{
  "cells": [
    {
      "cell_type": "markdown",
      "source": [
        "# Reductions"
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "slide"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Dependencies for this notebook: `graphviz` executable installed on the system and on the path, [networkx](https://anaconda.org/anaconda/networkx)   and [graphviz python](https://anaconda.org/conda-forge/python-graphviz) anaconda packages. (See [here](https://graphviz.readthedocs.io/en/stable/index.html) for the documentation of the latter package.)\n",
        "\nPlease run the utility code at the bottom of this notebook first."
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "skip"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "_Useful unicode symbols:_ φ   ∨   ∧   ¬   ≠    Ψ"
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "skip"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "__Announcements:__ New survey at [http://tiny.cc/cs121survey](http://tiny.cc/cs121survey), last two lectures' material. "
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "fragment"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "%%html\n",
        "<iframe src=\"http://free.timeanddate.com/countdown/i5vf6j5p/n43/cf11/cm0/cu4/ct1/cs1/ca0/co0/cr0/ss0/cac09f/cpc09f/pct/tcfff/fs100/szw576/szh243/iso2017-10-26T10:07:00\" allowTransparency=\"true\" frameborder=\"0\" width=\"177\" height=\"35\"></iframe>'"
      ],
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/html": [
              "<iframe src=\"http://free.timeanddate.com/countdown/i5vf6j5p/n43/cf11/cm0/cu4/ct1/cs1/ca0/co0/cr0/ss0/cac09f/cpc09f/pct/tcfff/fs100/szw576/szh243/iso2017-10-26T10:07:00\" allowTransparency=\"true\" frameborder=\"0\" width=\"177\" height=\"35\"></iframe>'"
            ],
            "text/plain": [
              "<IPython.core.display.HTML object>"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 119,
      "metadata": {
        "slideshow": {
          "slide_type": "fragment"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "reductions = Digraph()\n",
        "reductions.edges([(\"3SAT\",\"Quadratic Equations\"),(\"3SAT\",\"Independent Set\"),(\"3SAT\",\"Longest Path\"),(\"Independent Set\",\"Maximum Cut\")])"
      ],
      "outputs": [],
      "execution_count": null,
      "metadata": {
        "collapsed": true,
        "slideshow": {
          "slide_type": "skip"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "reductions"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 120,
          "data": {
            "image/svg+xml": [
              "<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\r\n",
              "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\r\n",
              " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\r\n",
              "<!-- Generated by graphviz version 2.38.0 (20140413.2041)\r\n",
              " -->\r\n",
              "<!-- Title: %3 Pages: 1 -->\r\n",
              "<svg width=\"471pt\" height=\"188pt\"\r\n",
              " viewBox=\"0.00 0.00 470.64 188.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\r\n",
              "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 184)\">\r\n",
              "<title>%3</title>\r\n",
              "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-184 466.637,-184 466.637,4 -4,4\"/>\r\n",
              "<!-- 3SAT -->\r\n",
              "<g id=\"node1\" class=\"node\"><title>3SAT</title>\r\n",
              "<ellipse fill=\"none\" stroke=\"black\" cx=\"258.142\" cy=\"-162\" rx=\"32.4942\" ry=\"18\"/>\r\n",
              "<text text-anchor=\"middle\" x=\"258.142\" y=\"-158.3\" font-family=\"Times New Roman,serif\" font-size=\"14.00\">3SAT</text>\r\n",
              "</g>\r\n",
              "<!-- Quadratic Equations -->\r\n",
              "<g id=\"node2\" class=\"node\"><title>Quadratic Equations</title>\r\n",
              "<ellipse fill=\"none\" stroke=\"black\" cx=\"85.1424\" cy=\"-90\" rx=\"85.2851\" ry=\"18\"/>\r\n",
              "<text text-anchor=\"middle\" x=\"85.1424\" y=\"-86.3\" font-family=\"Times New Roman,serif\" font-size=\"14.00\">Quadratic Equations</text>\r\n",
              "</g>\r\n",
              "<!-- 3SAT&#45;&gt;Quadratic Equations -->\r\n",
              "<g id=\"edge1\" class=\"edge\"><title>3SAT&#45;&gt;Quadratic Equations</title>\r\n",
              "<path fill=\"none\" stroke=\"black\" d=\"M232.616,-150.671C206.268,-140.01 164.563,-123.135 132.216,-110.047\"/>\r\n",
              "<polygon fill=\"black\" stroke=\"black\" points=\"133.517,-106.798 122.934,-106.292 130.891,-113.287 133.517,-106.798\"/>\r\n",
              "</g>\r\n",
              "<!-- Independent Set -->\r\n",
              "<g id=\"node3\" class=\"node\"><title>Independent Set</title>\r\n",
              "<ellipse fill=\"none\" stroke=\"black\" cx=\"258.142\" cy=\"-90\" rx=\"69.5877\" ry=\"18\"/>\r\n",
              "<text text-anchor=\"middle\" x=\"258.142\" y=\"-86.3\" font-family=\"Times New Roman,serif\" font-size=\"14.00\">Independent Set</text>\r\n",
              "</g>\r\n",
              "<!-- 3SAT&#45;&gt;Independent Set -->\r\n",
              "<g id=\"edge2\" class=\"edge\"><title>3SAT&#45;&gt;Independent Set</title>\r\n",
              "<path fill=\"none\" stroke=\"black\" d=\"M258.142,-143.697C258.142,-135.983 258.142,-126.712 258.142,-118.112\"/>\r\n",
              "<polygon fill=\"black\" stroke=\"black\" points=\"261.643,-118.104 258.142,-108.104 254.643,-118.104 261.643,-118.104\"/>\r\n",
              "</g>\r\n",
              "<!-- Longest Path -->\r\n",
              "<g id=\"node4\" class=\"node\"><title>Longest Path</title>\r\n",
              "<ellipse fill=\"none\" stroke=\"black\" cx=\"404.142\" cy=\"-90\" rx=\"58.4896\" ry=\"18\"/>\r\n",
              "<text text-anchor=\"middle\" x=\"404.142\" y=\"-86.3\" font-family=\"Times New Roman,serif\" font-size=\"14.00\">Longest Path</text>\r\n",
              "</g>\r\n",
              "<!-- 3SAT&#45;&gt;Longest Path -->\r\n",
              "<g id=\"edge3\" class=\"edge\"><title>3SAT&#45;&gt;Longest Path</title>\r\n",
              "<path fill=\"none\" stroke=\"black\" d=\"M281.925,-149.597C304.234,-138.901 338.022,-122.702 364.422,-110.044\"/>\r\n",
              "<polygon fill=\"black\" stroke=\"black\" points=\"366.275,-113.037 373.779,-105.558 363.249,-106.725 366.275,-113.037\"/>\r\n",
              "</g>\r\n",
              "<!-- Maximum Cut -->\r\n",
              "<g id=\"node5\" class=\"node\"><title>Maximum Cut</title>\r\n",
              "<ellipse fill=\"none\" stroke=\"black\" cx=\"258.142\" cy=\"-18\" rx=\"62.2891\" ry=\"18\"/>\r\n",
              "<text text-anchor=\"middle\" x=\"258.142\" y=\"-14.3\" font-family=\"Times New Roman,serif\" font-size=\"14.00\">Maximum Cut</text>\r\n",
              "</g>\r\n",
              "<!-- Independent Set&#45;&gt;Maximum Cut -->\r\n",
              "<g id=\"edge4\" class=\"edge\"><title>Independent Set&#45;&gt;Maximum Cut</title>\r\n",
              "<path fill=\"none\" stroke=\"black\" d=\"M258.142,-71.6966C258.142,-63.9827 258.142,-54.7125 258.142,-46.1124\"/>\r\n",
              "<polygon fill=\"black\" stroke=\"black\" points=\"261.643,-46.1043 258.142,-36.1043 254.643,-46.1044 261.643,-46.1043\"/>\r\n",
              "</g>\r\n",
              "</g>\r\n",
              "</svg>\r\n"
            ],
            "text/plain": [
              "<graphviz.dot.Digraph at 0x2e45a8a4ac8>"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 120,
      "metadata": {
        "slideshow": {
          "slide_type": "slide"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "__Def:__ $F \\leq_p G$ if there is poly-time $R:\\{0,1\\}^* \\rightarrow \\{0,1\\}^*$ such that $F(x)=G(R(x))$ for every $x\\in \\{0,1\\}^*$."
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "slide"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "\"$F$ is no harder than $G$\""
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "fragment"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "__Problem:__ Prove that $F \\leq_p G$ and $G \\leq_p H$ implies that $F \\leq_P H$"
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "fragment"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "__Proof:__ If $F(x)=G(R(x))$ and $G(y)=H(R'(y))$ then $F(x)=H(R'(R(x)))$"
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "fragment"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "__Problem:__ Let $SHORTPATH(G,s,t,k)=1$ iff there is path $s \\leadsto t$ of langth $\\leq k$.\n",
        "\nProve that $SHORTPATH \\leq_p 3SAT$"
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "fragment"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "## 3SAT "
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "slide"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "_\"The mother of all optimization problems\"_"
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "fragment"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "__Input:__ 3CNF formula:  AND of $m$ _clauses_.  \n",
        "\n",
        "Clause = OR of three _literals_\n",
        "\n",
        "Literal = variable or its negation.\n",
        "\n__Goal:__ Output 1 iif formula _satisfiable_."
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "fragment"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "φ = \"(x0 ∨ ¬x3 ∨ x2 ) ∧ (¬x0 ∨ x1 ∨ ¬x2 ) ∧ (x1 ∨ x2 ∨ ¬x3 )  \""
      ],
      "outputs": [],
      "execution_count": 121,
      "metadata": {
        "collapsed": true,
        "slideshow": {
          "slide_type": "slide"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Evaluate 3CNF φ on assignment x \n",
        "# Both are represented as strings\n",
        "def evalcnf(φ,x):\n",
        "\n",
        "    def varval(v):\n",
        "        return (1-int(x[int(v[2:])])  if v[0]==\"¬\" else int(x[int(v[1:])]))\n",
        "    \n",
        "    for (v0,v1,v2) in getclauses(φ):\n",
        "        # print(c+str([varval(v0),varval(v1),varval(v2)]))\n",
        "        if not varval(v0)+varval(v1)+varval(v2): return False\n",
        "    \n",
        "    return True\n",
        "\n",
        "# Clause list of a 3CNF φ\n",
        "def getclauses(φ):\n",
        "    clauses =  φ.split(\"∧\")\n",
        "    res = []\n",
        "    for c in clauses:\n",
        "        (v0,_,v1,_,v2) = c.strip()[1:-1].split()\n",
        "        res.append((v0.strip(),v1.strip(),v2.strip()))\n",
        "    return res\n",
        "    \n",
        "\n",
        "# number of variables of a formula φ\n",
        "def numvars(φ):\n",
        "    for n in range(len(φ)-1,0,-1):\n",
        "        if φ.find('x'+str(n))>= 0: return n+1\n",
        "    raise Exception\n"
      ],
      "outputs": [],
      "execution_count": null,
      "metadata": {
        "collapsed": true,
        "slideshow": {
          "slide_type": "skip"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "numvars(φ)"
      ],
      "outputs": [],
      "execution_count": null,
      "metadata": {
        "slideshow": {
          "slide_type": "skip"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "__Question:__ What is the value of φ on the assignment $x=0101$? __a__ True, __b__ False"
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "fragment"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "evalcnf(φ,\"0101\")"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 122,
          "data": {
            "text/plain": [
              "False"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 122,
      "metadata": {
        "slideshow": {
          "slide_type": "fragment"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "__Question:__ Is φ satisfiable?"
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "fragment"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "evalcnf(φ,\"1000\")"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 123,
          "data": {
            "text/plain": [
              "True"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 123,
      "metadata": {
        "slideshow": {
          "slide_type": "fragment"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Clique problem"
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "slide"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "__Input:__ Graph $G=(V,E)$ and $k\\in\\mathbb{N}$\n",
        "\n__Output:__ $1$ if there is $S \\subseteq V$ with $|S| \\geq k$ s.t. $\\{i,j \\} \\in E$ for every $i \\neq j \\in S$."
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "fragment"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "Image('clique.png' ,width=500)"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 124,
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              "<IPython.core.display.Image object>"
            ]
          },
          "metadata": {
            "image/png": {
              "width": 500
            }
          }
        }
      ],
      "execution_count": 124,
      "metadata": {
        "slideshow": {
          "slide_type": "fragment"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "__Question:__ Is $3SAT \\leq_p CLIQUE$ __a__ Yes __b__ No ,  __c__ I don't know "
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "slide"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "$CLIQUE(G,k)=ISET(\\overline{G},k)$"
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "fragment"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "(_independent set_:  $S$ containing no edges)"
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "fragment"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "## 3SAT to independent set"
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "slide"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "__Thm:__ $3SAT \\leq_p IS$"
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "slide"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "To prove need to show poly-time $R$ s.t. $R(\\varphi)=G$ s.t. \n",
        "\n",
        "* __completeness:__ If $\\varphi$ satisfiable then there is i.set of $G$ of size $\\geq k$\n",
        "\n* __soundness:__ If there is an i.set of $G$ with size $\\geq k$ then $\\varphi$ satisfiable."
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "fragment"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "__Pf:__ $2n$ vertices  \"$x_i=0$\" and \"$x_i=1$\""
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "slide"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "$3m$ vertices: $(x_7$ $\\vee$ $\\neg x_{10}$ $\\vee$ $x_{12})$ maps to triangle \"$x_7 \\neq 0$\", \"$x_{10} \\neq 1$\", \"$x_{12}\\neq 0$\""
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "fragment"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Edges between _inconsistent_ vertices"
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "fragment"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "assignment $\\Leftrightarrow$ independent set"
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "fragment"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Reduction φ ↦ G\n",
        "def SAT2IS_(φ):\n",
        "    n = numvars(φ) \n",
        "    G =Graph(engine='neato')\n",
        "    \n",
        "    # add pairs \"xi=0\" and \"xi=1\"\n",
        "    for i in range(n): G.edge(\"x\"+str(i)+\"=0\",\"x\"+str(i)+\"=1\")\n",
        "        \n",
        "    # map \"x7\" in clause 5 to \"5)x7≠0\", \"¬x12\" in clause 6 to \"6)x12≠1\"\n",
        "    def nodename(v,c): return str(c)+')'+(v[1:]+\"≠1\" if v[0]==\"¬\" else v+\"≠0\")\n",
        "    \n",
        "    #map \"5)x7≠0\" to its neighbor \"x7=0\"\n",
        "    def neighbor(n):   return n.split(')')[1].split('≠')[0]+\"=\"+n[-1]\n",
        "    \n",
        "    c = 0\n",
        "    for C in getclauses(φ):\n",
        "        (u,v,w) = (nodename(C[0],c),nodename(C[1],c),nodename(C[2],c))\n",
        "        \n",
        "        # add triangle of clause\n",
        "        G.edges([(u,v),(v,w),(u,w)])\n",
        "        \n",
        "        # connect each vertex to inconsistent neighbor\n",
        "        G.edges([(u,neighbor(u)),(v,neighbor(v)),(w,neighbor(w))])\n",
        "        c += 1\n",
        "    \n",
        "    return G"
      ],
      "outputs": [],
      "execution_count": null,
      "metadata": {
        "collapsed": true,
        "slideshow": {
          "slide_type": "slide"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "SAT2IS_(φ)"
      ],
      "outputs": [],
      "execution_count": null,
      "metadata": {
        "slideshow": {
          "slide_type": "skip"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# same reduction but taking care of colors and keeping track what happens to an assignment\n",
        "def SAT2IS(φ,x=\"\"):\n",
        "    S = []\n",
        "    n = numvars(φ) \n",
        "    G =Graph(engine='neato')\n",
        "    for i in range(n):\n",
        "        u = \"x\"+str(i)+\"=0\"\n",
        "        v = \"x\"+str(i)+\"=1\"\n",
        "        G.node(u,style='filled',fillcolor=('red' if x and x[i]==\"0\" else 'green'))\n",
        "        G.node(v,style='filled',fillcolor=('red' if x and x[i]==\"1\" else 'green'))\n",
        "        if x: S.append(u if x[i]=='0' else v)\n",
        "        G.edge(u,v,len='1')\n",
        "    \n",
        "    def nodename(v):\n",
        "        return ((v[1:],\"≠1\") if v[0]==\"¬\" else (v,\"≠0\"))\n",
        "    \n",
        "    cnum = 0\n",
        "    for C in getclauses(φ):\n",
        "        lastv = None\n",
        "        found = False\n",
        "        for v in C:\n",
        "            (var,cond) = nodename(v)\n",
        "            name = str(cnum)+\")\"+var+cond\n",
        "            color = 'lightblue'\n",
        "            if x and (not found) and x[int(var[1:])]!=cond[1]:\n",
        "                found = True\n",
        "                color = 'red'\n",
        "                S.append(name)\n",
        "            G.node(name,label=var+cond,style='filled',fillcolor=color)\n",
        "            if lastv: G.edge(lastv,name,len='1')\n",
        "            \n",
        "            lastv = name\n",
        "            G.edge(name,var+\"=\"+cond[1],len='1.7')\n",
        "        G.edge(str(cnum)+\")\"+\"\".join(nodename(C[0])),lastv,len='1')\n",
        "        cnum += 1\n",
        "    \n",
        "    return (G if not x else (G,S))"
      ],
      "outputs": [],
      "execution_count": null,
      "metadata": {
        "collapsed": true,
        "slideshow": {
          "slide_type": "skip"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "SAT2IS(φ,\"1010\")[0]"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 129,
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          "metadata": {}
        }
      ],
      "execution_count": 129,
      "metadata": {
        "slideshow": {
          "slide_type": "slide"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "__Completeness:__  $\\exists$ sat assign. $\\varphi$ $\\Rightarrow$ $\\exists$ $n+m$ iset in $G$ "
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "slide"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "__Pf:__ One vertex out of  pair $x_i=0$, $x_i=1$, one vertex out of triangle $x_i \\neq a$, $x_j \\neq b$, $x_k \\neq c$"
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "fragment"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "__Soundness:__ $\\exists$ $n+m$ iset in $G$ $\\Rightarrow$ $\\exists$ sat assign. $\\varphi$ "
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "fragment"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "__Pf:__ $n+m$ iset must have one vtx per pair, one vtx per triangle"
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "fragment"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "If \"$x_i=a$\" in  $I$ then set $x_i=a$."
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "fragment"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "$x$ must satisfy all clauses."
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "fragment"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Independent Set to Maximum Cut"
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "slide"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "__Def:__ If $G=(V,E)$ and $S \\subseteq V$ then $cut(S)$ is number of edges $\\{u,v\\}$ with $u\\in S$ and $v\\not\\in S$.\n",
        "\n$MAXCUT(G,k) = 1$ iff there is $S$ with $cut(S) \\geq k$."
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "slide"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "Image('maxcut.png')"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 130,
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        "slideshow": {
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    {
      "cell_type": "markdown",
      "source": [
        "__Thm:__ $ISET \\leq_p MAXCUT$"
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "slide"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "__Corollary:__ $3SAT \\leq_p MAXCUT$"
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "fragment"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "__Proof:__ $G=(V,E)$  maps to $H$ with: "
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "slide"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "* Special _source_ vertex\n",
        "\n",
        "* $n$ vertices corresponding to $V$\n",
        "\n* $2m$ extra vertices $e_0,e_1$ for each edge"
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "fragment"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "__Edges:__ $\\{s^*,v \\}$ for all $v\\in V$, $5$ edge gadget for every $e\\in E$."
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "fragment"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "__Claim:__ $G$ has $k$ iset $\\Leftrightarrow$ $H$ has cut of $k+4m$ edges."
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "fragment"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Reduction IS to MAXCUT\n",
        "def IS2MAXCUT_(G):\n",
        "    G =nxgraph(G)\n",
        "    H =Graph(engine='sfdp')\n",
        "    s =\"source\"\n",
        "    H.node(s) # create source node\n",
        "    \n",
        "    for v in G.nodes(): H.edge(s,v)\n",
        "    \n",
        "    j =0\n",
        "    for (u,v) in G.edges():\n",
        "        g1 = \"e\"+str(j)+\"a\"\n",
        "        g2 = \"e\"+str(j)+\"b\"\n",
        "        # add gadget\n",
        "        H.edges([(s,g1),(s,g2),(g1,g2),(u,g1),(v,g2)]) \n",
        "        j +=1\n",
        "        \n",
        "    \n",
        "    return H"
      ],
      "outputs": [],
      "execution_count": null,
      "metadata": {
        "collapsed": true,
        "slideshow": {
          "slide_type": "slide"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "IS2MAXCUT_(SAT2IS(φ))"
      ],
      "outputs": [],
      "execution_count": null,
      "metadata": {
        "slideshow": {
          "slide_type": "skip"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# same reduction as above but handling colors and showing the resulting cut when the original graph has independent set\n",
        "def IS2MAXCUT(G,I=[]):\n",
        "    G =nxgraph(G)\n",
        "    S = []\n",
        "    H =Graph(engine='sfdp')\n",
        "    H.node(\"source\",style='filled',fillcolor='blue')\n",
        "    S.append(\"source\")\n",
        "    for v in G.nodes():\n",
        "        color = ('red' if I and v in I else 'lightblue')\n",
        "        H.node(v,style='filled',fillcolor=color)\n",
        "        ecol = 'black'\n",
        "        pwidth = '1'\n",
        "        if I and v in I: \n",
        "            S.append(v)\n",
        "            ecol = 'red'\n",
        "            pwidth = '2'\n",
        "        H.edge(\"source\",v,len=\"2\",color=ecol,penwidth=pwidth) \n",
        "    \n",
        "    j =0\n",
        "    for (u,v) in G.edges():\n",
        "        g1 = \"e\"+str(j)+\"a\"\n",
        "        g2 = \"e\"+str(j)+\"b\"\n",
        "        c1 = 'green'\n",
        "        c2 = 'green'\n",
        "        if I and (not u in I):\n",
        "            c1 = 'red'\n",
        "            S.append(g1)\n",
        "        if I and (not v in I):\n",
        "            c2 = 'red'\n",
        "            S.append(g2)\n",
        "        gadget = Graph(\"gadget\"+str(j))\n",
        "        gadget.node(g1,style='filled',fillcolor=c1)\n",
        "        gadget.node(g2,style='filled',fillcolor=c2)\n",
        "        gadget.edge(g1,g2,len=\"1\",color=('red' if (g1 in S) != (g2 in I) else 'black'), penwidth=('2' if (g1 in S) != (g2 in I) else '1'))\n",
        "        gadget.edge(u,g1,len=\"1\", color=('red' if (g1 in S) != (u in I) else 'black'),penwidth =('2' if (g1 in S) != (u in S) else '1')  )\n",
        "        gadget.edge(v,g2,len=\"1\", color=('red' if (g2 in S) != (v in I) else 'black'),penwidth= ('2' if (g2 in S) != (v in S) else '1') )\n",
        "        \n",
        "        H.subgraph(gadget)\n",
        "        H.edge(g1,\"source\",len=\"2.5\",color=('red' if (g1 in S)  else 'black'),penwidth=('2' if (g1 in S)  else '1'))\n",
        "        H.edge(g2,\"source\",len=\"2.5\",color=('red' if (g2 in S)  else 'black'),penwidth= ('2' if (g2 in S)  else '1') )\n",
        "        j +=1\n",
        "        \n",
        "    \n",
        "    return (H,S) if I else H"
      ],
      "outputs": [],
      "execution_count": null,
      "metadata": {
        "collapsed": true,
        "slideshow": {
          "slide_type": "skip"
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      }
    },
    {
      "cell_type": "code",
      "source": [
        "IS2MAXCUT(*SAT2IS(φ,\"1000\"))[0]"
      ],
      "outputs": [
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          "metadata": {}
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      ],
      "execution_count": 132,
      "metadata": {
        "slideshow": {
          "slide_type": "slide"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "__Claim:__ $G$ has $k$ iset $\\Leftrightarrow$ $H$ has cut of $k+4m$ edges."
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "slide"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "Image('ISETtoMAXCUT.png')"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 133,
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              "wP8DR0I9+Bq/hH4AAAAASUVORK5CYII=\n"
            ],
            "text/plain": [
              "<IPython.core.display.Image object>"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 133,
      "metadata": {
        "inputHidden": false,
        "outputHidden": false,
        "slideshow": {
          "slide_type": "fragment"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "__Claim:__ $H$ has cut of $k+4m$ edges $\\Rightarrow$ $G$ has set $I$ with $|I|-|E(I,I)| \\geq k$."
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "fragment"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "## SIZE vs TIME"
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "slide"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "__Thm:__ For every $F \\in TIME_{++}(T(n))$ and large enough $n$, $F_n \\in SIZE(T(n))$\n",
        "\n($F_n$ = restriction of $F$ to $\\{0,1\\}^n$)"
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "slide"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "__Corollary:__ For every $F \\in TIME_{<<}(T(n))$ and large enough $n$, $F_n \\in SIZE(T(n)^5)$ "
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "fragment"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "__Thm:__ For every $F \\in TIME_{++}(T(n))$ and large enough $n$, $F_n \\in SIZE(T(n))$\n",
        "\n($F_n$ = restriction of $F$ to $\\{0,1\\}^n$)"
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "slide"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "#Proof by Python:\n",
        "\n",
        "def expand(P,T,n):\n",
        "    result = \"\"\n",
        "\n",
        "    for k in range(T):\n",
        "        i=index(k)\n",
        "        validx = ('one' if i<n else 'zero')\n",
        "        result += P.replace( 'validx_i',validx).replace(\n",
        "            'x_i',('x_i' if i<n else 'zero')).replace( '_i','_'+str(i))\n",
        "\n",
        "    return result\n",
        "\n\n",
        "def index(k):\n",
        "    r = math.floor(math.sqrt(k+1/4)-1/2)\n",
        "    return (k-r*(r+1) if k <= (r+1)*(r+1) else (r+1)*(r+2)-k)"
      ],
      "outputs": [],
      "execution_count": null,
      "metadata": {
        "collapsed": true,
        "slideshow": {
          "slide_type": "fragment"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "parity = r'''\n",
        "tmpa  := seen_i NAND seen_i\n",
        "tmpb  := x_i NAND tmpa\n",
        "val   :=  tmpb NAND tmpb\n",
        "ns   := s   NAND s\n",
        "y_0  := ns  NAND ns\n",
        "u    := val NAND s\n",
        "v    := s   NAND u\n",
        "w    := val NAND u\n",
        "s    := v   NAND w\n",
        "seen_i := zero NAND zero  \n",
        "stop := validx_i NAND validx_i\n",
        "loop := stop     NAND stop\n",
        "'''"
      ],
      "outputs": [],
      "execution_count": null,
      "metadata": {
        "collapsed": true,
        "slideshow": {
          "slide_type": "skip"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "prog = expand(parity,17,4)\n",
        "print(prog)"
      ],
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "\n",
            "tmpa  := seen_0 NAND seen_0\n",
            "tmpb  := x_0 NAND tmpa\n",
            "val   :=  tmpb NAND tmpb\n",
            "ns   := s   NAND s\n",
            "y_0  := ns  NAND ns\n",
            "u    := val NAND s\n",
            "v    := s   NAND u\n",
            "w    := val NAND u\n",
            "s    := v   NAND w\n",
            "seen_0 := zero NAND zero  \n",
            "stop := one NAND one\n",
            "loop := stop     NAND stop\n",
            "\n",
            "tmpa  := seen_1 NAND seen_1\n",
            "tmpb  := x_1 NAND tmpa\n",
            "val   :=  tmpb NAND tmpb\n",
            "ns   := s   NAND s\n",
            "y_0  := ns  NAND ns\n",
            "u    := val NAND s\n",
            "v    := s   NAND u\n",
            "w    := val NAND u\n",
            "s    := v   NAND w\n",
            "seen_1 := zero NAND zero  \n",
            "stop := one NAND one\n",
            "loop := stop     NAND stop\n",
            "\n",
            "tmpa  := seen_0 NAND seen_0\n",
            "tmpb  := x_0 NAND tmpa\n",
            "val   :=  tmpb NAND tmpb\n",
            "ns   := s   NAND s\n",
            "y_0  := ns  NAND ns\n",
            "u    := val NAND s\n",
            "v    := s   NAND u\n",
            "w    := val NAND u\n",
            "s    := v   NAND w\n",
            "seen_0 := zero NAND zero  \n",
            "stop := one NAND one\n",
            "loop := stop     NAND stop\n",
            "\n",
            "tmpa  := seen_1 NAND seen_1\n",
            "tmpb  := x_1 NAND tmpa\n",
            "val   :=  tmpb NAND tmpb\n",
            "ns   := s   NAND s\n",
            "y_0  := ns  NAND ns\n",
            "u    := val NAND s\n",
            "v    := s   NAND u\n",
            "w    := val NAND u\n",
            "s    := v   NAND w\n",
            "seen_1 := zero NAND zero  \n",
            "stop := one NAND one\n",
            "loop := stop     NAND stop\n",
            "\n",
            "tmpa  := seen_2 NAND seen_2\n",
            "tmpb  := x_2 NAND tmpa\n",
            "val   :=  tmpb NAND tmpb\n",
            "ns   := s   NAND s\n",
            "y_0  := ns  NAND ns\n",
            "u    := val NAND s\n",
            "v    := s   NAND u\n",
            "w    := val NAND u\n",
            "s    := v   NAND w\n",
            "seen_2 := zero NAND zero  \n",
            "stop := one NAND one\n",
            "loop := stop     NAND stop\n",
            "\n",
            "tmpa  := seen_1 NAND seen_1\n",
            "tmpb  := x_1 NAND tmpa\n",
            "val   :=  tmpb NAND tmpb\n",
            "ns   := s   NAND s\n",
            "y_0  := ns  NAND ns\n",
            "u    := val NAND s\n",
            "v    := s   NAND u\n",
            "w    := val NAND u\n",
            "s    := v   NAND w\n",
            "seen_1 := zero NAND zero  \n",
            "stop := one NAND one\n",
            "loop := stop     NAND stop\n",
            "\n",
            "tmpa  := seen_0 NAND seen_0\n",
            "tmpb  := x_0 NAND tmpa\n",
            "val   :=  tmpb NAND tmpb\n",
            "ns   := s   NAND s\n",
            "y_0  := ns  NAND ns\n",
            "u    := val NAND s\n",
            "v    := s   NAND u\n",
            "w    := val NAND u\n",
            "s    := v   NAND w\n",
            "seen_0 := zero NAND zero  \n",
            "stop := one NAND one\n",
            "loop := stop     NAND stop\n",
            "\n",
            "tmpa  := seen_1 NAND seen_1\n",
            "tmpb  := x_1 NAND tmpa\n",
            "val   :=  tmpb NAND tmpb\n",
            "ns   := s   NAND s\n",
            "y_0  := ns  NAND ns\n",
            "u    := val NAND s\n",
            "v    := s   NAND u\n",
            "w    := val NAND u\n",
            "s    := v   NAND w\n",
            "seen_1 := zero NAND zero  \n",
            "stop := one NAND one\n",
            "loop := stop     NAND stop\n",
            "\n",
            "tmpa  := seen_2 NAND seen_2\n",
            "tmpb  := x_2 NAND tmpa\n",
            "val   :=  tmpb NAND tmpb\n",
            "ns   := s   NAND s\n",
            "y_0  := ns  NAND ns\n",
            "u    := val NAND s\n",
            "v    := s   NAND u\n",
            "w    := val NAND u\n",
            "s    := v   NAND w\n",
            "seen_2 := zero NAND zero  \n",
            "stop := one NAND one\n",
            "loop := stop     NAND stop\n",
            "\n",
            "tmpa  := seen_3 NAND seen_3\n",
            "tmpb  := x_3 NAND tmpa\n",
            "val   :=  tmpb NAND tmpb\n",
            "ns   := s   NAND s\n",
            "y_0  := ns  NAND ns\n",
            "u    := val NAND s\n",
            "v    := s   NAND u\n",
            "w    := val NAND u\n",
            "s    := v   NAND w\n",
            "seen_3 := zero NAND zero  \n",
            "stop := one NAND one\n",
            "loop := stop     NAND stop\n",
            "\n",
            "tmpa  := seen_2 NAND seen_2\n",
            "tmpb  := x_2 NAND tmpa\n",
            "val   :=  tmpb NAND tmpb\n",
            "ns   := s   NAND s\n",
            "y_0  := ns  NAND ns\n",
            "u    := val NAND s\n",
            "v    := s   NAND u\n",
            "w    := val NAND u\n",
            "s    := v   NAND w\n",
            "seen_2 := zero NAND zero  \n",
            "stop := one NAND one\n",
            "loop := stop     NAND stop\n",
            "\n",
            "tmpa  := seen_1 NAND seen_1\n",
            "tmpb  := x_1 NAND tmpa\n",
            "val   :=  tmpb NAND tmpb\n",
            "ns   := s   NAND s\n",
            "y_0  := ns  NAND ns\n",
            "u    := val NAND s\n",
            "v    := s   NAND u\n",
            "w    := val NAND u\n",
            "s    := v   NAND w\n",
            "seen_1 := zero NAND zero  \n",
            "stop := one NAND one\n",
            "loop := stop     NAND stop\n",
            "\n",
            "tmpa  := seen_0 NAND seen_0\n",
            "tmpb  := x_0 NAND tmpa\n",
            "val   :=  tmpb NAND tmpb\n",
            "ns   := s   NAND s\n",
            "y_0  := ns  NAND ns\n",
            "u    := val NAND s\n",
            "v    := s   NAND u\n",
            "w    := val NAND u\n",
            "s    := v   NAND w\n",
            "seen_0 := zero NAND zero  \n",
            "stop := one NAND one\n",
            "loop := stop     NAND stop\n",
            "\n",
            "tmpa  := seen_1 NAND seen_1\n",
            "tmpb  := x_1 NAND tmpa\n",
            "val   :=  tmpb NAND tmpb\n",
            "ns   := s   NAND s\n",
            "y_0  := ns  NAND ns\n",
            "u    := val NAND s\n",
            "v    := s   NAND u\n",
            "w    := val NAND u\n",
            "s    := v   NAND w\n",
            "seen_1 := zero NAND zero  \n",
            "stop := one NAND one\n",
            "loop := stop     NAND stop\n",
            "\n",
            "tmpa  := seen_2 NAND seen_2\n",
            "tmpb  := x_2 NAND tmpa\n",
            "val   :=  tmpb NAND tmpb\n",
            "ns   := s   NAND s\n",
            "y_0  := ns  NAND ns\n",
            "u    := val NAND s\n",
            "v    := s   NAND u\n",
            "w    := val NAND u\n",
            "s    := v   NAND w\n",
            "seen_2 := zero NAND zero  \n",
            "stop := one NAND one\n",
            "loop := stop     NAND stop\n",
            "\n",
            "tmpa  := seen_3 NAND seen_3\n",
            "tmpb  := x_3 NAND tmpa\n",
            "val   :=  tmpb NAND tmpb\n",
            "ns   := s   NAND s\n",
            "y_0  := ns  NAND ns\n",
            "u    := val NAND s\n",
            "v    := s   NAND u\n",
            "w    := val NAND u\n",
            "s    := v   NAND w\n",
            "seen_3 := zero NAND zero  \n",
            "stop := one NAND one\n",
            "loop := stop     NAND stop\n",
            "\n",
            "tmpa  := seen_4 NAND seen_4\n",
            "tmpb  := zero NAND tmpa\n",
            "val   :=  tmpb NAND tmpb\n",
            "ns   := s   NAND s\n",
            "y_0  := ns  NAND ns\n",
            "u    := val NAND s\n",
            "v    := s   NAND u\n",
            "w    := val NAND u\n",
            "s    := v   NAND w\n",
            "seen_4 := zero NAND zero  \n",
            "stop := zero NAND zero\n",
            "loop := stop     NAND stop\n",
            "\n"
          ]
        }
      ],
      "execution_count": 134,
      "metadata": {
        "slideshow": {
          "slide_type": "slide"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "draw_graph(prog,\"both\")"
      ],
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "80 nodes\n"
          ]
        },
        {
          "output_type": "execute_result",
          "execution_count": 135,
          "data": {
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              "T5kqjxzeAAAAAElFTkSuQmCC\n"
            ],
            "text/plain": [
              "<IPython.core.display.Image object>"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 135,
      "metadata": {
        "slideshow": {
          "slide_type": "slide"
        }
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### Utility code (run this first)"
      ],
      "metadata": {
        "slideshow": {
          "slide_type": "skip"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "%pylab"
      ],
      "outputs": [],
      "execution_count": null,
      "metadata": {
        "slideshow": {
          "slide_type": "skip"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "from IPython.display import Image\n",
        "from IPython.core.display import HTML "
      ],
      "outputs": [],
      "execution_count": null,
      "metadata": {
        "collapsed": true,
        "inputHidden": false,
        "outputHidden": false,
        "slideshow": {
          "slide_type": "skip"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "import graphviz\n",
        "from graphviz import Graph\n",
        "from graphviz import Digraph"
      ],
      "outputs": [],
      "execution_count": null,
      "metadata": {
        "collapsed": true,
        "slideshow": {
          "slide_type": "skip"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "import networkx as nx\n",
        "\n",
        "import matplotlib\n",
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "from IPython.display import Image\n",
        "\n",
        "import warnings\n",
        "warnings.filterwarnings(\"ignore\")\n",
        "\n",
        "import math\n"
      ],
      "outputs": [],
      "execution_count": null,
      "metadata": {
        "collapsed": true,
        "slideshow": {
          "slide_type": "skip"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "import pydotplus\n",
        "\n",
        "def nxgraph(G):\n",
        "    P = pydotplus.graph_from_dot_data(G.source)\n",
        "    return nx.drawing.nx_pydot.from_pydot(P)\n"
      ],
      "outputs": [],
      "execution_count": null,
      "metadata": {
        "collapsed": true,
        "slideshow": {
          "slide_type": "skip"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# import matplotlib.cbook\n",
        "# warnings.filterwarnings(\"ignore\",category=matplotlib.cbook.mplDeprecation)\n",
        "\n",
        "def NAND2Graph(P):\n",
        "    n = max([int(var[2:]) for var in P.split() if var[:2]=='x_' ])+1 # no of inputs\n",
        "    m = max([int(var[2:]) for var in P.split() if var[:2]=='y_' ])+1 # no of outputs\n",
        "    nodes = {}\n",
        "    \n",
        "    def uniquenode(v):\n",
        "        idx = nodes.setdefault(v,-1)\n",
        "        nodes[v] = nodes[v]+1\n",
        "        return v+(\" \"*(idx+1))\n",
        "    \n",
        "    def lastnode(v):\n",
        "        idx = nodes.get(v,0)\n",
        "        return v+(\" \"*idx)\n",
        "        \n",
        "    \n",
        "    G = nx.DiGraph()\n",
        "    for line in P.split('\\n'):\n",
        "        if not line or line[0]=='#' or line[0]=='//': continue # ignore empty and commented out lines\n",
        "        (var1,assign,var2,op,var3) = line.split()\n",
        "        var1 = uniquenode(var1)\n",
        "        var2 = lastnode(var2)\n",
        "        var3 = lastnode(var3)\n",
        "        G.add_node(var1)\n",
        "        G.add_edge(var2,var1)\n",
        "        G.add_edge(var3,var1)\n",
        "    \n",
        "    return [G, [lastnode(\"y_\"+str(j)) for j in range(m)]]\n",
        " \n",
        "    \n",
        "def EVALgraph(L,n,m,x):\n",
        "    t = max([max(triple) for triple in L])+1\n",
        "    s = len(L)\n",
        "    \n",
        "    avars = [0]*t\n",
        "    avars[:n]=x\n",
        "    values = [0]*(n+s+m)\n",
        "    values[:n]=x\n",
        "\n",
        "    l=0\n",
        "    for (a,b,c) in L:\n",
        "        avars[a] =  1-avars[b]*avars[c]\n",
        "        values[l+n] = avars[a]\n",
        "        l+=1\n",
        "    \n",
        "        \n",
        "    y = avars[t-m:t]\n",
        "    values[n+s:n+s+m] = y\n",
        "    \n",
        "    \n",
        "    G = nx.DiGraph()\n",
        "    \n",
        "    \n",
        "    def getnode(i):\n",
        "        if i<n:\n",
        "            return 'x_'+ str(i)+'='+str(values[i])\n",
        "        elif i<n+s:\n",
        "            return 'l_'+str(i-n)+\"=\"+str(values[i])\n",
        "        else:\n",
        "            return 'y_'+str(i-n-s)+\"=\"+str(values[i])\n",
        "    \n",
        "    lines = [getnode(i) for i in range(n+s)] \n",
        "    \n",
        "    def last_written(var,l):\n",
        "        if var<n:\n",
        "            return var\n",
        "        z = -1\n",
        "        for i in range(l):\n",
        "            if L[i][0]==var:\n",
        "                z = i+n\n",
        "        return z\n",
        "    \n",
        "\n",
        "    for node in lines:\n",
        "        G.add_node(node)\n",
        "    \n",
        "    \n",
        "    l = 0\n",
        "    for (a,b,c) in L:\n",
        "        in1 = last_written(b,l)\n",
        "        in2 = last_written(c,l)\n",
        "        \n",
        "        if in1>=0:\n",
        "            G.add_edge(lines[in1],lines[l+n])\n",
        "        if in2>=0:\n",
        "            G.add_edge(lines[in2],lines[l+n])\n",
        "        l += 1\n",
        "    \n",
        "    for j in range(m):\n",
        "        G.add_edge(lines[last_written(t-m+j,s)],getnode(n+s+j))\n",
        "        \n",
        "    return [G, [getnode(n+s+j) for j in range(m)]]\n",
        "\n",
        "def draw_eval(prog,s,pruned=\"none\"):\n",
        "    x = [int(a) for a in s]\n",
        "    [n,m,t] = params(prog)\n",
        "    L =  triples(prog,n,m,t)\n",
        "    [G,O] = EVALgraph(L,n,m,x)\n",
        "    if pruned != \"none\":\n",
        "        prune(G,O,pruned)\n",
        "    return draw_DAG(G)\n",
        "    \n",
        "\n",
        "def draw_DAG(G):\n",
        "    print(str(len(G.nodes()))+\" nodes\")\n",
        "    D = nx.drawing.nx_pydot.to_pydot(G)\n",
        "    png_str = D.create_png()\n",
        "    return Image(data=png_str)\n",
        "\n\n",
        "def draw_graph(prog,pruned=\"none\"):\n",
        "    [G,O] = NAND2Graph(prog)\n",
        "    if pruned != \"none\":\n",
        "        prune(G,O,pruned)\n",
        "    return draw_DAG(G)"
      ],
      "outputs": [],
      "execution_count": null,
      "metadata": {
        "collapsed": true,
        "slideshow": {
          "slide_type": "skip"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "def prune(G,O,pruned):\n",
        "    \n",
        "    if pruned in [\"sinks\",\"both\"]:\n",
        "        found = True\n",
        "        while found:\n",
        "            found = False\n",
        "            nodes = G.nodes()\n",
        "            for n in nodes:\n",
        "                if len(G.successors(n))==0 and not(n in O):\n",
        "                    G.remove_node(n)\n",
        "                    found = True\n",
        "                \n",
        "    if pruned in [\"merge\",\"both\"]:    \n",
        "        found = True\n",
        "        while found:\n",
        "            found = False\n",
        "            for n in G.nodes():\n",
        "                for np in G.nodes():\n",
        "                    if (n != np) and (np[0]!='x') and (set(G.predecessors(n))== set(G.predecessors(np))):\n",
        "                        s = G.successors(np)\n",
        "                        for a in s:\n",
        "                            G.add_edge(n,a)\n",
        "                        G.remove_node(np)\n",
        "                        found = True\n",
        "                        break\n",
        "                if found:\n",
        "                    break\n",
        "    return G"
      ],
      "outputs": [],
      "execution_count": null,
      "metadata": {
        "collapsed": true,
        "slideshow": {
          "slide_type": "skip"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [
        "import networkx as nx\n",
        "\n",
        "import matplotlib\n",
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "from IPython.display import Image\n",
        "\n",
        "import warnings\n",
        "warnings.filterwarnings(\"ignore\")\n",
        "\n",
        "# import matplotlib.cbook\n",
        "# warnings.filterwarnings(\"ignore\",category=matplotlib.cbook.mplDeprecation)\n",
        "\n",
        "def NAND2Graph(P):\n",
        "    n = max([int(var[2:]) for var in P.split() if var[:2]=='x_' ])+1 # no of inputs\n",
        "    m = max([int(var[2:]) for var in P.split() if var[:2]=='y_' ])+1 # no of outputs\n",
        "    nodes = {}\n",
        "    \n",
        "    def uniquenode(v):\n",
        "        idx = nodes.setdefault(v,-1)\n",
        "        nodes[v] = nodes[v]+1\n",
        "        return v+(\" \"*(idx+1))\n",
        "    \n",
        "    def lastnode(v):\n",
        "        idx = nodes.get(v,0)\n",
        "        return v+(\" \"*idx)\n",
        "        \n",
        "    \n",
        "    G = nx.DiGraph()\n",
        "    for line in P.split('\\n'):\n",
        "        if not line or line[0]=='#' or line[0]=='//': continue # ignore empty and commented out lines\n",
        "        (var1,assign,var2,op,var3) = line.split()\n",
        "        var1 = uniquenode(var1)\n",
        "        var2 = lastnode(var2)\n",
        "        var3 = lastnode(var3)\n",
        "        G.add_node(var1)\n",
        "        G.add_edge(var2,var1)\n",
        "        G.add_edge(var3,var1)\n",
        "    \n",
        "    return [G, [lastnode(\"y_\"+str(j)) for j in range(m)]]\n",
        " \n",
        "    \n",
        "def EVALgraph(L,n,m,x):\n",
        "    t = max([max(triple) for triple in L])+1\n",
        "    s = len(L)\n",
        "    \n",
        "    avars = [0]*t\n",
        "    avars[:n]=x\n",
        "    values = [0]*(n+s+m)\n",
        "    values[:n]=x\n",
        "\n",
        "    l=0\n",
        "    for (a,b,c) in L:\n",
        "        avars[a] =  1-avars[b]*avars[c]\n",
        "        values[l+n] = avars[a]\n",
        "        l+=1\n",
        "    \n",
        "        \n",
        "    y = avars[t-m:t]\n",
        "    values[n+s:n+s+m] = y\n",
        "    \n",
        "    \n",
        "    G = nx.DiGraph()\n",
        "    \n",
        "    \n",
        "    def getnode(i):\n",
        "        if i<n:\n",
        "            return 'x_'+ str(i)+'='+str(values[i])\n",
        "        elif i<n+s:\n",
        "            return 'l_'+str(i-n)+\"=\"+str(values[i])\n",
        "        else:\n",
        "            return 'y_'+str(i-n-s)+\"=\"+str(values[i])\n",
        "    \n",
        "    lines = [getnode(i) for i in range(n+s)] \n",
        "    \n",
        "    def last_written(var,l):\n",
        "        if var<n:\n",
        "            return var\n",
        "        z = -1\n",
        "        for i in range(l):\n",
        "            if L[i][0]==var:\n",
        "                z = i+n\n",
        "        return z\n",
        "    \n",
        "\n",
        "    for node in lines:\n",
        "        G.add_node(node)\n",
        "    \n",
        "    \n",
        "    l = 0\n",
        "    for (a,b,c) in L:\n",
        "        in1 = last_written(b,l)\n",
        "        in2 = last_written(c,l)\n",
        "        \n",
        "        if in1>=0:\n",
        "            G.add_edge(lines[in1],lines[l+n])\n",
        "        if in2>=0:\n",
        "            G.add_edge(lines[in2],lines[l+n])\n",
        "        l += 1\n",
        "    \n",
        "    for j in range(m):\n",
        "        G.add_edge(lines[last_written(t-m+j,s)],getnode(n+s+j))\n",
        "        \n",
        "    return [G, [getnode(n+s+j) for j in range(m)]]\n",
        "\n",
        "def draw_eval(prog,s,pruned=\"none\"):\n",
        "    x = [int(a) for a in s]\n",
        "    [n,m,t] = params(prog)\n",
        "    L =  triples(prog,n,m,t)\n",
        "    [G,O] = EVALgraph(L,n,m,x)\n",
        "    if pruned != \"none\":\n",
        "        prune(G,O,pruned)\n",
        "    return draw_DAG(G)\n",
        "    \n",
        "\n",
        "def draw_DAG(G):\n",
        "    print(str(len(G.nodes()))+\" nodes\")\n",
        "    D = nx.drawing.nx_pydot.to_pydot(G)\n",
        "    png_str = D.create_png()\n",
        "    return Image(data=png_str)\n",
        "\n",
        "def display_graph(G):\n",
        "    print(str(len(G.nodes()))+\" nodes\")\n",
        "    D = nx.drawing.nx_pydot.to_pydot(G)\n",
        "    png_str = D.create_png()\n",
        "    return Image(data=png_str)\n",
        "\n\n\n",
        "def draw_graph(prog,pruned=\"none\"):\n",
        "    [G,O] = NAND2Graph(prog)\n",
        "    if pruned != \"none\":\n",
        "        prune(G,O,pruned)\n",
        "    return draw_DAG(G)\n",
        "\n",
        "def prune(G,O,pruned):\n",
        "    \n",
        "    if pruned in [\"sinks\",\"both\"]:\n",
        "        found = True\n",
        "        while found:\n",
        "            found = False\n",
        "            nodes = G.nodes()\n",
        "            for n in nodes:\n",
        "                if len(G.successors(n))==0 and not(n in O):\n",
        "                    G.remove_node(n)\n",
        "                    found = True\n",
        "                \n",
        "    if pruned in [\"merge\",\"both\"]:    \n",
        "        found = True\n",
        "        while found:\n",
        "            found = False\n",
        "            for n in G.nodes():\n",
        "                for np in G.nodes():\n",
        "                    if (n != np) and (np[0]!='x') and (set(G.predecessors(n))== set(G.predecessors(np))):\n",
        "                        s = G.successors(np)\n",
        "                        for a in s:\n",
        "                            G.add_edge(n,a)\n",
        "                        G.remove_node(np)\n",
        "                        found = True\n",
        "                        break\n",
        "                if found:\n",
        "                    break\n",
        "    return G"
      ],
      "outputs": [],
      "execution_count": null,
      "metadata": {
        "collapsed": true,
        "slideshow": {
          "slide_type": "skip"
        }
      }
    },
    {
      "cell_type": "code",
      "source": [],
      "outputs": [],
      "execution_count": null,
      "metadata": {
        "collapsed": true,
        "slideshow": {
          "slide_type": "skip"
        }
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    },
    {
      "cell_type": "code",
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      "outputs": [],
      "execution_count": null,
      "metadata": {
        "collapsed": true,
        "inputHidden": false,
        "outputHidden": false,
        "slideshow": {
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        }
      }
    }
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    "language_info": {
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