Changes

|}

|}

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>

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:

<br>

<br>

In Grammar&nbsp;3, the first three Rules say that a sentence (a string of type <math>S\!</math>), is a rune (a string of type <math>R\!</math>), a foil (a string of type <math>F\!</math>), or an arbitrary concatenation of strings of these two types.  Rules&nbsp;4 through 7 specify that a rune <math>R\!</math> is an empty string <math>\varepsilon,</math> a blank symbol <math>m_1,\!</math> a paint <math>p_j,\!</math> or any concatenation of strings of these three types.  Rule&nbsp;8 characterizes a foil <math>F\!</math> as 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.  The last two Rules say that a tract <math>T\!</math> is either a sentence <math>S\!</math> or else the concatenation of a tract, a comma, and a sentence, in that order.

In Grammar 3, the first three Rules say that a sentence (a string of type S),

is a rune (a string of type R), a foil (a string of type F), or an arbitrary

concatenation of strings of these two types.  Rules 4 through 7 specify that

a rune R is an empty string !e! = "", a blank symbol m_1 = " ", a paint p_j,

for j in J, or any concatenation of strings of these three types.  Rule 8

characterizes a foil F as a string of the form "-(" · T · ")-", where T is

a tract.  The last two Rules say that a tract T is either a sentence S or

else the concatenation of a tract, a comma, and a sentence, in that order.

At this point in the succession of grammars for !C!(!P!), the explicit

At this point in the succession of grammars for <math>\mathfrak{C} (\mathfrak{P}),</math> the explicit uses of indefinite iterations, like the kleene star operator, are now completely reduced to finite forms of concatenation, but the problems that some styles of analysis have with allowing non-terminal symbols to cover both themselves and the empty string are still present.

uses of indefinite iterations, like the kleene star operator, are now

completely reduced to finite forms of concatenation, but the problems

that some styles of analysis have with allowing non-terminal symbols

to cover both themselves and the empty string are still present.

Any degree of reflection on this difficulty raises the general question:

Any degree of reflection on this difficulty raises the general question: What is a practical strategy for accounting for the empty string in the organization of any formal language that counts it among its sentences? One answer that presents itself is this:  If the empty string belongs to a formal language, it suffices to count it once at the beginning of the formal account that enumerates its sentences and then to move on to more interesting materials.

What is a practical strategy for accounting for the empty string in the

organization of any formal language that counts it among its sentences?

One answer that presents itself is this:  If the empty string belongs to

a formal language, it suffices to count it once at the beginning of the

formal account that enumerates its sentences and then to move on to more

interesting materials.

Returning to the case of the cactus language !C!(!P!), that is,

Returning to the case of the cactus language !C!(!P!), that is,

the formal language of "painted and rooted cactus expressions",

the formal language of "painted and rooted cactus expressions",