{
"cells": [
{
"cell_type": "raw",
"id": "2990f581",
"metadata": {},
"source": [
"---\n",
"title: \"APL Competition 2022\"\n",
"author: \"Jeremy Howard\"\n",
"date: \"2022-07-05\"\n",
"categories: [Dyalog, APL, Glyphs]\n",
"---"
]
},
{
"cell_type": "markdown",
"id": "5d6af942",
"metadata": {},
"source": [
"This notebook covers useful glyphs and concepts useful for the 2022 Dyalog APL competition. Be sure to read the [Basics](01_APL.ipynb) and [Operators](03_operators.ipynb) notebooks first."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "84e8fa35",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→─────────────────────────────────────┐\n",
"│Was ON -style=max -trains=tree -fns=on│\n",
"└──────────────────────────────────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"]box on -style=max -trains=tree -fns=on"
]
},
{
"cell_type": "markdown",
"id": "da2674b7",
"metadata": {},
"source": [
"## Glpyhs"
]
},
{
"cell_type": "markdown",
"id": "426ab86a",
"metadata": {},
"source": [
"### `↑` (Up arrow)"
]
},
{
"cell_type": "markdown",
"id": "d7c5a6e2",
"metadata": {},
"source": [
"#### monadic `↑` (Mix)"
]
},
{
"cell_type": "markdown",
"id": "4ddbcb70-667a-4885-91b4-ae34a82441e7",
"metadata": {},
"source": [
"Mix takes elements of a complex array and stacks them along rows."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "3f02ff32",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→────────────┐\n",
"│ ┌→──┐ ┌→──┐ │\n",
"│ │Hip│ │Hop│ │\n",
"│ └───┘ └───┘ │\n",
"└∊────────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"'Hip' 'Hop'"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "3f4cf518",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──┐\n",
"↓Hip│\n",
"│Hop│\n",
"└───┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"↑ 'Hip' 'Hop'"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "c8195e4a",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──┐\n",
"│2 3│\n",
"└~──┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⍴↑ 'Hip' 'Hop'"
]
},
{
"cell_type": "markdown",
"id": "3606c3d8-4edc-4163-bcb0-44c275b0f6ca",
"metadata": {},
"source": [
"If elements are not all the same length, Mix will pad as needed"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "768e22de-e779-457e-9246-48159d63a2ab",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→────────────────────┐\n",
"│ ┌→──┐ ┌→──┐ ┌→────┐ │\n",
"│ │Hip│ │Hop│ │Dance│ │\n",
"│ └───┘ └───┘ └─────┘ │\n",
"└∊────────────────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"'Hip' 'Hop' 'Dance'"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "ec916625-a9af-4824-976f-987b695d158c",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→────┐\n",
"↓Hip │\n",
"│Hop │\n",
"│Dance│\n",
"└─────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"↑ 'Hip' 'Hop' 'Dance'"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "de720fe0-df52-47dc-9675-cefc4df54b0f",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──┐\n",
"│3 5│\n",
"└~──┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⍴↑ 'Hip' 'Hop' 'Dance'"
]
},
{
"cell_type": "markdown",
"id": "a63eeb4e-b6e7-4e8b-ab8f-3ffa2e056e2b",
"metadata": {},
"source": [
"Arrays can be mixed arrays containing both simple scalars and nested arrays"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "cd23ab13",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──────────┐\n",
"│ ┌→──┐ │\n",
"│ │6 4│ 5 3 │\n",
"│ └~──┘ │\n",
"└∊──────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
"text/html": [
"┌→──┐\n",
"↓6 4│\n",
"│5 0│\n",
"│3 0│\n",
"└~──┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(6 4) 5 3\n",
"↑ (6 4) 5 3"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "b510d101",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→────────────────┐\n",
"│ ┌→──┐ ┌→────┐ │\n",
"│ │1 2│ 3 │4 5 6│ │\n",
"│ └~──┘ └~────┘ │\n",
"└∊────────────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
"text/html": [
"┌→────┐\n",
"↓1 2 0│\n",
"│3 0 0│\n",
"│4 5 6│\n",
"└~────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(1 2) 3 (4 5 6)\n",
"↑(1 2) 3 (4 5 6)"
]
},
{
"cell_type": "markdown",
"id": "01db0c25",
"metadata": {},
"source": [
"#### dyadic `↑` (Take)"
]
},
{
"cell_type": "markdown",
"id": "8f3f149f-5ed7-4698-8371-31ce92b2c200",
"metadata": {},
"source": [
"If `⍵` (right argument) is positive, returns the first `⍵` (right argument) elements of `⍺` (left argument)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "b4cfe378",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→───┐\n",
"│Pine│\n",
"└────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"4 ↑ 'Pineapple'"
]
},
{
"cell_type": "markdown",
"id": "965b139e-f748-48ae-808a-4911514a4fe7",
"metadata": {},
"source": [
"If `⍺` (left argument) is longer than the length of `⍵` (right argument), `⍵` is padded with blanks"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "032e52ce",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→───────────┐\n",
"│Pineapple │\n",
"└────────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"12 ↑ 'Pineapple'"
]
},
{
"cell_type": "markdown",
"id": "eee3f026-d0ff-480a-9dff-a0378ee0bebc",
"metadata": {},
"source": [
