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"We already know that *every sentence of a propositional logic is logically equivalent to a conjunction of clauses*.\n",
"We will use this fact to our advantage and simplify the input sentence into the **conjunctive normal form** (CNF) which is a conjunction of disjunctions of literals.\n",
"For eg:\n",
"<br>\n",
"$$(A\\lor B)\\land (\\neg B\\lor C\\lor\\neg D)\\land (D\\lor\\neg E)$$\n",
"This is equivalent to the POS (Product of sums) form in digital electronics.\n",
"<br>\n",
"Here's an outline of how the conversion is done:\n",
"1. Convert bi-implications to implications\n",
"<br>\n",
"$\\alpha\\iff\\beta$ can be written as $(\\alpha\\implies\\beta)\\land(\\beta\\implies\\alpha)$\n",
"<br>\n",
"This also applies to compound sentences\n",
"<br>\n",
"$\\alpha\\iff(\\beta\\lor\\gamma)$ can be written as $(\\alpha\\implies(\\beta\\lor\\gamma))\\land((\\beta\\lor\\gamma)\\implies\\alpha)$\n",
"<br>\n",
"2. Convert implications to their logical equivalents\n",
"<br>\n",
"$\\alpha\\implies\\beta$ can be written as $\\neg\\alpha\\lor\\beta$\n",
"<br>\n",
"3. Move negation inwards\n",
"<br>\n",
"CNF requires atomic literals. Hence, negation cannot appear on a compound statement.\n",
"De Morgan's laws will be helpful here.\n",
"<br>\n",
"$\\neg(\\alpha\\land\\beta)\\equiv(\\neg\\alpha\\lor\\neg\\beta)$\n",
"<br>\n",
"$\\neg(\\alpha\\lor\\beta)\\equiv(\\neg\\alpha\\land\\neg\\beta)$\n",
"<br>\n",
"4. Distribute disjunction over conjunction\n",
"<br>\n",
"Disjunction and conjunction are distributive over each other.\n",
"Now that we only have conjunctions, disjunctions and negations in our expression, \n",
"we will distribute disjunctions over conjunctions wherever possible as this will give us a sentence which is a conjunction of simpler clauses, \n",
"which is what we wanted in the first place.\n",
"<br>\n",
"We need a term of the form\n",
"<br>\n",
"$(\\alpha_{1}\\lor\\alpha_{2}\\lor\\alpha_{3}...)\\land(\\beta_{1}\\lor\\beta_{2}\\lor\\beta_{3}...)\\land(\\gamma_{1}\\lor\\gamma_{2}\\lor\\gamma_{3}...)\\land...$\n",
"<br>\n",
"<br>\n",
"The `to_cnf` function executes this conversion using helper subroutines."
]
},
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"execution_count": 29,
"metadata": {},
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{
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"\n",
"<div class=\"highlight\"><pre><span></span><span class=\"k\">def</span> <span class=\"nf\">to_cnf</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Convert a propositional logical sentence to conjunctive normal form.</span>\n",
"<span class=\"sd\"> That is, to the form ((A | ~B | ...) & (B | C | ...) & ...) [p. 253]</span>\n",
"<span class=\"sd\"> >>> to_cnf('~(B | C)')</span>\n",
"<span class=\"sd\"> (~B & ~C)</span>\n",
"<span class=\"sd\"> """</span>\n",
" <span class=\"n\">s</span> <span class=\"o\">=</span> <span class=\"n\">expr</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">)</span>\n",
" <span class=\"k\">if</span> <span class=\"nb\">isinstance</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">,</span> <span class=\"nb\">str</span><span class=\"p\">):</span>\n",
" <span class=\"n\">s</span> <span class=\"o\">=</span> <span class=\"n\">expr</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">)</span>\n",
" <span class=\"n\">s</span> <span class=\"o\">=</span> <span class=\"n\">eliminate_implications</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">)</span> <span class=\"c1\"># Steps 1, 2 from p. 253</span>\n",
" <span class=\"n\">s</span> <span class=\"o\">=</span> <span class=\"n\">move_not_inwards</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">)</span> <span class=\"c1\"># Step 3</span>\n",
" <span class=\"k\">return</span> <span class=\"n\">distribute_and_over_or</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"p\">)</span> <span class=\"c1\"># Step 4</span>\n",
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"metadata": {},
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],
"source": [
"psource(to_cnf)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"`to_cnf` calls three subroutines.\n",
"<br>\n",
