logic.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "# Logic: `logic.py`; Chapters 6-8"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This notebook describes the [logic.py](https://github.com/aimacode/aima-python/blob/master/logic.py) module, which covers Chapters 6 (Logical Agents),  7 (First-Order Logic) and  8 (Inference in First-Order Logic) of *[Artificial Intelligence: A Modern Approach](http://aima.cs.berkeley.edu)*. See the [intro notebook](https://github.com/aimacode/aima-python/blob/master/intro.ipynb) for instructions.\n",
    "\n",
    "We'll start by looking at `Expr`, the data type for logical sentences, and the convenience function `expr`. We'll be covering two types of knowledge bases, `PropKB` - Propositional logic knowledge base and `FolKB` - First order logic knowledge base. We will construct a propositional knowledge base of a specific situation in the Wumpus World. We will next go through the `tt_entails` function and experiment with it a bit. The `pl_resolution` and `pl_fc_entails` functions will come next. We'll study forward chaining and backward chaining algorithms for `FolKB` and use them on `crime_kb` knowledge base.\n",
    "\n",
    "But the first step is to load the code:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from utils import *\n",
    "from logic import *"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## Logical Sentences"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The `Expr` class is designed to represent any kind of mathematical expression. The simplest type of `Expr` is a symbol, which can be defined with the function `Symbol`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Symbol('x')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Or we can define multiple symbols at the same time with the function `symbols`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "(x, y, P, Q, f) = symbols('x, y, P, Q, f')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can combine `Expr`s with the regular Python infix and prefix operators. Here's how we would form the logical sentence \"P and not Q\":"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(P & ~Q)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P & ~Q"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This works because the `Expr` class overloads the `&` operator with this definition:\n",
    "\n",
    "```python\n",
    "def __and__(self, other): return Expr('&',  self, other)```\n",
    "     \n",
    "and does similar overloads for the other operators. An `Expr` has two fields: `op` for the operator, which is always a string, and `args` for the arguments, which is a tuple of 0 or more expressions. By \"expression,\" I mean either an instance of `Expr`, or a number. Let's take a look at the fields for some `Expr` examples:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'&'"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sentence = P & ~Q\n",
    "\n",
    "sentence.op"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(P, ~Q)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sentence.args"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'P'"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P.op"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "()"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P.args"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'P'"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Pxy = P(x, y)\n",
    "\n",
    "Pxy.op"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(x, y)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Pxy.args"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is important to note that the `Expr` class does not define the *logic* of Propositional Logic sentences; it just gives you a way to *represent* expressions. Think of an `Expr` as an [abstract syntax tree](https://en.wikipedia.org/wiki/Abstract_syntax_tree).  Each of the `args` in an `Expr` can be either a symbol, a number, or a nested `Expr`. We can nest these trees to any depth. Here is a deply nested `Expr`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(((3 * f(x, y)) + (P(y) / 2)) + 1)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "3 * f(x, y) + P(y) / 2 + 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Operators for Constructing Logical Sentences\n",
    "\n",
    "Here is a table of the operators that can be used to form sentences. Note that we have a problem: we want to use Python operators to make sentences, so that our programs (and our interactive sessions like the one here) will show simple code. But Python does not allow implication arrows as operators, so for now we have to use a more verbose notation that Python does allow: `|'==>'|` instead of just `==>`. Alternately, you can always use the more verbose `Expr` constructor forms:\n",
    "\n",
    "| Operation                | Book | Python Infix Input | Python Output | Python `Expr` Input\n",
