{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Square Spiral Example Joy Code"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "Here is the example of Joy code from the `README` file:\n",
    "\n",
    "    [[[abs]ii <=][[<>][pop !-]||]&&][[!-][[++]][[--]]ifte dip][[pop !-][--][++]ifte]ifte\n",
    "\n",
    "It might seem unreadable but with a little familiarity it becomes just as\n",
    "legible as any other notation.  Some layout helps:\n",
    "\n",
    "    [   [[abs] ii <=]\n",
    "        [\n",
    "            [<>] [pop !-] ||\n",
    "        ] &&\n",
    "    ]\n",
    "    [[    !-] [[++]] [[--]] ifte dip]\n",
    "    [[pop !-]  [--]   [++]  ifte    ]\n",
    "    ifte\n",
    "\n",
    "This function accepts two integers on the stack and increments or\n",
    "decrements one of them such that the new pair of numbers is the next\n",
    "coordinate pair in a square spiral (like the kind used to construct an\n",
    "Ulam Spiral).  \n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Original Form\n",
    "\n",
    "It's adapted from [the original code on StackOverflow](https://stackoverflow.com/questions/398299/looping-in-a-spiral/31864777#31864777):\n",
    "\n",
    "\n",
    "> If all you're trying to do is generate the first N points in the spiral\n",
    "> (without the original problem's constraint of masking to an N x M\n",
    "> region), the code becomes very simple:\n",
    "\n",
    "    void spiral(const int N)\n",
    "    {\n",
    "        int x = 0;\n",
    "        int y = 0;\n",
    "        for(int i = 0; i < N; ++i)\n",
    "        {\n",
    "            cout << x << '\\t' << y << '\\n';\n",
    "            if(abs(x) <= abs(y) && (x != y || x >= 0))\n",
    "                x += ((y >= 0) ? 1 : -1);\n",
    "            else\n",
    "                y += ((x >= 0) ? -1 : 1);\n",
    "        }\n",
    "    }"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Translation to Joy\n",
    "\n",
    "I'm going to make a function that take two ints (`x` and `y`) and\n",
    "generates the next pair, we'll turn it into a generator later using the\n",
    "`x` combinator."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### First Boolean Predicate\n",
    "\n",
    "We need a function that computes `abs(x) <= abs(y)`, we can use `ii` to\n",
    "apply `abs` to both values and then compare them\n",
    "with `<=`:\n",
    "\n",
    "    [abs] ii <="
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": []
    }
   ],
   "source": [
    "[_p [abs] ii <=] inscribe"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "23 -18"
     ]
    }
   ],
   "source": [
    "clear 23 -18"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "      23 -18 • _p\n",
      "      23 -18 • [abs] ii <=\n",
      "23 -18 [abs] • ii <=\n",
      "          23 • abs -18 abs <=\n",
      "          23 • -18 abs <=\n",
      "      23 -18 • abs <=\n",
      "       23 18 • <=\n",
      "       false • \n",
      "\n",
      "false"
     ]
    }
   ],
   "source": [
    "[_p] trace"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": []
    }
   ],
   "source": [
    "clear"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Short-Circuiting Boolean Combinators\n",
    "\n",
    "I've defined two short-circuiting Boolean combinators `&&` and `||` that\n",
    "each accept two quoted predicate programs, run the first, and\n",
    "conditionally run the second only if required (to compute the final\n",
    "Boolean value).  They run their predicate arguments `nullary`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": []
    }
   ],
   "source": [
    "[&& [nullary] cons [nullary [false]] dip branch] inscribe\n",
    "[|| [nullary] cons [nullary] dip [true] branch] inscribe"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "false"
     ]
    }
   ],
   "source": [
    "clear \n",
    "[true] [false] &&"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "false"
     ]
    }
   ],
   "source": [
    "clear \n",
    "[false] [true] &&"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "true"
     ]
    }
   ],
   "source": [
    "clear \n",
    "[true] [false] ||"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "true"
