Changes

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.

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.

Returning to the case of the cactus language <math>\mathfrak{C} (\mathfrak{P}),</math> in other words, the formal language <math>\operatorname{PARCE}</math> of ''painted and rooted cactus expressions'', it serves the purpose of efficient accounting to partition the language into the following couple of sublanguages:

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

 

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

it serves the purpose of efficient accounting to partition the

 

language PARCE into the following couple of sublanguages:

<p><math>\operatorname{EPARCE} \ = \ \underline\varepsilon \ = \ \{ \varepsilon \}</math></p></li>

    make up the language EPARCE that consists of

    EPARCE  =  {""}.

<p>The ''significantly painted and rooted cactus expressions'' make up the language <math>\operatorname{SPARCE}</math> that consists of everything else, namely, all of the non-empty strings in the language <math>\operatorname{PARCE}.</math>  In sum:</p>

<p><math>\operatorname{SPARCE} \ = \ \operatorname{PARCE} \setminus \varepsilon</math></p></li>

    make up the language SPARCE that consists of everything else,

    namely, all of the non-empty strings in the language PARCE.

As a result of marking the distinction between empty and significant sentences,

As a result of marking the distinction between empty and significant sentences,

that is, by categorizing each of these three classes of strings as an entity

that is, by categorizing each of these three classes of strings as an entity