555 lines
20 KiB
Plaintext
555 lines
20 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 1,
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"metadata": {},
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"outputs": [],
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"source": [
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"from notebook_preamble import J, V, define"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Square Spiral Example Joy Code"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"\n",
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"Here is the example of Joy code from the `README` file:\n",
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"\n",
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" [[[abs]ii <=][[<>][pop !-]||]&&][[!-][[++]][[--]]ifte dip][[pop !-][--][++]ifte]ifte\n",
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"\n",
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"It might seem unreadable but with a little familiarity it becomes just as\n",
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"legible as any other notation. Some layout helps:\n",
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"\n",
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" [ [[abs] ii <=]\n",
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" [\n",
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" [<>] [pop !-] ||\n",
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" ] &&\n",
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" ]\n",
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" [[ !-] [[++]] [[--]] ifte dip]\n",
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" [[pop !-] [--] [++] ifte ]\n",
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" ifte\n",
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"\n",
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"This function accepts two integers on the stack and increments or\n",
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"decrements one of them such that the new pair of numbers is the next\n",
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"coordinate pair in a square spiral (like the kind used to construct an\n",
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"Ulam Spiral). \n",
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"\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Original Form\n",
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"\n",
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"It's adapted from [the original code on StackOverflow](https://stackoverflow.com/questions/398299/looping-in-a-spiral/31864777#31864777):\n",
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"\n",
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"\n",
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"> If all you're trying to do is generate the first N points in the spiral\n",
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"> (without the original problem's constraint of masking to an N x M\n",
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"> region), the code becomes very simple:\n",
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"\n",
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" void spiral(const int N)\n",
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" {\n",
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" int x = 0;\n",
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" int y = 0;\n",
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" for(int i = 0; i < N; ++i)\n",
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" {\n",
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" cout << x << '\\t' << y << '\\n';\n",
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" if(abs(x) <= abs(y) && (x != y || x >= 0))\n",
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" x += ((y >= 0) ? 1 : -1);\n",
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" else\n",
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" y += ((x >= 0) ? -1 : 1);\n",
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" }\n",
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" }"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Translation to Joy\n",
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"\n",
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"I'm going to make a function that take two ints (`x` and `y`) and\n",
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"generates the next pair, we'll turn it into a generator later using the\n",
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"`x` combinator."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### First Boolean Predicate\n",
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"\n",
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"We need a function that computes `abs(x) <= abs(y)`, we can use `ii` to\n",
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"apply `abs` to both values and then compare them\n",
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"with `<=`:\n",
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"\n",
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" [abs] ii <=\n",
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"\n",
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"I've defined two short-circuiting Boolean combinators `&&` and `||` that\n",
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"each accept two quoted predicate programs, run the first, and\n",
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"conditionally run the second only if required (to compute the final\n",
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"Boolean value). They run their predicate arguments `nullary`."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {},
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"outputs": [],
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"source": [
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"define('&& [nullary] cons [nullary [0]] dip branch')\n",
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"define('|| [nullary] cons [nullary] dip [1] branch')"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Given those, we can define `x != y || x >= 0` as:\n",
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"\n",
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" [<>] [pop 0 >=] ||\n",
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"\n",
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"And `(abs(x) <= abs(y) && (x != y || x >= 0))` as:\n",
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"\n",
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" [[abs] ii <=] [[<>] [pop 0 >=] ||] &&\n",
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"\n",
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"It's a little rough, but, as I say, with a little familiarity it becomes\n",
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"legible."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### The Increment / Decrement Branches\n",
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"\n",