"If `⍵` (right argument) is negative, it returns the last elements rather than the first elements."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "83c4b6c7",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→────┐\n",
"│apple│\n",
"└─────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"¯5 ↑ 'Pineapple'"
]
},
{
"cell_type": "markdown",
"id": "b35d6a60-a867-4b7b-9a17-4cf96dc13e47",
"metadata": {},
"source": [
"When `⍺` (left argument) is a scalar `⍵` (right argument) indexes on the first axis. The same rules apply to padding and negative numbers with higher dimension arrays."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "359d90ed",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→─────────┐\n",
"↓1 2 3 4│\n",
"│5 6 7 8│\n",
"│9 10 11 12│\n",
"└~─────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⎕←mat←3 4⍴⍳12"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "6f0d38ba",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──────┐\n",
"↓1 2 3 4│\n",
"│5 6 7 8│\n",
"└~──────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"2 ↑ mat"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "71b7b051-1be6-4420-bafa-34889ce1eda1",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→─────────┐\n",
"↓5 6 7 8│\n",
"│9 10 11 12│\n",
"└~─────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"¯2 ↑ mat"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "a67fd7b2",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→─────────┐\n",
"↓1 2 3 4│\n",
"│5 6 7 8│\n",
"│9 10 11 12│\n",
"│0 0 0 0│\n",
"│0 0 0 0│\n",
"└~─────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"5 ↑ mat"
]
},
{
"cell_type": "markdown",
"id": "9f2068d5-1bd6-4e0d-a12e-8ccaf5c5310c",
"metadata": {},
"source": [
"When an array is passed to `⍺` (left argument) you can slice on more than just the first axis. For example, `2 ¯3 ↑ mat` will give the first 2 rows (1st axis) and the last 3 columns (2nd axis) of mat."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "d25a75ba",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→────┐\n",
"↓2 3 4│\n",
"│6 7 8│\n",
"└~────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"2 ¯3 ↑ mat"
]
},
{
"cell_type": "markdown",
"id": "ccce31ab",
"metadata": {},
"source": [
"### `↓` (Down arrow)"
]
},
{
"cell_type": "markdown",
"id": "b13ca055",
"metadata": {},
"source": [
"#### monadic `↓` (Split)"
]
},
{
"cell_type": "markdown",
"id": "74e5497d-a008-4150-87e3-e51d982d67ce",
"metadata": {},
"source": [
"Split takes an array and reshapes it to be all 1 row. That row will contain nested arrays that each contain the values in 1 of the rows."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "0a82ac8d",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→─────────┐\n",
"↓1 2 3 4│\n",
"│5 6 7 8│\n",
"│9 10 11 12│\n",
"└~─────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⎕←mat←3 4⍴⍳12"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "c37f5e72",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→─────────────────────────────────┐\n",
"│ ┌→──────┐ ┌→──────┐ ┌→─────────┐ │\n",
"│ │1 2 3 4│ │5 6 7 8│ │9 10 11 12│ │\n",
"│ └~──────┘ └~──────┘ └~─────────┘ │\n",
"└∊─────────────────────────────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"↓mat"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "87d57b9b-2bd9-4f13-a6a9-ec41eb8a39ae",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→────────┐\n",
"↓ ┌→──┐ │\n",
"│ │Hip│ │\n",
"│ └───┘ │\n",
"│ ┌→──┐ │\n",
"│ │Hop│ │\n",
"│ └───┘ │\n",
"│ ┌→────┐ │\n",
"│ │Dance│ │\n",
"│ └─────┘ │\n",
"└∊────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⎕ ← mat ← 3 1⍴'Hip' 'Hop' 'Dance'"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "2a27177a-63cc-4997-aa9a-dc8a94dc967f",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→────────────────────────────────┐\n",
"│ ┌→──────┐ ┌→──────┐ ┌→────────┐ │\n",
"│ │ ┌→──┐ │ │ ┌→──┐ │ │ ┌→────┐ │ │\n",
"│ │ │Hip│ │ │ │Hop│ │ │ │Dance│ │ │\n",
"│ │ └───┘ │ │ └───┘ │ │ └─────┘ │ │\n",
"│ └∊──────┘ └∊──────┘ └∊────────┘ │\n",
"└∊────────────────────────────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"↓mat"
]
},
{
"cell_type": "markdown",
"id": "17c9279a",
"metadata": {},
"source": [
"#### dyadic `↓` (Drop)"
]
},
{
"cell_type": "markdown",
"id": "6313de4d-63b9-4406-8e04-d4fcf1a08966",
"metadata": {},
"source": [
"If `⍵` (right argument) is positive, removes the first `⍵` (right argument) elements of `⍺` (left argument)."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "07a1a03c",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→────┐\n",
"│apple│\n",
"└─────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"4 ↓ 'Pineapple'"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "17186dd4",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌⊖┐\n",
"│ │\n",
"└─┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"12 ↓ 'Pineapple'"
]
},
{
"cell_type": "markdown",
"id": "a8233efd-8aef-4610-9956-7110caa943ff",
"metadata": {},
"source": [
"If `⍵` is negative, it removes the last elements rather than the first elements."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "bd5cbaac",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→───┐\n",