     ]
    }
   ],
   "source": [
    "clear \n",
    "[false] [true] ||"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": []
    }
   ],
   "source": [
    "clear"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Translating the Conditionals\n",
    "\n",
    "Given those, we can define `x != y || x >= 0` as:\n",
    "\n",
    "    _a == [!=] [pop 0 >=] ||"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": []
    }
   ],
   "source": [
    "[_a [!=] [pop 0 >=] ||] inscribe"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And `(abs(x) <= abs(y) && (x != y || x >= 0))` as:\n",
    "\n",
    "    _b == [_p] [_a] &&"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": []
    }
   ],
   "source": [
    "[_b [_p] [_a] &&] inscribe"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It's a little rough, but, as I say, with a little familiarity it becomes\n",
    "legible."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "23 -18"
     ]
    }
   ],
   "source": [
    "clear 23 -18"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                                      23 -18 • _b\n",
      "                                      23 -18 • [_p] [_a] &&\n",
      "                                 23 -18 [_p] • [_a] &&\n",
      "                            23 -18 [_p] [_a] • &&\n",
      "                            23 -18 [_p] [_a] • [nullary] cons [nullary [false]] dip branch\n",
      "                  23 -18 [_p] [_a] [nullary] • cons [nullary [false]] dip branch\n",
      "                  23 -18 [_p] [[_a] nullary] • [nullary [false]] dip branch\n",
      "23 -18 [_p] [[_a] nullary] [nullary [false]] • dip branch\n",
      "                                 23 -18 [_p] • nullary [false] [[_a] nullary] branch\n",
      "                                 23 -18 [_p] • [stack] dinfrirst [false] [[_a] nullary] branch\n",
      "                         23 -18 [_p] [stack] • dinfrirst [false] [[_a] nullary] branch\n",
      "                         23 -18 [_p] [stack] • dip infrst [false] [[_a] nullary] branch\n",
      "                                      23 -18 • stack [_p] infrst [false] [[_a] nullary] branch\n",
      "                             23 -18 [-18 23] • [_p] infrst [false] [[_a] nullary] branch\n",
      "                        23 -18 [-18 23] [_p] • infrst [false] [[_a] nullary] branch\n",
      "                        23 -18 [-18 23] [_p] • infra first [false] [[_a] nullary] branch\n",
      "                                      23 -18 • _p [-18 23] swaack first [false] [[_a] nullary] branch\n",
      "                                      23 -18 • [abs] ii <= [-18 23] swaack first [false] [[_a] nullary] branch\n",
      "                                23 -18 [abs] • ii <= [-18 23] swaack first [false] [[_a] nullary] branch\n",
      "                                          23 • abs -18 abs <= [-18 23] swaack first [false] [[_a] nullary] branch\n",
      "                                          23 • -18 abs <= [-18 23] swaack first [false] [[_a] nullary] branch\n",
      "                                      23 -18 • abs <= [-18 23] swaack first [false] [[_a] nullary] branch\n",
      "                                       23 18 • <= [-18 23] swaack first [false] [[_a] nullary] branch\n",
      "                                       false • [-18 23] swaack first [false] [[_a] nullary] branch\n",
      "                              false [-18 23] • swaack first [false] [[_a] nullary] branch\n",
      "                              23 -18 [false] • first [false] [[_a] nullary] branch\n",
      "                                23 -18 false • [false] [[_a] nullary] branch\n",
      "                        23 -18 false [false] • [[_a] nullary] branch\n",
      "         23 -18 false [false] [[_a] nullary] • branch\n",
      "                                      23 -18 • false\n",
      "                                23 -18 false • \n",
      "\n",
      "23 -18 false"
     ]
    }
   ],
   "source": [
    "[_b] trace"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": []
    }
   ],
   "source": [
    "clear"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The Increment / Decrement Branches\n",
