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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> |
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| 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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| <br> | | <br> |
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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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