"`eliminate_implications` converts bi-implications and implications to their logical equivalents.\n",
"<br>\n",
"`move_not_inwards` removes negations from compound statements and moves them inwards using De Morgan's laws.\n",
"<br>\n",
"`distribute_and_over_or` distributes disjunctions over conjunctions.\n",
"<br>\n",
"Run the cells below for implementation details.\n"
]
},
{
"cell_type": "code",
"execution_count": 30,
"outputs": [],
"source": [
"%psource eliminate_implications"
]
},
{
"cell_type": "code",
"execution_count": 31,
"%psource move_not_inwards"
]
},
{
"cell_type": "code",
"execution_count": 32,
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"outputs": [],
"source": [
"%psource distribute_and_over_or"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's convert some sentences to see how it works\n"
]
},
{
"cell_type": "code",
"execution_count": 33,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"((A | ~B) & (B | ~A))"
]
},
"execution_count": 33,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A, B, C, D = expr('A, B, C, D')\n",
"to_cnf(A |'<=>'| B)"
]
},
{
"cell_type": "code",
"execution_count": 34,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"((A | ~B | ~C) & (B | ~A) & (C | ~A))"
]
},
"execution_count": 34,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"to_cnf(A |'<=>'| (B & C))"
]
},
{
"cell_type": "code",
"execution_count": 35,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(A & (C | B) & (D | B))"
]
},
"execution_count": 35,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"to_cnf(A & (B | (C & D)))"
]
},
{
"cell_type": "code",
"execution_count": 36,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"((B | ~A | C | ~D) & (A | ~A | C | ~D) & (B | ~B | C | ~D) & (A | ~B | C | ~D))"
]
},
"execution_count": 36,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"to_cnf((A |'<=>'| ~B) |'==>'| (C | ~D))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Coming back to our resolution problem, we can see how the `to_cnf` function is utilized here"
]
},
{
"cell_type": "code",
"execution_count": 37,
"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\">pl_resolution</span><span class=\"p\">(</span><span class=\"n\">KB</span><span class=\"p\">,</span> <span class=\"n\">alpha</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Propositional-logic resolution: say if alpha follows from KB. [Figure 7.12]"""</span>\n",
" <span class=\"n\">clauses</span> <span class=\"o\">=</span> <span class=\"n\">KB</span><span class=\"o\">.</span><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=\"o\">~</span><span class=\"n\">alpha</span><span class=\"p\">))</span>\n",
" <span class=\"n\">new</span> <span class=\"o\">=</span> <span class=\"nb\">set</span><span class=\"p\">()</span>\n",
" <span class=\"k\">while</span> <span class=\"bp\">True</span><span class=\"p\">:</span>\n",
" <span class=\"n\">n</span> <span class=\"o\">=</span> <span class=\"nb\">len</span><span class=\"p\">(</span><span class=\"n\">clauses</span><span class=\"p\">)</span>\n",
" <span class=\"n\">pairs</span> <span class=\"o\">=</span> <span class=\"p\">[(</span><span class=\"n\">clauses</span><span class=\"p\">[</span><span class=\"n\">i</span><span class=\"p\">],</span> <span class=\"n\">clauses</span><span class=\"p\">[</span><span class=\"n\">j</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\">n</span><span class=\"p\">)</span> <span class=\"k\">for</span> <span class=\"n\">j</span> <span class=\"ow\">in</span> <span class=\"nb\">range</span><span class=\"p\">(</span><span class=\"n\">i</span><span class=\"o\">+</span><span class=\"mi\">1</span><span class=\"p\">,</span> <span class=\"n\">n</span><span class=\"p\">)]</span>\n",
" <span class=\"k\">for</span> <span class=\"p\">(</span><span class=\"n\">ci</span><span class=\"p\">,</span> <span class=\"n\">cj</span><span class=\"p\">)</span> <span class=\"ow\">in</span> <span class=\"n\">pairs</span><span class=\"p\">:</span>\n",
" <span class=\"n\">resolvents</span> <span class=\"o\">=</span> <span class=\"n\">pl_resolve</span><span class=\"p\">(</span><span class=\"n\">ci</span><span class=\"p\">,</span> <span class=\"n\">cj</span><span class=\"p\">)</span>\n",
