Lec_05_sugar.ipynb

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    "%%html\n",
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   "source": [
    "# CS 121 Lecture 5 : Syntactic sugar and computing every function"
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   "source": [
    "# utility code \n",
    "%run \"Utilities.ipynb\"\n",
    "from IPython.display import clear_output\n",
    "clear_output()"
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   "source": [
    "def AND(a,b): return a*b\n",
    "\n",
    "def OR(a,b): return 1 if a+b else 0\n",
    "\n",
    "def NOT(a): return 1-a"
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    "def IF(cond,a,b):\n",
    "    notcond = NAND(cond,cond)\n",
    "    temp = NAND(b,notcond)\n",
    "    temp1 = NAND(a,cond)\n",
    "    return NAND(temp,temp1)\n",
    "mystery = circuit2prog(circuit(IF))"
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      "gate_0 = NAND(X[0],X[0])\n",
      "gate_1 = NAND(X[2],gate_0)\n",
      "gate_2 = NAND(X[1],X[0])\n",
      "Y[0] = NAND(gate_1,gate_2)\n",
      "\n"
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   ],
   "source": [
    "# What function does the following NAND-CIRC program compute?\n",
    "print(mystery)\n",
    "circuit(mystery)"
   ]
  },
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     "data": {
      "text/markdown": [
       "_Input_ | _Output_\n",
       "--------|---------\n",
       "  $000$  | $0$ \n",
       "  $001$  | $1$ \n",
       "  $010$  | $0$ \n",
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   "source": [
    "table(mystery)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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   },
   "source": [
    "__Theorem:__ Let NAND-CIRC++ be the language that has on \n",
    "top of NAND-CIRC the following construct:\n",
    "```Python\n",
    "if (cond) {\n",
    "    do something\n",
    "}``` \n",
    "then for every $f:\\{0,1\\}^n \\rightarrow \\{0,1\\}^m$, if $f$ is computable by a NAND-CIRC++ program, then $f$ \n",
    "is computable by a NAND-CIRC program."
   ]
  },
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   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
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   },
   "source": [
    "__Proof:__ First, can use `foo = IF(cond,bar,blah)` since we can compute `IF`. <br>\n",
    "Transform a line `foo = NAND(bar,blah)` in `do something` to\n",
    "```Python\n",
    "temp = NAND(bar,blah)\n",
    "foo  = IF(cond,temp,foo)\n",
    "```"
   ]
  },
  {
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   "metadata": {
    "slideshow": {
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   },
   "source": [
    "## The lookup function"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "We define $LOOKUP_\\ell(x,i)=x_i$ where $x\\in \\{0,1\\}^{2^\\ell}$ and  $i\\in \\{0,1\\}^\\ell$.\n",
    "\n",
    "$$LOOKUP_1(x_0,x_1,i_0) = x_{i_0}$$\n",
    "\n",
    "$$LOOKUP_2(x_0,x_1,x_2,x_3,i_0,i_1) = x_{i_0 i_1} = \n",
    "\\begin{cases}x_0, & i_0=0,i_1 =0 \\\\ x_1, & i_0=0,i_1=1 \\\\\n",
    "x_2, & i_0 =1, i_1 =0 \\\\\n",
    "x_3, & i_0=1, i_1=1 \\end{cases}\n",
    "$$\n",
    "\n",
    "__Q:__ Compute $LOOKUP_1$, $LOOKUP_2$ using NANDs or AND/OR/NOTs - you can use `IF`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
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   },
   "outputs": [],
   "source": [
    "def LOOKUP1(x_0,x_1,i_0): return IF(i_0,x_1,x_0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0 1 1 0\n"
     ]
    }
   ],
   "source": [
    "def LOOKUP2(x_0,x_1,x_2,x_3,i_0,i_1):\n",
    "    return IF(i_0,\n",
    "             IF(i_1,x_3,x_2),\n",
    "              IF(i_1,x_1,x_0)\n",
    "             )\n",
    "\n",
    "x_0,x_1,x_2,x_3 = 0,1,1,0\n",
    "print(LOOKUP2(x_0,x_1,x_2,x_3,0,0),\n",
    "      LOOKUP2(x_0,x_1,x_2,x_3,0,1),\n",
    "      LOOKUP2(x_0,x_1,x_2,x_3,1,0),\n",
    "      LOOKUP2(x_0,x_1,x_2,x_3,1,1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
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   "source": [
    "circuit(LOOKUP2)"
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   "execution_count": 48,
   "metadata": {
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   "outputs": [
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   "source": [
    "def LOOKUP3(x_0,x_1,x_2,x_3,x_4,x_5,x_6,x_7,i_0,i_1,i_2):\n",
    "    return IF(i_0,\n",
    "      LOOKUP2(x_4,x_5,x_6,x_7,i_1,i_2),\n",
    "      LOOKUP2(x_0,x_1,x_2,x_3,i_1,i_2))\n",
    "x_0,x_1,x_2,x_3,x_4,x_5,x_6,x_7 = 1,1,0,0,1,1,1,0\n",
    "print(LOOKUP3(x_0,x_1,x_2,x_3,x_4,x_5,x_6,x_7,0,1,0),\n",
    "      LOOKUP3(x_0,x_1,x_2,x_3,x_4,x_5,x_6,x_7,1,0,0),\n",
    "      LOOKUP3(x_0,x_1,x_2,x_3,x_4,x_5,x_6,x_7,1,1,1))"
   ]
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   "source": [
    "circuit(LOOKUP3)"
   ]
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  {
   "cell_type": "markdown",
   "metadata": {
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     "slide_type": "slide"
    }
   },
   "source": [
    "__Q:__ Write a recursive equation for $LOOKUP_{\\ell+1}$ in terms of $LOOKUP_{\\ell}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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   "source": [
    "$$LOOKUP_{\\ell+1}(x,i) = \\begin{cases} LOOKUP_\\ell(x_0,\\ldots,x_{2^{\\ell-1}-1},i_1,\\ldots,i_{\\ell-1}), & i_0 = 0 \\\\\n",
    "LOOKUP_\\ell(x_{2^{\\ell-1}},\\ldots,x_{2^{\\ell}-1},i_1,\\ldots,i_{\\ell-1}), & i_0 = 1  \\end{cases}$$"
   ]
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  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
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     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4 12 28\n"
     ]
    }
   ],
   "source": [
    "print(circuit(IF).size(), circuit(LOOKUP2).size(), circuit(LOOKUP3).size())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "__Cor:__ $size(LOOKUP_{\\ell+1}) \\leq 2\\cdot size(LOOKUP_\\ell) + 4$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "__Cor:__ $LOOKUP_\\ell \\in SIZE(4 \\cdot 2^\\ell)$ for every $\\ell$. <br>\n",
    "$SIZE(s) := \\{ f \\;|\\; f \\text{ computable by circuit of $\\leq s$ gates } \\}$"
   ]
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  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "<center><img src=\"./funcvscircs.png\" width=\"800px\" height=\"600px\"></img></center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "![](cucumber.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## More syntactic sugar"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "__Q:__ Let $one:\\{0,1\\} \\rightarrow \\{0,1\\}$ be the function $one(a)=1$ for $a\\in \\{0,1\\}$, and let $zero:\\{0,1\\} \\rightarrow \\{0,1\\}$\n",
    "be function $zero(a)=0$ for $a\\in \\{0,1\\}$. Give NAND circuits to compute $one$ and $zero$."
   ]
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  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
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   },
   "outputs": [
    {
     "data": {
      "text/markdown": [
       "_Input_ | _Output_\n",
       "--------|---------\n",
       "  $0$  | $1$ \n",
       "  $1$  | $1$ \n"
      ],
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/markdown": [
       "_Input_ | _Output_\n",
       "--------|---------\n",
       "  $0$  | $0$ \n",
       "  $1$  | $0$ \n"
      ],
      "text/plain": [
       "<IPython.core.display.Markdown object>"
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   "source": [
    "def one(a):\n",
    "    return NAND(a,NAND(a,a))\n",
    "def zero(a):\n",
    "    b = one(a)\n",
    "    return NAND(b,b)\n",
    "table(one)\n",
    "table(zero)"
   ]
  },
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   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
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     "slide_type": "slide"
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   "outputs": [
    {
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   ],
   "source": [
    "circuit(zero)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "__Corollary:__ Can add the function `LOOKUP` and the constants `0,1` to NAND-CIRC and get equivalent power."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "__Corollary:__ For every $f:\\{0,1\\}^\\ell \\rightarrow \\{0,1\\}$, can compute $f$ using \n",
    "a circuit of size at most $10\\cdot 2^\\ell$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Even more  sugar"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Define NAND-CIRC-PROC to be NAND-CIRC + procedure defining code\n",
    "\n",
    "Then can transform every NAND-CIRC-PROC program to NAND-CIRC program"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "import re"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "def strip_empty_lines(code):\n",
    "    lines = [l for l in code.split('\\n') if l and not l.isspace()]\n",
    "    return \"\\n\".join(lines)+\"\\n\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "def inline_proc(code, proc_name, proc_args,proc_body):\n",
    "    code = strip_empty_lines(code)\n",
    "    arglist = \",\".join([r\"([a-zA-Z0-9\\_\\[\\]]+)\" for i in range(len(proc_args))])\n",
    "    regexp = fr'([a-zA-Z0-9\\_\\[\\]]+)\\s*=\\s*{proc_name}\\({arglist}\\)\\s*$'\n",
    "    #  captures \"variable = func_name(arguments)\"    \n",
    "    while True:\n",
    "        m = re.search(regexp, code, re.MULTILINE)\n",
    "        if not m: break\n",
    "        newcode = proc_body \n",
    "        for i in range(len(proc_args)): \n",
    "            newcode = newcode.replace(proc_args[i], m.group(i+2))\n",
    "        newcode = newcode.replace('return', m.group(1) + \" = \")\n",
    "        code = code[:m.start()] + newcode + code[m.end()+1:]\n",
    "    return code"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "blah =  r'''temp = NAND(a,b)\n",
    "return NAND(b,c)\n",
    "'''\n",
    "\n",
    "code = r'''\n",
    "z = blah(x,y)\n",
    "w = blah(z,x)\n",
    "'''\n",
    "\n",
    "print(inline_proc(code,\"blah\",['a','b'],blah))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "def parse_procs(code):\n",
    "    \"\"\"Parse code that contain procedure definitions into a list of\n",
    "    triples (name, arguments, body)\"\"\"\n",
    "    code = strip_empty_lines(code)\n",
    "    lines = code.split('\\n') \n",
    "    regexp = r'def\\s+([a-zA-Z\\_0-9]+)\\(([\\sa-zA-Z0-9\\_,]+)\\)\\s*:\\s*'\n",
    "    procs = []\n",
    "    current_line = 0\n",
    "    rest = \"\"\n",
    "    while current_line < len(lines):\n",
    "        m = re.match(regexp,lines[current_line])\n",
    "        if m: \n",
    "            current_line+= 1\n",
    "            code = \"\"\n",
    "            while current_line < len(lines) and lines[current_line][0]==' ':\n",
    "                code += lines[current_line].strip()+'\\n'\n",
    "                current_line += 1\n",
    "            procs.append((m.group(1) , m.group(2).split(','), code))\n",
    "        else:\n",
    "            rest += lines[current_line]+'\\n'\n",
    "            current_line += 1\n",
    "    return rest, procs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "def desugar(code):\n",
    "    code, procs = parse_procs(code)\n",
    "    for name,args,body in reversed(procs):\n",
    "        code = inline_proc(code,name,args,body)\n",
    "    return code"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "# What does this do?\n",
    "sweetcode = r'''\n",
    "def NOT(a): \n",
    "    return NAND(a,a)\n",
    "    \n",
    "def AND(a,b):\n",
    "    temp = NAND(a,b) \n",
    "    return NOT(temp)\n",
    "\n",
    "def OR(a,b):\n",
    "    temp1 = NOT(a)\n",
    "    temp2 = NOT(b) \n",
    "    return NAND(temp1,temp2)\n",
    "\n",
    "and1 = AND(X[0],X[1])\n",
    "and2 = AND(X[0],X[2])\n",
    "and3 = AND(X[1],X[2])\n",
    "or1  = OR(and1,and2)\n",
    "Y[0] = OR(or1,and3)\n",
    "'''"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/markdown": [
       "_Input_ | _Output_\n",
       "--------|---------\n",
       "  $000$  | $0$ \n",
       "  $001$  | $0$ \n",
       "  $010$  | $0$ \n",
       "  $011$  | $1$ \n",
       "  $100$  | $0$ \n",
       "  $101$  | $1$ \n",
       "  $110$  | $1$ \n",
       "  $111$  | $1$ \n"
      ],
      "text/plain": [
       "<IPython.core.display.Markdown object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
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   ],
   "source": [
    "sugarfree  =  desugar(sweetcode)\n",
    "table(sugarfree)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": []
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     "output_type": "execute_result"
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     "metadata": {
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   ],
   "source": [
    "circuit(sugarfree)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "temp = NAND(X[0],X[1])\n",
      "and1 =  NAND(temp,temp)\n",
      "temp = NAND(X[0],X[2])\n",
      "and2 =  NAND(temp,temp)\n",
      "temp = NAND(X[1],X[2])\n",
      "and3 =  NAND(temp,temp)\n",
      "temp1 =  NAND(and1,and1)\n",
      "temp2 =  NAND(and2,and2)\n",
      "or1 =  NAND(temp1,temp2)\n",
      "temp1 =  NAND(or1,or1)\n",
      "temp2 =  NAND(and3,and3)\n",
      "Y[0] =  NAND(temp1,temp2)\n",
      "\n"
     ]
    }
   ],
   "source": [
    "print(sugarfree)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Sugar all the way "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "def NOT(a): return NAND(a,a)\n",
    "def AND(a,b): return NOT(NAND(a,b))\n",
    "def OR(a,b): return NAND(NOT(a),NOT(b))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "GLOBAL_NANDONLY = True"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "def MAJ(a,b,c):\n",
    "    u = NAND(a,b)\n",
    "    v = NAND(a,c)\n",
    "    w = NAND(b,c)\n",
    "    z = NAND(u,v)\n",
    "    return NAND(w,NOT(z))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
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    "def XOR(a,b):\n",
    "    u = NAND(a,b)\n",
    "    v = NAND(a,u)\n",
    "    w = NAND(b,u)\n",
    "    return NAND(v,w)\n",
    "\n",
    "def XOR3(a,b,c):\n",
    "    return XOR(XOR(a,b),c)"
   ]
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     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def ADD1(a,b):\n",
    "    return AND(a,b),XOR(a,b)\n",
    "\n",
    "circuit(ADD1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "<center><img src=\"nandgame.png\" width =\"1000\" height =\"800\"></img></center>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1, 1, 0]"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def addnumbers(X,Y):\n",
    "    n = len(X)\n",
    "    Z = [0]*(n+1)\n",
    "    carry = zero(X[0])\n",
    "    for i in range(n-1,-1,-1):\n",
    "        Z[i+1] = XOR3(X[i],Y[i],carry)\n",
    "        carry = MAJ(carry,X[i],Y[i])\n",
    "    Z[0] = carry\n",
    "    return Z\n",
    "\n",
    "addnumbers([1,1],[1,1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": []
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    },
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    "add2circ = circuit(addnumbers,2,2)\n",
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     "metadata": {
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     "output_type": "display_data"
    }
   ],
   "source": [
    "X = list(range(10,50,5))\n",
    "Y = [circuit(addnumbers,n,n).size() for n in X]\n",
    "import matplotlib.pyplot as plt\n",
    "plt.plot(X,Y)\n",
    "plt.ylabel('circuit size')\n",
    "plt.xlabel('input length')\n",
    "plt.show()"
   ]
  }
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