Changes

There is nothing wrong with the more expansive pan of the covered equation, since it follows straightforwardly from the definition of the kleene star operation, but the covering statement to the effect that <math>S :> S^*\!</math> is not a very productive piece of information, in the sense of telling very much about the language that falls under the type of a sentence <math>S.\!</math>  In particular, since it implies that <math>S :> \underline\varepsilon,</math> and since <math>\underline\varepsilon \cdot \mathfrak{L} \, = \, \mathfrak{L} \cdot \underline\varepsilon \, = \, \mathfrak{L},</math> for any formal language <math>\mathfrak{L},</math> the empty string <math>\varepsilon</math> is counted over and over in every term of the union, and every non-empty sentence under <math>S\!</math> appears again and again in every term of the union that follows the initial appearance of <math>S.\!</math>  As a result, this style of characterization has to be classified as ''true but not very informative''.  If at all possible, one prefers to partition the language of interest into a disjoint union of subsets, thereby accounting for each sentence under its proper term, and one whose place under the sum serves as a useful parameter of its character or its complexity.  In general, this form of description is not always possible to achieve, but it is usually worth the trouble to actualize it whenever it is.

There is nothing wrong with the more expansive pan of the covered equation, since it follows straightforwardly from the definition of the kleene star operation, but the covering statement to the effect that <math>S :> S^*\!</math> is not a very productive piece of information, in the sense of telling very much about the language that falls under the type of a sentence <math>S.\!</math>  In particular, since it implies that <math>S :> \underline\varepsilon,</math> and since <math>\underline\varepsilon \cdot \mathfrak{L} \, = \, \mathfrak{L} \cdot \underline\varepsilon \, = \, \mathfrak{L},</math> for any formal language <math>\mathfrak{L},</math> the empty string <math>\varepsilon</math> is counted over and over in every term of the union, and every non-empty sentence under <math>S\!</math> appears again and again in every term of the union that follows the initial appearance of <math>S.\!</math>  As a result, this style of characterization has to be classified as ''true but not very informative''.  If at all possible, one prefers to partition the language of interest into a disjoint union of subsets, thereby accounting for each sentence under its proper term, and one whose place under the sum serves as a useful parameter of its character or its complexity.  In general, this form of description is not always possible to achieve, but it is usually worth the trouble to actualize it whenever it is.

Suppose that one tries to deal with this problem by eliminating each use of the kleene star operation, by reducing it to a purely finitary set of steps, or by finding an alternative way to cover the sublanguage that it is used to generate.  This amounts, in effect, to ''recognizing a type'', a complex process that involves the following steps:

Suppose that one tries to deal with this problem by eliminating each use of

the kleene star operation, by reducing it to a purely finitary set of steps,

or by finding an alternative way to cover the sublanguage that it is used to

generate.  This amounts, in effect, to "recognizing a type", a complex process

that involves the following steps:

1.  Noticing a category of strings that

# '''Noticing''' a category of strings that is generated by iteration or recursion.

    is generated by iteration or recursion.

# '''Making a note of it''' by instituting an explicitly-named grammatical category.

In sum, one introduces a non-terminal symbol for each type of sentence and each ''part of speech'' or sentential component that is generated by means of iteration or recursion under the ruling constraints of the grammar.  In order to do this one needs to analyze the iteration of each grammatical operation in a way that is analogous to a mathematically inductive definition, but further in a way that is not forced explicitly to recognize a distinct and separate type of expression merely to account for and to recount every increment in the parameter of iteration.

 

 

In sum, one introduces a non-terminal symbol for each type of sentence and

each "part of speech" or sentential component that is generated by means of

iteration or recursion under the ruling constraints of the grammar.  In order

to do this one needs to analyze the iteration of each grammatical operation in

a way that is analogous to a mathematically inductive definition, but further in

a way that is not forced explicitly to recognize a distinct and separate type of

expression merely to account for and to recount every increment in the parameter

of iteration.

Returning to the case of the cactus language, the process of recognizing an

Returning to the case of the cactus language, the process of recognizing an

iterative type or a recursive type can be illustrated in the following way.

iterative type or a recursive type can be illustrated in the following way.