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The converses of these decomposition relations are tantamount to the corresponding forms of composition operations, making it possible for these complementary forms of analysis and synthesis to articulate the structures of strings and sentences in two directions.

The converses of these decomposition relations are tantamount to the corresponding forms of composition operations, making it possible for these complementary forms of analysis and synthesis to articulate the structures of strings and sentences in two directions.

The ''painted cactus language'' with paints in the set <math>\mathfrak{P} = \{ p_j : j \in J \}</math> is the formal language <math>\mathfrak{L} = \mathfrak{C} (\mathfrak{P}) \subseteq \mathfrak{A}^* = (\mathfrak{M} \cup \mathfrak{P})^*</math> that is defined as follows:

The "painted cactus language" with paints in the

set !P! = {p_j : j in J} is the formal language

!L! = !C!(!P!) c !A!* = (!M! |_| !P!)* that is

defined as follows:

PC 1. The blank symbol m_1 is a sentence.

 

PC 2. The paint p_j is a sentence, for each j in J.

| PC 1. || The blank symbol <math>m_1\!</math> is a sentence.

 

PC 3. Conc^0 and Surc^0 are sentences.

| PC 2. || The paint <math>p_j\!</math> is a sentence, for each <math>j\!</math> in <math>J.\!</math>

 

PC 4. For each positive integer k,

| PC 3. || <math>\operatorname{Conc}^0</math> and <math>\operatorname{Surc}^0</math> are sentences.

 

      if   z_1, ..., z_k are sentences,

| PC 4. || For each positive integer <math>k,\!</math>

 

      then Conc^k_j  z_j is a sentence,

| &nbsp; || if <math>s_1, \ldots, s_k\!</math> are sentences,

 

      and   Surc^k_j  z_j is a sentence.

| &nbsp; || then <math>\operatorname{Conc}_{j=1}^k s_j</math> is a sentence,

| &nbsp; || and <math>\operatorname{Surc}_{j=1}^k s_j</math> is a sentence.

As usual, saying that z is a sentence is just a conventional way of

As usual, saying that z is a sentence is just a conventional way of

stating that the string z belongs to the relevant formal language !L!.

stating that the string z belongs to the relevant formal language !L!.