"│Pine│\n",
"└────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"¯5 ↓ 'Pineapple'"
]
},
{
"cell_type": "markdown",
"id": "752c7060-7a79-4d54-8e62-8f0f7ff44108",
"metadata": {},
"source": [
"When `⍺` (left argument) is a scalar `⍵` (right argument) indexes on the first axis. The same rules apply to padding and negative numbers with higher dimension arrays."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "0bbd6fb2",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→─────────┐\n",
"↓1 2 3 4│\n",
"│5 6 7 8│\n",
"│9 10 11 12│\n",
"└~─────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⎕←mat←3 4⍴⍳12"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "74e2fbcb",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→─────────┐\n",
"↓9 10 11 12│\n",
"└~─────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"2 ↓ mat"
]
},
{
"cell_type": "markdown",
"id": "fea13e80-85e0-4527-ab23-26a60d43db50",
"metadata": {},
"source": [
"When an array is passed to `⍺` (left argument) you can slice on more than just the first axis. For example, `2 ¯3 ↑ mat` will remove the first 2 rows (1st axis) and the last 3 columns (2nd axis) of mat."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "6179ae17",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→┐\n",
"↓9│\n",
"└~┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"2 ¯3 ↓ mat"
]
},
{
"cell_type": "markdown",
"id": "b3a82aff",
"metadata": {},
"source": [
"### `⌸` (Quad equal)"
]
},
{
"cell_type": "markdown",
"id": "08eb109e",
"metadata": {},
"source": [
"#### monadic `⌸` (Key operator)"
]
},
{
"cell_type": "markdown",
"id": "92134238-c4d0-4374-93fa-882eac9d5311",
"metadata": {},
"source": [
"The `key operator` gives the index locations of each unique element in a vector. For example `,⌸ 'banana'` has a `b` as the 1st element. `a` is the 2nd, 4th, and 6th elements. `n` is the 3rd and 5th elements."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "2ddcb1ea",
"metadata": {},
"outputs": [],
"source": [
"a ← 'banana'"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "548f40f2",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──────┐\n",
"↓b 1 │\n",
"│a 2 4 6│\n",
"│n 3 5 │\n",
"└+──────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
",⌸ a"
]
},
{
"cell_type": "markdown",
"id": "260f8205-ee48-487a-8ce8-4b00c59746f8",
"metadata": {},
"source": [
"You can also combine this with other functions to do operations such as sum those index locations together."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "c4293cfd",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→───┐\n",
"↓b 1│\n",
"│a 12│\n",
"│n 8│\n",
"└+───┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a {⍺,+/⍵}⌸ ⍳6"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "d5b2cb64",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──────┐\n",
"↓b 1 │\n",
"│a 2 4 6│\n",
"│n 3 5 │\n",
"└+──────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a ,⌸ ⍳6"
]
},
{
"cell_type": "markdown",
"id": "d6e4c907",
"metadata": {},
"source": [
"### `⊂` (Left shoe)"
]
},
{
"cell_type": "markdown",
"id": "7148c091",
"metadata": {},
"source": [
"#### monadic `⊂` (Enclose)"
]
},
{
"cell_type": "markdown",
"id": "e8e8d8d0-42eb-45ef-8cf6-95db63922b33",
"metadata": {},
"source": [
"Enclose is used to turn an array into a nested array with just one element. For example `1 2 3` is an array of shape 3. "
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "5973fe9f",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→────┐\n",
"│1 2 3│\n",
"└~────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"1 2 3"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "bd386b4c",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→┐\n",
"│3│\n",
"└~┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⍴1 2 3"
]
},
{
"cell_type": "markdown",
"id": "cec54d4c-8487-4019-9153-71ad37bd1174",
"metadata": {},
"source": [
"When we enclose `1 2 3` it becomes a scalar, meaning the shape is the empty list."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "c1bf4da8",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌─────────┐\n",
"│ ┌→────┐ │\n",
"│ │1 2 3│ │\n",
"│ └~────┘ │\n",
"└∊────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⊂1 2 3"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "ac73f91f",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌⊖┐\n",
"│0│\n",
"└~┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⍴⊂1 2 3"
]
},
{
"cell_type": "markdown",
"id": "d40ca852-b6f1-4dae-9aec-5018b6c39d70",
"metadata": {},
"source": [
"This can be used in conjunction with `,` in order to created arrays with nested elements."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "b3a8f422",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→────────┐\n",
"│ ┌→──┐ │\n",
"│ 1 │2 3│ │\n",
"│ └~──┘ │\n",
"└∊────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"1(2 3)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "5174b653",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→┐\n",
"│2│\n",
"└~┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⍴ 1(2 3)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "2a417898",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→────────┐\n",
