are compatible. To do that, a prospective grammar must be capable of maintaining a high level of intermediate organization, like that exhibited by Grammar 2, while respecting the principle of intermediate significance, thus accumulating the benefits of the context‑free style in Grammar 5. A plausible synthesis of those features is given in Grammar 6.

Resources

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

	Survey of Semiotics,… on Sign Relations • Semiotic Equi…
	Survey of Semiotics,… on Sign Relations • Semiotic Equi…
	Sign Relations • Sem… on Survey of Semiotics, Semiosis,…
	Survey of Semiotics,… on Sign Relations • Semiotic Equi…
	Sign Relations • Sem… on Survey of Semiotics, Semiosis,…
	Sign Relations • Sem… on Survey of Semiotics, Semiosis,…
	Survey of Semiotics,… on Sign Relations • Semiotic Equi…
	Sign Relations • Sem… on Survey of Semiotics, Semiosis,…
	Survey of Semiotics,… on Systems of Interpretation •…
	Survey of Semiotics,… on Sign Relations, Triadic Relati…

M	T	W	T	F	S	S
						1
2	3	4	5	6	7	8
9	10	11	12	13	14	15
16	17	18	19	20	21	22
23	24	25	26	27	28	29
30