    "|--------------------------|----------------------|-------------------------|---|---|\n",
    "| Negation                 | &not; P      | `~P`                       | `~P` | `Expr('~', P)`\n",
    "| And                      | P &and; Q       | `P & Q`                     | `P & Q` | `Expr('&', P, Q)`\n",
    "| Or                       | P &or; Q | `P`<tt> &#124; </tt>`Q`| `P`<tt> &#124; </tt>`Q` | `Expr('`&#124;`', P, Q)`\n",
    "| Inequality (Xor)         | P &ne; Q     | `P ^ Q`                | `P ^ Q`  | `Expr('^', P, Q)`\n",
    "| Implication                  | P &rarr; Q    | `P` <tt>&#124;</tt>`'==>'`<tt>&#124;</tt> `Q`   | `P ==> Q` | `Expr('==>', P, Q)`\n",
    "| Reverse Implication      | Q &larr; P     | `Q` <tt>&#124;</tt>`'<=='`<tt>&#124;</tt> `P`  |`Q <== P` | `Expr('<==', Q, P)`\n",
    "| Equivalence            | P &harr; Q   | `P` <tt>&#124;</tt>`'<=>'`<tt>&#124;</tt> `Q`   |`P <=> Q` | `Expr('<=>', P, Q)`\n",
    "\n",
    "Here's an example of defining a sentence with an implication arrow:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(~(P & Q) ==> (~P | ~Q))"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "~(P & Q)  |'==>'|  (~P | ~Q)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## `expr`: a Shortcut for Constructing Sentences\n",
    "\n",
    "If the `|'==>'|` notation looks ugly to you, you can use the function `expr` instead:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(~(P & Q) ==> (~P | ~Q))"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr('~(P & Q)  ==>  (~P | ~Q)')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`expr` takes a string as input, and parses it into an `Expr`. The string can contain arrow operators: `==>`, `<==`, or `<=>`, which are handled as if they were regular Python infix operators. And `expr` automatically defines any symbols, so you don't need to pre-define them:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sqrt(((b ** 2) - ((4 * a) * c)))"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr('sqrt(b ** 2 - 4 * a * c)')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For now that's all you need to know about `expr`. If you are interested, we explain the messy details of how `expr` is implemented and how `|'==>'|` is handled in the appendix."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Propositional Knowledge Bases: `PropKB`\n",
    "\n",
    "The class `PropKB` can be used to represent a knowledge base of propositional logic sentences.\n",
    "\n",
    "We see that the class `KB` has four methods, apart from `__init__`. A point to note here: the `ask` method simply calls the `ask_generator` method. Thus, this one has already been implemented and what you'll have to actually implement when you create your own knowledge base class (if you want to, though I doubt you'll ever need to; just use the ones we've created for you), will be the `ask_generator` function and not the `ask` function itself.\n",
    "\n",
    "The class `PropKB` now.\n",
    "* `__init__(self, sentence=None)` : The constructor `__init__` creates a single field `clauses` which will be a list of all the sentences of the knowledge base. Note that each one of these sentences will be a 'clause' i.e. a sentence which is made up of only literals and `or`s.\n",
    "* `tell(self, sentence)` : When you want to add a sentence to the KB, you use the `tell` method. This method takes a sentence, converts it to its CNF, extracts all the clauses, and adds all these clauses to the `clauses` field. So, you need not worry about `tell`ing only clauses to the knowledge base. You can `tell` the knowledge base a sentence in any form that you wish; converting it to CNF and adding the resulting clauses will be handled by the `tell` method.\n",
    "* `ask_generator(self, query)` : The `ask_generator` function is used by the `ask` function. It calls the `tt_entails` function, which in turn returns `True` if the knowledge base entails query and `False` otherwise. The `ask_generator` itself returns an empty dict `{}` if the knowledge base entails query and `None` otherwise. This might seem a little bit weird to you. After all, it makes more sense just to return a `True` or a `False` instead of the `{}` or `None` But this is done to maintain consistency with the way things are in First-Order Logic, where, an `ask_generator` function, is supposed to return all the substitutions that make the query true. Hence the dict, to return all these substitutions. I will be mostly be using the `ask` function which returns a `{}` or a `False`, but if you don't like this, you can always use the `ask_if_true` function which returns a `True` or a `False`.\n",
    "* `retract(self, sentence)` : This function removes all the clauses of the sentence given, from the knowledge base. Like the `tell` function, you don't have to pass clauses to remove them from the knowledge base; any sentence will do fine. The function will take care of converting that sentence to clauses and then remove those."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Wumpus World KB\n",
