Changes

In this rendition, a string of type <math>T\!</math> is not in general a sentence itself but a proper ''part of speech'', that is, a strictly ''lesser'' component of a sentence in any suitable ordering of sentences and their components.  In order to see how the grammatical category <math>T\!</math> gets off the ground, that is, to detect its minimal strings and to discover how its ensuing generations get started from these, it is useful to observe that the covering rule <math>T :> S\!</math> means that <math>T\!</math> ''inherits'' all of the initial conditions of <math>S,\!</math> namely, <math>T \, :> \, \varepsilon, m_1, p_j.</math>  In accord with these simple beginnings it comes to parse that the rule <math>T \, :> \, T \, \cdot \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} \, \cdot \, S,</math> with the substitutions <math>T = \varepsilon</math> and <math>S = \varepsilon</math> on the covered side of the rule, bears the germinal implication that <math>T \, :> \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime}.</math>

In this rendition, a string of type <math>T\!</math> is not in general a sentence itself but a proper ''part of speech'', that is, a strictly ''lesser'' component of a sentence in any suitable ordering of sentences and their components.  In order to see how the grammatical category <math>T\!</math> gets off the ground, that is, to detect its minimal strings and to discover how its ensuing generations get started from these, it is useful to observe that the covering rule <math>T :> S\!</math> means that <math>T\!</math> ''inherits'' all of the initial conditions of <math>S,\!</math> namely, <math>T \, :> \, \varepsilon, m_1, p_j.</math>  In accord with these simple beginnings it comes to parse that the rule <math>T \, :> \, T \, \cdot \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime} \, \cdot \, S,</math> with the substitutions <math>T = \varepsilon</math> and <math>S = \varepsilon</math> on the covered side of the rule, bears the germinal implication that <math>T \, :> \, ^{\backprime\backprime} \operatorname{,} ^{\prime\prime}.</math>

Grammar&nbsp;2 achieves a portion of its success through a higher degree of intermediate organization.  Roughly speaking, the level of organization can be seen as reflected in the cardinality of the intermediate alphabet <math>\mathfrak{Q} = \{ ^{\backprime\backprime} \operatorname{T} ^{\prime\prime} \}</math> but it is clearly not explained by this simple circumstance alone, since it is taken for granted that the intermediate symbols serve a purpose, a purpose that is easily recognizable but that may not be so easy to pin down and to specify exactly.  Nevertheless, it is worth the trouble of exploring this aspect of organization and this direction of development a little further. Although it is not strictly necessary to do so, it is possible to organize the materials of the present grammar in a slightly better fashion by recognizing two recurrent types of strings that appear in the typical cactus expression. In doing this, one arrives at the following two definitions:

Grammar&nbsp;2 achieves a portion of its success through a higher degree of intermediate organization.  Roughly speaking, the level of organization can be seen as reflected in the cardinality of the intermediate alphabet <math>\mathfrak{Q} = \{ ^{\backprime\backprime} \operatorname{T} ^{\prime\prime} \}</math> but it is clearly not explained by this simple circumstance alone, since it is taken for granted that the intermediate symbols serve a purpose, a purpose that is easily recognizable but that may not be so easy to pin down and to specify exactly.  Nevertheless, it is worth the trouble of exploring this aspect of organization and this direction of development a little further.

 

 

Although it is not strictly necessary to do so, it is possible to organize the materials of our developing grammar in a slightly better fashion by recognizing two recurrent types of strings that appear in the typical cactus expression. In doing this, one arrives at the following two definitions:

A ''rune'' is a string of blanks and paints concatenated together.  Thus, a typical rune <math>R\!</math> is a string over <math>\{ m_1 \} \cup \mathfrak{P},</math> possibly the empty string:

A ''rune'' is a string of blanks and paints concatenated together.  Thus, a typical rune <math>R\!</math> is a string over <math>\{ m_1 \} \cup \mathfrak{P},</math> possibly the empty string:

This is just the surcatenation of the sentences <math>S_1, \ldots, S_k.\!</math>  Given the possibility that this sequence of sentences is empty, and thus that the tract <math>T\!</math> is the empty string, the minimum foil <math>F\!</math> is the expression <math>^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime}.</math>  Explicitly marking each foil <math>F\!</math> that is embodied in a cactus expression is tantamount to recognizing another intermediate symbol, <math>^{\backprime\backprime} F ^{\prime\prime} \in \mathfrak{Q},</math> further articulating the structures of sentences and expanding the grammar for the language

This is just the surcatenation of the sentences <math>S_1, \ldots, S_k.\!</math>  Given the possibility that this sequence of sentences is empty, and thus that the tract <math>T\!</math> is the empty string, the minimum foil <math>F\!</math> is the expression <math>^{\backprime\backprime} \, \operatorname{()} \, ^{\prime\prime}.</math>  Explicitly marking each foil <math>F\!</math> that is embodied in a cactus expression is tantamount to recognizing another intermediate symbol, <math>^{\backprime\backprime} F ^{\prime\prime} \in \mathfrak{Q},</math> further articulating the structures of sentences and expanding the grammar for the language

<math>\mathfrak{C} (\mathfrak{P}).</math>  All of the same remarks about the versatile uses of the intermediate symbols, as string variables and as type names, apply again to the letter <math>^{\backprime\backprime} F ^{\prime\prime}.</math>

<math>\mathfrak{C} (\mathfrak{P}).</math>  All of the same remarks about the versatile uses of the intermediate symbols, as string variables and as type names, apply again to the letter <math>^{\backprime\backprime} F ^{\prime\prime}.</math>

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