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| ==Note 6== | | ==Note 6== |
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− | <pre>
| + | One more example may serve to suggest just how much dynamic complexity can be built on a universe of discourse that has but a single logical feature at its base. |
− | One more example may serve to suggest just how much dynamic | + | |
− | complexity can be built on a universe of discourse that has | + | But first, let me introduce a few more elements of general notation that I'll be using to describe finite dimensional universes of discourse and the qualitative dynamics that we envision occurring in them. |
− | but a single logical feature at its base. | |
| | | |
− | But first, let me introduce a few more elements of general
| + | Let <math>\mathcal{X} = \{ x_1, \ldots, x_n \}</math> be the ''alphabet'' of logical ''features'' or ''variables'' that we use to describe the n-dimensional universe of discourse <math>X^\circ = [\mathcal{X}] = [ x_1, \ldots, x_n ].</math> Picturesquely viewed, one may think of a venn diagram with n overlapping "circles" that are labeled with the feature names in the set <math>\mathcal{X}.</math> Staying with this picture, one visualizes the universe of discourse <math>X^\circ = [\mathcal{X}]</math> as having two layers: (1) the set <math>X = \langle \mathcal{X} \rangle = \langle x_1, \dots, x_n \rangle</math> of ''points'' or ''cells'' — in another sense of the word than when we speak of ''cellular automata'' — (2) the set <math>X^\uparrow = (X \to \mathbb{B})</math> of ''propositions'', boolean-valued functions, or maps from <math>X\!</math> to <math>\mathbb{B}.</math> |
− | notation that I'll be using to describe finite dimensional
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− | universes of discourse and the qualitative dynamics that
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− | we envision occurring in them. | |
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− | Let !X! = {x_1, ..., x_n} be the "alphabet" of logical "features"
| + | Thus, we may speak of the universe of discourse <math>X^\circ</math> as being an ordered pair <math>(X, X^\uparrow),</math> with <math>2^n\!</math> points in the underlying space <math>X\!</math> and <math>2^{2^n}</math> propositions in the function space <math>X^\uparrow.</math> |
− | or "variables" that we use to describe the n-dimensional universe
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− | of discourse X% = [!X!] = [x_1, ..., x_n]. Picturesquely viewed,
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− | one may think of a venn diagram with n overlapping "circles" that
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− | are labeled with the feature names in the set !X!. Staying with
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− | this picture, one visualizes the universe of discourse X% = [!X!]
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− | as having two layers: (1) the set X = <|!X!|> = <|x_1, ..., x_n|>
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− | of "points" or "cells" -- in another sense of the word than when
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− | we speak of "cellular automata" -- (2) the set X^ = (X -> B) of
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− | "propositions", boolean-valued functions, or maps from X to B.
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− | Thus, we may speak of the universe of discourse X% as being
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− | an ordered pair <X, X^>, with 2^n points in the underlying | |
− | space X and 2^(2^n) propositions in the function space X^. | |
− | That's just life in Ascii-land. It ain't Chicago.
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| A more complete table setting out these notations can be found here: | | A more complete table setting out these notations can be found here: |
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− | DLOG D2. http://stderr.org/pipermail/inquiry/2003-May/000480.html
| + | :* [http://stderr.org/pipermail/inquiry/2003-May/000480.html DLOG D2] |
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| Now, to the Example. | | Now, to the Example. |
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| + | <pre> |
| Once again, let us begin with a 1-feature alphabet !X! = {x_1} = {x}. | | Once again, let us begin with a 1-feature alphabet !X! = {x_1} = {x}. |
| In the discussion that follows I will consider a class of trajectories | | In the discussion that follows I will consider a class of trajectories |