    "Let us create a `PropKB` for the wumpus world with the sentences mentioned in `section 7.4.3`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "wumpus_kb = PropKB()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We define the symbols we use in our clauses.<br/>\n",
    "$P_{x, y}$ is true if there is a pit in `[x, y]`.<br/>\n",
    "$B_{x, y}$ is true if the agent senses breeze in `[x, y]`.<br/>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "P11, P12, P21, P22, P31, B11, B21 = expr('P11, P12, P21, P22, P31, B11, B21')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we tell sentences based on `section 7.4.3`.<br/>\n",
    "There is no pit in `[1,1]`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "wumpus_kb.tell(~P11)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A square is breezy if and only if there is a pit in a neighboring square. This has to be stated for each square but for now, we include just the relevant squares."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "wumpus_kb.tell(B11 | '<=>' | ((P12 | P21)))\n",
    "wumpus_kb.tell(B21 | '<=>' | ((P11 | P22 | P31)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we include the breeze percepts for the first two squares leading up to the situation in `Figure 7.3(b)`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "wumpus_kb.tell(~B11)\n",
    "wumpus_kb.tell(B21)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can check the clauses stored in a `KB` by accessing its `clauses` variable"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[~P11,\n",
       " (~P12 | B11),\n",
       " (~P21 | B11),\n",
       " (P12 | P21 | ~B11),\n",
       " (~P11 | B21),\n",
       " (~P22 | B21),\n",
       " (~P31 | B21),\n",
       " (P11 | P22 | P31 | ~B21),\n",
       " ~B11,\n",
       " B21]"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "wumpus_kb.clauses"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We see that the equivalence $B_{1, 1} \\iff (P_{1, 2} \\lor P_{2, 1})$ was automatically converted to two implications which were inturn converted to CNF which is stored in the `KB`.<br/>\n",
    "$B_{1, 1} \\iff (P_{1, 2} \\lor P_{2, 1})$ was split into $B_{1, 1} \\implies (P_{1, 2} \\lor P_{2, 1})$ and $B_{1, 1} \\Longleftarrow (P_{1, 2} \\lor P_{2, 1})$.<br/>\n",
    "$B_{1, 1} \\implies (P_{1, 2} \\lor P_{2, 1})$ was converted to $P_{1, 2} \\lor P_{2, 1} \\lor \\neg B_{1, 1}$.<br/>\n",
    "$B_{1, 1} \\Longleftarrow (P_{1, 2} \\lor P_{2, 1})$ was converted to $\\neg (P_{1, 2} \\lor P_{2, 1}) \\lor B_{1, 1}$ which becomes $(\\neg P_{1, 2} \\lor B_{1, 1}) \\land (\\neg P_{2, 1} \\lor B_{1, 1})$ after applying De Morgan's laws and distributing the disjunction.<br/>\n",
    "$B_{2, 1} \\iff (P_{1, 1} \\lor P_{2, 2} \\lor P_{3, 2})$ is converted in similar manner."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Inference in Propositional Knowlwdge Base\n",
    "In this section we will look at two algorithms to check if a sentence is entailed by the `KB`. Our goal is to decide whether $\\text{KB} \\vDash \\alpha$ for some sentence $\\alpha$.\n",
    "### Truth Table Enumeration\n",
    "It is a model-checking approach which, as the name suggests, enumerates all possible models in which the `KB` is true and checks if $\\alpha$ is also true in these models. We list the $n$ symbols in the `KB` and enumerate the $2^{n}$ models in a depth-first manner and check the truth of `KB` and $\\alpha$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%psource tt_check_all"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that `tt_entails()` takes an `Expr` which is a conjunction of clauses as the input instead of the `KB` itself. You can use the `ask_if_true()` method of `PropKB` which does all the required conversions. Let's check what `wumpus_kb` tells us about $P_{1, 1}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(True, False)"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "wumpus_kb.ask_if_true(~P11), wumpus_kb.ask_if_true(P11)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Looking at Figure 7.9 we see that in all models in which the knowledge base is `True`, $P_{1, 1}$ is `False`. It makes sense that `ask_if_true()` returns `True` for $\\alpha = \\neg P_{1, 1}$ and `False` for $\\alpha = P_{1, 1}$. This begs the question, what if $\\alpha$ is `True` in only a portion of all models. Do we return `True` or `False`? This doesn't rule out the possibility of $\\alpha$ being `True` but it is not entailed by the `KB` so we return `False` in such cases. We can see this is the case for $P_{2, 2}$ and $P_{3, 1}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(False, False)"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "wumpus_kb.ask_if_true(~P22), wumpus_kb.ask_if_true(P22)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Proof by Resolution\n",
