{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# The NAND++ Programming language\n",
    "\n",
    "_Version: 0.2_"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The NAND++ programming language was designed to accompany the upcoming book [\"Introduction to Theoretical Computer Science\"](http://introtcs.org). This is an appendix to this book, which is also available online as a Jupyter notebook in the  [boazbk/nandnotebooks](https://github.com/boazbk/nandnotebooks) on Github. You can also try the [live binder version](https://hub.mybinder.org/user/boazbk-nandnotebooks-221ipnov/notebooks/NANDpp_language.ipynb).\n",
    "\n",
    "The NAND++ programming language is defined in [Chapter 6: \"Loops and Infinity\"](https://introtcs.netlify.com/public/lec_06_loops.html)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The NAND programming language we saw before corresponds to _non uniform_, _finite_ computation.\n",
    "\n",
    "NAND++ captures uniform computation and is equivalent to Turing Machines.\n",
    "\n",
    "One way to think about NAND++ is\n",
    "\n",
    "$$NAND++ = NAND \\;+\\;  \\text{loops} \\;+\\; \\text{arrays}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Enhanced NAND++\n",
    "\n",
    "We start by describing \"enhanced NAND++ programs\", and later we will describe \"vanilla\" or \"standard\" NAND++.\n",
    "\n",
    "Enhanced NAND++ programs have the following form: every line is either of the form\n",
    "\n",
    "`foo = NAND(bar,blah)`\n",
    "\n",
    "or\n",
    "\n",
    "`i += foo`\n",
    "\n",
    "or \n",
    "\n",
    "`i -= foo`\n",
    "\n",
    "where `foo` is a variable identifier that is either a _scalar variable_, which is a sequence of letters, numbers and underscopres, or an _array element_, which starts with a capital letter, and ends with `[i]`\n",
    "\n",
    "We have a special variable `loop`. If `loop` is set to $1$ at the end of the program then execution goes back to the beginning.\n",
    "\n",
    "We have the special input and output arrays `X[.]` and  `Y[.]` but because their length is not fixed in advance, we also have  `Xvalid[.]` and `Yvalid[.]` arrays. The input is  `X[` $0$ `]` , ... , `X[` $n-1$ `]` where $n$ is the smallest integer such that `Xvalid[` $n$ `]` $=0$. The output is `Y[` $0$ `]` , ..., `Y[` $m-1$ `]` where $m$ is the smallest integer such that `Yvalid[` $m$ `]` $=0$.\n",
    "\n",
    "The default value of every variable in NAND++ is zero."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Ignore in first read: utility code\n",
    "\n",
    "We use some utility code, which you can safely ignore in first read, to allow us to write NAND++ code in Python"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# utility code \n",
    "%run \"NAND programming language.ipynb\"\n",
    "from IPython.display import clear_output\n",
    "clear_output()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Ignore this utility function in first and even second and third read\n",
    "import inspect\n",
    "import ast\n",
    "import astor\n",
    "\n",
    "def noop(f):\n",
    "    return f;\n",
    "\n",
    "def runwithstate(f):\n",
    "    \"\"\"Modify a function f to take and return an argument state and make all names relative to state.\"\"\"\n",
    "    tree = ast.parse(inspect.getsource(f))\n",
    "    tmp = ast.parse(\"def _temp(state):\\n    pass\\n\").body[0]\n",
    "    args = tmp.args\n",
    "    name = tmp.name\n",
    "    tree.body[0].args = args\n",
    "    tree.body[0].name = name\n",
    "    tree.body[0].decorator_list = []\n",
    "    \n",
    "\n",
    "    class AddState(ast.NodeTransformer):\n",
    "        def visit_Name(self, node: ast.Name):\n",
    "            if node.id == \"enandpp\": return ast.Name(id=\"noop\", ctx=Load())\n",
