{
"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 | ¬ P | `~P` | `~P` | `Expr('~', P)`\n",
"| And | P ∧ Q | `P & Q` | `P & Q` | `Expr('&', P, Q)`\n",
"| Or | P ∨ Q | `P` | `Q`| `P` | `Q` | `Expr('`|`', P, Q)`\n",
"| Inequality (Xor) | P ≠ Q | `P ^ Q` | `P ^ Q` | `Expr('^', P, Q)`\n",
"| Implication | P → Q | `P` |`'==>'`| `Q` | `P ==> Q` | `Expr('==>', P, Q)`\n",
"| Reverse Implication | Q ← 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: `==>`, ` ~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"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
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