    "Recall that our goal is to check whether $\\text{KB} \\vDash \\alpha$ i.e. is $\\text{KB} \\implies \\alpha$ true in every model. Suppose we wanted to check if $P \\implies Q$ is valid. We check the satisfiability of $\\neg (P \\implies Q)$, which can be rewritten as $P \\land \\neg Q$. If $P \\land \\neg Q$ is unsatisfiable, then $P \\implies Q$ must be true in all models. This gives us the result \"$\\text{KB} \\vDash \\alpha$ <em>if and only if</em> $\\text{KB} \\land \\neg \\alpha$ is unsatisfiable\".<br/>\n",
    "This technique corresponds to <em>proof by <strong>contradiction</strong></em>, a standard mathematical proof technique. We assume $\\alpha$ to be false and show that this leads to a contradiction with known axioms in $\\text{KB}$. We obtain a contradiction by making valid inferences using inference rules. In this proof we use a single inference rule, <strong>resolution</strong> which states $(l_1 \\lor \\dots \\lor l_k) \\land (m_1 \\lor \\dots \\lor m_n) \\land (l_i \\iff \\neg m_j) \\implies l_1 \\lor \\dots \\lor l_{i - 1} \\lor l_{i + 1} \\lor \\dots \\lor l_k \\lor m_1 \\lor \\dots \\lor m_{j - 1} \\lor m_{j + 1} \\lor \\dots \\lor m_n$. Applying the resolution yeilds us a clause which we add to the KB. We keep doing this until:\n",
    "<ul>\n",
    "<li>There are no new clauses that can be added, in which case $\\text{KB} \\nvDash \\alpha$.</li>\n",
    "<li>Two clauses resolve to yield the <em>empty clause</em>, in which case $\\text{KB} \\vDash \\alpha$.</li>\n",
    "</ul>\n",
    "The <em>empty clause</em> is equivalent to <em>False</em> because it arises only from resolving two complementary\n",
    "unit clauses such as $P$ and $\\neg P$ which is a contradiction as both $P$ and $\\neg P$ can't be <em>True</em> at the same time."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%psource pl_resolution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(True, False)"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pl_resolution(wumpus_kb, ~P11), pl_resolution(wumpus_kb, P11)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(False, False)"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pl_resolution(wumpus_kb, ~P22), pl_resolution(wumpus_kb, P22)"
   ]
  },
  {
   "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",
    "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. Fortnately we have a small paragraph and can do extraction and conversion manually. We'll store the clauses in list aptly named `clauses`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "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",
    "<ul>\n",
    "<li>`Criminal(x)`: `x` is a criminal</li>\n",
    "<li>`American(x)`: `x` is an American</li>\n",
    "<li>`Sells(x ,y, z)`: `x` sells `y` to `z`</li>\n",
    "<li>`Weapon(x)`: `x` is a weapon</li>\n",
    "<li>`Hostile(x)`: `x` is a hostile nation</li>\n",
    "</ul>\n",
    "Let us now combine them with appropriate variable naming 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",
    "\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",
   "execution_count": 28,
   "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",
   "execution_count": 29,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "clauses.append(expr(\"Enemy(Nono, America)\"))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<em>\"Nono ... has some missiles\"</em><br/>\n",
    "This states the existance 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",
    "\n",
    "$\\text{Owns}(\\text{Nono}, \\text{M1}), \\text{Missile}(\\text{M1})$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "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",
   "execution_count": 31,
   "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",
   "execution_count": 32,
   "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",
   "execution_count": 33,
   "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",
   "execution_count": 34,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "crime_kb = FolKB(clauses)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Inference in First-Order Logic\n",
    "In this section we look at a forward chaining and a backward chaining algorithm for `FolKB`. Both the aforementioned algorithms rely on a process called <strong>unification</strong>, a key component of all first-order inference algorithms."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Unification\n",
