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If there exists a derivation <math>(W_1, \ldots, W_k)\!</math> in <math>\mathfrak{G},</math> one says that <math>W_1\!</math> ''derives'' <math>W_k\!</math> in <math>\mathfrak{G}</math> or that <math>W_k\!</math> is ''derivable'' from <math>W_1\!</math> in <math>\mathfrak{G},</math> and one

If there exists a derivation (W_1, ..., W_k) in !G!,

one says that W_1 "derives" W_k in !G!, conversely,

that W_k is "derivable" from W_1 in !G!, and one

typically summarizes the derivation by writing:

typically summarizes the derivation by writing:

W_1 :*:> W_k.

| <math>W_1 :\!*\!:> W_k.\!</math>

The language !L! = !L!(!G!) = <!G!> that is "generated"

The language <math>\mathfrak{L} = \mathfrak{L} (\mathfrak{G}) = \langle \mathfrak{G} \rangle</math> that is ''generated'' by the formal grammar <math>\mathfrak{G} = ( \, ^{\backprime\backprime} S \, ^{\prime\prime}, \, \mathfrak{Q}, \, \mathfrak{A}, \, \mathfrak{K} \, )</math> is the set of strings over the terminal alphabet <math>\mathfrak{A}</math> that are derivable from the initial symbol <math>^{\backprime\backprime} S \, ^{\prime\prime}</math> by way of the intermediate symbols in <math>\mathfrak{Q}</math> according to the characterizations in <math>\mathfrak{K}.</math> In sum:

by the formal grammar !G! = ("S", !Q!, !A!, !K!) is the

set of strings over the terminal alphabet !A! that are

derivable from the initial symbol "S" by way of the

intermediate symbols in !Q! according to the

characterizations in K.  In sum:

!L!(!G!) = <!G!>  = {W in !A!* : "S" :*:> W}.

| <math>\mathfrak{L} (\mathfrak{G}) \ = \ \langle \mathfrak{G} \rangle \ = \ \{ W \in \mathfrak{A}^* \, : \, ^{\backprime\backprime} S \, ^{\prime\prime} \, :\!*\!:> \, W \}.</math>

Finally, a string W is called a "word", a "sentence", or so on,

Finally, a string <math>W\!</math> is called a ''word'', a ''sentence'', or so on, of the language generated by <math>\mathfrak{G}</math> if and only if <math>W\!</math> is in <math>\mathfrak{L} (\mathfrak{G}).</math>

of the language generated by !G! if and only if W is in !L!(!G!).

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==The Cactus Language : Stylistics==

==The Cactus Language : Stylistics==