" <span class=\"k\">if</span> <span class=\"bp\">False</span> <span class=\"ow\">in</span> <span class=\"n\">resolvents</span><span class=\"p\">:</span>\n",
" <span class=\"k\">return</span> <span class=\"bp\">True</span>\n",
" <span class=\"n\">new</span> <span class=\"o\">=</span> <span class=\"n\">new</span><span class=\"o\">.</span><span class=\"n\">union</span><span class=\"p\">(</span><span class=\"nb\">set</span><span class=\"p\">(</span><span class=\"n\">resolvents</span><span class=\"p\">))</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">new</span><span class=\"o\">.</span><span class=\"n\">issubset</span><span class=\"p\">(</span><span class=\"nb\">set</span><span class=\"p\">(</span><span class=\"n\">clauses</span><span class=\"p\">)):</span>\n",
" <span class=\"k\">return</span> <span class=\"bp\">False</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">c</span> <span class=\"ow\">in</span> <span class=\"n\">new</span><span class=\"p\">:</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">c</span> <span class=\"ow\">not</span> <span class=\"ow\">in</span> <span class=\"n\">clauses</span><span class=\"p\">:</span>\n",
" <span class=\"n\">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",
"</pre></div>\n",
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"### Forward and backward chaining\n",
"Previously, we said we will look at two algorithms to check if a sentence is entailed by the `KB`, \n",
"but here's a third one. \n",
"The difference here is that our goal now is to determine if a knowledge base of definite clauses entails a single proposition symbol *q* - the query.\n",
"There is a catch however, the knowledge base can only contain **Horn clauses**.\n",
"<br>\n",
"#### Horn Clauses\n",
"Horn clauses can be defined as a *disjunction* of *literals* with **at most** one positive literal. \n",
"<br>\n",
"A Horn clause with exactly one positive literal is called a *definite clause*.\n",
"<br>\n",
"A Horn clause might look like \n",
"<br>\n",
"$\\neg a\\lor\\neg b\\lor\\neg c\\lor\\neg d... \\lor z$\n",
"<br>\n",
"This, coincidentally, is also a definite clause.\n",
"<br>\n",
"Using De Morgan's laws, the example above can be simplified to \n",
"<br>\n",
"$a\\land b\\land c\\land d ... \\implies z$\n",
"<br>\n",
"This seems like a logical representation of how humans process known data and facts. \n",
"Assuming percepts `a`, `b`, `c`, `d` ... to be true simultaneously, we can infer `z` to also be true at that point in time. \n",
"There are some interesting aspects of Horn clauses that make algorithmic inference or *resolution* easier.\n",
"- Definite clauses can be written as implications:\n",
"<br>\n",
"The most important simplification a definite clause provides is that it can be written as an implication.\n",
"The premise (or the knowledge that leads to the implication) is a conjunction of positive literals.\n",
"The conclusion (the implied statement) is also a positive literal.\n",
"The sentence thus becomes easier to understand.\n",
"The premise and the conclusion are conventionally called the *body* and the *head* respectively.\n",
"A single positive literal is called a *fact*.\n",
"- Forward chaining and backward chaining can be used for inference from Horn clauses:\n",
"<br>\n",
"Forward chaining is semantically identical to `AND-OR-Graph-Search` from the chapter on search algorithms.\n",
"Implementational details will be explained shortly.\n",
"- Deciding entailment with Horn clauses is linear in size of the knowledge base:\n",
"<br>\n",
"Surprisingly, the forward and backward chaining algorithms traverse each element of the knowledge base at most once, greatly simplifying the problem.\n",
"<br>\n",
"<br>\n",
"The function `pl_fc_entails` implements forward chaining to see if a knowledge base `KB` entails a symbol `q`.\n",
"<br>\n",
"Before we proceed further, note that `pl_fc_entails` doesn't use an ordinary `KB` instance. \n",
"The knowledge base here is an instance of the `PropDefiniteKB` class, derived from the `PropKB` class, \n",
"but modified to store definite clauses.\n",
"<br>\n",
"The main point of difference arises in the inclusion of a helper method to `PropDefiniteKB` that returns a list of clauses in KB that have a given symbol `p` in their premise."