    "We sometimes require finding substitutions that make different logical expressions look identical. This process, called unification, is done by the `unify` algorithm. It takes as input two sentences and returns a <em>unifier</em> for them if one exists. A unifier is a dictionary which stores the substitutions required to make the two sentences identical. It does so by recursively unifying the components of a sentence, where the unification of a variable symbol `var` with a constant symbol `Const` is the mapping `{var: Const}`. Let's look at a few examples."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{x: 3}"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "unify(expr('x'), 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{x: B}"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "unify(expr('A(x)'), expr('A(B)'))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{x: Bella, y: Dobby}"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "unify(expr('Cat(x) & Dog(Dobby)'), expr('Cat(Bella) & Dog(y)'))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In cases where there is no possible substitution that unifies the two sentences the function return `None`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "None\n"
     ]
    }
   ],
   "source": [
    "print(unify(expr('Cat(x)'), expr('Dog(Dobby)')))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We also need to take care we do not unintentionally use same variable name. Unify treats them as a single variable which prevents it from taking multiple value."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "None\n"
     ]
    }
   ],
   "source": [
    "print(unify(expr('Cat(x) & Dog(Dobby)'), expr('Cat(Bella) & Dog(x)')))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Forward Chaining Algorithm\n",
    "We consider the simple forward-chaining algorithm presented in <em>Figure 9.3</em>. We look at each rule in the knoweldge base and see if the premises can be satisfied. This is done by finding a substitution which unifies the each of the premise with a clause in the `KB`. If we are able to unify the premises the conclusion (with the corresponding substitution) is added to the `KB`. This inferencing process is repeated until either the query can be answered or till no new sentences can be aded. We test if the newly added clause unifies with the query in which case the substitution yielded by `unify` is an answer to the query. If we run out of sentences to infer, this means the query was a failure.\n",
    "\n",
    "The function `fol_fc_ask` is a generator which yields all substitutions which validate the query."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%psource fol_fc_ask"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's find out all the hostile nations. Note that we only told the `KB` that Nono was an enemy of America, not that it was hostile."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[{x: Nono}]\n"
     ]
    }
   ],
   "source": [
    "answer = fol_fc_ask(crime_kb, expr('Hostile(x)'))\n",
    "print(list(answer))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The generator returned a single substitution which says that Nono is a hostile nation. See how after adding another enemy nation the generator returns two substitutions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[{x: Nono}, {x: JaJa}]\n"
     ]
    }
   ],
   "source": [
    "crime_kb.tell(expr('Enemy(JaJa, America)'))\n",
    "answer = fol_fc_ask(crime_kb, expr('Hostile(x)'))\n",
    "print(list(answer))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<strong><em>Note</em>:</strong> `fol_fc_ask` makes changes to the `KB` by adding sentences to it."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Backward Chaining Algorithm\n",
    "This algorithm works backward from the goal, chaining through rules to find known facts that support the proof. Suppose `goal` is the query we want to find the substitution for. We find rules of the form $\\text{lhs} \\implies \\text{goal}$ in the `KB` and try to prove `lhs`. There may be multiple clauses in the `KB` which give multiple `lhs`. It is sufficient to prove only one of these. But to prove a `lhs` all the conjuncts in the `lhs` of the clause must be proved. This makes it similar to <em>And/Or</em> search."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### OR\n",
    "The <em>OR</em> part of the algorithm comes from our choice to select any clause of the form $\\text{lhs} \\implies \\text{goal}$. Looking at all rules's `lhs` whose `rhs` unify with the `goal`, we yield a substitution which proves all the conjuncts in the `lhs`. We use `parse_definite_clause` to attain `lhs` and `rhs` from a clause of the form $\\text{lhs} \\implies \\text{rhs}$. For atomic facts the `lhs` is an empty list."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%psource fol_bc_or"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### AND\n",
    "The <em>AND</em> corresponds to proving all the conjuncts in the `lhs`. We need to find a substitution which proves each <em>and</em> every clause in the list of conjuncts."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%psource fol_bc_and"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now the main function `fl_bc_ask` calls `fol_bc_or` with substitution initialized as empty. The `ask` method of `FolKB` uses `fol_bc_ask` and fetches the first substitution returned by the generator to answer query. Let's query the knowledge base we created from `clauses` to find hostile nations."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Rebuild KB because running fol_fc_ask would add new facts to the KB\n",