"│ ┌→──┐ │\n",
"│ 1 │2 3│ │\n",
"│ └~──┘ │\n",
"└∊────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"1,⊂2 3"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "8430caa1",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌─────────────┐\n",
"│ ┌→────────┐ │\n",
"│ │ ┌→──┐ │ │\n",
"│ │ 1 │2 3│ │ │\n",
"│ │ └~──┘ │ │\n",
"│ └∊────────┘ │\n",
"└∊────────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⊂ 1(2 3)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "97f5641a",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌⊖┐\n",
"│0│\n",
"└~┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⍴⊂ 1(2 3)"
]
},
{
"cell_type": "markdown",
"id": "a11682e8-5df0-41a6-9ade-bcd15d9ad98f",
"metadata": {},
"source": [
"A good way to see what enclose does is to do multiple of them, which continues to nest your original array deeper the more you add."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "f271d213",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌─────────────────┐\n",
"│ ┌─────────────┐ │\n",
"│ │ ┌→────────┐ │ │\n",
"│ │ │ ┌→──┐ │ │ │\n",
"│ │ │ 1 │2 3│ │ │ │\n",
"│ │ │ └~──┘ │ │ │\n",
"│ │ └∊────────┘ │ │\n",
"│ └∊────────────┘ │\n",
"└∊────────────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⊂⊂ 1(2 3)"
]
},
{
"cell_type": "markdown",
"id": "5be67cfa-da94-47b9-b691-da86d26ae32e",
"metadata": {},
"source": [
"An enclosed simple scalar is still the original simple scalar."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "e55cbaac",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"1\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⊂1"
]
},
{
"cell_type": "markdown",
"id": "0510c31d",
"metadata": {},
"source": [
"This adds `4 5 6` to each element of the LHS (1, 2, and 3):"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "90e97812",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→────────────────────────┐\n",
"│ ┌→────┐ ┌→────┐ ┌→────┐ │\n",
"│ │5 6 7│ │6 7 8│ │7 8 9│ │\n",
"│ └~────┘ └~────┘ └~────┘ │\n",
"└∊────────────────────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"1 2 3+⊂4 5 6"
]
},
{
"cell_type": "markdown",
"id": "31548044",
"metadata": {},
"source": [
"It's basically the same idea as this, since `(⊂ 4 5 6)` and `1` are both scalars:"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "fd7c7b48",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→────┐\n",
"│2 3 4│\n",
"└~────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"1 2 3+1"
]
},
{
"cell_type": "markdown",
"id": "58d72f9c",
"metadata": {},
"source": [
"#### dyadic `⊂` (Partitioned enclose)"
]
},
{
"cell_type": "markdown",
"id": "adb07307-c5a4-463d-af3d-bccfc1d8e739",
"metadata": {},
"source": [
"`partitioned enclose` is used to split an array into separate enclosures (created nested arrays). \n",
"\n",
"`⍺` (left argument) dictates where to split the array up. For example, `1 0 1 0 0 0 0` has a positive integer in the 1st and 3rd position and so we will split at the 1st and 3rd element"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "8272a0ec-7474-4355-8ee1-1c6e4963bcb9",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──────────────────┐\n",
"│ ┌→──┐ ┌→────────┐ │\n",
"│ │1 2│ │3 4 5 6 7│ │\n",
"│ └~──┘ └~────────┘ │\n",
"└∊──────────────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"1 0 1 0 0 0 0⊂ 1 2 3 4 5 6 7"
]
},
{
"cell_type": "markdown",
"id": "2b7eb786-0c9a-492d-99f7-1d12e301fb13",
"metadata": {},
"source": [
"This can also be done with textual arrays"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "b2e43612",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→─────────────┐\n",
"│ ┌→─┐ ┌→────┐ │\n",
"│ │Hi│ │Earth│ │\n",
"│ └──┘ └─────┘ │\n",
"└∊─────────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"1 0 1 0 0 0 0⊂'HiEarth'"
]
},
{
"cell_type": "markdown",
"id": "e471002f-6e7f-4f41-afcb-d6593192cd26",
"metadata": {},
"source": [
"The left argument of `partitioned enclose` controls the positioning of the newly created enclosures."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "2869efa5-9ea8-41ec-b5eb-2442b25f9a1a",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──────────────────────────────┐\n",
"│ ┌⊖┐ ┌→──┐ ┌→──┐ ┌⊖┐ ┌⊖┐ ┌→──┐ │\n",
"│ │0│ │1 2│ │3 4│ │0│ │0│ │5 6│ │\n",
"│ └~┘ └~──┘ └~──┘ └~┘ └~┘ └~──┘ │\n",
"└∊──────────────────────────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"2 0 1 0 3 0 ⊂ 1 2 3 4 5 6"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "2c277ea0-ac6e-40a9-833d-8a9317c86d65",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→─────────────────┐\n",
"│ ┌→─┐ ┌⊖┐ ┌→────┐ │\n",
"│ │Hi│ │ │ │Earth│ │\n",
"│ └──┘ └─┘ └─────┘ │\n",
"└∊─────────────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"1 0 2 0 0 0 0⊂'HiEarth'"
]
},
{
"cell_type": "markdown",
"id": "129ede39",
"metadata": {},
"source": [
"### `⊆` (Left shoe underbar)"
]
},
{
"cell_type": "markdown",
"id": "83bd3a74",
"metadata": {},
"source": [