    "            new_node = ast.Attribute(ast.copy_location(ast.Name('state', ast.Load()), node), node.id,\n",
    "                                     ast.copy_location(ast.Load(), node))\n",
    "            return ast.copy_location(new_node, node)\n",
    "        \n",
    "    tree = AddState().visit(tree)\n",
    "    tree.body[0].body = tree.body[0].body + [ast.parse(\"return state\")]\n",
    "    tree = ast.fix_missing_locations(tree)\n",
    "    src = astor.to_source(tree)\n",
    "    # print(src)\n",
    "    exec(src,globals())\n",
    "    _temp.original_func = f\n",
    "    return _temp\n",
    "\n",
    "    \n",
    "def enandpp(f):\n",
    "    g = runwithstate(f)\n",
    "    def _temp1(X):\n",
    "        nonlocal g\n",
    "        return ENANDPPEVAL(g,X)\n",
    "    _temp1.original_func = f\n",
    "    _temp1.transformed_func = g\n",
    "    return _temp1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# ignore utility class in first and even second or third read\n",
    "\n",
    "from  collections import defaultdict\n",
    "class NANDPPstate:\n",
    "    \"\"\"State of a NAND++ program.\"\"\"\n",
    "    \n",
    "    def __init__(self):\n",
    "        self.scalars = defaultdict(int)\n",
    "        self.arrays  = defaultdict(lambda: defaultdict(int))\n",
    "        # eventually should make self.i non-negative integer type\n",
    "        \n",
    "    def __getattr__(self,var):\n",
    "        g =  globals()\n",
    "        if var in g and callable(g[var]): return g[var]\n",
    "        if var[0].isupper():\n",
    "            return self.arrays[var]\n",
    "        else:\n",
    "            return self.scalars[var]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def ENANDPPEVAL(f,X):\n",
    "    \"\"\"Evaluate an enhanced NAND++ function on input X\"\"\"\n",
    "    s = NANDPPstate()\n",
    "    for i in range(len(X)):\n",
    "        s.X[i] = X[i]\n",
    "        s.Xvalid[i] = 1\n",
    "    while True:\n",
    "        s = f(s)\n",
    "        if not s.loop: break\n",
    "    res = []\n",
    "    i = 0\n",
    "    while s.Yvalid[i]: \n",
    "        res += [s.Y[i]]\n",
    "        i+= 1\n",
    "    return res"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "def rreplace(s, old, new, occurrence=1): # from stackoverflow\n",
    "    li = s.rsplit(old, occurrence)\n",
    "    return new.join(li)\n",
    "\n",
    "    \n",
    "        \n",
    "def ENANDPPcode(P):\n",
    "    \"\"\"Return ENAND++ code of given function\"\"\"\n",
    "    \n",
    "    code = ''\n",
    "    counter = 0\n",
    "    \n",
    "    class CodeENANDPPcounter:\n",
    "        def __init__(self,name=\"i\"): \n",
    "            self.name = name\n",
    "        \n",
    "        def __iadd__(self,var):\n",
    "            nonlocal code\n",
    "            code += f'\\ni += {var}'\n",
    "            return self\n",
    "        \n",
    "        def __isub__(self,var):\n",
    "            nonlocal code\n",
    "            code += f'\\ni -= {var}'\n",
    "            return self\n",
    "        \n",
    "        def __str__(self): return self.name\n",
    "    \n",
    "    class CodeNANDPPstate:\n",
    "    \n",
    "    \n",
    "        def __getattribute__(self,var):\n",
    "            # print(f\"getting {var}\")\n",
    "            if var=='i': return CodeENANDPPcounter()\n",
    "            g =  globals()\n",
    "            if var in g and callable(g[var]): return g[var]\n",
    "            if var[0].isupper():  \n",
    "                class Temp:\n",
    "                    def __getitem__(self,k):  return f\"{var}[{str(k)}]\"\n",
    "                    def __setitem__(s,k,v): setattr(self,f\"{var}[{str(k)}]\",v)            \n",
    "                return Temp()\n",
    "            return var\n",
