design/tpl (Classification System): Introduce linear algebra notation
I find this provides a visualization that is likely to be significantly more intuitive for others. It even holds when the matrix is not rectangular (yes, I know, it's not really a matrix then), so long as all matrices share the same respective K_j.master
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@ -570,6 +570,117 @@ More subtly,
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\label{f:ex:class-match-all-ranks}
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\end{figure}
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Visually,
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the one-dimensional construction of \axmref{class-pred} does not lend
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itself well to how intuitive the behavior of the system actually is.
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We therefore establish a relationship to the notation of linear algebra
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to emphasize the relationship between each of the inputs.
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\newcommand\matseqsup[1]{%
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\begin{bmatrix}
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M^{#1}_{0_0} & \dots & M^{#1}_{0_k} \\
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\vdots & \ddots & \vdots \\
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M^{#1}_{j_0} & \dots & M^{#1}_{j_k} \\
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\end{bmatrix}%
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}
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\newcommand\vecseqsup[1]{%
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\begin{bmatrix}
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v^{#1}_0 \\
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\vdots \\
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v^{#1}_j \\
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\end{bmatrix}%
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}
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% This must be an axiom because it defines how the connectives operate; see
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% the remark.
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\begin{axiom}[Classification Matrix Notation]\axmlabel{class-mat-not}
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Let $\Gamma^2$ be defined by \axmref{class-yield}.
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Then,
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\begin{equation*}
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\Gamma^2 =
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\matseqsup{0}\monoidops\matseqsup{l}
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\monoidop
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\vecseqsup{0}\monoidops\vecseqsup{m}
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\monoidop
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s^0\monoidops s^n,
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\end{equation*}
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from which $\Gamma^1$, $\Gamma^0$, and $\gamma$ can be derived.
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\end{axiom}
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\begin{remark}[Logical Connectives With Matrix Notation]
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From the definition of \axmref{class-mat-not},
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it should be clear that the logical connective $\monoidop$ necessarily
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acts like a Hadamard product with regards to how individual elements are
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combined.
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\end{remark}
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\axmref{class-mat-not} makes it easy to visualize classification
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operations simply by drawing horizontal boxes across the predicates,
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as demonstrated by \spref{f:class-mat-boxes}.
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% NB: Give this formatting extra attention if the document's formatting is
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% substantially changed, since it's not exactly responsible with it's
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% hard-coded units.
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\begingroup
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\begin{figure}[ht]
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\def\classmatraise#1{%
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\begin{aligned}
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#1 \\ {} \\ #1
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\end{aligned}
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}
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\def\classmateq{%
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\matseqsup{0}
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\classmatraise{\monoidop\cdots\monoidop}
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\matseqsup{l}
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\classmatraise\monoidop
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\vecseqsup{0}
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\classmatraise{\monoidop\cdots\monoidop}
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\vecseqsup{m}
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\classmatraise{%
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{}\monoidop s^0\monoidop\cdots\monoidop s^n%
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}
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}
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\begin{align*}
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&\;\raisebox{-3mm}[0mm]{%
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\begin{turn}{45}
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$\equiv$
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\end{turn}%
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} \;\fbox{$
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\left(M^0_{0_0} \monoidops M^0_{0_k}\right)
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\monoidops
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\left(M^l_{0_0} \monoidops M^l_{0_k}\right)
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\monoidop
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v^0_0 \monoidops v^m_0
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\monoidop
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s^0 \monoidops s^n
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$} &\Gamma^2_0 \\[-2mm]
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&\fbox{\raisebox{0mm}[0mm][6mm]{\hphantom{$\classmateq$}}} \\[-8mm]
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%
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&\classmateq &\vdots\; \\[-10mm]
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%
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&\fbox{\raisebox{0mm}[0mm][6mm]{\hphantom{$\classmateq$}}} \\
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&\;\raisebox{3mm}[0mm]{%
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\begin{turn}{-45}
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$\equiv$
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\end{turn}%
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} \;\fbox{$
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\left(M^0_{j_0} \monoidops M^0_{j_k}\right)
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\monoidops
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\left(M^l_{j_0} \monoidops M^l_{j_k}\right)
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\monoidop
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v^0_j \monoidops v^m_j
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\monoidop
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s^0 \monoidops s^n
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$} &\Gamma^2_j
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\end{align*}
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\caption{Visual interpretation of classification by \axmref{class-mat-not}.
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The adjacent frames represent equivalencies between the first-order
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logic of \axmref{class-yield} and the matrix notation.}
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\label{f:class-mat-boxes}
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\end{figure}
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\endgroup
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\index{classification!as proposition|(}
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\begin{lemma}[Match As Proposition]\lemlabel{match-prop}
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Matches can be represented using propositional logic provided that
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@ -23,6 +23,7 @@
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\usepackage{marginnote} % Notes in the margin
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\usepackage{ccicons} % CC license icons
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\usepackage{manfnt} % Dangerous Bend symbols
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\usepackage{rotating} % Rotating objects
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@ -123,6 +124,8 @@
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% Group theory
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\newcommand\Monoid[3]{\left({#1},{#2},{#3}\right)}
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\let\monoidop\bullet
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\newcommand\monoidops{\monoidop\cdots\monoidop}
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% Closed binary function
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\newcommand\cbif[1]{#1\times#1\rightarrow#1}
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