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"<div class=\"highlight\"><pre><span></span> <span class=\"k\">def</span> <span class=\"nf\">clauses_with_premise</span><span class=\"p\">(</span><span class=\"bp\">self</span><span class=\"p\">,</span> <span class=\"n\">p</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Return a list of the clauses in KB that have p in their premise.</span>\n",
"<span class=\"sd\"> This could be cached away for O(1) speed, but we'll recompute it."""</span>\n",
" <span class=\"k\">return</span> <span class=\"p\">[</span><span class=\"n\">c</span> <span class=\"k\">for</span> <span class=\"n\">c</span> <span class=\"ow\">in</span> <span class=\"bp\">self</span><span class=\"o\">.</span><span class=\"n\">clauses</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">c</span><span class=\"o\">.</span><span class=\"n\">op</span> <span class=\"o\">==</span> <span class=\"s1\">'==>'</span> <span class=\"ow\">and</span> <span class=\"n\">p</span> <span class=\"ow\">in</span> <span class=\"n\">conjuncts</span><span class=\"p\">(</span><span class=\"n\">c</span><span class=\"o\">.</span><span class=\"n\">args</span><span class=\"p\">[</span><span class=\"mi\">0</span><span class=\"p\">])]</span>\n",
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"<body>\n",
"<h2></h2>\n",
"\n",
"<div class=\"highlight\"><pre><span></span><span class=\"k\">def</span> <span class=\"nf\">pl_fc_entails</span><span class=\"p\">(</span><span class=\"n\">KB</span><span class=\"p\">,</span> <span class=\"n\">q</span><span class=\"p\">):</span>\n",
" <span class=\"sd\">"""Use forward chaining to see if a PropDefiniteKB entails symbol q.</span>\n",
"<span class=\"sd\"> [Figure 7.15]</span>\n",
"<span class=\"sd\"> >>> pl_fc_entails(horn_clauses_KB, expr('Q'))</span>\n",
"<span class=\"sd\"> True</span>\n",
"<span class=\"sd\"> """</span>\n",
" <span class=\"n\">count</span> <span class=\"o\">=</span> <span class=\"p\">{</span><span class=\"n\">c</span><span class=\"p\">:</span> <span class=\"nb\">len</span><span class=\"p\">(</span><span class=\"n\">conjuncts</span><span class=\"p\">(</span><span class=\"n\">c</span><span class=\"o\">.</span><span class=\"n\">args</span><span class=\"p\">[</span><span class=\"mi\">0</span><span class=\"p\">]))</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">c</span> <span class=\"ow\">in</span> <span class=\"n\">KB</span><span class=\"o\">.</span><span class=\"n\">clauses</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">c</span><span class=\"o\">.</span><span class=\"n\">op</span> <span class=\"o\">==</span> <span class=\"s1\">'==>'</span><span class=\"p\">}</span>\n",
" <span class=\"n\">inferred</span> <span class=\"o\">=</span> <span class=\"n\">defaultdict</span><span class=\"p\">(</span><span class=\"nb\">bool</span><span class=\"p\">)</span>\n",
" <span class=\"n\">agenda</span> <span class=\"o\">=</span> <span class=\"p\">[</span><span class=\"n\">s</span> <span class=\"k\">for</span> <span class=\"n\">s</span> <span class=\"ow\">in</span> <span class=\"n\">KB</span><span class=\"o\">.</span><span class=\"n\">clauses</span> <span class=\"k\">if</span> <span class=\"n\">is_prop_symbol</span><span class=\"p\">(</span><span class=\"n\">s</span><span class=\"o\">.</span><span class=\"n\">op</span><span class=\"p\">)]</span>\n",
" <span class=\"k\">while</span> <span class=\"n\">agenda</span><span class=\"p\">:</span>\n",