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"Turning to the branches of the main `if` statement:\n",
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"\n",
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" x += ((y >= 0) ? 1 : -1);\n",
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"\n",
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"Rewrite as a hybrid (pseudo-code) `ifte` expression:\n",
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"\n",
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" [y >= 0] [x += 1] [X -= 1] ifte\n",
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"\n",
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"Change each C phrase to Joy code:\n",
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"\n",
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" [0 >=] [[++] dip] [[--] dip] ifte\n",
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"\n",
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"Factor out the dip from each branch:\n",
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"\n",
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" [0 >=] [[++]] [[--]] ifte dip\n",
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"\n",
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"Similar logic applies to the other branch:\n",
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"\n",
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" y += ((x >= 0) ? -1 : 1);\n",
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"\n",
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" [x >= 0] [y -= 1] [y += 1] ifte\n",
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"\n",
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" [pop 0 >=] [--] [++] ifte"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### \"Not Negative\""
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"metadata": {},
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"outputs": [],
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"source": [
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"define('!- 0 >=')"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Putting the Pieces Together\n",
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"\n",
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"We can assemble the three functions we just defined in quotes and give\n",
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"them them to the `ifte` combinator. With some arrangement to show off\n",
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"the symmetry of the two branches, we have:\n",
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"\n",
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" [[[abs] ii <=] [[<>] [pop !-] ||] &&]\n",
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" [[ !-] [[++]] [[--]] ifte dip]\n",
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" [[pop !-] [--] [++] ifte ]\n",
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" ifte\n",
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"\n",
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"As I was writing this up I realized that, since the `&&` combinator\n",
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"doesn't consume the stack (below its quoted args), I can unquote the\n",
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"predicate, swap the branches, and use the `branch` combinator instead of\n",
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"`ifte`:\n",
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"\n",
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" [[abs] ii <=] [[<>] [pop !-] ||] &&\n",
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" [[pop !-] [--] [++] ifte ]\n",
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" [[ !-] [[++]] [[--]] ifte dip]\n",
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" branch"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"metadata": {},
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"outputs": [],
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"source": [
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"define('spiral_next [[[abs] ii <=] [[<>] [pop !-] ||] &&] [[!-] [[++]] [[--]] ifte dip] [[pop !-] [--] [++] ifte] ifte')"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Let's try it out:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"1 0\n"
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]
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}
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],
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"source": [
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"J('0 0 spiral_next')"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"1 -1\n"
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]
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}
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],
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"source": [
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"J('1 0 spiral_next')"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"0 -1\n"
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]
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}
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],
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"source": [
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"J('1 -1 spiral_next')"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"-1 -1\n"
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]
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}
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],
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"source": [
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"J('0 -1 spiral_next')"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Turning it into a Generator with `x`\n",
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"\n",
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"It can be used with the x combinator to make a kind of generator for\n",
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"spiral square coordinates.\n",
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"\n",
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"\n",
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"We can use `codireco` to make a generator\n",
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"\n",
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" codireco ::= cons dip rest cons\n",
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"\n",
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"It will look like this:\n",
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"\n",
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" [value [F] codireco]\n",
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"\n",
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"Here's a trace of how it works:\n",
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"\n",
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" [0 [dup ++] codireco] . x\n",
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" [0 [dup ++] codireco] . 0 [dup ++] codireco\n",
