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When there is no possibility of confusion, the letter <math>^{\backprime\backprime} 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} 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} 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>

When there is no possibility of confusion, the letter <math>^{\backprime\backprime} 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} 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} 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>

A ''foil'' is a string of the form <math>^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime},</math> where <math>T\!</math> is a tract.  Thus, a typical foil <math>F\!</math> has the form:

A ''foil'' is a string of the form <math>^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} \, \cdot \, T \, \cdot \, ^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime},</math> where <math>T\!</math> is a tract.  Thus, a typical foil <math>F\!</math> has the form:

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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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<math>\mathfrak{C} (\mathfrak{P}) : \text{Grammar 3}\!</math>

<math>\mathfrak{C} (\mathfrak{P}) : \text{Grammar 3}\!</math>

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<math>\mathfrak{Q} = \{ \, ^{\backprime\backprime} \, F \, ^{\prime\prime}, \, ^{\backprime\backprime} \, R \, ^{\prime\prime}, \, ^{\backprime\backprime} \, T \, ^{\prime\prime} \, \}</math>

<math>\mathfrak{Q} = \{ \, ^{\backprime\backprime} F \, ^{\prime\prime}, \, ^{\backprime\backprime} R \, ^{\prime\prime}, \, ^{\backprime\backprime} T \, ^{\prime\prime} \, \}</math>

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