"#### monadic `⊆` (Nest)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "e05aaf18",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"1\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(⊆1) ≡ ⊂1"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "63181909",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"1\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(⊆1 2 3) ≡ ⊂1 2 3"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "f63a328c",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"1\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(⊆ 1 (1 2 3)) ≡ 1 (1 2 3)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "1cac6b3b-5649-453a-ba3d-bfa8487e678b",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──────────┐\n",
"│ ┌→────┐ │\n",
"│ 1 │1 2 3│ │\n",
"│ └~────┘ │\n",
"└∊──────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
"text/html": [
"┌→──────────┐\n",
"│ ┌→────┐ │\n",
"│ 1 │1 2 3│ │\n",
"│ └~────┘ │\n",
"└∊──────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⊆ 1 (1 2 3)\n",
"\n",
"1 (1 2 3)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "47b45de8",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌───────────────┐\n",
"│ ┌→──────────┐ │\n",
"│ │ ┌→────┐ │ │\n",
"│ │ 1 │1 2 3│ │ │\n",
"│ │ └~────┘ │ │\n",
"│ └∊──────────┘ │\n",
"└∊──────────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⊂ 1 (1 2 3)"
]
},
{
"cell_type": "markdown",
"id": "620edc7a",
"metadata": {},
"source": [
"#### dyadic `⊆` (Partition)"
]
},
{
"cell_type": "markdown",
"id": "235c21d5-0ad0-4570-a74e-bb005cc07630",
"metadata": {},
"source": [
"Partition breaks an array (`⍵`) up based on the numbers provided in the left argument (`⍺`).\n",
"\n",
"A new partition is created anytime the integer in the left argument changes. 0 means that element is not included in the output. For example:\n",
"\n",
"`1 1 1 0 1 1 1 0 1 1 1 1⊆'How are you?'`:\n",
"+ The first 3 elements are in the first partition because there are 3 of the same integers `1 1 1`. \n",
"+ The first space is excluded because it is the fourth element in the right argument and the fourth element in the left argument is 0.\n",
"+ The second partition starts on the 5th element because it's a non-zero integer that is not the same as the previous number."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "feca5be8",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→───────────────────┐\n",
"│ ┌→──┐ ┌→──┐ ┌→───┐ │\n",
"│ │How│ │are│ │you?│ │\n",
"│ └───┘ └───┘ └────┘ │\n",
"└∊───────────────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"1 1 1 0 1 1 1 0 1 1 1 1⊆'How are you?'"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "adfd2d1e",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→─────────────┐\n",
"│ ┌→─┐ ┌→────┐ │\n",
"│ │Hi│ │Earth│ │\n",
"│ └──┘ └─────┘ │\n",
"└∊─────────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"1 1 2 2 2 2 2⊆'HiEarth'"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "ef923d85",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→───────────┐\n",
"│ ┌→─┐ ┌→──┐ │\n",
"│ │Hi│ │Ear│ │\n",
"│ └──┘ └───┘ │\n",
"└∊───────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"1 1 2 2 2 0 0⊆'HiEarth'"
]
},
{
"cell_type": "markdown",
"id": "f15d3d80",
"metadata": {},
"source": [
"### `⊃` (Right shoe)"
]
},
{
"cell_type": "markdown",
"id": "e288df04",
"metadata": {},
"source": [
"#### monadic `⊃` (Disclose;First)"
]
},
{
"cell_type": "markdown",
"id": "55141b18-a802-49c9-ba35-eb4a56048a10",
"metadata": {},
"source": [
"Disclose removes the nesting around arrays. For example, in simple arrays disclose with reverse and enclose. This is because it is getting the first element in a list, and when an array is enclosed the first element is the whole original array."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "bec34bd9",
"metadata": {},
"outputs": [],
"source": [
"a←1 2"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "ab4fc081",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌───────┐\n",
"│ ┌→──┐ │\n",
"│ │1 2│ │\n",
"│ └~──┘ │\n",
"└∊──────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⊂a"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "fd538116",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──┐\n",
"│1 2│\n",
"└~──┘\n",
" \n",
"1\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a ≡ ⎕ ← ⊃⊂1 2"
]
},
{
"cell_type": "markdown",
"id": "063a8f05-4a94-43dc-8e41-50cd6358bccb",
"metadata": {},
"source": [
"What disclose/first is doing is getting the first element in the array, regardless of whether it is nested or not."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "4b3c0639",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"W\n",
"-\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⊃ 'Word'"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "e5a4bb34-4bcd-4f59-af73-2d35788426f9",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──────────┐\n",
"│ ┌→──┐ │\n",
"│ 1 │1 2│ 3 │\n",
"│ └~──┘ │\n",
"└∊──────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⊃ (1 (1 2) 3) 4"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "38f38a51",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──┐\n",
"│1 2│\n",
"└~──┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⊃ (1 2)(3 4 5)"
]
},
{
"cell_type": "markdown",
"id": "1fd8817b",
"metadata": {},
"source": [
"#### dyadic `⊃` (Pick)"
]
},
{
"cell_type": "markdown",
"id": "247bbd4a-20cb-46c9-b58e-44daeeebcf40",
"metadata": {},
"source": [
"Pick is a way of indexing into an array to get particular elements. For example `3 ⊃ 'Word'` will give the 3rd element in `Word`."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "37075d37",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"r\n",
"-\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"3 ⊃ 'Word'"
]
},
{
"cell_type": "markdown",
"id": "47eef0aa-64dc-4523-a8d6-4a9e6ea316b0",
"metadata": {},
"source": [
"If the element you are selecting is an array, that array will be returned"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "e36eafb9",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→────┐\n",
"│3 4 5│\n",
"└~────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"2 ⊃ (1 2)(3 4 5)"
]
},
{
"cell_type": "markdown",
"id": "b2c5ff09-eaff-44fd-a02c-83e264c0edca",
"metadata": {},
"source": [
"If you pass an array to the left argument it will index further. For example, `2 1 ⊃ (1 2)(3 4 5)` will first go to the 2nd element (`3 4 5`) and then output the first element inside of that (`3`)."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "092ff790",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"3\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"2 1 ⊃ (1 2)(3 4 5)"
]
},
{
"cell_type": "markdown",
"id": "1a20f498-87d4-43d7-b0c3-32a634a42062",
"metadata": {},
"source": [
"In order to index into a multi-dimensional array you need to specify the different dimensions as a scalar. For example if you are looking at a matrix (2d array), `(⊂2 1)` as the left argument will output the value in the 2nd row, 1st column."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "deae5a82",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→────┐\n",
"↓1 2 3│\n",
"│4 5 6│\n",
"└~────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⎕←mat ← 2 3⍴⍳6"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "267aa753",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"4\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(⊂2 1) ⊃ 2 3⍴⍳6"
]
},
{
"cell_type": "markdown",
"id": "c8c2b52d-10fc-43c2-ab5a-4eff4caed036",
"metadata": {},
"source": [
"These above concepts of indexing into subarrays and indexing into multi-dimensional arrays can be combined in one function call."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "a9e0b49e",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→────────────────────────────────────┐\n",
"↓ ┌→────────┐ ┌→────────┐ ┌→────────┐ │\n",
"│ │ ┌→──┐ │ │ ┌→──┐ │ │ ┌→──┐ │ │\n",
"│ │ │ABC│ 1 │ │ │DEF│ 2 │ │ │GHI│ 3 │ │\n",
"│ │ └───┘ │ │ └───┘ │ │ └───┘ │ │\n",
"│ └∊────────┘ └∊────────┘ └∊────────┘ │\n",
"│ ┌→────────┐ ┌→────────┐ ┌→────────┐ │\n",
"│ │ ┌→──┐ │ │ ┌→──┐ │ │ ┌→──┐ │ │\n",
"│ │ │JKL│ 4 │ │ │MNO│ 5 │ │ │PQR│ 6 │ │\n",
"│ │ └───┘ │ │ └───┘ │ │ └───┘ │ │\n",
"│ └∊────────┘ └∊────────┘ └∊────────┘ │\n",
"└∊────────────────────────────────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"G←2 3⍴('ABC' 1)('DEF' 2)('GHI' 3)('JKL' 4),('MNO' 5)('PQR' 6)\n",
"G"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "67b8a341",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──┐\n",
"│JKL│\n",
"└───┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"((⊂2 1),1) ⊃ G"
]
},
{
"cell_type": "markdown",
"id": "8b0326cf",
"metadata": {},
"source": [
"### `⊢` (Right tack)"
]
},
{
"cell_type": "markdown",
"id": "0a34369a",
"metadata": {},
"source": [
"#### monadic `⊢` (Same)"
]
},
{
"cell_type": "markdown",
"id": "5b2a1f38-663e-4282-a644-1d6824c74ba2",
"metadata": {},
"source": [
"Return the argument back unchanged"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "b81b6d0f",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"1\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⊢1"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "00691415",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──┐\n",
"│abc│\n",
"└───┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⊢'abc'"
]
},
{
"cell_type": "markdown",
"id": "97d0971d",
"metadata": {},
"source": [
"#### dyadic `⊢` (Right)"
]
},
{
"cell_type": "markdown",
"id": "5a0e1516-7b4b-4dc9-a169-38e4341ab45a",
"metadata": {},
"source": [
"Returns the right argument back unchanged"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "461ac21e",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"1\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"'abc'⊢1"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "bfe47293",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──┐\n",
"│abc│\n",
"└───┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"1⊢'abc'"
]
},
{
"cell_type": "markdown",
"id": "0c0ad5b4",
"metadata": {},
"source": [
"### `⊣` (Left tack)"
]
},
{
"cell_type": "markdown",
"id": "9813f4e1",
"metadata": {},
"source": [
"#### monadic `⊣` (Same)"
]
},
{
"cell_type": "markdown",
"id": "b825c9f3-bab7-495a-9330-bd2d1a73a804",
"metadata": {},
"source": [
"Returns the argument back unchanged"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "de77f334",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──┐\n",
"│abc│\n",
"└───┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⊣'abc'"
]
},
{
"cell_type": "markdown",
"id": "dff7a3c9",
"metadata": {},
"source": [
"#### dyadic `⊣` (Left)"
]
},
{
"cell_type": "markdown",
"id": "363c0077-700d-4c33-a509-c69ece47aa11",
"metadata": {},
"source": [
"Returns the left argument back unchanged"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "89558f1d",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──┐\n",
"│abc│\n",
"└───┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"'abc'⊣1"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "2d004ac1",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"1\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"1⊣'abc'"
]
},
{
"cell_type": "markdown",
"id": "a44cd73e",
"metadata": {},
"source": [
"### `⊥` (Up tack)"
]
},
{
"cell_type": "markdown",
"id": "d51a72f8",
"metadata": {},
"source": [
"#### dyadic `⊥` (Decode)"
]
},
{
"cell_type": "markdown",
"id": "9a83c65a-76ad-4c1e-b180-52b0fb8f4133",
"metadata": {},
"source": [
"Decode decodes the right argument using the encode definition on the left.\n",
"\n",
"In the example of `2 ⊥ 1 1 0 1`:\n",
"\n",
"+ 2 on the left means that all values are encoded with 2 values, in other words binary\n",
"+ `1 1 0 1` in binary is equal to 13, so decode will return 13."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "c018adfc",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"13\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"2 ⊥ 1 1 0 1 ⍝ binary decode"
]
},
{
"cell_type": "markdown",
"id": "1f268ef5-1519-451c-9a7e-557d3d1d2b92",
"metadata": {},
"source": [
"An array can also be decoded if each element is encoded differently. For example, if you have an array of 3 elements representing hours, minutes, and seconds you can use decode to convert it into seconds.\n",
"\n",
"Hours would be decoded with 24, because there's 24 hours in a day. Minutes would be decoded with 60, because there's 60 minutes in an hour. Seconds would be decoded with 60 because there are 60 seconds in a minute."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "d9b3afe1",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"10000\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⍝ mixed radix: conversion of hours,\n",
"⍝ minutes and seconds to seconds:\n",
"24 60 60 ⊥ 2 46 40"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "2c76d681",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"10000\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(2×60×60) + (46×60) + 40"
]
},
{
"cell_type": "markdown",
"id": "0660fdf2-db3f-44fe-adf9-01f5ff0ba5a8",
"metadata": {},
"source": [
"Below are a few examples of a few other examples of ways decode can be used.\n",
"\n",
"A decimal decode:"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "2bce48db",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"1101\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"10 ⊥ 1 1 0 1 ⍝ decimal decode"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "6463996b",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"3416\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"10 ⊥ 3 4 1 6 ⍝ decimal decode"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "d5b1df93",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"5J9\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"1j1⊥1 2 3 4"
]
},
{
"cell_type": "markdown",
"id": "c6dfee67-faf5-48a6-b92d-1ad1cdaccf71",
"metadata": {},
"source": [
"`1⊥ `is sum over the first axis (+⌿)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "d75ca7b1",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"23\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"1⊥3 1 4 1 5 9"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "6ca32a66",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→─────────┐\n",
"↓1 2 3 4│\n",
"│5 6 7 8│\n",
"│9 10 11 12│\n",
"└~─────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⎕←M←3 4⍴⍳12"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "c32aa91e",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──────────┐\n",
"│15 18 21 24│\n",
"└~──────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"1⊥M"
]
},
{
"cell_type": "markdown",
"id": "154054f8",
"metadata": {},
"source": [
"### `⊤` (Down tack)"
]
},
{
"cell_type": "markdown",
"id": "144cf08f",
"metadata": {},
"source": [
"#### dyadic `⊤` (Encode)"
]
},
{
"cell_type": "markdown",
"id": "e7b22cb5-4a71-4b1e-a49e-c418ddcc6473",
"metadata": {},
"source": []
},
{
"cell_type": "code",
"execution_count": null,
"id": "e37a3491",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→────────┐\n",
"│0 3 6 5 8│\n",
"└~────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"10 10 10 10 10 ⊤ 3658 ⍝ decimal encode"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "e38e6791",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→────┐\n",
"│6 5 8│\n",
"└~────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"10 10 10 ⊤ 3658 ⍝ truncated decimal encode"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "99af88f7",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→─────┐\n",
"│36 5 8│\n",
"└~─────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"0 10 10 ⊤ 3658"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "117dfea6",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──────┐\n",
"│0 1 1 1│\n",
"└~──────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"2 2 2 2 ⊤ 7 ⍝ binary encode"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "8930de81",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──┐\n",
"│1 1│\n",
"└~──┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"2 2 ⊤ 7 ⍝ truncated binary encode"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "0edabb74",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→────┐\n",
"↓0 0 1│\n",
"│1 1 1│\n",
"│0 1 0│\n",
"│1 1 0│\n",
"└~────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"2 2 2 2 ⊤ 5 7 12 ⍝ binary encode"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "77aac657",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──────┐\n",
"│2 46 40│\n",
"└~──────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⍝ mixed radix: encode of 10000 seconds\n",
"⍝ to hours, minutes and seconds:\n",
"24 60 60 ⊤ 10000"
]
},
{
"cell_type": "markdown",
"id": "a4c28485",
"metadata": {},
"source": [
"## Forks"
]
},
{
"cell_type": "markdown",
"id": "51b3f84c",
"metadata": {},
"source": [
"In traditional mathematical notation (TMN): `(f+g)(x)=f(x)+g(x)`. Forks (3-trains) are just a generalisation of this pattern to all functions (though the middle one has to be dyadic)."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "520f5e4e",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"20.41887026\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(÷3)+(*3)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "b2245603",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"20.41887026\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(÷+*)3"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "21fa36a1",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"6\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
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],
"source": [
"(+/÷≢) 2 5 8 9"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "4b012428",
"metadata": {},
"outputs": [],
"source": [
"mean ← +/÷≢"
]
},
{
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"id": "2bf8d6db",
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" \n",
"6\n",
" \n",
""
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"source": [
"mean 2 5 8 9"
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},
{
"cell_type": "markdown",
"id": "0b876337",
"metadata": {},
"source": [
"For a dyadic fork, each of `f` and `g` are passed the LHS and the RHS:"
]
},
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"cell_type": "code",
"execution_count": null,
"id": "7a616a6d",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──────────────────────┐\n",
"│1 1 1 0 1 1 1 0 1 1 1 1│\n",
"└~──────────────────────┘\n",
""
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},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
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],
"source": [
"' '≠'How are you?'"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "52630585",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→───────────┐\n",
"│How are you?│\n",
"└────────────┘\n",
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"output_type": "execute_result"
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"source": [
"' '⊢'How are you?'"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "7e788e65",
"metadata": {},
"outputs": [
{
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"┌→───────────────────┐\n",
"│ ┌→──┐ ┌→──┐ ┌→───┐ │\n",
"│ │How│ │are│ │you?│ │\n",
"│ └───┘ └───┘ └────┘ │\n",
"└∊───────────────────┘\n",
""
]
},
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"metadata": {},
"output_type": "execute_result"
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],
"source": [
"(' '≠'How are you?') ⊆ (' '⊢'How are you?')"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "71bcd3d6",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→───────────────────┐\n",
"│ ┌→──┐ ┌→──┐ ┌→───┐ │\n",
"│ │How│ │are│ │you?│ │\n",
"│ └───┘ └───┘ └────┘ │\n",
"└∊───────────────────┘\n",
""
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},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
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"source": [
"' '(≠⊆⊢)'How are you?'"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "8dc97773",
"metadata": {},
"outputs": [],
"source": [
"split ← {(⍺≠⍵) ⊆ (⍺⊢⍵)}"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "75a14063",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→───────────────────┐\n",
"│ ┌→──┐ ┌→──┐ ┌→───┐ │\n",
"│ │How│ │are│ │you?│ │\n",
"│ └───┘ └───┘ └────┘ │\n",
"└∊───────────────────┘\n",
""
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},
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"metadata": {},
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"source": [
"' ' split 'How are you?'"
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"id": "7c8db425",
"metadata": {},
"outputs": [],
"source": []
}
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