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" [0 [dup ++] codireco] 0 . [dup ++] codireco\n",
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" [0 [dup ++] codireco] 0 [dup ++] . codireco\n",
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" [0 [dup ++] codireco] 0 [dup ++] . cons dip rest cons\n",
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" [0 [dup ++] codireco] [0 dup ++] . dip rest cons\n",
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" . 0 dup ++ [0 [dup ++] codireco] rest cons\n",
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" 0 . dup ++ [0 [dup ++] codireco] rest cons\n",
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" 0 0 . ++ [0 [dup ++] codireco] rest cons\n",
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" 0 1 . [0 [dup ++] codireco] rest cons\n",
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" 0 1 [0 [dup ++] codireco] . rest cons\n",
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" 0 1 [[dup ++] codireco] . cons\n",
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" 0 [1 [dup ++] codireco] . "
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"But first we have to change the `spiral_next` function to work on a\n",
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"quoted pair of integers, and leave a copy of the pair on the stack.\n",
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"From:\n",
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"\n",
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" y x spiral_next\n",
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" ---------------------\n",
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" y' x'\n",
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"\n",
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"to:\n",
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"\n",
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" [x y] [spiral_next] infra\n",
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" -------------------------------\n",
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" [x' y']"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 9,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"[0 1]\n"
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]
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}
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],
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"source": [
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"J('[0 0] [spiral_next] infra')"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"So our generator is:\n",
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"\n",
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" [[x y] [dup [spiral_next] infra] codireco]\n",
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"\n",
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"Or rather:\n",
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"\n",
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" [[0 0] [dup [spiral_next] infra] codireco]\n",
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"\n",
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"There is a function `make_generator` that will build the generator for us\n",
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"out of the value and stepper function:\n",
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"\n",
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" [0 0] [dup [spiral_next] infra] make_generator\n",
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" ----------------------------------------------------\n",
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" [[0 0] [dup [spiral_next] infra] codireco]"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Here it is in action:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 10,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"[0 0] [0 1] [-1 1] [-1 0]\n"
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]
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}
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],
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"source": [
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"J('[0 0] [dup [spiral_next] infra] make_generator x x x x pop')"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Four `x` combinators, four pairs of coordinates."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Conclusion\n",
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"\n",
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"So that's an example of Joy code. It's a straightforward translation of\n",
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"the original. It's a little long for a single definition, you might\n",
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"break it up like so:\n",
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"\n",
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" _spn_P ::= [[abs] ii <=] [[<>] [pop !-] ||] &&\n",
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"\n",
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" _spn_T ::= [ !-] [[++]] [[--]] ifte dip\n",
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" _spn_E ::= [pop !-] [--] [++] ifte\n",
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"\n",
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" spiral_next ::= _spn_P [_spn_E] [_spn_T] branch\n",
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"\n",
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"This way it's easy to see that the function is a branch with two\n",
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"quasi-symmetrical paths.\n",
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"\n",
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"We then used this function to make a simple generator of coordinate\n",
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"pairs, where the next pair in the series can be generated at any time by\n",
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"using the `x` combinator on the generator (which is just a quoted\n",
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"expression containing a copy of the current pair and the \"stepper\n",
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"function\" to generate the next pair from that.)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 11,
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"metadata": {},
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"outputs": [],
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"source": [
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"define('_spn_P [[abs] ii <=] [[<>] [pop !-] ||] &&')\n",
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"define('_spn_T [!-] [[++]] [[--]] ifte dip')\n",
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"define('_spn_E [pop !-] [--] [++] ifte')\n",
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"define('spiral_next _spn_P [_spn_E] [_spn_T] branch')"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 12,
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"metadata": {
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"scrolled": true
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},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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" . 23 18 spiral_next\n",
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" 23 . 18 spiral_next\n",
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" 23 18 . spiral_next\n",
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" 23 18 . _spn_P [_spn_E] [_spn_T] branch\n",
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" 23 18 . [[abs] ii <=] [[<>] [pop !-] ||] && [_spn_E] [_spn_T] branch\n",
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" 23 18 [[abs] ii <=] . [[<>] [pop !-] ||] && [_spn_E] [_spn_T] branch\n",
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" 23 18 [[abs] ii <=] [[<>] [pop !-] ||] . && [_spn_E] [_spn_T] branch\n",
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" 23 18 [[abs] ii <=] [[<>] [pop !-] ||] . [nullary] cons [nullary [0]] dip branch [_spn_E] [_spn_T] branch\n",
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" 23 18 [[abs] ii <=] [[<>] [pop !-] ||] [nullary] . cons [nullary [0]] dip branch [_spn_E] [_spn_T] branch\n",
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" 23 18 [[abs] ii <=] [[[<>] [pop !-] ||] nullary] . [nullary [0]] dip branch [_spn_E] [_spn_T] branch\n",
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"23 18 [[abs] ii <=] [[[<>] [pop !-] ||] nullary] [nullary [0]] . dip branch [_spn_E] [_spn_T] branch\n",
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" 23 18 [[abs] ii <=] . nullary [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch\n",
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" 23 18 [[abs] ii <=] . [stack] dinfrirst [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch\n",
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" 23 18 [[abs] ii <=] [stack] . dinfrirst [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch\n",
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" 23 18 [[abs] ii <=] [stack] . dip infra first [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch\n",
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" 23 18 . stack [[abs] ii <=] infra first [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch\n",
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" 23 18 [18 23] . [[abs] ii <=] infra first [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch\n",
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" 23 18 [18 23] [[abs] ii <=] . infra first [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch\n",
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" 23 18 . [abs] ii <= [18 23] swaack first [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch\n",
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" 23 18 [abs] . ii <= [18 23] swaack first [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch\n",
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" 23 18 [abs] . [dip] dupdip i <= [18 23] swaack first [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch\n",
|
|
" 23 18 [abs] [dip] . dupdip i <= [18 23] swaack first [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch\n",
|
|
" 23 18 [abs] . dip [abs] i <= [18 23] swaack first [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch\n",
|
|
" 23 . abs 18 [abs] i <= [18 23] swaack first [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch\n",
|
|
" 23 . 18 [abs] i <= [18 23] swaack first [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch\n",
|
|
" 23 18 . [abs] i <= [18 23] swaack first [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch\n",
|
|
" 23 18 [abs] . i <= [18 23] swaack first [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch\n",
|
|
" 23 18 . abs <= [18 23] swaack first [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch\n",
|
|
" 23 18 . <= [18 23] swaack first [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch\n",
|
|
" False . [18 23] swaack first [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch\n",
|
|
" False [18 23] . swaack first [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch\n",
|
|
" 23 18 [False] . first [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch\n",
|
|
" 23 18 False . [0] [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch\n",
|
|
" 23 18 False [0] . [[[<>] [pop !-] ||] nullary] branch [_spn_E] [_spn_T] branch\n",
|
|
" 23 18 False [0] [[[<>] [pop !-] ||] nullary] . branch [_spn_E] [_spn_T] branch\n",
|
|
" 23 18 . 0 [_spn_E] [_spn_T] branch\n",
|
|
" 23 18 0 . [_spn_E] [_spn_T] branch\n",
|
|
" 23 18 0 [_spn_E] . [_spn_T] branch\n",
|
|
" 23 18 0 [_spn_E] [_spn_T] . branch\n",
|
|
" 23 18 . _spn_E\n",
|
|
" 23 18 . [pop !-] [--] [++] ifte\n",
|
|
" 23 18 [pop !-] . [--] [++] ifte\n",
|
|
" 23 18 [pop !-] [--] . [++] ifte\n",
|
|
" 23 18 [pop !-] [--] [++] . ifte\n",
|
|
" 23 18 [pop !-] [--] [++] . [nullary not] dipd branch\n",
|
|
" 23 18 [pop !-] [--] [++] [nullary not] . dipd branch\n",
|
|
" 23 18 [pop !-] . nullary not [--] [++] branch\n",
|
|
" 23 18 [pop !-] . [stack] dinfrirst not [--] [++] branch\n",
|
|
" 23 18 [pop !-] [stack] . dinfrirst not [--] [++] branch\n",
|
|
" 23 18 [pop !-] [stack] . dip infra first not [--] [++] branch\n",
|
|
" 23 18 . stack [pop !-] infra first not [--] [++] branch\n",
|
|
" 23 18 [18 23] . [pop !-] infra first not [--] [++] branch\n",
|
|
" 23 18 [18 23] [pop !-] . infra first not [--] [++] branch\n",
|
|
" 23 18 . pop !- [18 23] swaack first not [--] [++] branch\n",
|
|
" 23 . !- [18 23] swaack first not [--] [++] branch\n",
|
|
" 23 . 0 >= [18 23] swaack first not [--] [++] branch\n",
|
|
" 23 0 . >= [18 23] swaack first not [--] [++] branch\n",
|
|
" True . [18 23] swaack first not [--] [++] branch\n",
|
|
" True [18 23] . swaack first not [--] [++] branch\n",
|
|
" 23 18 [True] . first not [--] [++] branch\n",
|
|
" 23 18 True . not [--] [++] branch\n",
|
|
" 23 18 False . [--] [++] branch\n",
|
|
" 23 18 False [--] . [++] branch\n",
|
|
" 23 18 False [--] [++] . branch\n",
|
|
" 23 18 . --\n",
|
|
" 23 17 . \n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"V('23 18 spiral_next')"
|
|
]
|
|
}
|
|
],
|
|
"metadata": {
|
|
"kernelspec": {
|
|
"display_name": "Python 2",
|
|
"language": "python",
|
|
"name": "python2"
|
|
},
|
|
"language_info": {
|
|
"codemirror_mode": {
|
|
"name": "ipython",
|
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"version": 3
|
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},
|
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"file_extension": ".py",
|
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"mimetype": "text/x-python",
|
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"name": "python",
|
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"nbconvert_exporter": "python",
|
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"pygments_lexer": "ipython3",
|
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"version": "3.8.3"
|
|
}
|
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},
|
|
"nbformat": 4,
|
|
"nbformat_minor": 2
|
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}
|