Changes

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 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:

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 R is a string over {m_1} |_| !P!, possibly

the empty string.

R in ({m_1} |_| !P!)*.

| <math>R\ \in\ ( \{ m_1 \} \cup \mathfrak{P} )^*</math>

When there is no possibility of confusion, the letter "R" can be used

When there is no possibility of confusion, the letter <math>^{\backprime\backprime} \operatorname{R} ^{\prime\prime}</math> can be used either as a string variable that ranges over the set of runes or else as a type name for the class of runes.  The latter reading amounts to the enlistment of a fresh intermediate symbol, <math>^{\backprime\backprime} \operatorname{R} ^{\prime\prime} \in \mathfrak{Q},</math> as a part of a new grammar for <math>\mathfrak{C} (\mathfrak{P}).</math> In effect, <math>^{\backprime\backprime} \operatorname{R} ^{\prime\prime}</math> affords a grammatical recognition for any rune that forms a part of a sentence in <math>\mathfrak{C} (\mathfrak{P}).</math>  In situations where these variant usages are likely to be confused, the types of strings can be indicated by means of expressions like <math>r <: R\!</math> and <math>W <: R.\!</math>

either as a string variable that ranges over the set of runes or else

as a type name for the class of runes.  The latter reading amounts to

the enlistment of a fresh intermediate symbol, "R" in !Q!, as a part

of a new grammar for !C!(!P!).  In effect, "R" affords a grammatical

recognition for any rune that forms a part of a sentence in !C!(!P!).

In situations where these variant usages are likely to be confused,

the types of strings can be indicated by means of expressions like

"r <: R" and "W <: R".

A "foil" is a string of the form "-(" · T · ")-", where T is a tract.

A "foil" is a string of the form "-(" · T · ")-", where T is a tract.

Thus, a typical foil F has the form:

Thus, a typical foil F has the form: