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Added DPLL and WalkSAT sections (#823)  
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    "and for the sentence to be true, P _has_ to be true and Q _has_ to be false.\n",
    "The pure symbol heuristic thus simplifies the problem a bit.\n",
    "3. Unit clause heuristic:\n",
    "<br>\n",
    "In the context of DPLL, clauses with just one literal and clauses with all but one _false_ literals are called unit clauses.\n",
    "If a clause is a unit clause, it can only be satisfied by assigning the necessary value to make the last literal true.\n",
    "We have no other choice.\n",
    "<br>\n",
    "Assigning one unit clause can create another unit clause.\n",
    "For example, when P is false, $(P\\lor Q)$ becomes a unit clause, causing _true_ to be assigned to Q.\n",
    "A series of forced assignments derived from previous unit clauses is called _unit propagation_.\n",
    "In this way, this heuristic simplifies the problem further.\n",
    "<br>\n",
    "The algorithm often employs other tricks to scale up to large problems.\n",
    "However, these tricks are currently out of the scope of this notebook. Refer to section 7.6 of the book for more details.\n",
    "<br>\n",
    "<br>\n",
    "Let's have a look at the algorithm."
   ]
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       "\n",
       "<div class=\"highlight\"><pre><span></span><span class=\"k\">def</span> <span class=\"nf\">dpll</span><span class=\"p\">(</span><span class=\"n\">clauses</span><span class=\"p\">,</span> <span class=\"n\">symbols</span><span class=\"p\">,</span> <span class=\"n\">model</span><span class=\"p\">):</span>\n",
       "    <span class=\"sd\">&quot;&quot;&quot;See if the clauses are true in a partial model.&quot;&quot;&quot;</span>\n",
       "    <span class=\"n\">unknown_clauses</span> <span class=\"o\">=</span> <span class=\"p\">[]</span>  <span class=\"c1\"># clauses with an unknown truth value</span>\n",
       "    <span class=\"k\">for</span> <span class=\"n\">c</span> <span class=\"ow\">in</span> <span class=\"n\">clauses</span><span class=\"p\">:</span>\n",
       "        <span class=\"n\">val</span> <span class=\"o\">=</span> <span class=\"n\">pl_true</span><span class=\"p\">(</span><span class=\"n\">c</span><span class=\"p\">,</span> <span class=\"n\">model</span><span class=\"p\">)</span>\n",
       "        <span class=\"k\">if</span> <span class=\"n\">val</span> <span class=\"ow\">is</span> <span class=\"bp\">False</span><span class=\"p\">:</span>\n",
       "            <span class=\"k\">return</span> <span class=\"bp\">False</span>\n",
       "        <span class=\"k\">if</span> <span class=\"n\">val</span> <span class=\"ow\">is</span> <span class=\"ow\">not</span> <span class=\"bp\">True</span><span class=\"p\">:</span>\n",
       "            <span class=\"n\">unknown_clauses</span><span class=\"o\">.</span><span class=\"n\">append</span><span class=\"p\">(</span><span class=\"n\">c</span><span class=\"p\">)</span>\n",
       "    <span class=\"k\">if</span> <span class=\"ow\">not</span> <span class=\"n\">unknown_clauses</span><span class=\"p\">:</span>\n",
       "        <span class=\"k\">return</span> <span class=\"n\">model</span>\n",
       "    <span class=\"n\">P</span><span class=\"p\">,</span> <span class=\"n\">value</span> <span class=\"o\">=</span> <span class=\"n\">find_pure_symbol</span><span class=\"p\">(</span><span class=\"n\">symbols</span><span class=\"p\">,</span> <span class=\"n\">unknown_clauses</span><span class=\"p\">)</span>\n",
       "    <span class=\"k\">if</span> <span class=\"n\">P</span><span class=\"p\">:</span>\n",
       "        <span class=\"k\">return</span> <span class=\"n\">dpll</span><span class=\"p\">(</span><span class=\"n\">clauses</span><span class=\"p\">,</span> <span class=\"n\">removeall</span><span class=\"p\">(</span><span class=\"n\">P</span><span class=\"p\">,</span> <span class=\"n\">symbols</span><span class=\"p\">),</span> <span class=\"n\">extend</span><span class=\"p\">(</span><span class=\"n\">model</span><span class=\"p\">,</span> <span class=\"n\">P</span><span class=\"p\">,</span> <span class=\"n\">value</span><span class=\"p\">))</span>\n",
       "    <span class=\"n\">P</span><span class=\"p\">,</span> <span class=\"n\">value</span> <span class=\"o\">=</span> <span class=\"n\">find_unit_clause</span><span class=\"p\">(</span><span class=\"n\">clauses</span><span class=\"p\">,</span> <span class=\"n\">model</span><span class=\"p\">)</span>\n",
       "    <span class=\"k\">if</span> <span class=\"n\">P</span><span class=\"p\">:</span>\n",
       "        <span class=\"k\">return</span> <span class=\"n\">dpll</span><span class=\"p\">(</span><span class=\"n\">clauses</span><span class=\"p\">,</span> <span class=\"n\">removeall</span><span class=\"p\">(</span><span class=\"n\">P</span><span class=\"p\">,</span> <span class=\"n\">symbols</span><span class=\"p\">),</span> <span class=\"n\">extend</span><span class=\"p\">(</span><span class=\"n\">model</span><span class=\"p\">,</span> <span class=\"n\">P</span><span class=\"p\">,</span> <span class=\"n\">value</span><span class=\"p\">))</span>\n",
       "    <span class=\"k\">if</span> <span class=\"ow\">not</span> <span class=\"n\">symbols</span><span class=\"p\">:</span>\n",
       "        <span class=\"k\">raise</span> <span class=\"ne\">TypeError</span><span class=\"p\">(</span><span class=\"s2\">&quot;Argument should be of the type Expr.&quot;</span><span class=\"p\">)</span>\n",
       "    <span class=\"n\">P</span><span class=\"p\">,</span> <span class=\"n\">symbols</span> <span class=\"o\">=</span> <span class=\"n\">symbols</span><span class=\"p\">[</span><span class=\"mi\">0</span><span class=\"p\">],</span> <span class=\"n\">symbols</span><span class=\"p\">[</span><span class=\"mi\">1</span><span class=\"p\">:]</span>\n",
       "    <span class=\"k\">return</span> <span class=\"p\">(</span><span class=\"n\">dpll</span><span class=\"p\">(</span><span class=\"n\">clauses</span><span class=\"p\">,</span> <span class=\"n\">symbols</span><span class=\"p\">,</span> <span class=\"n\">extend</span><span class=\"p\">(</span><span class=\"n\">model</span><span class=\"p\">,</span> <span class=\"n\">P</span><span class=\"p\">,</span> <span class=\"bp\">True</span><span class=\"p\">))</span> <span class=\"ow\">or</span>\n",
       "            <span class=\"n\">dpll</span><span class=\"p\">(</span><span class=\"n\">clauses</span><span class=\"p\">,</span> <span class=\"n\">symbols</span><span class=\"p\">,</span> <span class=\"n\">extend</span><span class=\"p\">(</span><span class=\"n\">model</span><span class=\"p\">,</span> <span class=\"n\">P</span><span class=\"p\">,</span> <span class=\"bp\">False</span><span class=\"p\">)))</span>\n",
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   "source": [
    "The algorithm uses the ideas described above to check satisfiability of a sentence in propositional logic.\n",
    "It recursively calls itself, simplifying the problem at each step. It also uses helper functions `find_pure_symbol` and `find_unit_clause` to carry out steps 2 and 3 above.\n",
    "<br>\n",
    "The `dpll_satisfiable` helper function converts the input clauses to _conjunctive normal form_ and calls the `dpll` function with the correct parameters."
   ]
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       "<body>\n",
       "<h2></h2>\n",
       "\n",
       "<div class=\"highlight\"><pre><span></span><span class=\"k\">def</span> <span class=\"nf\">dpll_satisfiable</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">):</span>\n",
       "    <span class=\"sd\">&quot;&quot;&quot;Check satisfiability of a propositional sentence.</span>\n",
       "<span class=\"sd\">    This differs from the book code in two ways: (1) it returns a model</span>\n",
       "<span class=\"sd\">    rather than True when it succeeds; this is more useful. (2) The</span>\n",
       "<span class=\"sd\">    function find_pure_symbol is passed a list of unknown clauses, rather</span>\n",
       "<span class=\"sd\">    than a list of all clauses and the model; this is more efficient.&quot;&quot;&quot;</span>\n",
       "    <span class=\"n\">clauses</span> <span class=\"o\">=</span> <span class=\"n\">conjuncts</span><span class=\"p\">(</span><span class=\"n\">to_cnf</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">))</span>\n",
       "    <span class=\"n\">symbols</span> <span class=\"o\">=</span> <span class=\"nb\">list</span><span class=\"p\">(</span><span class=\"n\">prop_symbols</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">))</span>\n",
       "    <span class=\"k\">return</span> <span class=\"n\">dpll</span><span class=\"p\">(</span><span class=\"n\">clauses</span><span class=\"p\">,</span> <span class=\"n\">symbols</span><span class=\"p\">,</span> <span class=\"p\">{})</span>\n",
       "</pre></div>\n",
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   ],
   "source": [
    "psource(dpll_satisfiable)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's see a few examples of usage."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [],
   "source": [
    "A, B, C, D = expr('A, B, C, D')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{C: False, A: True, D: True, B: True}"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dpll_satisfiable(A & B & ~C & D)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is a simple case to highlight that the algorithm actually works."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{C: True, D: False, B: True}"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dpll_satisfiable((A & B) | (C & ~A) | (B & ~D))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If a particular symbol isn't present in the solution, \n",
    "it means that the solution is independent of the value of that symbol.\n",
    "In this case, the solution is independent of A."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{A: True, B: True}"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dpll_satisfiable(A |'<=>'| B)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{C: True, A: True, B: False}"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dpll_satisfiable((A |'<=>'| B) |'==>'| (C & ~A))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{C: True, A: True}"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dpll_satisfiable((A | (B & C)) |'<=>'| ((A | B) & (A | C)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2. WalkSAT algorithm\n",
    "This algorithm is very similar to Hill climbing.\n",
    "On every iteration, the algorithm picks an unsatisfied clause and flips a symbol in the clause.\n",
    "This is similar to finding a neighboring state in the `hill_climbing` algorithm.\n",
    "<br>\n",
    "The symbol to be flipped is decided by an evaluation function that counts the number of unsatisfied clauses.\n",
    "Sometimes, symbols are also flipped randomly, to avoid local optima. A subtle balance between greediness and randomness is required. Alternatively, some versions of the algorithm restart with a completely new random assignment if no solution has been found for too long, as a way of getting out of local minima of numbers of unsatisfied clauses.\n",
    "<br>\n",
    "<br>\n",
    "Let's have a look at the algorithm."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
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       "<body>\n",
       "<h2></h2>\n",
       "\n",
       "<div class=\"highlight\"><pre><span></span><span class=\"k\">def</span> <span class=\"nf\">WalkSAT</span><span class=\"p\">(</span><span class=\"n\">clauses</span><span class=\"p\">,</span> <span class=\"n\">p</span><span class=\"o\">=</span><span class=\"mf\">0.5</span><span class=\"p\">,</span> <span class=\"n\">max_flips</span><span class=\"o\">=</span><span class=\"mi\">10000</span><span class=\"p\">):</span>\n",
       "    <span class=\"sd\">&quot;&quot;&quot;Checks for satisfiability of all clauses by randomly flipping values of variables</span>\n",
       "<span class=\"sd\">    &quot;&quot;&quot;</span>\n",
       "    <span class=\"c1\"># Set of all symbols in all clauses</span>\n",
       "    <span class=\"n\">symbols</span> <span class=\"o\">=</span> <span class=\"p\">{</span><span class=\"n\">sym</span> <span class=\"k\">for</span> <span class=\"n\">clause</span> <span class=\"ow\">in</span> <span class=\"n\">clauses</span> <span class=\"k\">for</span> <span class=\"n\">sym</span> <span class=\"ow\">in</span> <span class=\"n\">prop_symbols</span><span class=\"p\">(</span><span class=\"n\">clause</span><span class=\"p\">)}</span>\n",
       "    <span class=\"c1\"># model is a random assignment of true/false to the symbols in clauses</span>\n",
       "    <span class=\"n\">model</span> <span class=\"o\">=</span> <span class=\"p\">{</span><span class=\"n\">s</span><span class=\"p\">:</span> <span class=\"n\">random</span><span class=\"o\">.</span><span class=\"n\">choice</span><span class=\"p\">([</span><span class=\"bp\">True</span><span class=\"p\">,</span> <span class=\"bp\">False</span><span class=\"p\">])</span> <span class=\"k\">for</span> <span class=\"n\">s</span> <span class=\"ow\">in</span> <span class=\"n\">symbols</span><span class=\"p\">}</span>\n",
       "    <span class=\"k\">for</span> <span class=\"n\">i</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"n\">max_flips</span><span class=\"p\">):</span>\n",
       "        <span class=\"n\">satisfied</span><span class=\"p\">,</span> <span class=\"n\">unsatisfied</span> <span class=\"o\">=</span> <span class=\"p\">[],</span> <span class=\"p\">[]</span>\n",
       "        <span class=\"k\">for</span> <span class=\"n\">clause</span> <span class=\"ow\">in</span> <span class=\"n\">clauses</span><span class=\"p\">:</span>\n",
       "            <span class=\"p\">(</span><span class=\"n\">satisfied</span> <span class=\"k\">if</span> <span class=\"n\">pl_true</span><span class=\"p\">(</span><span class=\"n\">clause</span><span class=\"p\">,</span> <span class=\"n\">model</span><span class=\"p\">)</span> <span class=\"k\">else</span> <span class=\"n\">unsatisfied</span><span class=\"p\">)</span><span class=\"o\">.</span><span class=\"n\">append</span><span class=\"p\">(</span><span class=\"n\">clause</span><span class=\"p\">)</span>\n",
       "        <span class=\"k\">if</span> <span class=\"ow\">not</span> <span class=\"n\">unsatisfied</span><span class=\"p\">:</span>  <span class=\"c1\"># if model satisfies all the clauses</span>\n",
       "            <span class=\"k\">return</span> <span class=\"n\">model</span>\n",
       "        <span class=\"n\">clause</span> <span class=\"o\">=</span> <span class=\"n\">random</span><span class=\"o\">.</span><span class=\"n\">choice</span><span class=\"p\">(</span><span class=\"n\">unsatisfied</span><span class=\"p\">)</span>\n",
       "        <span class=\"k\">if</span> <span class=\"n\">probability</span><span class=\"p\">(</span><span class=\"n\">p</span><span class=\"p\">):</span>\n",
       "            <span class=\"n\">sym</span> <span class=\"o\">=</span> <span class=\"n\">random</span><span class=\"o\">.</span><span class=\"n\">choice</span><span class=\"p\">(</span><span class=\"nb\">list</span><span class=\"p\">(</span><span class=\"n\">prop_symbols</span><span class=\"p\">(</span><span class=\"n\">clause</span><span class=\"p\">)))</span>\n",
       "        <span class=\"k\">else</span><span class=\"p\">:</span>\n",
       "            <span class=\"c1\"># Flip the symbol in clause that maximizes number of sat. clauses</span>\n",
       "            <span class=\"k\">def</span> <span class=\"nf\">sat_count</span><span class=\"p\">(</span><span class=\"n\">sym</span><span class=\"p\">):</span>\n",
       "                <span class=\"c1\"># Return the the number of clauses satisfied after flipping the symbol.</span>\n",
       "                <span class=\"n\">model</span><span class=\"p\">[</span><span class=\"n\">sym</span><span class=\"p\">]</span> <span class=\"o\">=</span> <span class=\"ow\">not</span> <span class=\"n\">model</span><span class=\"p\">[</span><span class=\"n\">sym</span><span class=\"p\">]</span>\n",
       "                <span class=\"n\">count</span> <span class=\"o\">=</span> <span class=\"nb\">len</span><span class=\"p\">([</span><span class=\"n\">clause</span> <span class=\"k\">for</span> <span class=\"n\">clause</span> <span class=\"ow\">in</span> <span class=\"n\">clauses</span> <span class=\"k\">if</span> <span class=\"n\">pl_true</span><span class=\"p\">(</span><span class=\"n\">clause</span><span class=\"p\">,</span> <span class=\"n\">model</span><span class=\"p\">)])</span>\n",
       "                <span class=\"n\">model</span><span class=\"p\">[</span><span class=\"n\">sym</span><span class=\"p\">]</span> <span class=\"o\">=</span> <span class=\"ow\">not</span> <span class=\"n\">model</span><span class=\"p\">[</span><span class=\"n\">sym</span><span class=\"p\">]</span>\n",
       "                <span class=\"k\">return</span> <span class=\"n\">count</span>\n",
       "            <span class=\"n\">sym</span> <span class=\"o\">=</span> <span class=\"n\">argmax</span><span class=\"p\">(</span><span class=\"n\">prop_symbols</span><span class=\"p\">(</span><span class=\"n\">clause</span><span class=\"p\">),</span> <span class=\"n\">key</span><span class=\"o\">=</span><span class=\"n\">sat_count</span><span class=\"p\">)</span>\n",
       "        <span class=\"n\">model</span><span class=\"p\">[</span><span class=\"n\">sym</span><span class=\"p\">]</span> <span class=\"o\">=</span> <span class=\"ow\">not</span> <span class=\"n\">model</span><span class=\"p\">[</span><span class=\"n\">sym</span><span class=\"p\">]</span>\n",
       "    <span class=\"c1\"># If no solution is found within the flip limit, we return failure</span>\n",
       "    <span class=\"k\">return</span> <span class=\"bp\">None</span>\n",
       "</pre></div>\n",
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   ],
   "source": [
    "psource(WalkSAT)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function takes three arguments:\n",
    "<br>\n",
    "1. The `clauses` we want to satisfy.\n",
    "<br>\n",
    "2. The probability `p` of randomly changing a symbol.\n",
    "<br>\n",
    "3. The maximum number of flips (`max_flips`) the algorithm will run for. If the clauses are still unsatisfied, the algorithm returns `None` to denote failure.\n",
    "<br>\n",
    "The algorithm is identical in concept to Hill climbing and the code isn't difficult to understand.\n",
    "<br>\n",
    "<br>\n",
    "Let's see a few examples of usage."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "A, B, C, D = expr('A, B, C, D')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{C: False, A: True, D: True, B: True}"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "WalkSAT([A, B, ~C, D], 0.5, 100)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is a simple case to show that the algorithm converges."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{C: True, A: True, B: True}"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "WalkSAT([A & B, A & C], 0.5, 100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{C: True, A: True, D: True, B: True}"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "WalkSAT([A & B, C & D, C & B], 0.5, 100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [],
   "source": [
    "WalkSAT([A & B, C | D, ~(D | B)], 0.5, 1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This one doesn't give any output because WalkSAT did not find any model where these clauses hold. We can solve these clauses to see that they together form a contradiction and hence, it isn't supposed to have a solution."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "One point of difference between this algorithm and the `dpll_satisfiable` algorithms is that both these algorithms take inputs differently. \n",
    "For WalkSAT to take complete sentences as input, \n",
    "we can write a helper function that converts the input sentence into conjunctive normal form and then calls WalkSAT with the list of conjuncts of the CNF form of the sentence."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def WalkSAT_CNF(sentence, p=0.5, max_flips=10000):\n",
    "    return WalkSAT(conjuncts(to_cnf(sentence)), 0, max_flips)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can call `WalkSAT_CNF` and `DPLL_Satisfiable` with the same arguments."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{A: False, D: False, C: True, B: False}"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "WalkSAT_CNF((A & B) | (C & ~A) | (B & ~D), 0.5, 1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It works!\n",
    "<br>\n",
    "Notice that the solution generated by WalkSAT doesn't omit variables that the sentence doesn't depend upon. \n",
    "If the sentence is independent of a particular variable, the solution contains a random value for that variable because of the stochastic nature of the algorithm.\n",
    "<br>\n",
    "<br>\n",
    "Let's compare the runtime of WalkSAT and DPLL for a few cases. We will use the `%%timeit` magic to do this."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "sentence_1 = A |'<=>'| B\n",
    "sentence_2 = (A & B) | (C & ~A) | (B & ~D)\n",
    "sentence_3 = (A | (B & C)) |'<=>'| ((A | B) & (A | C))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "100 loops, best of 3: 2.46 ms per loop\n"
     ]
    }
   ],
   "source": [
    "%%timeit\n",
    "dpll_satisfiable(sentence_1)\n",
    "dpll_satisfiable(sentence_2)\n",
    "dpll_satisfiable(sentence_3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "100 loops, best of 3: 1.91 ms per loop\n"
     ]
    }
   ],
   "source": [
    "%%timeit\n",
    "WalkSAT_CNF(sentence_1)\n",
    "WalkSAT_CNF(sentence_2)\n",
    "WalkSAT_CNF(sentence_3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On an average, for solvable cases, `WalkSAT` is quite faster than `dpll` because, for a small number of variables, \n",
    "`WalkSAT` can reduce the search space significantly. \n",
    "Results can be different for sentences with more symbols though.\n",
    "Feel free to play around with this to understand the trade-offs of these algorithms better."
   ]
  },
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## First-Order Logic Knowledge Bases: `FolKB`\n",
    "\n",
    "The class `FolKB` can be used to represent a knowledge base of First-order logic sentences. You would initialize and use it the same way as you would for `PropKB` except that the clauses are first-order definite clauses. We will see how to write such clauses to create a database and query them in the following sections."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Criminal KB\n",
    "In this section we create a `FolKB` based on the following paragraph.<br/>\n",
    "<em>The law says that it is a crime for an American to sell weapons to hostile nations. The country Nono, an enemy of America, has some missiles, and all of its missiles were sold to it by Colonel West, who is American.</em><br/>\n",
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    "The first step is to extract the facts and convert them into first-order definite clauses. Extracting the facts from data alone is a challenging task. Fortunately, we have a small paragraph and can do extraction and conversion manually. We'll store the clauses in list aptly named `clauses`."