    "\n",
    "Turning to the branches of the main `if` statement:\n",
    "\n",
    "    x += ((y >= 0) ? 1 : -1);\n",
    "\n",
    "Rewrite as a hybrid (pseudo-code) `ifte` expression:\n",
    "\n",
    "    [y >= 0] [x += 1] [X -= 1] ifte\n",
    "\n",
    "Change each C phrase to Joy code:\n",
    "\n",
    "    [0 >=] [[++] dip] [[--] dip] ifte\n",
    "\n",
    "Factor out the dip from each branch:\n",
    "\n",
    "    [0 >=] [[++]] [[--]] ifte dip\n",
    "\n",
    "Similar logic applies to the other branch:\n",
    "\n",
    "    y += ((x >= 0) ? -1 : 1);\n",
    "\n",
    "    [x >= 0] [y -= 1] [y += 1] ifte\n",
    "\n",
    "    [pop 0 >=] [--] [++] ifte"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Putting the Pieces Together\n",
    "\n",
    "We can assemble the three functions we just defined in quotes and give\n",
    "them them to the `ifte` combinator.  With some arrangement to show off\n",
    "the symmetry of the two branches, we have:\n",
    "\n",
    "    [[[abs] ii <=] [[<>] [pop !-] ||] &&]\n",
    "    [[    !-] [[++]] [[--]] ifte dip]\n",
    "    [[pop !-]  [--]   [++]  ifte    ]\n",
    "    ifte"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": []
    }
   ],
   "source": [
    "[spiral_next\n",
    "\n",
    "[_b]\n",
    "[[    !-] [[++]] [[--]] ifte dip]\n",
    "[[pop !-]  [--]   [++]  ifte    ]\n",
    "ifte\n",
    "\n",
    "] inscribe"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As I was writing this up I realized that, since the `&&` combinator\n",
    "doesn't consume the stack (below its quoted args), I can unquote the\n",
    "predicate, swap the branches, and use the `branch` combinator instead of\n",
    "`ifte`:\n",
    "\n",
    "    [[abs] ii <=] [[<>] [pop !-] ||] &&\n",
    "    [[pop !-]  [--]   [++]  ifte    ]\n",
    "    [[    !-] [[++]] [[--]] ifte dip]\n",
    "    branch"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's try it out:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0 0"
     ]
    }
   ],
   "source": [
    "clear 0 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1 0"
     ]
    }
   ],
   "source": [
    "spiral_next"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1 -1"
     ]
    }
   ],
   "source": [
    "spiral_next"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0 -1"
     ]
    }
   ],
   "source": [
    "spiral_next"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-1 -1"
     ]
    }
   ],
   "source": [
    "spiral_next"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-1 0"
     ]
    }
   ],
   "source": [
    "spiral_next"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-1 1"
     ]
    }
   ],
   "source": [
    "spiral_next"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0 1"
     ]
    }
   ],
   "source": [
    "spiral_next"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1 1"
     ]
    }
   ],
   "source": [
    "spiral_next"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2 1"
     ]
    }
   ],
   "source": [
    "spiral_next"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2 0"
     ]
    }
   ],
   "source": [
    "spiral_next"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2 -1"
     ]
    }
   ],
   "source": [
    "spiral_next"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2 -2"
     ]
    }
   ],
   "source": [
    "spiral_next"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1 -2"
     ]
    }
   ],
   "source": [
    "spiral_next"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0 -2"
     ]
    }
   ],
   "source": [
    "spiral_next"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-1 -2"
     ]
    }
   ],
   "source": [
    "spiral_next"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Turning it into a Generator with `x`\n",
    "\n",
    "It can be used with the x combinator to make a kind of generator for\n",
    "spiral square coordinates.\n",
    "\n",
    "\n",
    "We can use `codireco` to make a generator\n",
    "\n",
    "    codireco == cons dip rest cons\n",
    "\n",
    "It will look like this:\n",
    "\n",
    "    [value [F] codireco]\n",
    "\n",
    "Here's a trace of how it works:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "           [0 [dup ++] codireco] • x\n",