    "    \n",
    "        def __init__(self):\n",
    "            pass\n",
    "    \n",
    "        def __setattr__(self,var,val):\n",
    "            nonlocal code\n",
    "            if var=='i': return\n",
    "            if code.find(val)==-1:\n",
    "                code += f'\\n{var} = {val}'\n",
    "            else:\n",
    "                code = rreplace(code,val,var)\n",
    "    \n",
    "    s = CodeNANDPPstate()\n",
    "    \n",
    "    def myNAND(a,b):\n",
    "        nonlocal code, counter\n",
    "        var = f'temp_{counter}'\n",
    "        counter += 1\n",
    "        code += f'\\n{var} = NAND({a},{b})'\n",
    "        return var\n",
    "    \n",
    "        \n",
    "    \n",
    "    \n",
    "    \n",
    "    s = runwith(lambda : P.transformed_func(s),\"NAND\",myNAND) \n",
    "    \n",
    "    return code"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Our first NAND++ program\n",
    "\n",
    "Here is an enhanced NAND++ program to increment a number:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "@enandpp\n",
    "def inc():\n",
    "    carry = IF(started,carry,one(started))\n",
    "    started = one(started)\n",
    "    Y[i] = XOR(X[i],carry)\n",
    "    carry = AND(X[i],carry)\n",
    "    Yvalid[i] = one(started)\n",
    "    loop = COPY(Xvalid[i])\n",
    "    i += loop"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0, 0, 1, 0, 1, 0]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "inc([1,1,0,0,1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "temp_0 = NAND(started,started)\n",
      "temp_1 = NAND(started,temp_0)\n",
      "temp_2 = NAND(started,started)\n",
      "temp_3 = NAND(temp_1,temp_2)\n",
      "temp_4 = NAND(carry,started)\n",
      "carry = NAND(temp_3,temp_4)\n",
      "temp_6 = NAND(started,started)\n",
      "started = NAND(started,temp_6)\n",
      "temp_8 = NAND(X[i],carry)\n",
      "temp_9 = NAND(X[i],temp_8)\n",
      "temp_10 = NAND(carry,temp_8)\n",
      "Y[i] = NAND(temp_9,temp_10)\n",
      "temp_12 = NAND(X[i],carry)\n",
      "carry = NAND(temp_12,temp_12)\n",
      "temp_14 = NAND(started,started)\n",
      "Yvalid[i] = NAND(started,temp_14)\n",
      "temp_16 = NAND(Xvalid[i],Xvalid[i])\n",
      "loop = NAND(temp_16,temp_16)\n",
      "i += loop\n"
     ]
    }
   ],
   "source": [
    "print(ENANDPPcode(inc))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And here is an enhanced NAND++ program to compute the `XOR` function on unbounded length inputs (it uses `XOR` on two variables as a subroutine):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "@enandpp\n",
    "def UXOR():\n",
    "    Yvalid[0] = one(X[0])\n",
    "    Y[0] = XOR(X[i],Y[0])\n",
    "    loop = Xvalid[i]\n",
    "    i += Xvalid[i]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0]"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "UXOR([1,1,0,0,1,1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "temp_0 = NAND(X[0],X[0])\n",
      "Yvalid[0] = NAND(X[0],temp_0)\n",
      "temp_2 = NAND(X[i],Y[0])\n",
      "temp_3 = NAND(X[i],temp_2)\n",
      "temp_4 = NAND(Y[0],temp_2)\n",
      "Y[0] = NAND(temp_3,temp_4)\n",
      "loop = Xvalid[i]\n",
      "i += Xvalid[i]\n"
     ]
    }
   ],
   "source": [
    "print(ENANDPPcode(UXOR))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## \"Vanilla\" NAND++\n",
    "\n",
    "In \"vanilla\" NAND++ we do not have the commands `i += foo` and `i -= foo` but rather `i` travels obliviously according to the sequence $0,1,0,1,2,1,0,1,2,3,2,1,0,1,2,\\ldots$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0, 1, 0, 1, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 3, 4, 3, 2, 1]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def index():\n",
    "    \"\"\"Generator for the values of i in the NAND++ sequence\"\"\"\n",
    "    i = 0\n",
    "    last = 0\n",
    "    direction  = 1\n",
    "    while True:\n",
    "        yield i\n",
    "        i += direction\n",
    "        if i> last: \n",
    "            direction = -1\n",
    "            last = i\n",
    "        if i==0: direction = +1\n",
    "            \n",
    "a = index()\n",
    "[next(a) for i in range(20)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "def NANDPPEVAL(f,X):\n",
    "    \"\"\"Evaluate a NAND++ function on input X\"\"\"\n",
    "    s = NANDPPstate() # intialize state\n",
    "    \n",
    "    # copy input:\n",
    "    for i in range(len(X)): \n",
    "        s.X[i] = X[i]\n",
    "        s.Xvalid[i] = 1\n",
    "        \n",
    "    # main loop:\n",
    "    for  i in index(): \n",
    "        s.i = i\n",
    "        s = f(s)\n",
    "        if not s.loop: break\n",
    "    \n",
    "    # copy output:\n",
    "    res = [] \n",
    "    i = 0\n",
    "    while s.Yvalid[i]: \n",
    "        res += [s.Y[i]]\n",
    "        i+= 1\n",
    "    return res\n",
    "\n",
    "\n",
    "\n",
    "def nandpp(f):\n",
    "    \"\"\"Modify python code to obtain NAND++ program\"\"\"\n",
    "    g = runwithstate(f)\n",
    "    def _temp1(X):\n",
    "        return NANDPPEVAL(g,X)\n",
    "    _temp1.original_func = f\n",
    "    _temp1.transformed_func = g\n",
    "    return _temp1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here is the increment function in vanilla NAND++. Note that we need to keep track of an Array `Visited` to make sure we only add the carry once per location."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "@nandpp\n",
    "def inc():\n",
    "    carry = IF(started,carry,one(started))\n",
    "    started = one(started)\n",
    "    Y[i] = IF(Visited[i],Y[i],XOR(X[i],carry))\n",
    "    Visited[i] = one(started)\n",
    "    carry = AND(X[i],carry)\n",
    "    Yvalid[i] = one(started)\n",
    "    loop = Xvalid[i]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0, 0, 1, 1, 1, 0]"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "inc([1,1,0,1,1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And here is the \"vanilla NAND++\" version of XOR:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "@nandpp\n",
    "def vuXOR():\n",
    "    Yvalid[0] = one(X[0])\n",
    "    Y[0] = IF(Visited[i],Y[0],XOR(X[i],Y[0]))\n",
    "    Visited[i] = one(X[0])\n",
    "    loop = Xvalid[i]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0]"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vuXOR([1,0,0,1,0,1,1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "def NANDPPcode(P):\n",
    "    \"\"\"Return NAND++ code of given function\"\"\"\n",
    "    \n",
    "    code = ''\n",
    "    counter = 0\n",
    "    \n",
    "    \n",
    "    class CodeNANDPPstate:\n",
    "    \n",
    "    \n",
    "        def __getattribute__(self,var):\n",
    "            # print(f\"getting {var}\")\n",
    "            g =  globals()\n",
    "            if var in g and callable(g[var]): return g[var]\n",
    "            if var[0].isupper():  \n",
    "                class Temp:\n",
    "                    def __getitem__(self,k):  return var+\"[i]\"\n",
    "                    def __setitem__(s,k,v):   \n",
    "                        setattr(self,var+\"[i]\",v)            \n",
    "                return Temp()\n",
    "            return var\n",
    "    \n",
    "        def __init__(self):\n",
    "            pass\n",
    "    \n",
    "        def __setattr__(self,var,val):\n",
    "            nonlocal code\n",
    "            # print(f\"setting {var} to {val}\")\n",
    "            if code.find(val)==-1:\n",
    "                code += f'\\n{var} = {val}'\n",
    "            else:\n",
    "                code = rreplace(code,val,var)\n",
