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The ''painted cactus language'' <math>\mathfrak{C}</math> is actually a parameterized family of languages, consisting of one language <math>\mathfrak{C}(\mathfrak{P})</math> for each set <math>\mathfrak{P}</math> of ''paints''.

The ''painted cactus language'' <math>\mathfrak{C}</math> is actually a parameterized family of languages, consisting of one language <math>\mathfrak{C}(\mathfrak{P})</math> for each set <math>\mathfrak{P}</math> of ''paints''.

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The alphabet <math>\mathfrak{A} = \mathfrak{M} \cup \mathfrak{P}</math> is the disjoint union of two sets of symbols:

The alphabet !A!  = !M! |_| !P! is the disjoint union of two sets of symbols:

1. !M! is the alphabet of "measures", the set of "punctuation marks",

    or the collection of "syntactic constants" that is common to all

    of the languages !C!(!P!).  This set of signs is given as follows:

| valign="top" | 1.

 

| <math>\mathfrak{M}</math> is the alphabet of ''measures'', the set of ''punctuation marks'', or the collection of ''syntactic constants'' that is common to all of the languages <math>\mathfrak{C}(\mathfrak{P}).</math> This set of signs is given as follows:

    !M!  = {m_1, m_2, m_3, m_4}

 

        = {" ", "-(", ",", ")-"}

 

        = {blank, links, comma, right}.

\mathfrak{M}

 

& = &

2. !P! is the "palette", the alphabet of "paints", or the collection

    of "syntactic variables" that is peculiar to the language !C!(!P!).

\mathfrak{m}_1 & , &

    This set of signs is given as follows:

\mathfrak{m}_2 & , &

 

\mathfrak{m}_3 & , &

    !P!  = {p_j :  j in J}.

\mathfrak{m}_4 &

& = &

^{\backprime\backprime} \, \operatorname{(} \, ^{\prime\prime} & , &

^{\backprime\backprime} \, \operatorname{,} \, ^{\prime\prime} & , &

^{\backprime\backprime} \, \operatorname{)} \, ^{\prime\prime} &

& = &

\operatorname{blank} & , &

\operatorname{links} & , &

\operatorname{comma} & , &

\operatorname{right} &

\}. \\

| valign="top" | 2.

| <math>\mathfrak{P}</math> is the ''palette'', the alphabet of ''paints'', or the collection of ''syntactic variables'' that is peculiar to the language <math>\mathfrak{C}(\mathfrak{P}).</math>  This set of signs is given as follows:

| <math>\mathfrak{P} = \{ \mathfrak{p}_j :  j \in J \}.</math>

The easiest way to define the language !C!(!P!) is to indicate the general sorts

The easiest way to define the language !C!(!P!) is to indicate the general sorts

of operations that suffice to construct the greater share of its sentences from

of operations that suffice to construct the greater share of its sentences from