      "           [0 [dup ++] codireco] • 0 [dup ++] codireco\n",
      "         [0 [dup ++] codireco] 0 • [dup ++] codireco\n",
      "[0 [dup ++] codireco] 0 [dup ++] • codireco\n",
      "[0 [dup ++] codireco] 0 [dup ++] • codi reco\n",
      "[0 [dup ++] codireco] 0 [dup ++] • cons dip reco\n",
      "[0 [dup ++] codireco] [0 dup ++] • dip reco\n",
      "                                 • 0 dup ++ [0 [dup ++] codireco] reco\n",
      "                               0 • dup ++ [0 [dup ++] codireco] reco\n",
      "                             0 0 • ++ [0 [dup ++] codireco] reco\n",
      "                             0 1 • [0 [dup ++] codireco] reco\n",
      "       0 1 [0 [dup ++] codireco] • reco\n",
      "       0 1 [0 [dup ++] codireco] • rest cons\n",
      "         0 1 [[dup ++] codireco] • cons\n",
      "         0 [1 [dup ++] codireco] • \n",
      "\n",
      "0 [1 [dup ++] codireco]"
     ]
    }
   ],
   "source": [
    "clear\n",
    "\n",
    "[0 [dup ++] codireco] [x] trace"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": []
    }
   ],
   "source": [
    "clear"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But first we have to change the `spiral_next` function to work on a\n",
    "quoted pair of integers, and leave a copy of the pair on the stack.\n",
    "From:\n",
    "\n",
    "       y x spiral_next\n",
    "    ---------------------\n",
    "            y' x'\n",
    "\n",
    "to:\n",
    "\n",
    "       [x y] [spiral_next] infra\n",
    "    -------------------------------\n",
    "               [x' y']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0 1]"
     ]
    }
   ],
   "source": [
    "[0 0] [spiral_next] infra"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So our generator is:\n",
    "\n",
    "    [[x y] [dup [spiral_next] infra] codireco]\n",
    "\n",
    "Or rather:\n",
    "\n",
    "    [[0 0] [dup [spiral_next] infra] codireco]\n",
    "\n",
    "There is a function `make_generator` that will build the generator for us\n",
    "out of the value and stepper function:\n",
    "\n",
    "       [0 0] [dup [spiral_next] infra] make_generator\n",
    "    ----------------------------------------------------\n",
    "         [[0 0] [dup [spiral_next] infra] codireco]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": []
    }
   ],
   "source": [
    "clear"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here it is in action:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0 0] [0 1] [-1 1] [-1 0]"
     ]
    }
   ],
   "source": [
    "[0 0] [dup [spiral_next] infra] make_generator x x x x pop"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Four `x` combinators, four pairs of coordinates.\n",
    "\n",
    "Or you can leave out `dup` and let the value stay in the generator until you want it:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[2 4]"
     ]
    }
   ],
   "source": [
    "clear\n",
    "\n",
    "[0 0] [[spiral_next] infra] make_generator 50 [x] times first"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Conclusion\n",
    "\n",
    "So that's an example of Joy code.  It's a straightforward translation of\n",
    "the original.  It's a little long for a single definition, you might\n",
    "break it up like so:\n",
    "\n",
    "    _spn_Pa == [abs] ii <=\n",
    "    _spn_Pb == [!=] [pop 0 >=] ||\n",
    "    _spn_P  == [_spn_Pa] [_spn_Pb] &&\n",
    "    \n",
    "    _spn_T == [    !-] [[++]] [[--]] ifte dip\n",
    "    _spn_E == [pop !-]  [--]   [++]  ifte\n",
    "\n",
    "    spiral_next == _spn_P [_spn_E] [_spn_T] branch\n",
    "\n",
    "This way it's easy to see that the function is a branch with two\n",
    "quasi-symmetrical paths.\n",
    "\n",
    "We then used this function to make a simple generator of coordinate\n",
    "pairs, where the next pair in the series can be generated at any time by\n",
    "using the `x` combinator on the generator (which is just a quoted\n",
    "expression containing a copy of the current pair and the \"stepper\n",
    "function\" to generate the next pair from that.)"
   ]
  }
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