" <span class=\"n\">p</span> <span class=\"o\">=</span> <span class=\"n\">agenda</span><span class=\"o\">.</span><span class=\"n\">pop</span><span class=\"p\">()</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">p</span> <span class=\"o\">==</span> <span class=\"n\">q</span><span class=\"p\">:</span>\n",
" <span class=\"k\">return</span> <span class=\"bp\">True</span>\n",
" <span class=\"k\">if</span> <span class=\"ow\">not</span> <span class=\"n\">inferred</span><span class=\"p\">[</span><span class=\"n\">p</span><span class=\"p\">]:</span>\n",
" <span class=\"n\">inferred</span><span class=\"p\">[</span><span class=\"n\">p</span><span class=\"p\">]</span> <span class=\"o\">=</span> <span class=\"bp\">True</span>\n",
" <span class=\"k\">for</span> <span class=\"n\">c</span> <span class=\"ow\">in</span> <span class=\"n\">KB</span><span class=\"o\">.</span><span class=\"n\">clauses_with_premise</span><span class=\"p\">(</span><span class=\"n\">p</span><span class=\"p\">):</span>\n",
" <span class=\"n\">count</span><span class=\"p\">[</span><span class=\"n\">c</span><span class=\"p\">]</span> <span class=\"o\">-=</span> <span class=\"mi\">1</span>\n",
" <span class=\"k\">if</span> <span class=\"n\">count</span><span class=\"p\">[</span><span class=\"n\">c</span><span class=\"p\">]</span> <span class=\"o\">==</span> <span class=\"mi\">0</span><span class=\"p\">:</span>\n",
" <span class=\"n\">agenda</span><span class=\"o\">.</span><span class=\"n\">append</span><span class=\"p\">(</span><span class=\"n\">c</span><span class=\"o\">.</span><span class=\"n\">args</span><span class=\"p\">[</span><span class=\"mi\">1</span><span class=\"p\">])</span>\n",
" <span class=\"k\">return</span> <span class=\"bp\">False</span>\n",
"</pre></div>\n",
"</body>\n",
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"psource(pl_fc_entails)"
]
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"cell_type": "markdown",
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"source": [
"The function accepts a knowledge base `KB` (an instance of `PropDefiniteKB`) and a query `q` as inputs.\n",
"<br>\n",
"<br>\n",
"`count` initially stores the number of symbols in the premise of each sentence in the knowledge base.\n",
"<br>\n",
"The `conjuncts` helper function separates a given sentence at conjunctions.\n",
"<br>\n",
"`inferred` is initialized as a *boolean* defaultdict. \n",
"This will be used later to check if we have inferred all premises of each clause of the agenda.\n",
"<br>\n",
"`agenda` initially stores a list of clauses that the knowledge base knows to be true.\n",
"The `is_prop_symbol` helper function checks if the given symbol is a valid propositional logic symbol.\n",
"<br>\n",
"<br>\n",
"We now iterate through `agenda`, popping a symbol `p` on each iteration.\n",
"If the query `q` is the same as `p`, we know that entailment holds.\n",
"<br>\n",
"The agenda is processed, reducing `count` by one for each implication with a premise `p`.\n",
"A conclusion is added to the agenda when `count` reaches zero. This means we know all the premises of that particular implication to be true.\n",
"<br>\n",
"`clauses_with_premise` is a helpful method of the `PropKB` class.\n",
"It returns a list of clauses in the knowledge base that have `p` in their premise.\n",
"<br>\n",
"<br>\n",
"Now that we have an idea of how this function works, let's see a few examples of its usage, but we first need to define our knowledge base. We assume we know the following clauses to be true."
