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As usual, saying that <math>s\!</math> is a sentence is just a conventional way of stating that the string <math>s\!</math> belongs to the relevant formal language <math>\mathfrak{L}.</math> An individual sentence of <math>\mathfrak{C} (\mathfrak{P}),</math> for any palette <math>\mathfrak{P},</math> is referred to as a ''painted and rooted cactus expression'' (PARCE) on the palette <math>\mathfrak{P},</math> or a ''cactus expression'', for short. Anticipating the forms that the parse graphs of these PARCE's will take, to be described in the next Subsection, the language <math>\mathfrak{L} = \mathfrak{C} (\mathfrak{P})</math> is also described as the set <math>\operatorname{PARCE} (\mathfrak{P})</math> of PARCE's on the palette <math>\mathfrak{P},</math> more generically, as the PARCE's that constitute the language <math>\operatorname{PARCE}.</math>
As usual, saying that <math>s\!</math> is a sentence is just a conventional way of stating that the string <math>s\!</math> belongs to the relevant formal language <math>\mathfrak{L}.</math> An individual sentence of <math>\mathfrak{C} (\mathfrak{P}),</math> for any palette <math>\mathfrak{P},</math> is referred to as a ''painted and rooted cactus expression'' (PARCE) on the palette <math>\mathfrak{P},</math> or a ''cactus expression'', for short. Anticipating the forms that the parse graphs of these PARCE's will take, to be described in the next Subsection, the language <math>\mathfrak{L} = \mathfrak{C} (\mathfrak{P})</math> is also described as the set <math>\operatorname{PARCE} (\mathfrak{P})</math> of PARCE's on the palette <math>\mathfrak{P},</math> more generically, as the PARCE's that constitute the language <math>\operatorname{PARCE}.</math>
A ''bare'' PARCE, a bit loosely referred to as a ''bare cactus expression'', is a PARCE on the empty palette <math>\mathfrak{P} = \emptyset.</math> A bare PARCE is a sentence in the ''bare cactus language'', <math>\mathfrak{C}^0 = \mathfrak{C} (\emptyset) = \operatorname{PARCE}^0 = \operatorname{PARCE} (\emptyset).</math> This set of strings, regarded as a formal language in its own right, is a sublanguage of every cactus language <math>\mathfrak{C} (\mathfrak{P}).</math> A bare cactus expression is commonly encountered in practice when one has occasion to start with an arbitrary PARCE and then finds a reason to delete or to erase all of its paints.
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Only one thing remains to cast this description of the cactus language
Only one thing remains to cast this description of the cactus language
into a form that is commonly found acceptable. As presently formulated,
into a form that is commonly found acceptable. As presently formulated,
