higher ranked args in encode

xpqz · Aug 5, 2022 · 392efe4 · 392efe4
1 parent cc8766a
commit 392efe4
Showing 1 changed file with 253 additions and 2 deletions.
diff --git a/contents/decode.ipynb b/contents/decode.ipynb
@@ -262,7 +262,7 @@
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- "Going the other way, how many soldiers are there in a legion?"
+ "Going the other way, how many soldiers are there in a legion? "
  ]
  },
  {
@@ -290,7 +290,7 @@
  "cell_type": "markdown",
  "metadata": {},
  "source": [
- "There are some less obvious uses, too. For example, we can use `1⊥` to sum vectors:"
+ "_Decode_ has some less obvious uses, too. For example, we can use `1⊥` to sum vectors:"
  ]
  },
  {
@@ -492,6 +492,257 @@
  "source": [
  "⎕ ← i j←(⍴m)⊤k ⍝ 1D to 2D"
  ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Higher ranks\n",
+ "\n",
+ "So far, we only used vectors as arguments to _Encode_ and _Decode_. However, they can take arguments of higher ranks, too. It may not be immediately obvious what this means, so let's look at that.\n",
+ "\n",
+ "_Encode_ returns an array where the shape is the combined shape of the left and right arguments:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<pre class=\"language-APL\"> 1 2 4 5\n",
+ "23 46 10 33\n",
+ "20 40 0 20\n",
+ "</pre>"
+ ]
+ },
+ "execution_count": 1,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "24 60 60⊤5000 10000 15000 20000 ⍝ Convert 5000 10000 15000 20000 to HMS"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "So the shape is `3 4`. If we apply the same operation to a matrix, the same relationship holds:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 3,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<pre class=\"language-APL\"> 5000 10000\n",
+ "15000 20000\n",
+ "</pre>"
+ ]
+ },
+ "execution_count": 3,
+ "metadata": {},
+ "output_type": "execute_result"
+ },
+ {
+ "data": {
+ "text/html": [
+ "<pre class=\"language-APL\"> 1 2\n",
+ " 4 5\n",
+ "\n",
+ "23 46\n",
+ "10 33\n",
+ "\n",
+ "20 40\n",
+ " 0 20\n",
+ "</pre>"
+ ]
+ },
+ "execution_count": 3,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "seconds ← ⎕ ← 2 2⍴5000 10000 15000 20000\n",
+ "24 60 60⊤seconds"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Again, the shape is 3 layers (as `3=≢24 60 60`) of shape `2 2`. The leading axis holds the results which is easier to see with a transpose applied:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 7,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<pre class=\"language-APL\">1 23 20\n",
+ "2 46 40\n",
+ "\n",
+ "4 10 0\n",
+ "5 33 20\n",
+ "</pre>"
+ ]
+ },
+ "execution_count": 7,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "⎕IO←0\n",
+ "2 0 1⍉24 60 60⊤seconds"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Perhaps more surprisingly, we can use a higher-ranked argument to the _left_, too. It then provides _multiple_ radix vectors into which to encode the right hand side. Let's say we want to convert a set of numbers to bases 10 and 16:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 8,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<pre class=\"language-APL\">10 16\n",
+ "10 16\n",
+ "10 16\n",
+ "</pre>"
+ ]
+ },
+ "execution_count": 8,
+ "metadata": {},
+ "output_type": "execute_result"
+ },
+ {
+ "data": {
+ "text/html": [
+ "<pre class=\"language-APL\">1 2\n",
+ "0 1\n",
+ "\n",
+ "2 5\n",
+ "8 0\n",
+ "\n",
+ "8 6\n",
+ "0 0\n",
+ "</pre>"
+ ]
+ },
+ "execution_count": 8,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "radix ← ⎕ ← 3 2⍴10 16\n",
+ "radix⊤128 256"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We expected the shape `3 2 2`, but perhaps it's tricky to untangle why the result looks like it does. It's essentially an outer product, with a different axis order:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 9,
+ "metadata": {},
+ "outputs": [
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+ "data": {
+ "text/html": [
+ "<pre class=\"language-APL\">1 2\n",
+ "0 1\n",
+ "\n",
+ "2 5\n",
+ "8 0\n",
+ "\n",
+ "8 6\n",
+ "0 0\n",
+ "</pre>"
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+ "execution_count": 9,
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+ }
+ ],
+ "source": [
+ "⍉↑⍉(10 10 10)(16 16 16)∘.⊤128 256"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 10,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<pre class=\"language-APL\">1 2 8\n",
+ "0 8 0\n",
+ "\n",
+ "2 5 6\n",
+ "1 0 0\n",
+ "</pre>"
+ ]
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+ "execution_count": 10,
+ "metadata": {},
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+ ],
+ "source": [
+ "⍉(3 2⍴10 16)⊤128 256"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "which is\n",
+ "\n",
+ " 128 = (1,2,8)₁₀\n",
+ " 128 = (0,8,0)₁₆\n",
+ "\n",
+ " 256 = (2,5,6)₁₀\n",
+ " 256 = (1,0,0)₁₆"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Higher ranks for both left and right left as an exercise for the interested reader."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": []
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