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   ]
  },
  {
   "cell_type": "code",
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   "execution_count": 48,
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   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "clauses = []"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<em>“... it is a crime for an American to sell weapons to hostile nations”</em><br/>\n",
    "The keywords to look for here are 'crime', 'American', 'sell', 'weapon' and 'hostile'. We use predicate symbols to make meaning of them.\n",
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    "\n",
    "* `Criminal(x)`: `x` is a criminal\n",
    "* `American(x)`: `x` is an American\n",
    "* `Sells(x ,y, z)`: `x` sells `y` to `z`\n",
    "* `Weapon(x)`: `x` is a weapon\n",
    "* `Hostile(x)`: `x` is a hostile nation\n",
    "\n",
    "Let us now combine them with appropriate variable naming to depict the meaning of the sentence. The criminal `x` is also the American `x` who sells weapon `y` to `z`, which is a hostile nation.\n",
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    "\n",
    "$\\text{American}(x) \\land \\text{Weapon}(y) \\land \\text{Sells}(x, y, z) \\land \\text{Hostile}(z) \\implies \\text{Criminal} (x)$"
   ]
  },
  {
   "cell_type": "code",
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   "execution_count": 49,
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   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "clauses.append(expr(\"(American(x) & Weapon(y) & Sells(x, y, z) & Hostile(z)) ==> Criminal(x)\"))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<em>\"The country Nono, an enemy of America\"</em><br/>\n",
    "We now know that Nono is an enemy of America. We represent these nations using the constant symbols `Nono` and `America`. the enemy relation is show using the predicate symbol `Enemy`.\n",
    "\n",
    "$\\text{Enemy}(\\text{Nono}, \\text{America})$"
   ]
  },
  {
   "cell_type": "code",
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   "execution_count": 50,
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   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "clauses.append(expr(\"Enemy(Nono, America)\"))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<em>\"Nono ... has some missiles\"</em><br/>\n",
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    "This states the existence of some missile which is owned by Nono. $\\exists x \\text{Owns}(\\text{Nono}, x) \\land \\text{Missile}(x)$. We invoke existential instantiation to introduce a new constant `M1` which is the missile owned by Nono.\n",
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    "\n",
    "$\\text{Owns}(\\text{Nono}, \\text{M1}), \\text{Missile}(\\text{M1})$"
   ]
  },
  {
   "cell_type": "code",
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   "execution_count": 51,
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   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "clauses.append(expr(\"Owns(Nono, M1)\"))\n",
    "clauses.append(expr(\"Missile(M1)\"))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "<em>\"All of its missiles were sold to it by Colonel West\"</em><br/>\n",
    "If Nono owns something and it classifies as a missile, then it was sold to Nono by West.\n",
    "\n",
    "$\\text{Missile}(x) \\land \\text{Owns}(\\text{Nono}, x) \\implies \\text{Sells}(\\text{West}, x, \\text{Nono})$"
   ]
  },
  {
   "cell_type": "code",
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   "execution_count": 52,
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   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "clauses.append(expr(\"(Missile(x) & Owns(Nono, x)) ==> Sells(West, x, Nono)\"))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<em>\"West, who is American\"</em><br/>\n",
    "West is an American.\n",
    "\n",
    "$\\text{American}(\\text{West})$"
   ]
  },
  {
   "cell_type": "code",
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   "execution_count": 53,
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   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "clauses.append(expr(\"American(West)\"))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We also know, from our understanding of language, that missiles are weapons and that an enemy of America counts as “hostile”.\n",
    "\n",
    "$\\text{Missile}(x) \\implies \\text{Weapon}(x), \\text{Enemy}(x, \\text{America}) \\implies \\text{Hostile}(x)$"
   ]
  },
  {
   "cell_type": "code",
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   "execution_count": 54,
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   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "clauses.append(expr(\"Missile(x) ==> Weapon(x)\"))\n",
    "clauses.append(expr(\"Enemy(x, America) ==> Hostile(x)\"))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now that we have converted the information into first-order definite clauses we can create our first-order logic knowledge base."
   ]
  },
  {
   "cell_type": "code",
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   "execution_count": 55,
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   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "crime_kb = FolKB(clauses)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Inference in First-Order Logic\n",
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    "In this section we look at a forward chaining and a backward chaining algorithm for `FolKB`. Both aforementioned algorithms rely on a process called <strong>unification</strong>, a key component of all first-order inference algorithms."
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   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
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