    "crime_kb = FolKB(clauses)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{v_5: x, x: Nono}"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "crime_kb.ask(expr('Hostile(x)'))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You may notice some new variables in the substitution. They are introduced to standardize the variable names to prevent naming problems as discussed in the [Unification section](#Unification)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Appendix: The Implementation of `|'==>'|`\n",
    "\n",
    "Consider the `Expr` formed by this syntax:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(P ==> ~Q)"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P |'==>'| ~Q"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What is the funny `|'==>'|` syntax? The trick is that \"`|`\" is just the regular Python or-operator, and so is exactly equivalent to this: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(P ==> ~Q)"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(P | '==>') | ~Q"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In other words, there are two applications of or-operators. Here's the first one:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "PartialExpr('==>', P)"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P | '==>'"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What is going on here is that the `__or__` method of `Expr` serves a dual purpose. If the right-hand-side is another `Expr` (or a number), then the result is an `Expr`, as in `(P | Q)`. But if the right-hand-side is a string, then the string is taken to be an operator, and we create a node in the abstract syntax tree corresponding to a partially-filled  `Expr`, one where we know the left-hand-side is `P` and the operator is `==>`, but we don't yet know the right-hand-side.\n",
    "\n",
    "The `PartialExpr` class has an `__or__` method that says to create an `Expr` node with the right-hand-side filled in. Here we can see the combination of the `PartialExpr` with `Q` to create a complete `Expr`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(P ==> ~Q)"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "partial = PartialExpr('==>', P) \n",
    "partial | ~Q"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This  [trick](http://code.activestate.com/recipes/384122-infix-operators/) is due to [Ferdinand Jamitzky](http://code.activestate.com/recipes/users/98863/), with a modification by [C. G. Vedant](https://github.com/Chipe1),\n",
    "who suggested using a string inside the or-bars.\n",
    "\n",
    "## Appendix: The Implementation of `expr`\n",
    "\n",
    "How does `expr` parse a string into an `Expr`? It turns out there are two tricks (besides the Jamitzky/Vedant trick):\n",
    "\n",
    "1. We do a string substitution, replacing \"`==>`\" with \"`|'==>'|`\" (and likewise for other operators).\n",
    "2. We `eval` the resulting string in an environment in which every identifier\n",
    "is bound to a symbol with that identifier as the `op`.\n",
    "\n",
    "In other words,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(~(P & Q) ==> (~P | ~Q))"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr('~(P & Q)  ==>  (~P | ~Q)')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "is equivalent to doing:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(~(P & Q) ==> (~P | ~Q))"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P, Q = symbols('P, Q')\n",
    "~(P & Q)  |'==>'|  (~P | ~Q)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "One thing to beware of: this puts `==>` at the same precedence level as `\"|\"`, which is not quite right. For example, we get this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(((P & Q) ==> P) | Q)"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P & Q  |'==>'|  P | Q"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "which is probably not what we meant; when in doubt, put in extra parens:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "((P & Q) ==> (P | Q))"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(P & Q)  |'==>'|  (P | Q)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Examples"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from notebook import Canvas_fol_bc_ask\n",
    "canvas_bc_ask = Canvas_fol_bc_ask('canvas_bc_ask', crime_kb, expr('Criminal(x)'))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "# Authors\n",
    "\n",
    "This notebook by [Chirag Vartak](https://github.com/chiragvartak) and [Peter Norvig](https://github.com/norvig).\n",
    "\n"
   ]
  }
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