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4 + link to further discussion

Feeling a vague inclination to keep the topic warm, but failing a clear sense of the word "random" that is wanted here, maybe it would serve to mention a few concepts that I use for thinking about "arbitrary" transitions in bit spaces, for example, a boolean space of abstract type $\mathbb{B}^k$.

First off, given $k$ boolean variables, $x_1, \ldots, x_k$, we have two options for pointing out a point $\mathbf{x}$ in $\mathbb{B}^k$, namely, by giving its $k$-tuple $(x_1, \ldots, x_k)$ in the coordinate space, or by giving a "singular proposition" that singles it out, that is, a logical proposition equivalent to a conjunction of $k$ literals, written either as a boolean product, $e_1 \cdot \ldots \cdot e_k$, or with the symbol "$\land$" for "and", as $e_1 \land \ldots \land e_k$, where $e_j = x_j$ or $e_j = \lnot x_j$ for each $j \in [1, k]$.

Notice that the $k$ coordinate projections are maps of the form $x_j : \mathbb{B}^k \to \mathbb{B}$, which puts them in a special case among the $2^{2^k}$ boolean functions or logical propositions of the form $f : \mathbb{B}^k \to \mathbb{B}$.

Another important special case is the set of linear propositions, indicated here as $\mathbb{B}^k \overset{\ell}{\to} \mathbb{B}$, and you know there must be $2^k$ of those. They are in fact all the boolean sums that can be formed from subsets of the $k$ variables, where "+" is the field operation, in other words, $\operatorname{xor}$.

Further information about special classes of propositions in boolean function spaces may be found here:

3 + coordinate projections + linear propositions

Feeling a vague inclination to keep the topic warm, but failing a clear sense of the word "random" that is wanted here, maybe it would serve to mention a few concepts that I use for thinking about "arbitrary" transitions in bit spaces, for example, a boolean space of abstract type $\mathbb{B}^k$.

First off, given $k$ boolean variables, $x_1, \ldots, x_k$, we have two options for pointing out a point $\mathbf{x}$ in $\mathbb{B}^k$, namely, by giving its $k$-tuple $(x_1, \ldots, x_k)$ in the coordinate space, or by giving a "singular proposition" that singles it out, that is, a logical proposition equivalent to a conjunction of $k$ literals, written either as a boolean product, $e_1 \cdot \ldots \cdot e_k$, or with the symbol "$\land$" for "and", as $e_1 \land \ldots \land e_k$, where $e_j = x_j$ or $e_j = \lnot x_j$ for each $j \in [1, k]$.

I need

Notice that the $k$ coordinate projections are maps of the form $x_j : \mathbb{B}^k \to stop here, \mathbb{B}$, which puts them in part a special case among the $2^{2^k}$ boolean functions or logical propositions of the form $f : \mathbb{B}^k \to see if any \mathbb{B}$.

Another important special case is the set of this parses

Back later linear propositions, indicated here as $\mathbb{B}^k \overset{\ell}{\to} \mathbb{B}$, and you know there must be $2^k$ of those. They are in fact all the boolean sums that can be formed from subsets of the $k$ variables, where "+" is the field operation, in other words, $\operatorname{xor}$.

2 clarity, grammar

Feeling a vague inclination to keep this the topic warm, but failing a clear sense of the word "random" that is wanted here, maybe it would serve to mention a few concepts that I use for thinking about "arbitrary" transitions in bit spaces, for example, a boolean space of abstract type $\mathbb{B}^k$.

First off, given $k$ boolean variables, $x_1, \ldots, x_k$, we have two options for pointing out a point in $\mathbb{B}^k$, namely, by giving its $k$-tuple $(x_1, \ldots, x_k)$ in the coordinate space, or by giving a "singular proposition" that singles it out, that is, a logical proposition that is equivalent to a conjunction of $k$ literals, written either as a boolean product, $e_1 \cdot \ldots \cdot e_k$, or with the symbol "$\land$" for "and", as $e_1 \land \ldots \land e_k$, where $e_j = x_j$ or $e_j = \lnot x_j$ for each $j \in [1, k]$.

I need to stop here, in part to see if any of this parses …

Back later …

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