]
},
{
"cell_type": "code",
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"source": [
"clauses = ['(B & F)==>E', \n",
" '(A & E & F)==>G', \n",
" '(B & C)==>F', \n",
" '(A & B)==>D', \n",
" '(E & F)==>H', \n",
" '(H & I)==>J',\n",
" 'A', \n",
" 'B', \n",
" 'C']"
]
},
{
"cell_type": "markdown",
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"source": [
"We will now `tell` this information to our knowledge base."
]
},
{
"cell_type": "code",
"execution_count": 43,
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"source": [
"definite_clauses_KB = PropDefiniteKB()\n",
"for clause in clauses:\n",
" definite_clauses_KB.tell(expr(clause))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can now check if our knowledge base entails the following queries."
]
},
{
"cell_type": "code",
"execution_count": 44,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"True"
"pl_fc_entails(definite_clauses_KB, expr('G'))"
]
},
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"source": [
"pl_fc_entails(definite_clauses_KB, expr('H'))"
]
},
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"cell_type": "code",
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{
"data": {
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"source": [
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"pl_fc_entails(definite_clauses_KB, expr('I'))"
]
},
{
"cell_type": "code",
"execution_count": 47,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"False"
]
},
"execution_count": 47,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"pl_fc_entails(definite_clauses_KB, expr('J'))"
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{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Effective Propositional Model Checking\n",
"\n",
"The previous segments elucidate the algorithmic procedure for model checking. \n",
"In this segment, we look at ways of making them computationally efficient.\n",
"<br>\n",
"The problem we are trying to solve is conventionally called the _propositional satisfiability problem_, abbreviated as the _SAT_ problem.\n",
"In layman terms, if there exists a model that satisfies a given Boolean formula, the formula is called satisfiable.\n",
"<br>\n",
"The SAT problem was the first problem to be proven _NP-complete_.\n",
"The main characteristics of an NP-complete problem are:\n",
"- Given a solution to such a problem, it is easy to verify if the solution solves the problem.\n",
"- The time required to actually solve the problem using any known algorithm increases exponentially with respect to the size of the problem.\n",
"<br>\n",
"<br>\n",
"Due to these properties, heuristic and approximational methods are often applied to find solutions to these problems.\n",
"<br>\n",
"It is extremely important to be able to solve large scale SAT problems efficiently because \n",
"many combinatorial problems in computer science can be conveniently reduced to checking the satisfiability of a propositional sentence under some constraints.\n",
"<br>\n",
"We will introduce two new algorithms that perform propositional model checking in a computationally effective way.\n",
"<br>\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### 1. DPLL (Davis-Putnam-Logeman-Loveland) algorithm\n",
"This algorithm is very similar to Backtracking-Search.\n",
"It recursively enumerates possible models in a depth-first fashion with the following improvements over algorithms like `tt_entails`:\n",
"1. Early termination:\n",
"<br>\n",
"In certain cases, the algorithm can detect the truth value of a statement using just a partially completed model.\n",
"For example, $(P\\lor Q)\\land(P\\lor R)$ is true if P is true, regardless of other variables.\n",
"This reduces the search space significantly.\n",
"2. Pure symbol heuristic:\n",
"<br>\n",
"A symbol that has the same sign (positive or negative) in all clauses is called a _pure symbol_.\n",
"It isn't difficult to see that any satisfiable model will have the pure symbols assigned such that its parent clause becomes _true_.\n",
"For example, $(P\\lor\\neg Q)\\land(\\neg Q\\lor\\neg R)\\land(R\\lor P)$ has P and Q as pure symbols\n",
"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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"execution_count": 48,
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