    "    \n",
    "    s = CodeNANDPPstate()\n",
    "    \n",
    "    def myNAND(a,b):\n",
    "        nonlocal code, counter\n",
    "        var = f'temp_{counter}'\n",
    "        counter += 1\n",
    "        code += f'\\n{var} = NAND({a},{b})'\n",
    "        return var\n",
    "    \n",
    "        \n",
    "    \n",
    "    \n",
    "    \n",
    "    s = runwith(lambda : P.transformed_func(s),\"NAND\",myNAND) \n",
    "    \n",
    "    return code\n",
    "\n",
    "\n",
    "# utility code - replace string from right, taken from stackoverflow\n",
    "def rreplace(s, old, new, occurrence=1): \n",
    "    li = s.rsplit(old, occurrence)\n",
    "    return new.join(li)\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "temp_0 = NAND(started,started)\n",
      "temp_1 = NAND(started,temp_0)\n",
      "temp_2 = NAND(started,started)\n",
      "temp_3 = NAND(temp_1,temp_2)\n",
      "temp_4 = NAND(carry,started)\n",
      "carry = NAND(temp_3,temp_4)\n",
      "temp_6 = NAND(started,started)\n",
      "started = NAND(started,temp_6)\n",
      "temp_8 = NAND(X[i],carry)\n",
      "temp_9 = NAND(X[i],temp_8)\n",
      "temp_10 = NAND(carry,temp_8)\n",
      "temp_11 = NAND(temp_9,temp_10)\n",
      "temp_12 = NAND(Visited[i],Visited[i])\n",
      "temp_13 = NAND(temp_11,temp_12)\n",
      "temp_14 = NAND(Y[i],Visited[i])\n",
      "Y[i] = NAND(temp_13,temp_14)\n",
      "temp_16 = NAND(started,started)\n",
      "Visited[i] = NAND(started,temp_16)\n",
      "temp_18 = NAND(X[i],carry)\n",
      "carry = NAND(temp_18,temp_18)\n",
      "temp_20 = NAND(started,started)\n",
      "Yvalid[i] = NAND(started,temp_20)\n",
      "loop = Xvalid[i]\n"
     ]
    }
   ],
   "source": [
    "print(NANDPPcode(inc))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Tranforming Enhanced NAND++ to NAND++"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Eventually we will have here code to automatically transform an enhanced NAND++ program into a NAND++ program.\n",
    "At the moment, let us just give the high level ideas. See [Chapter 6 in the book](http://introtcs.org/public/lec_06_loops.html) for more details.\n",
    "\n",
    "\n",
    "To transform an enhanced NAND++ program to a standard NAND++ program we do the following:\n",
    "\n",
    "1. We make all our operations \"guarded\" in the sense that there is a special variable `noop` such that if `noop` equals $1$ then we do not make any writes.\n",
    "\n",
    "2. We use a `Visited` array to keep track of all locations we visited, and use that to keep track of an `decreasing` variable that is equal to $1$ if and only the value of `i` in the next step will be one smaller.\n",
    "\n",
    "3. If we have an operation of the form `i += foo` or `i -= foo` at line $\\ell$ then we replace it with lines of code that do the following: \n",
    "\n",
    "    a.  (Guarded) set `temp_`$\\ell$ ` = foo`\n",
    "    \n",
    "    b.  (Unguarded) If `Waitingline_` $\\ell$ and `Restart[i]` : set `noop`=$0$ if  `increasing` is equal to `wait_increasing`. (Otherwise `noop` stays the same.)\n",
    "    \n",
    "    c.  (Guarded) set `Restart[i]` to $1$.\n",
    "    \n",
    "    d.  (Guarded) set `Waitingline_` $\\ell$  to $1$.\n",
    "    \n",
    "    e.  (Guarded) set `wait_increasing` to $1$ if the operation is `i += foo` and to $0$ if it's `i -= foo`\n",
    "    \n",
    "    f.  (Guarded) set `noop = temp_` $\\ell$\n",
    "    \n",
    "    g.  (Unguarded) set `temp_` $\\ell$ `= 0`\n",
    "    \n",
    "    h.  (Guarded) set `Restart[i]` to $0$.\n",
    "    \n",
    "    i.  (Guarded) set `Waitingline_`$\\ell$ to $0$."
   ]
  },
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   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
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