Work ䷻

Cactus Language • Syntax

Cactus Language • Grammar 1

Definition 1 • PNG

Display 1.1 • PNG

Display 1.1 • LaTeX

$\begin{array}{llll} 1. & \mathrm{Conc}_{j=1}^0 & = & ``" \\[8pt] & \mathrm{Conc}_{j=1}^k S_j & = & \mathrm{Prec} (\mathrm{Conc}_{j=1}^{k-1} S_j, S_k) \\[12pt] 2. & \mathrm{Surc}_{j=1}^0 & = & ``()" \\[8pt] & \mathrm{Surc}_{j=1}^k S_j & = & \mathrm{Subc} (\mathrm{Surc}_{j=1}^{k-1} S_j, S_k) \end{array}$

Display 1.2 • PNG

Display 1.2 • LaTeX

$\begin{array}{llll} 1. & \mathrm{Conc} & :> & ``" \\[8pt] & \mathrm{Conc} & :> & \mathrm{Conc} \cdot S \\[12pt] 2. & \mathrm{Surc} & :> & ``()" \\[8pt] & \mathrm{Surc} & :> & ``(" \cdot S \cdot ``)" \\[8pt] & \mathrm{Surc} & :> & \mathrm{Surc} \cdot ( \, ``)" \, )^{-1} \cdot ``," \cdot S \cdot ``)" \end{array}$

Cactus Language • Grammar 2

Definition 2 • PNG

Cactus Language • Grammar 3

Definition 3 • PNG

Cactus Language • Grammar 4

Definition 4 • PNG

Display 4.1 • PNG

Display 4.2 • PNG

Cactus Language • Grammar 5

Definition 5 • PNG

Cactus Language • Grammar 6

Definition 6 • PNG

General Rule • PNG

Cactus Language • Pragmatics

Stricture, Strait, Constraint, Information, Complexity

Stricture Table 1 • PNG

Stricture Table 1 • LaTeX

$\begin{matrix} ``X" & ``P" & ``Q" \\[4pt] ``X \times X" & ``X \times P" & ``X \times Q" \\[4pt] ``P \times X" & ``P \times P" & ``P \times Q" \\[4pt] ``Q \times X" & ``Q \times P" & ``Q \times Q" \end{matrix}$

Stricture Table 2 • PNG

Stricture Table 2 • LaTeX

$\begin{array}{lcccc} \text{High:} & ``P \times P" & ``P \times Q" & ``Q \times P" & ``Q \times Q" \\[4pt] \text{Med:} & ``P" & ``X \times P" & ``P \times X" \\[4pt] \text{Med:} & ``Q" & ``X \times Q" & ``Q \times X" \\[4pt] \text{Low:} & ``X" & ``X \times X" \end{array}$

Cactus Language • Mechanics

Cactus Language • Mechanics 4

Display 1 • PNG

Display 2 • PNG

Display 2 • LaTeX

The Concatenation $\mathrm{Conc}_{j=1}^k$ of the $k$ sentences $(s_j)_{j=1}^k$ has a Parse defined as follows.

$\mathrm{Parse} (\mathrm{Conc}^0) ~=~ \mathrm{Node}^0.$
For $k > 0,$
$\mathrm{Parse} (\mathrm{Conc}_{j=1}^k s_j) ~=~ \mathrm{Node}_{j=1}^k \mathrm{Parse} (s_j).$

The Surcatenation $\mathrm{Surc}_{j=1}^k$ of the $k$ sentences $(s_j)_{j=1}^k$ has a Parse defined as follows.

$\mathrm{Parse} (\mathrm{Surc}^0) ~=~ \mathrm{Lobe}^0.$
For $k > 0,$
$\mathrm{Parse} (\mathrm{Surc}_{j=1}^k s_j) ~=~ \mathrm{Lobe}_{j=1}^k \mathrm{Parse} (s_j).$

Cactus Language • Stretching Exercises